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New York State Common Core

6

GRADE

Mathematics Curriculum
GRADE 6 • MODULE 5

Table of Contents1

Area, Surface Area, and Volume Problems
Module Overview .................................................................................................................................................. 3
Topic A: Area of Triangles, Quadrilaterals, and Polygons (6.G.A.1) .................................................................... 13
Lesson 1: The Area of Parallelograms Through Rectangle Facts ............................................................ 15
Lesson 2: The Area of Right Triangles ..................................................................................................... 31
Lesson 3: The Area of Acute Triangles Using Height and Base ............................................................... 41
Lesson 4: The Area of All Triangles Using Height and Base .................................................................... 56
Lesson 5: The Area of Polygons Through Composition and Decomposition .......................................... 67
Lesson 6: Area in the Real World............................................................................................................ 87
Topic B: Polygons on the Coordinate Plane (6.G.A.3) ......................................................................................... 95
Lesson 7: Distance on the Coordinate Plane .......................................................................................... 96
Lesson 8: Drawing Polygons in the Coordinate Plane .......................................................................... 107
Lesson 9: Determining Perimeter and Area of Polygons on the Coordinate Plane.............................. 120
Lesson 10: Distance, Perimeter, and Area in the Real World............................................................... 136
Mid-Module Assessment and Rubric ................................................................................................................ 145
Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day)
Topic C: Volume of Right Rectangular Prisms (6.G.A.2) .................................................................................... 153
Lesson 11: Volume with Fractional Edge Lengths and Unit Cubes ....................................................... 154
Lesson 12: From Unit Cubes to the Formulas for Volume.................................................................... 174
Lesson 13: The Formulas for Volume ................................................................................................... 190
Lesson 14: Volume in the Real World................................................................................................... 203
Topic D: Nets and Surface Area (6.G.A.2, 6.G.A.4)............................................................................................ 214
Lesson 15: Representing Three-Dimensional Figures Using Nets ........................................................ 216
Lesson 16: Constructing Nets ............................................................................................................... 249
Lesson 17: From Nets to Surface Area.................................................................................................. 265
1

Each lesson is ONE day, and ONE day is considered a 45-minute period.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

6•5

Lesson 18: Determining Surface Area of Three-Dimensional Figures .................................................. 280
Lesson 19: Surface Area and Volume in the Real World ...................................................................... 291
Lesson 19a: Addendum Lesson for Modeling―Applying Surface Area and Volume to Aquariums
(Optional) .......................................................................................................................... 303
End-of-Module Assessment and Rubric ............................................................................................................ 317
Topics C through D (assessment 1 day, return 1 day, remediation or further applications 1 day)

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

6•5

Grade 6 • Module 5

Area, Surface Area, and Volume Problems
OVERVIEW
Starting in Grade 1, students compose and decompose plane and solid figures (1.G.A.2). They move to spatial structuring of rectangular arrays in Grade 2 (2.G.A.2) and continually build upon their understanding of arrays to ultimately apply their knowledge to two- and three-dimensional figures in Grade 4 (4.MD.A.3) and Grade 5
(5.MD.C.3, 5.MD.C.5). Students move from building arrays to using arrays to find area and eventually move to decomposing three-dimensional shapes into layers that are arrays of cubes. In this module, students utilize their previous experiences in shape composition and decomposition in order to understand and develop formulas for area, volume, and surface area.
In Topic A, students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. They determine that area is additive. Students learn through exploration that the area of a triangle is exactly half of the area of its corresponding rectangle. In Lesson 1, students discover through composition that the area of a parallelogram is the same as a rectangle. In Lesson 2, students compose rectangles using two copies of a right triangle. They extend their previous knowledge about the area formula for rectangles (4.MD.A.3) to evaluate the area of the rectangle using = ℎ and discover through manipulation that the area of a right triangle is exactly half that of its corresponding rectangle. In Lesson 3, students discover that any triangle may be decomposed into right triangles, and in Lesson 4, students further explore all triangles and discover through manipulation that the area of all triangles is exactly half the area of its corresponding rectangle. During this discovery process, students become aware that triangles have altitude, which is the length of the height of the triangle. The altitude is the perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. Students understand that any side of the triangle can be a base, but the altitude always determines the base. They move from recognizing right triangles as categories (4.G.A.2) to determining that right triangles are constructed when altitudes are perpendicular and meet the base at one endpoint. Acute triangles are constructed when the altitude is perpendicular and meets within the length of the base, and obtuse triangles are constructed when the altitude is perpendicular and lies outside the length of the base. Students use this information to cut triangular pieces and rearrange them to fit exactly within one half of the corresponding rectangle to determine that the area formula for any triangle can be determined using =

Module 5:
Date:

1
2

ℎ.

Area, Surface Area, and Volume Problems
11/6/14

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Module Overview

6•5

In Lesson 5, students apply their knowledge of the area of a triangular region, where they deconstruct parallelograms, trapezoids, and other quadrilaterals and polygons into triangles or rectangles in order to determine area. They intuitively decompose rectangles to determine the area of polygons. Topic A closes with Lesson 6 where students apply their learning from the topic to find areas of composite figures in real-life contexts, as well as to determine the area of missing regions (6.G.A.1).
In Module 3, students used coordinates and absolute value to find distances between points on a coordinate plane (6.NS.C.8). In Topic B, students extend this learning to Lessons 7 and 8 where they find edge lengths of polygons (the distance between two vertices using absolute value) and draw polygons given coordinates
(6.G.A.3). From these drawings, students determine the area of polygons on the coordinate plane by composing and decomposing into polygons with known area formulas. In Lesson 9, students further investigate and calculate the area of polygons on the coordinate plane and also calculate the perimeter. They note that finding perimeter is simply finding the sum of the polygon’s edge lengths (or finding the sum of the distances between vertices). Topic B concludes with students determining distance, perimeter, and area on the coordinate plane in real-world contexts.
In Grade 5, students recognized volume as an attribute of solid figures. They measured volume by packing right rectangular prisms with unit cubes and found that determining volume was the same as multiplying the edge lengths of the prism (5.MD.C.3, 5.MD.C.4). Students extend this knowledge to Topic C where they continue packing right rectangular prisms with unit cubes; however, this time the right rectangular prism has fractional lengths (6.G.A.2). In Lesson 11, students decompose a one cubic unit prism in order to conceptualize finding the volume of a right rectangular prism with fractional edge lengths using unit cubes.
They connect those findings to apply the formula = ℎ and multiply fractional edge lengths (5.NF.B.4). In
Lessons 12 and 13, students extend and apply the volume formula to = The area of the base × height or simply = ℎ, where represents the area of the base. In Lesson 12, students explore the bases of right rectangular prisms and find the area of the base first, then multiply by the height. They determine that two formulas can be used to find the volume of a right rectangular prism. In Lesson 13, students apply both formulas to application problems. Topic C concludes with real-life application of the volume formula where students extend the notion that volume is additive (5.MD.C.5c) and find the volume of composite solid figures. They apply volume formulas and use their previous experience with solving equations (6.EE.B.7) to find missing volumes and missing dimensions.
Module 5 concludes with deconstructing the faces of solid figures to determine surface area. Students note the difference between finding the volume of right rectangular prisms and finding the surface area of such prisms. In Lesson 15, students build solid figures using nets. They note which nets compose specific solid figures and also understand when nets cannot compose a solid figure. From this knowledge, students deconstruct solid figures into nets to identify the measurement of the solids’ face edges. With this knowledge from Lesson 16, students are prepared to use nets to determine the surface area of solid figures in
Lesson 17. They find that adding the areas of each face of the solid will result in a combined surface area. In
Lesson 18, students find that each right rectangular prism has a front, a back, a top, a bottom, and two sides.
They determine that surface area is obtained by adding the areas of all the faces. They understand that the front and back of the prism have the same surface area, the top and bottom have the same surface area, and the sides have the same surface area. Thus, students develop the formula = 2 + 2ℎ + 2ℎ (6.G.A.4).
To wrap up the module, students apply the surface area formula to real-life contexts and distinguish between the need to find surface area or volume within contextual situations.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

6•5

Focus Standards
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas
V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.A.3

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 6.G.A.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Foundational Standards
Reason with shapes and their attributes.
1.G.A.2

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. 2

2.G.A.2

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

3.G.A.2

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
4.MD.A.3

2

Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. Students do not need to learn formal names such as “right rectangular prism.”

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

6•5

Draw and identify lines and angles, and classify shapes by properties of their lines and angles. 4.G.A.2

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.4

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a.

5.NF.B.7

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 3

Geometric measurement: understand conceptual concepts of volume and relate volume to multiplication and to addition.
5.MD.C.3

Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a.

A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

b.

A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5.MD.C.4

Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.C.5

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a.

Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b.

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

3

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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Module Overview

NYS COMMON CORE MATHEMATICS CURRICULUM

c.

6•5

Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Graph points on a coordinate plane to solve real-world and mathematical problems.
5.G.A.1

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.
Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond
(e.g., x-axis and x-coordinate, y-axis and y-coordinate).

5.G.A.2

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Classify two-dimensional figures into categories based on their properties.
5.G.B.3

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.8

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Reason about and solve one-variable equations and inequalities.
6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Focus Standards for Mathematical Practice
MP.1

Make sense of problems and persevere in solving them. Students make sense of realworld problems that involve area, volume, and surface area. One problem will involve multiple steps without breaking the problem into smaller, simpler questions. To solve surface area problems, students will have to find the area of different parts of the polygon before calculating the total area.

MP.3

Construct viable arguments and critique the reasoning of others. Students will develop different arguments as to why area formulas work for different polygons. Through this development, students may discuss and question their peers’ thinking process. When students draw nets to represent right rectangular prisms, their representations may be

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Area, Surface Area, and Volume Problems
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Module Overview

6•5

different from their peers’. Although more than one answer may be correct, students will have an opportunity to defend their answers as well as question their peers. Students may also solve real-world problems using different methods; therefore, they may have to explain their thinking and critique their peers.
MP.4

Model with mathematics. Models will be used to demonstrate why the area formulas for different quadrilaterals are accurate. Students will use unit cubes to build right rectangular prisms and use these to calculate volume. The unit cubes will be used to model that = ℎ and = ℎ, where represents the area of the base, are both accurate formulas to calculate the volume of a right rectangular prism. Students will use nets to model the process of calculating the surface area of a right rectangular prism.

MP.6

Attend to precision. Students will understand and use labels correctly throughout the module. For example, when calculating the area of a triangle, the answer will be labeled units2 because the area is the product of two dimensions. When two different units are given within a problem, students know to use previous knowledge of conversions to make the units match before solving the problem. In multi-step problems, students solve each part of the problem separately and know when to round in order to calculate the most precise answer. Students will attend to precision of language when describing exactly how a region may be composed or decomposed to determine its area.

Terminology
New or Recently Introduced Terms







Altitude and Base of a Triangle (An altitude of a triangle is a perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. For every triangle, there are three choices for the altitude, and hence there are three base-altitude pairs. The height of a triangle is the length of the altitude. The length of the base is called either the base length or, more commonly, the base. Usually, context makes it clear whether the base refers to a number or a segment. These terms can mislead students: base suggests the bottom, while height usually refers to vertical distances. Do not reinforce these impressions by consistently displaying all triangles with horizontal bases.)
Cube (A cube is a right rectangular prism all of whose edges are of equal length.)
Hexagon (Given 6 different points , , , , , and in the plane, a 6-sided polygon, or hexagon, is the union of 6 segments , , , , , and such that (1) the segments intersect only at their endpoints, and (2) no two adjacent segments are collinear. For both pentagons and hexagons, the segments are called the sides, and their endpoints are called the vertices. Like quadrilaterals, pentagons and hexagons can be denoted by the order of vertices defining the segments. For example, the pentagon has vertices , , , , and that define the 5 segments in the definition above. Similar to quadrilaterals, pentagons and hexagons also have interiors, which can be described using pictures in elementary school.)
Line Perpendicular to a Plane (A line intersecting a plane at a point is said to be perpendicular to the plane if is perpendicular to every line that (1) lies in and (2) passes through the point .
A segment is said to be perpendicular to a plane if the line that contains the segment is
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Area, Surface Area, and Volume Problems
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NYS COMMON CORE MATHEMATICS CURRICULUM










Module Overview

6•5

perpendicular to the plane. In Grade 6, a line perpendicular to a plane can be described using a picture.) Parallel Planes (Two planes are parallel if they do not intersect. In Euclidean geometry, a useful test for checking whether two planes are parallel is if the planes are different and if there is a line that is perpendicular to both planes.)
Pentagon (Given 5 different points , , , , and in the plane, a 5-sided polygon, or pentagon, is the union of 5 segments , , , , and such that (1) the segments intersect only at their endpoints, and (2) no two adjacent segments are collinear.)
Right Rectangular Prism (Let and ′ be two parallel planes. Let be a rectangular region 4 in the plane . At each point of , consider the segment ′ perpendicular to , joining to a point ′ of the plane ′. The union of all these segments is called a right rectangular prism. It can be shown that the region ′ in ′ corresponding to the region is also a rectangular region whose sides are equal in length to the corresponding sides of . The regions and ′ are called the base faces (or just bases) of the prism. It can also be shown that the planar region between two corresponding sides of the bases is also a rectangular region called the lateral face of the prism. In all, the boundary of a right rectangular prism has 6 faces: the 2 base faces and 4 lateral faces. All adjacent faces intersect along segments called edges—base edges and lateral edges.)
Surface of a Prism (The surface of a prism is the union of all of its faces—the base faces and lateral faces.) Triangular Region (A triangular region is the union of the triangle and its interior.)

Familiar Terms and Symbols 5















4

5

Angle
Area
Length of a Segment
Parallel
Parallelogram
Perimeter
Perpendicular
Quadrilateral
Rectangle
Segment
Square
Trapezoid
Triangle
Volume

A rectangular region is the union of a rectangle and its interior.
These are terms and symbols students have seen previously.

Module 5:
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Area, Surface Area, and Volume Problems
11/6/14

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Module Overview

6•5

Suggested Tools and Representations





Coordinate Planes
Nets
Prisms
Rulers

Rapid White Board Exchanges
Implementing an RWBE requires that each student be provided with a personal white board, a white board marker, and a means of erasing his or her work. An economic choice for these materials is to place sheets of card stock inside sheet protectors to use as the personal white boards and to cut sheets of felt into small squares to use as erasers.
An RWBE consists of a sequence of 10 to 20 problems on a specific topic or skill that starts out with a relatively simple problem and progressively gets more difficult. The teacher should prepare the problems in a way that allows him or her to reveal them to the class one at a time. A flip chart or PowerPoint presentation can be used, or the teacher can write the problems on the board and either cover some with paper or simply write only one problem on the board at a time.
The teacher reveals, and possibly reads aloud, the first problem in the list and announces, “Go.” Students work the problem on their personal white boards as quickly as possible and hold their work up for their teacher to see their answers as soon as they have the answer ready. The teacher gives immediate feedback to each student, pointing and/or making eye contact with the student and responding with an affirmation for correct work such as, “Good job!”, “Yes!”, or “Correct!”, or responding with guidance for incorrect work such as “Look again,” “Try again,” “Check your work,” etc. In the case of the RWBE, it is not recommended that the feedback include the name of the student receiving the feedback.
If many students have struggled to get the answer correct, go through the solution of that problem as a class before moving on to the next problem in the sequence. Fluency in the skill has been established when the class is able to go through each problem in quick succession without pausing to go through the solution of each problem individually. If only one or two students have not been able to successfully complete a problem, it is appropriate to move the class forward to the next problem without further delay; in this case find a time to provide remediation to that student before the next fluency exercise on this skill is given.

Sprints
Sprints are designed to develop fluency. They should be fun, adrenaline-rich activities that intentionally build energy and excitement. A fast pace is essential. During Sprint administration, teachers assume the role of athletic coaches. A rousing routine fuels students’ motivation to do their personal best. Student recognition of increasing success is critical, and so every improvement is acknowledged. (See the Sprint Delivery Script for the suggested means of acknowledging and celebrating student success.)
One Sprint has two parts with closely-related problems on each. Students complete the two parts of the
Sprint in quick succession with the goal of improving on the second part, even if only by one more.

Module 5:
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Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Sprints are not to be used for a grade. Thus, there is no need for students to write their names on the Sprints.
The low-stakes nature of the exercise means that even students with allowances for extended time can participate. When a particular student finds the experience undesirable, it is recommended that the student be allowed to opt-out and take the Sprint home. In this case, it is ideal if the student has a regular opportunity to express the desire to opt-in.
With practice, the Sprint routine takes about 8 minutes.

Sprint Delivery Script
Gather the following: stopwatch, a copy of Sprint A for each student, a copy of Sprint B for each student, answers for Sprint A and Sprint B. The following delineates a script for delivery of a pair of Sprints.
This sprint covers: topic.
Do not look at the Sprint; keep it turned face down on your desk.
There are xx problems on the Sprint. You will have 60 seconds. Do as many as you can. I do not expect any of you to finish.
On your mark, get set, GO.
60 seconds of silence.
STOP. Circle the last problem you completed.
I will read the answers. You say “YES” if your answer matches. Mark the ones you have wrong. Don’t try to correct them.
Energetically, rapid-fire call the answers ONLY.
Stop reading answers after there are no more students answering, “Yes.”
Fantastic! Count the number you have correct, and write it on the top of the page. This is your personal goal for Sprint B.
Raise your hand if you have 1 or more correct. 2 or more, 3 or more...
Let us all applaud our runner-up, [insert name], with x correct. And let us applaud our winner, [insert name], with x correct.
You have a few minutes to finish up the page and get ready for the next Sprint.
Students are allowed to talk and ask for help; let this part last as long as most are working seriously.
Stop working. I will read the answers again so you can check your work. You say “YES” if your answer matches. Energetically, rapid-fire call the answers ONLY.
Optionally, ask students to stand, and lead them in an energy-expanding exercise that also keeps the brain going. Examples are jumping jacks or arm circles, etc., while counting by 15’s starting at 15, going up to 150 and back down to 0. You can follow this first exercise with a cool down exercise of a similar nature, such as calf raises with counting by one-sixths � , , , , , 1 … �.
1 1 1 2 5
6 3 2 3 6

Hand out the second Sprint, and continue reading the script.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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Module Overview

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Keep the Sprint face down on your desk.
There are xx problems on the Sprint. You will have 60 seconds. Do as many as you can. I do not expect any of you to finish.
On your mark, get set, GO.
60 seconds of silence.
STOP. Circle the last problem you completed.
I will read the answers. You say “YES” if your answer matches. Mark the ones you have wrong. Don’t try to correct them.
Quickly read the answers ONLY.
Count the number you have correct, and write it on the top of the page.
Raise your hand if you have 1 or more correct. 2 or more, 3 or more, ...
Let us all applaud our runner-up, [insert name], with x correct. And let us applaud our winner, [insert name], with x correct.
Write the amount by which your score improved at the top of the page.
Raise your hand if you improved your score by 1 or more. 2 or more, 3 or more, ...
Let us all applaud our runner-up for most improved, [insert name]. And let us applaud our winner for most improved, [insert name].
You can take the Sprint home and finish it if you want.

Assessment Summary
Assessment Type Administered

Format

Standards Addressed

Mid-Module
Assessment Task

After Topic B

Constructed response with rubric

6.G.A.1, 6.G.A.3

End-of-Module
Assessment Task

After Topic D

Constructed response with rubric

6.G.A.1, 6.G.A.2,
6.G.A.3, 6.G.A.4

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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New York State Common Core

6

Mathematics Curriculum

GRADE

GRADE 6 • MODULE 5

Topic A:

Area of Triangles, Quadrilaterals, and
Polygons
6.G.A.1
Focus Standard:

6.G.A.1

Instructional Days:

6

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Lesson 1: The Area of Parallelograms Through Rectangle Facts (S) 1
Lesson 2: The Area of Right Triangles (E)
Lesson 3: The Area of Acute All Triangles Using Height and Base (M)
Lesson 4: The Area of All Triangles Using Height and Base (E)
Lesson 5: The Area of Polygons Through Composition and Decomposition (S)
Lesson 6: Area in the Real World (E)

In Topic A, students discover the area of triangles, quadrilaterals, and other polygons through composition and decomposition. In Lesson 1, students discover through composition that the area of a parallelogram is the same as the area of a rectangle with the same base and height measurements. Students show the area formula for the region bound by a parallelogram by composing it into rectangles and determining that the area formula for rectangles and parallelograms is = ℎ. In Lesson 2, students justify the area formula for a right triangle by viewing the right triangle as part of a rectangle composed of two right triangles. They discover that a right triangle is exactly half of a rectangle, thus proving that the area of a triangle is

1
2

ℎ.

Students further explore the area formula for all triangles in Lessons 3 and 4. They decompose triangles into right triangles and deconstruct triangles to discover that the area of a triangle is exactly one half the area of a parallelogram. Using known area formulas for rectangles, triangles, and parallelograms, students find area
1

Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

Topic A:
Date:

Area of Triangles, Quadrilaterals, and Polygons
11/3/14

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13

Topic A

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

formulas for polygons by decomposing the regions into triangles, rectangles, and parallelograms. Specifically, students use right triangles to develop an understanding of the area of all triangles. They decompose the region of a trapezoid into two triangles and determine the area. The topic closes with Lesson 6, where students determine the area of composite figures in real-life contextual situations using composition and decomposition of polygons. They determine the area of a missing region using composition and decomposition of polygons.

Topic A:
Date:

Area of Triangles, Quadrilaterals, and Polygons
11/3/14

© 2014 Common Core, Inc. Some rights reserved. commoncore.org

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

14

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 1: The Area of Parallelograms Through Rectangle
Facts
Student Outcomes


Students show the area formula for the region bounded by a parallelogram by composing it into rectangles.
They understand that the area of a parallelogram is the area of the region bounded by the parallelogram.

Lesson Notes
In order to participate in the discussions, each student will need the parallelogram templates attached to this lesson, along with the following: scissors, glue, ruler, and paper on which to glue their shapes.

Classwork
Fluency Exercise (5 minutes): Multiplication of Fractions
Sprint: Refer to the Sprints and the Sprint Delivery Script sections in the Module Overview for directions to administer a
Sprint.

Opening Exercise (4 minutes)

Scaffolding:

Students name the given shapes.

Some students may not know this vocabulary yet, so creating a poster or chart for student desks may help them to remember these terms.

Opening Exercise
Name each shape.

Parallelogram

Acute Triangle

Right Triangle

Rectangle



Trapezoid

Identify the shape that is commonly referred to as a parallelogram. How do you know it is a parallelogram?

Note: A rectangle is considered a parallelogram, but is commonly called a rectangle because it is a more specific name.


The shape is a quadrilateral (4-sided) and has two sets of parallel lines.

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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15

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM



What are some quadrilaterals that you know?




6•5

Answers will vary.

Today, we are going to find the area of one of these quadrilaterals: the parallelogram. We are going to use our knowledge of the area of rectangles to help us. Who can remind us what we mean by area?


The number of square units that make up the inside of the shape.

Note: English language learners would benefit from a further discussion of area that relates to things they have personal connections to.


Talk to your neighbor about how to calculate the area of a rectangle.

Pick someone who can clearly explain how to find the area of a rectangle.


Count the number of square units inside the shape (if that is given), or multiply the base by the height.

Discussion (10 minutes)
Provide each student with the picture of a parallelogram provided as an attachment to this lesson.


What shape do you have in front of you?




A parallelogram.

Work with a partner to make a prediction of how we would calculate the area of the shape.


Answers will vary.



Cut out the parallelogram.



Since we know how to find the area of a rectangle, how can we change the parallelogram into a rectangle?




Cut off a right triangle on one side of the parallelogram, and glue it to the other side.

Draw a dotted line, perpendicular to the base, to show the triangle you will cut. Fold your paper along this line. 

MP.7



Check to make sure all students have drawn the dotted line in the correct place before instructing them to cut.
Explain that the fold on the line shows that the two right angles form a 180° angle.
Could the dotted line be drawn in a different location? If so, where?


The dotted line can be drawn in a different location. It could be drawn on the other side of the parallelogram, displayed below.



The base and height of a parallelogram form a right angle.



Measure, in inches, the base and height of the parallelogram using the correct mathematical tools.


The base is 7 inches, and the height is 3 inches.
Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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16

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM



Cut along the dotted line.



6•5

Glue both parts of the parallelogram onto a piece of paper to make a rectangle.




Why is the new shape classified as a rectangle?




The new shape is a rectangle because it is a quadrilateral that has four right angles.

Use the correct mathematical tool to measure, in inches, and label each side of the rectangle created from the original parallelogram.


MP.7



7 in.
How do these measurements compare to the base and height of the parallelogram?




They are the same.

When we moved the right triangle, did the area inside the shape change? Explain.




3 in.

The area did not change because both shapes are the same size. The original quadrilateral just looks different. 21 square inches or 21 inches squared or 21 in2 .

What is the area of the rectangle?


Note: English language learners would benefit from a discussion on why all three of these answers represent the same value. If the area of the rectangle is 21 square inches, what is the area of the original parallelogram? Why?






The area of the original parallelogram is also 21 square inches because both shapes have the same amount of space inside.





The formula to calculate the area of a parallelogram would be the same as a rectangle, = ℎ.

We know the formula for the area of a rectangle is Area = base × height, or = ℎ. What is the formula to calculate the area of a parallelogram?
Examine the given parallelogram, and label the base and height.


Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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17

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM


MP.7

6•5

Why is the height the vertical line and not the slanted edge?

Note: English language learners may need a further explanation of the meaning of the slanted edge.
If we look back to the rectangle we created, the base and height of both the rectangle and the original parallelogram are perpendicular to each other. Therefore, the height of a parallelogram is the perpendicular line segment drawn from the top base to the bottom base.



Exercise 1 (5 minutes)
Students work individually to complete the following problems.
Exercises
1.

Find the area of each parallelogram below. Note that the figures are not drawn to scale.
a.

b.

c.

.

=

.

= ( )
=

. .

Scaffolding:
English language learners may need some clarification about what it means to not be drawn to scale and why this may be the case.

=

= ( )
=

=

= . ( . )
=

Discussion (8 minutes)
Give each student a copy of the slanted parallelogram shown below.


How could we construct a rectangle from this parallelogram?



Why can’t we use the same method we used previously?




Answers will vary.
The vertical dotted line does not go through the entire parallelogram.

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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18

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Students may struggle drawing the height because they will not be sure whether part of the height can be outside of the parallelogram. 

Cut out the shape.



To solve this problem, we are actually going to cut the parallelogram horizontally into four equal pieces. Use the appropriate measurement tool to determine where to make the cuts.

Allow time for students to think about how to approach this problem. If time allows, have students share their thoughts before the teacher demonstrates how to move forward.
Demonstrate these cuts before allowing students to make the cuts.



We have four parallelograms. How can we use them to calculate the area of the original parallelogram?




How can we make these parallelograms into rectangles?




Cut a right triangle off of every parallelogram, and move the right triangle to the other side of the parallelogram. How can we show that the original parallelogram forms a rectangle?




Turn each of the parallelograms into rectangles.

If we push all the rectangles together, they will form one rectangle.

Therefore, it does not matter how tilted a parallelogram is. The formula to calculate the area will always be the same as the area formula of a rectangle.

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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19

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM



6•5

Draw and label the height of the parallelogram below.

Students should connect two dotted lines as below:



height

base
Exercise 2 (5 minutes)
Students complete the exercises individually.
2.

Draw and label the height of each parallelogram. Use the correct mathematical tool to measure (in inches) the base and height, and calculate the area of each parallelogram.
a.

base

height base = = (. . )( . ) =
b.

height

base

= = (. . )( . ) =

Lesson 1:
Date:

base

The Area of Parallelograms Through Rectangle Facts
11/4/14

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20

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

c.

base height = = ( . )( . ) =
3.

If the area of a parallelogram is

and the height is

=

÷ =

=

=



� � ÷

area of the parallelogram. Solve the equation. �

, write an equation that relates the height, base, and

Scaffolding:
English language learners may benefit from a sentence starter such as “The formulas are the same because …”

Closing (3 minutes)


Why are the area formulas for rectangles and parallelograms the same?


The area formulas for rectangles and parallelograms are the same because a parallelogram can be changed to a rectangle. By cutting a right triangle from one side of the parallelogram and connecting it to the other side of the parallelogram, a rectangle is formed.

Lesson Summary

The formula to calculate the area of a parallelogram is = , where represents the base and represents the height of the parallelogram.
The height of a parallelogram is the line segment perpendicular to the base. The height is usually drawn from a vertex that is opposite the base.

Exit Ticket (5 minutes)

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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21

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 1: The Area of Parallelograms Through Rectangle Facts
Exit Ticket
Calculate the area of each parallelogram. Note that the figures are not drawn to scale.
1.
10 ft.

12 ft.

20 ft.

2.
42 cm
5 cm

35 cm
15 cm

3.

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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22

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Calculate the area of each parallelogram. Note that the figures are not drawn to scale.
1.

.

.

.

= = . ( . ) =
2.

= = ( ) =
3.

= = ( ) =

Problem Set Sample Solutions
Draw and label the height of each parallelogram.
1.

height base base

2. height base

base

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

23

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Calculate the area of each parallelogram. The figures are not drawn to scale.
3.

=

= ( )
=

4.

. .

=

. .

= . . (. . )
= .

. .
5.

.

.

.

6.

Lesson 1:
Date:

.

m

=

. � . �

= . � . �

=

=

=

=

� �

= � �

=

=

=

The Area of Parallelograms Through Rectangle Facts
11/4/14

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24

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

7.

6•5

Brittany and Sid were both asked to draw the height of a parallelogram. Their answers are below.
Brittany

Sid

height

height

base

base

Are both Brittany and Sid correct? If not, who is correct? Explain your answer.
Both Brittany and Sid are correct because both of their heights represent a line segment that is perpendicular to the base and whose endpoint is on the opposite side of the parallelogram.
8.

Do the rectangle and parallelogram below have the same area? Explain why or why not.

.

.

.

.

.

Yes, the rectangle and parallelogram have the same area because if we cut off the right triangle on the left side of the parallelogram, we can move it over to the right side and make the parallelogram into a rectangle. At this time, both rectangles would have the same dimensions; therefore, their areas would be the same.
9.

A parallelogram has an area of . . and a base of . . Write an equation that relates the area to the base and height, . Solve the equation to determine the length of the height.
. = . ()

. ÷ . = . () ÷ .

. =

Therefore, the height of the parallelogram is . .

Lesson 1:
Date:

The Area of Parallelograms Through Rectangle Facts
11/4/14

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25

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Number Correct: ______

Multiplication of Fractions—Round 1
1 3
×
2 4

8 3
×
9 4

Directions: Determine the product of the fractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

5 5
×
6 7

16.

1 3
×
4 7

19.

2 7
×
9 9

22.

2 9
×
3 10

25.

5 3
×
8 10

28.

3 7
×
4 8
5 4
×
7 9

20.

1 2
×
3 5

23.

3 1
×
5 6

26.

4 7
×
5 8

3 4
×
4 7

17.

29.

4 8
×
5 9
3 1
×
5 8

21.

3 5
×
7 8

24.

2 3
×
7 4

1 8
×
4 9

18.

27.

Lesson 1:
Date:

3 10
×
5 11

8
7
×
13 24

1
3
2 ×3
2
4
4
1
1 ×6
5
3
2
5
8 ×4
7
6
2
1
5 ×2
5
8
6
1
4 ×1
7
4
2
2
2 ×4
3
5

6

9
1
×7
10
3

3
2
1 ×4
8
5

5
4
3 ×2
6
15
1
4 ×5
3

30.

The Area of Parallelograms Through Rectangle Facts
11/4/14

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26

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Multiplication of Fractions—Round 1 [KEY]
1 3
×
2 4

8 3
×
9 4

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

22.

=

25.

=

28.

20.

23.

=

26.

=

29.

3 1
×
5 8
2 7
×
9 9
1 2
×
3 5
3 5
×
7 8

2 9
×
3 10
3 1
×
5 6
2 3
×
7 4

5 3
×
8 10
4 7
×
5 8

24.

=

5 4
×
7 9

21.

1 3
×
4 7

27.

30.

=

1 8
×
4 9

18.

4 8
×
5 9

3 4
×
4 7

17.

3 7
×
4 8

Lesson 1:
Date:

19.

5 5
×
6 7

16.

=

3 10
×
5 11

Directions: Determine the product of the fractions.

8
7
×
13 24

1
3
2 ×3
2
4
4
1
1 ×6
5
3

=

=

=

=

=

2
5
8 ×4
7
6

=

2
2
2 ×4
3
5

=

2
1
5 ×2
5
8
6
1
4 ×1
7
4

6

9
1
×7
10
3

3
2
1 ×4
8
5

5
4
3 ×2
6
15
1
4 ×5
3

=

=

=

=

=

=

The Area of Parallelograms Through Rectangle Facts
11/4/14

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27

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

Number Correct: ______
Improvement: ______

Multiplication of Fractions—Round 2
5 1
×
6 4

3 2
×
7 9

Directions: Determine the product of the fractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

2 5
×
3 7

16.

3 7
×
8 9

19.

1 3
×
4 4

22.

6 5
×
7 8

25.

5 2
×
6 3

28.

1 2
×
3 5
3 5
×
4 6

20.

5 3
×
8 10

23.

1 9
×
6 10

26.

1 8
×
4 11

4 10
×
5 13

17.

29.

5 5
×
7 8
2 3
×
7 8

21.

6 1
×
11 2

24.

3 8
×
4 9

2 3
×
9 8

18.

27.

Lesson 1:
Date:

6•5

1 4
×
8 5

3 2
×
7 15

1
3
1 ×4
2
4
5
3
2 ×3
6
8
7
1
1 ×5
8
5
2
3
6 ×2
3
8
1
6
7 ×3
2
7
3×4

1
3

3
1
2 ×5
5
6
2
4 ×7
5

4
1
1 ×2
7
2
5 3
3 ×
6 10

30.

The Area of Parallelograms Through Rectangle Facts
11/4/14

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28

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Multiplication of Fractions—Round 2 [KEY]
5 1
×
6 4

3 2
×
7 9

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

=

22.

=

25.

=

28.

20.

=

23.

=

26.

=

29.

2 3
×
7 8
1 3
×
4 4

5 3
×
8 10
6 1
×
11 2
6 5
×
7 8

1 9
×
6 10
3 8
×
4 9
5 2
×
6 3

1 8
×
4 11

24.

=

3 5
×
4 6

21.

=

3 7
×
8 9

=

2 3
×
9 8

18.

=

5 5
×
7 8

4 10
×
5 13

17.

=

1 2
×
3 5

Lesson 1:
Date:

19.

2 5
×
3 7

16.

=

1 4
×
8 5

Directions: Determine the product of the fractions.

27.

3 2
×
7 15

1
3
1 ×4
2
4

=

=

=

=

5
3
2 ×3
6
8

=

1
6
7 ×3
2
7

=

2
4 ×7
5

=

7
1
1 ×5
8
5
2
3
6 ×2
3
8
3×4

1
3

3
1
2 ×5
5
6
4
1
1 ×2
7
2
5 3
3 ×
6 10

30.

=

=

=

=

=

=

The Area of Parallelograms Through Rectangle Facts
11/4/14

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29

Lesson 1

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1:
Date:

6•5

The Area of Parallelograms Through Rectangle Facts
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30

Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 2: The Area of Right Triangles
Student Outcomes


Students justify the area formula for a right triangle by viewing the right triangle as part of a rectangle composed of two right triangles.

Lesson Notes
For students to complete the Exploratory Challenge, they will need the attached templates to this lesson, as well as scissors, a ruler, and glue. Students may need more than one copy of each triangle.
Students will use the attached template to develop the formula necessary to calculate the area of a right triangle. The templates will also allow students to visualize why the area of a right triangle is exactly half of the area of a rectangle with the same dimensions. They will calculate the area of two different right triangles to see that the formula works for more than just the first triangle given. Once students develop the formula, they can use substitution and the given dimensions to calculate the area.

Classwork
Discussion (1 minute)


What are some properties of a right triangle?

One interior angle must be exactly 90°.
Three-sided polygon.




Exploratory Challenge (14 minutes)
Students work in groups of 2–3 to discover the formula that can be used to calculate the area of a right triangle. Each group will need the templates attached to this lesson, glue, a ruler, and scissors.
Exploratory Challenge
a.

MP.1

Use the shapes labeled with an X to predict the formula needed to calculate the area of a right triangle. Explain your prediction. ×

= × × or =

Formula for the area of right triangles:

× . × . =

Area of the given triangle: =

Lesson 2:
Date:

Scaffolding:
It students are struggling, use some guiding questions:
 What do you know about the area of a rectangle?
 How are the area of a triangle and rectangle related?  Can you fit the triangle inside the rectangle?

The Area of Right Triangles
11/3/14

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31
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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

Use the shapes labeled with a Y to determine if the formula you discovered in part (a) is correct.
Does your area formula for triangle Y match the formula you got for triangle X?
Answers will vary; however, the area formulas should be the same if students discovered the correct area formula. If so, do you believe you have the correct formula needed to calculate the area of a right triangle? Why or why not?

MP.1

Answers will vary.
If not, which formula do you think is correct? Why?
Answers will vary.

× .× . = .

Area of the given triangle: =

Discussion (5 minutes)





1

ℎ, or = .
2
2

Each right triangle represents half of a rectangle. The area formula of a rectangle is = ℎ, but since a right triangle only covers half the area of a rectangle, we take the area of the rectangle and multiply it by half, or divide by 2.

How do we know this formula is correct?




The area formula of a right triangle is =

What is the area formula for right triangles?

How can we determine which side of a right triangle is the base and which side is the height?


Similar to a parallelogram, the base and the height of a right triangle are perpendicular to each other, so they form the right angle of the triangle. However, it does not matter which of these two sides are labeled the base and which is labeled the height. The commutative property of multiplication allows us to calculate the formula in any order.

Exercises (15 minutes)
Students complete each exercise independently. Students may use a calculator.
Exercises
Calculate the area of each right triangle below. Each figure is not drawn to scale.
1.

.

.

= ( . )( . )

=

.

Lesson 2:
Date:

=

The Area of Right Triangles
11/3/14

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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

2.

.

.

3.

.

(. )(. )

= .

= ( .)( .)

=

.

=

=

.

.

4.

=

6•5

=

=
=
=
=

� � � �


� �


or

5.

.

.

Lesson 2:
Date:

= (. )(. )

=

= .

The Area of Right Triangles
11/3/14

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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

6.

6•5

Mr. Jones told his students they each need a half of a piece of paper. Calvin cut his piece of paper horizontally and
Matthew cut his piece of paper diagonally. Which student has the larger area on their half piece of paper? Explain.
Calvin’s Paper

Matthew’s Paper

After cutting the paper, both Calvin and Matthew have the same area. Calvin cut his into two rectangles that are each half the area of the original piece of paper. Matthew cut his paper into two equivalent right triangles that are also half the area of the original piece of paper.
7.

Ben requested that the rectangular stage be split into two equal sections for the upcoming school play. The only instruction he gave was that he needed the area of each section to be half of the original size. If Ben wants the stage to be split into two right triangles, did he provide enough information? Why or why not?
Ben did not provide enough information because the stage may be split horizontally or vertically through the middle of the rectangle. This would result in two equal pieces, but they would not be right triangles.
If the area of a right triangle is . . . and its base is . ., write an equation that relates the area to the height, , and the base. Solve the equation to determine the height.

8.

(. .)

. = (. .)
. =

. ÷ . . = (. .) ÷ . . . =

Therefore, the height of the right triangle is .

Closing (5 minutes)


How are the area formulas of rectangles and right triangles related?


When a rectangle and a right triangle have the same dimensions, the area of the right triangle is exactly half of the area of the rectangle. Therefore, the area formulas of rectangles and right triangles are related because the area formula of a rectangle is divided by 2 (or multiplied by a half) in order to translate it to the area formula for a right triangle.

Exit Ticket (5 minutes)

Lesson 2:
Date:

The Area of Right Triangles
11/3/14

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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 2: The Area of Right Triangles
Exit Ticket
1.

Calculate the area of the right triangle. Each figure is not drawn to scale.
6 in.

2.

8 in.

10 in.

Dan and Joe are responsible for cutting the grass on the local high school soccer field. Joe cuts a diagonal line through the field, as shown in the diagram below, and says that each person is responsible for cutting the grass on one side of the line. Dan says that this is not fair because he will have to cut more grass than Joe. Is Dan correct?
Why or why not?

SOCCER FIELD

Lesson 2:
Date:

The Area of Right Triangles
11/3/14

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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
1.

2.

= ( .)( .) =

Calculate the area of the right triangle. Each figure is not drawn to scale. =

.

. .

Dan and Joe are responsible for cutting the grass on the local high school soccer field. Joe cuts a diagonal line through the field and says that each person is responsible for cutting the grass on one side of the line. Dan says that this is not fair because he will have to cut more grass than Joe. Is Dan correct? Why or why not?
SOCCER FIELD

Dan is not correct. The diagonal line Joe cut in the grass would split the field into two right triangles. The area of each triangle is exactly half the area of the entire field because the area formula for a right triangle is =

× × .

Problem Set Sample Solutions
.

Calculate the area of each right triangle below. Note that the figures are not drawn to scale.
1.

=
2.

= (. )(. ) = .

=

.

.

= ( ) � � = � � �
� = =

Lesson 2:
Date:

The Area of Right Triangles
11/3/14

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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

3.

. .

4.

.

. .

= = (. .)(. .) = .

=
5.

= ( )( ) =

.

=
6.

.

.

= � .� ( .) = �
.� �
.� = =

Elania has two congruent rugs at her house. She cut one vertically down the middle, and she cut diagonally through the other one.

C
A

B

D

After making the cuts, which rug (labeled A, B, C, or D) has the larger area? Explain.
All of the rugs are the same size after making the cuts. The vertical line goes down the center of the rectangle, making two congruent parts. The diagonal line also splits the rectangle in two congruent parts because the area of a right triangle is exactly half the area of the rectangle.

Lesson 2:
Date:

The Area of Right Triangles
11/3/14

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Lesson 2

NYS COMMON CORE MATHEMATICS CURRICULUM

7.

6•5

Give the dimensions of a right triangle and a parallelogram with the same area. Explain how you know.
Answers will vary.

8.

. . and the height is

, and the height. Solve the equation to determine the base.

If the area of a right triangle is

= � .�

= � .�

÷ . = � .� ÷ .

. =

. =

Therefore, the base of the right triangle is

Lesson 2:
Date:

., write an equation that relates the area to the base,

.

The Area of Right Triangles
11/3/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 2:
Date:

Lesson 2

6•5

The Area of Right Triangles
11/3/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 2:
Date:

Lesson 2

6•5

The Area of Right Triangles
11/3/14

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Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 3: The Area of Acute Triangles Using Height and
Base
Student Outcomes


Students show the area formula for a triangular region by decomposing a triangle into right triangles. For a given triangle, the height of the triangle is the length of the altitude. The length of the base is called either the length base or, more commonly, the base.



Students understand that the height of the triangle is the perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. Students understand that any side of a triangle can be considered a base and that the choice of base determines the height.

Lesson Notes
For this lesson, students will need the triangle template attached to this lesson and a ruler.
Throughout the lesson, students will determine if the area formula for right triangles is the same as the formula used to calculate the area of acute triangles.

Classwork
Fluency Exercise (5 minutes): Multiplication of Decimals
Sprint: Refer to the Sprints and the Sprint Delivery Script sections of the Module Overview for directions to administer a
Sprint.

Discussion (5 minutes)


What is different between the two triangles below?




One triangle is a right triangle because it has one right angle; the other does not have a right angle, so it is not a right triangle.
1
2

= × base × height

How do we find the area of the right triangle?


Lesson 3:
Date:

The Area of Acute Triangles Using Height and Base
11/4/14

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41

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM



How do we know which side of the right triangle is the base and which is the height?




6•5

If you choose one of the two shorter sides to be the base, then the side that is perpendicular to this side will be the height.

How do we calculate the area of the other triangle?


We do not know how to calculate the area of the other triangle because we do not know its height.

Mathematical Modeling Exercise (10 minutes)
Students will need the triangle template found at the end of the lesson and a ruler to complete this example. To save class time, cut out the triangles ahead of time.


The height of a triangle does not always have to be a side of the triangle. The height of a triangle is also called the altitude, which is a line segment from a vertex of the triangle and perpendicular to the opposite side.

Note: English language learners may benefit from a poster showing each part of a right triangle and acute triangle (and eventually an obtuse triangle) labeled, so they can see the height and altitude and develop a better understanding of the new vocabulary words.
Model how to draw the altitude of the given triangle.


MP.3

Fold the paper to show where the altitude would be located, and then draw the altitude, or the height, of the triangle. 

Notice that by drawing the altitude we have created two right triangles. Using the knowledge we gained yesterday, can we calculate the area of the entire triangle?




We can calculate the area of the entire triangle by calculating the area of the two right triangles and then adding these areas together.

Measure and label each base and height. Round your measurements to the nearest half inch.




Outline or shade each right triangle with a different color to help students see the two different triangles.

1.5 in.

1
1
ℎ = (5 in.)(3 in.) = 7.5 in2
2
2
1
1 = ℎ = (1.5 in.)(3 in.) = 2.25 in2
2
2

=

Calculate the area of each right triangle.





5 in.

3 in.

Scaffolding:

Now that we know the area of each right triangle, how can we calculate the area of the entire triangle?


To calculate the area of the entire triangle, we can add the two areas together.

Lesson 3:
Date:

The Area of Acute Triangles Using Height and Base
11/4/14

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42

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM



= 7.5 in2 + 2.25 in2 = 9.75 in2

Calculate the area of the entire triangle.




6•5

Talk to your neighbor, and try to determine a more efficient way to calculate the area of the entire triangle.

Allow students some time to discuss their thoughts.

Answers will vary. Allow a few students to share their thoughts.



Test a few of the students’ predictions on how to find the area of the entire triangle faster. The last prediction you should try is the correct one shown below.
1
2

= × base × height. Some of you believe we can still use this same formula for the given triangle.



In the previous lesson, we said that the area of right triangles can be calculated using the formula



Draw a rectangle around the given triangle.


5 in.

MP.3


1.5 in.

Does the triangle represent half of the area of the rectangle? Why or why not?




3 in.

The triangle does represent half of the area of the rectangle. If the altitude of the triangle splits the rectangle into two separate rectangles, then the slanted sides of the triangle split these rectangles into two equal parts.
The length of the base is 6.5 inches because we have to add the two parts together.

What is the length of the base?


The height is 3 inches because that is the length of the line segment that is perpendicular to the base.



What is the length of the altitude (the height)?



Calculate the area of the triangle using the formula we discovered yesterday, =






1
1
ℎ = (6.5 in.)(3 in.) = 9.75 in2
2
2

Is this the same area we got when we split the triangle into two right triangles?




=

1
× base × height.
2

Yes.

It is important to determine if this is true for more than just this one example.

Exercises (15 minutes) formula, =

1
ℎ, is always correct. One partner calculates the area of the given triangle by calculating the area of two
2

Students work with partners on the exercises below. The purpose of the first exercise is to determine if the area

right triangles, and the other partner calculates the area just as one triangle. Partners should switch who finds each area in order to provide every student with a chance to practice both methods. Students may use a calculator as long as they record their work on their paper as well.

Lesson 3:
Date:

The Area of Acute Triangles Using Height and Base
11/4/14

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43

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

Exercises
1.

Work with a partner on the exercises below. Determine if the area formula =

6•5

is always correct. You may

use a calculator, but be sure to record your work on your paper as well. Figures are not drawn to scale.

.

( )( )

= =

.

. .

. .

Lesson 3:
Date:

= + . = .

(. . )(. . )

= .

= . + . . = . .

=

(. . )(. . )

= . =

( . )(. . )

= .

=

( . ) � . �

= � . � � . �

=
=

=

= . +

� . � � . �

= � . � � . �

=
=

= +
=
+
=

( )( )

= =

. = .

� . � � . �

= � . � � . �

=
=

=

=

.

(. )( )

= .

=

= . + . = .

.

= + . = .

(. )( )

= .

MP.2

.

Area of Entire Triangle

=

.

. .

Area of Two Right Triangles

( )( )

= =

= + =

= + =

( )( )

=

=

The Area of Acute Triangles Using Height and Base
11/4/14

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44

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

2.

Can we use the formula =

Explain your thinking.
Yes, the formula =

6•5

× × to calculate the area of triangles that are not right triangles?

× × can be used for more than just right triangles. We just need to be able to

determine the height, when it is not the length of one of the sides.
3.

Examine the given triangle and expression.

.

MP.2

.

( . )( . )

.

. represents the base of the triangle because . + . = .

Explain what each part of the expression represents according to the triangle. . represents the altitude of the triangle because this length is perpendicular to the base.

4.

= ( .)( .) + ( .)( .). Explain how each student approached the problem.

Joe found the area of a triangle by writing = ( .)( .), while Kaitlyn found the area by writing
Joe combined the two bases of the triangle first, and then calculated the area of the entire triangle, whereas Kaitlyn calculated the area of two smaller right triangles, and then added these areas together.

5.

The triangle below has an area of . . . If the base is . ., let be the height in inches.

a.

Explain how the equation . = (. . ) represents the situation.

b.

Solve the equation.

The equation shows the area, . , is one half the base, . ., times the height, in inches, .

(. .)

. = (. .)
. =

. ÷ . . = (. .) ÷ . .
. . =

Lesson 3:
Date:

The Area of Acute Triangles Using Height and Base
11/4/14

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45

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Closing (5 minutes)


When a triangle is not a right triangle, how can you determine its base and height?




The height of a triangle is the length of the altitude. The altitude is the line segment from a vertex of a triangle to the line containing the opposite side (or the base) that is perpendicular to the base.

How can you use your knowledge of area to calculate the area of more complex shapes? Show students the shape to the right.


Decompose the shape into smaller shapes for which we know how to calculate the area, and then add all the areas together.

Exit Ticket (5 minutes)

Lesson 3:
Date:

The Area of Acute Triangles Using Height and Base
11/4/14

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46

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 3: The Area of Acute Triangles Using Height and Base
Exit Ticket
Calculate the area of each triangle using two different methods. Figures are not drawn to scale.
1.

8 ft.

3 ft.

7 ft.

12 ft.

2.

9 in.

32 in.
18 in.

Lesson 3:
Date:

36 in.

The Area of Acute Triangles Using Height and Base
11/4/14

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47

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Calculate the area of each triangle. Figures are not drawn to scale.
1.

.

.

.

2.

.

.

.

.

.

=

=

( . )( . ) = .

( . )( . ) =

= . + = .

OR

= =

=

( . )( . ) = .

( . )( . ) =

( . )( . ) =

= + =

OR

=

( . )( . ) =

Problem Set Sample Solutions
Calculate the area of each shape below. Figures are not drawn to scale.
1.
. .

=

. .

. .

Lesson 3:
Date:

. .

=

(. . )(. . ) = .

(. . )(. . ) = .

= . + . = .

OR

=

(. . )(. . ) = .

The Area of Acute Triangles Using Height and Base
11/4/14

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48

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

2.

3. . .

=

=

6•5

( )( ) =

( )( ) =

= + =

OR

.

.

=

.

. .

4.

=

( )( ) =

( . )( . ) =

= ( . )( . ) =

=

( . )( . ) =

= + + = =

( )( ) =

= ( ) = =

( )( ) =

= + + =

5.

Immanuel is building a fence to make an enclosed play area for his dog. The enclosed area will be in the shape of a triangle with a base of . and an altitude of . How much space does the dog have to play? =

= ( . )( . ) =

The dog will have to play.
6.

Chauncey is building a storage bench for his son’s playroom. The storage bench will fit into the corner and against two walls to form a triangle. Chauncey wants to buy a cover for the bench.
If the storage bench is

. along one wall and . along the other wall, how big will

� . � � . � = � . � � . � = =

the cover have to be to cover the entire bench? =

Chauncey would have to buy a cover that has an area of

Lesson 3:
Date:

to cover the entire bench.

The Area of Acute Triangles Using Height and Base
11/4/14

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49

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

7.

6•5

Examine the triangle to the right.
a.

b.

( . )( . ) + ( . )( . ) or ( . )( . )

Write an expression to show how you would calculate the area.

If students wrote the first expression, then . and . represent the two parts of the base, and . is the height, or the altitude, of the triangle.

Identify each part of your expression as it relates to the triangle.

.

.

.

.

If students wrote the second expression, then . represents the base because . + . = ., and . represents the height, or the altitude, of the triangle.

8.

A triangular room has an area of

. . If the height is , write an equation to determine the length of

the base, , in meters. Then solve the equation.

Lesson 3:
Date:

= � �

= �


÷ = �
� ÷

=

=

The Area of Acute Triangles Using Height and Base
11/4/14

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50

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Number Correct: ______

Multiplication of Decimals—Round 1
Directions: Determine the products of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

4.5 × 3
7.2 × 8
8.3 × 4

22.

5.9 × 10

25.

6 × 2.8

28.

4 × 8.9

31.

1.8 × 8

3.5 × 4.1

19.

34.

9.4 × 6
5.8 × 2

23.

3.4 × 3

26.

9.7 × 3

29.

3.9 × 7

32.

9 × 2.3

9.3 × 1.7

20.

35.

10.2 × 7
7.1 × 9

24.

3.2 × 4

27.

8 × 10.2

30.

6 × 5.5

10.4 × 7.6

21.

33.

Lesson 3:
Date:

2.7 × 8.3
1.8 × 7.8

7.5 × 10.1
7.2 × 6.3
1.9 × 8.3
9.8 × 5.1

18.2 × 12
13.4 × 22
92.3 × 45
86.1 × 16

29.7 × 8.2
56.8 × 9.5

110.3 × 20.2
256.6 × 54.9
312.8 × 16.5

36.

The Area of Acute Triangles Using Height and Base
11/6/14

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51

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Multiplication of Decimals—Round 1 [KEY]
Directions: Determine the products of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

4.5 × 3

.
.
.

.

28.

.

31.

.

34.

.

23.

.

26.

.

29.

.

32.

.

3.4 × 3
3.2 × 4
6 × 2.8
9.7 × 3

8 × 10.2
4 × 8.9
3.9 × 7
6 × 5.5
1.8 × 8

35.

9 × 2.3

30.

.

5.9 × 10

27.

.

7.1 × 9

24.

.

5.8 × 2

10.4 × 7.6

21.

.

8.3 × 4

.

9.3 × 1.7

20.

.

10.2 × 7

Lesson 3:
Date:

25.

.

9.4 × 6

.

2.7 × 8.3

22.

7.2 × 8

3.5 × 4.1

19.

33.

1.8 × 7.8

7.5 × 10.1
7.2 × 6.3
1.9 × 8.3
9.8 × 5.1

18.2 × 12
13.4 × 22

.
.
.
.
.
.
.
.
.

92.3 × 45

.

56.8 × 9.5

.

86.1 × 16

29.7 × 8.2

110.3 × 20.2
256.6 × 54.9
312.8 × 16.5

36.

.

.

.

.
.

The Area of Acute Triangles Using Height and Base
11/6/14

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52

Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

Number Correct: ______
Improvement: ______

Multiplication of Decimals—Round 2
Directions: Determine the products of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

3.7 × 8

9.2 × 10
3.3 × 5

22.

1.9 × 2

25.

4 × 9.8

28.

3 × 10.2

31.

8.2 × 6

4.6 × 5.2

19.

34.

2.1 × 3
7.4 × 4

23.

5.6 × 7

26.

5 × 8.7

29.

2.8 × 6

32.

4.5 × 9

6.8 × 1.9

20.

35.

4.8 × 9
8.1 × 9

24.

3.6 × 8

27.

1.4 × 7

30.

3.9 × 9

7.8 × 10.4

21.

33.

Lesson 3:
Date:

6•5

3.8 × 3.9
9.3 × 4.2
1.4 × 9.5
9.4 × 2.7
5.6 × 4.2
8.6 × 3.1

14.5 × 19

33 × 10.2
51 × 32.4
45 × 17.6

15.2 × 6.7
39.5 × 8.4

96.8 × 31.7

189.1 × 72.9
302.4 × 13.1

36.

The Area of Acute Triangles Using Height and Base
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Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Multiplication of Decimals—Round 2 [KEY]
Directions: Determine the products of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

3.7 × 8

.

.

.

28.

.

31.

.

34.

.

23.

.

26.

.

29.

.

32.

.

5.6 × 7
3.6 × 8
4 × 9.8
5 × 8.7
1.4 × 7

3 × 10.2
2.8 × 6
3.9 × 9
8.2 × 6

35.

4.5 × 9

30.

.

1.9 × 2

27.

.

8.1 × 9

24.

.

7.4 × 4

7.8 × 10.4

21.

.

3.3 × 5

.

6.8 × 1.9

20.

.

4.8 × 9

Lesson 3:
Date:

25.

.

2.1 × 3

.

3.8 × 3.9

22.

.

9.2 × 10

4.6 × 5.2

19.

33.

9.4 × 4.2
1.4 × 9.5
9.4 × 2.7
5.6 × 4.2
8.6 × 3.1

14.5 × 19

33 × 10.2

.
.
.
.

.
.
.
.
.

51 × 32.4

.

39.5 × 8.4

.

45 × 17.6

15.2 × 6.7

96.8 × 31.7

189.1 × 72.9
302.4 × 13.1

36.

.

.

.

.
.

The Area of Acute Triangles Using Height and Base
11/6/14

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Lesson 3

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 3:
Date:

6•5

The Area of Acute Triangles Using Height and Base
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55

Lesson 4

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 4: The Area of All Triangles Using Height and Base
Student Outcomes


Students construct the altitude for three different cases: an altitude that is a side of a right angle, an altitude that lies over the base, and an altitude that is outside the triangle.



Students deconstruct triangles to justify that the area of a triangle is exactly one half the area of a parallelogram. Lesson Notes
Students will need the attached templates, scissors, a ruler, and glue to complete the Exploratory Challenge.

Classwork
Opening Exercise (5 minutes)
Opening Exercise
Draw and label the altitude of each triangle below.

altitude

altitude

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Date:

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6•5

altitude

Discussion (3 minutes)


to show that the formula works for three different types of triangles.



Examine the triangles in the Opening Exercise. What is different about them?

The height, or altitude, is in a different location for each triangle. The first triangle has an altitude inside the triangle. The second triangle has a side length that is the altitude, and the third triangle has an altitude outside of the triangle.





1
2

The last few lessons showed that the area formula for triangles is = × base × height. Today we are going

1
2

We will use = × base × height because that is the area formula we have used for both right

If we wanted to calculate the area of these triangles, what formula do you think we would use? Explain.


triangles and acute triangles.

Exploratory Challenge/Exercises 1–5 (22 minutes)
Students work in small groups to show that the area formula is the same for all three types of triangles shown in the
Opening Exercise. Each group will need the attached templates, scissors, a ruler, and glue. Each exercise comes with steps that might be useful to provide for students who work better with such scaffolds.
Exploratory Challenge
1.

MP.1

=

× × .

Use rectangle X and the triangle with the altitude inside (triangle x) to show that the area formula for the triangle is
a.

b.

= . × . . = .

Step One: Find the area of rectangle X.

Half of the area of the rectangle is . ÷ = . .

Step Two: What is half the area of rectangle X?

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Date:

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Lesson 4

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c.

6•5

Step Three: Prove, by decomposing triangle X, that it is the same as half of rectangle X. Please glue your decomposed triangle onto a separate sheet of paper. Glue it next to rectangle X. What conclusions can you make about the triangle’s area compared to the rectangle’s area?

Students will cut their triangle and glue it into half of the rectangle. This may take more than one try, so extra copies of the triangles may be necessary.
Because the triangle fits inside half of the rectangle, we know the triangle’s area is half of the rectangle’s area. 2.

is =

× × .

Use rectangle Y and the triangle with a side that is the altitude (triangle Y) to show the area formula for the triangle = . × . =

a.

Step One: Find the area of rectangle Y.

b.

Half the area of the rectangle is ÷ = . .

c.

Step Three: Prove, by decomposing triangle Y, that it is the same as half of rectangle Y. Please glue your decomposed triangle onto a separate sheet of paper. Glue it next to rectangle Y. What conclusions can you make about the triangle’s area compared to the rectangle’s area?

Step Two: What is half the area of rectangle Y?

Students will again cut triangle Y and glue it into the rectangle. This may take more than one try, so extra copies of the triangles may be necessary.
MP.1

The right triangle also fits in exactly half of the rectangle, so the triangle’s area is once again half the size of the rectangle’s area.
3.

formula for the triangle is =

× × .

Use rectangle Z and the triangle with the altitude outside (triangle Z) to show that the area
a.

= . × . . = .

Step One: Find the area of rectangle Z.

Scaffolding:
 Students may struggle with this step since they have yet to see an obtuse angle. The teacher may want to model this step if he or she feels students may become confused.
 After modeling, the students can then try this step on their own.

Half of the area of the rectangle is . ÷ = . .

b.

Step Two: What is half the area of rectangle Z?

c.

Step Three: Prove, by decomposing triangle Z, that it is the same as half of rectangle Z. Please glue your decomposed triangle onto a separate sheet of paper. Glue it next to rectangle Z. What conclusions can you make about the triangle’s area compared to the rectangle’s area?

Students will cut their triangle and glue it onto the rectangle to show that obtuse triangles also have an area that is half the size of a rectangle that has the same dimensions. This may take more than one try, so extra copies of the triangles may be necessary.

Lesson 4:
Date:

The Area of All Triangles Using Height and Base
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6•5

Note: In order for students to fit an obtuse triangle into half of a rectangle, they will need to cut the triangle into three separate triangles.
Similar to the other two triangles, when the altitude is outside the triangle, the area of the triangle is exactly half of the area of the rectangle.

MP.1

4.

When finding the area of a triangle, does it matter where the altitude is located? =

× × .

It does not matter where the altitude is located. To find the area of a triangle the formula is always

5.

How can you determine which part of the triangle is the base and which is the height?
The base and the height of any triangle form a right angle because the altitude is always perpendicular to the base.

Take time to show how other groups may have calculated the area of the triangle using a different side for the base and how this still results in the same area.
After discussing how any side of a triangle can be labeled the base, students write a summary to explain the outcomes of the Exploratory Challenge.

Exercises 6–8 (5 minutes)
Exercises 6–8
Calculate the area of each triangle. Figures are not drawn to scale. .

6.

.

= ( .)( .) =

.

=

.

.

.

.

.

.

� . � � . � = �
. � �
. � = =

Draw three triangles (acute, right, and obtuse) that have the same area. Explain how you know they have the same area. Answers will vary.

Lesson 4:
Date:

The Area of All Triangles Using Height and Base
11/4/14

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6•5

Closing (5 minutes)


Different groups share their Exploratory Challenge and discuss the outcomes.



Why does the area formula for a triangle work for every triangle?


Every type of triangle fits inside exactly half of a rectangle that has the same base and height lengths.

Exit Ticket (5 minutes)

Lesson 4:
Date:

The Area of All Triangles Using Height and Base
11/4/14

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60

Lesson 4

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 4: The Area of All Triangles Using Height and Base
Exit Ticket
Find the area of each triangle. Figures are not drawn to scale.
1.

21 cm

12.6 cm

2.

16.8 cm

25 in.

20 in.

15 in.

17 in.

8 in.

3.
29 ft.
12 ft.

8 ft.

Lesson 4:
Date:

21 ft.

The Area of All Triangles Using Height and Base
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61

Lesson 4

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Find the area of each triangle. Figures are not drawn to scale.
1.

.

.

= (. )(. ) = .

.

=

.

.

.

.

( .)( .) =

.

.

.

.

= ( .)( .) =

Problem Set Sample Solutions
Calculate the area of each triangle below. Figures are not drawn to scale.
1.

.

.

.

.

.

= ( .)( .) =

Lesson 4:
Date:

The Area of All Triangles Using Height and Base
11/4/14

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62

Lesson 4

NYS COMMON CORE MATHEMATICS CURRICULUM

=

( )( ) =

. =

6•5

.
.

.

(. )(. ) = .

( )( ) =

= ( )( ) = .

=

= ( ) =

= + + . + = .

5.

The Andersons are going on a long sailing trip during the summer. However, one of the sails on their sailboat ripped, and they have to replace it. The sail is pictured below.

If the sailboat sails are on sale for $ per square foot, how much will the new sail cost?

= ( .)( .)

=

=

$
× = $

The cost of the new sail is $.

Lesson 4:
Date:

The Area of All Triangles Using Height and Base
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6•5

Darnell and Donovan are both trying to calculate the area of an obtuse triangle. Examine their calculations below.

Darnell’s Work

× . × .

=

Donovan’s Work

× . × .

=

=

=

Which student calculated the area correctly? Explain why the other student is not correct.
Donovan calculated the area correctly. Although Darnell did use the altitude of the triangle, he used the length between the altitude and the base rather than the length of the actual base.
Russell calculated the area of the triangle below. His work is shown.

× ×

= . =

Although Russell was told his work is correct, he had a hard time explaining why it is correct. Help Russell explain why his calculations are correct.
The formula for the area of the a triangle is =

. Russell followed this formula because is the height of

the triangle, and is the base of the triangle.

8.

The larger triangle below has a base of . ; the gray triangle has an area of . .

a.

b.

Determine the area of the larger triangle if it has a height of . .

(. )(. )

= .

=

Let be the area of the unshaded (white) triangle in square meters. Write and solve an equation to determine the value of , using the areas of the larger triangle and the gray triangle.
. + = .

. + − . = . − . = .

Lesson 4:
Date:

The Area of All Triangles Using Height and Base
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Lesson 4:
Date:

6•5

The Area of All Triangles Using Height and Base
11/4/14

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Lesson 4

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Lesson 4:
Date:

6•5

The Area of All Triangles Using Height and Base
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66

Lesson 5

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 5: The Area of Polygons Through Composition and
Decomposition
Student Outcomes


Students show the area formula for the region bounded by a polygon by decomposing the region into triangles and other polygons. They understand that the area of a polygon is actually the area of the region bounded by the polygon.



Students find the area for the region bounded by a trapezoid by decomposing the region into two triangles.
They understand that the area of a trapezoid is actually the area of the region bounded by the trapezoid.
Students decompose rectangles to determine the area of other quadrilaterals.

Lesson Notes
This graphic can be displayed for students to make sense of the second part of each Student Outcome.

Decomposing irregularly shaped polygons into rectangles involves making a choice of where to separate the figure. This very often involves calculating the length of unknown sides of the new figures. This may be more intuitive for some students than others.
MP.2 Mastering missing length problems will make the objectives of this lesson more easily
& achieved.
MP.7
When decomposing irregularly shaped polygons into triangles and other polygons, identifying the base and height of the triangle also sometimes requires missing length skills. Classwork
Opening Exercise (5 minutes): Missing Length Problems
There are extra copies of this figure at the end of this lesson. Project this image with a document camera or interactive white board, if desired. Specify the length of two horizontal lengths and two vertical lengths, and have students find the missing side lengths. Highlighting vertical sides in one color and horizontal sides in another color is valuable for many students.

Lesson 5:
Date:

Scaffolding:
The words composition and decomposition are likely new words. The base word, compose, is a verb that means the act of joining or putting together. Decompose means the opposite, to take apart. In this lesson, the words composition and decomposition are used to describe how irregular figures can be separated into triangles and other polygons. The area of these parts can then be added together to calculate the area of the whole figure.

The Area of Polygons Through Composition and Decomposition
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67

Lesson 5

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6•5

Opening Exercise
Here is an aerial view of a woodlot.
A

F

B

E

D

C

If = units, = units, = units, and = units, find the lengths of both other sides. = units

= units

If = units, = units, = units, and = units, find the lengths of both other sides. = units

= units

Lesson 5:
Date:

Scaffolding:
If students have difficulty seeing these relationships, it can be helpful to show progressions of figures, such as those below, which move gradually from the sides of a rectangle to an irregular rectilinear figure. Consistent use of visuals, as well as manipulatives, such as cut-outs of these figures, will aid in understanding. The Area of Polygons Through Composition and Decomposition
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6•5

Discussion (5 minutes)
If students are struggling to see this relationship, it might be helpful for them to complete the rectangle that encloses the figure:



How do you know which operation to use when finding missing side lengths?





If we know two short sides (vertical or horizontal), we add to find the longer side.
If we know the long side and one short side (vertical or horizontal), we subtract.

These examples used whole numbers for the lengths of the sides. What would you do if there were decimal lengths? 


MP.7

Would the process be the same for deciding whether to add or subtract?




One critical step is making sure you add and subtract numbers that have the same place value by lining up the decimal points.

What if the lengths were given as fractions or mixed numbers?




Yes.

When adding or subtracting decimals, what is one step that is critical to arriving at the correct answer?




We would add or subtract the decimal numbers.

We would add or subtract the fractions or the mixed numbers.

Would the process be the same for deciding whether to add or subtract?


Yes.

Ask students to find the next diagram on their classwork page. Work through the scenario with them. The area of this figure can be found in at least three ways: using two horizontal cuts, using two vertical cuts, or subtracting the missing area from the larger rectangle (using overall length and width). There is a drawing included at the end of this lesson that has the grid line inserted.

Lesson 5:
Date:

The Area of Polygons Through Composition and Decomposition
11/4/14

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Lesson 5

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6•5

MP.1



How could we determine the total area?




Let’s divide the figure using two horizontal lines. Will that make any rectangles with two known sides?





No, the entire 7 m side cannot be used. Part has been removed, two 2 m segments, leaving only 3 m.

2 m by 9 m, 2 m by 9 m, and 3 m by 5 m

2 m × 9 m = 18 m2 , 2 m × 9 m = 18 m2 , 3 m × 5 m = 15 m2

Calculate and mark each of these areas.




Yes, it makes two 2 by 9 rectangles.

What are the dimensions of the three resulting rectangles?




51 m2

Can we then use this 7 m measure directly? Why or why not?




2 m × 4 m = 8 m2 , 2 m × 4 m = 8 m2 , 7 m × 5 m = 35 m2

Divide the next figure using two vertical lines. Will that make any rectangles with two known sides?




2 m by 4 m, 2 m by 4 m, and 7 m by 5 m.

What is the total area of the figure?




Some students will benefit from actually cutting the irregularly shaped polygons before marking the dimensions on the student pages. If needed, there are reproducible copies included at the end of the lesson.

Calculate and mark each of these areas.




The entire 9 m side cannot be used. Part has been removed, 4 m, leaving only 5 m. We use subtraction.

Scaffolding:

What are the dimensions of the three resulting rectangles?




No. The 9 m includes the top part of the figure, but we have already found the dimensions of this part.

What is the height of that third rectangle and how do you find it?




Yes, it makes two 2 by 4 rectangles.

Can we then use this 9 m measure directly? Why or why not?




Using two horizontal lines, two vertical lines, or one of each.

51 m2

What is the total area of the figure?


Lesson 5:
Date:

The Area of Polygons Through Composition and Decomposition
11/4/14

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Lesson 5

NYS COMMON CORE MATHEMATICS CURRICULUM



Yes, both sides of the 5 by 5 rectangle had to be found by decomposing the other measures.

Divide the last figure using one vertical line and one horizontal line. Are there missing sides to calculate?















MP.2

6•5

What are the dimensions of the three resulting rectangles?
 2 m by 9 m, 2 m by 4 m, and 5 m by 5 m
Calculate and mark each of these areas.
 2 m × 9 m = 18 m2 , 2 m × 4 m = 8 m2 , 5 m × 5 m = 25 m2
What is the total area of the figure?
 51 m2
Finally, if we look at this as a large rectangle with a piece removed, what are the dimensions of the large rectangle?  9 m by 7 m
What are the dimensions of the missing piece that looks like it was cut out?
 3 m by 4 m
Calculate these two areas.
 9 m × 7 m = 63 m2 , 3 m × 4 m = 12 m2
How can we use these two areas to find the area of the original figure?
 Subtract the smaller area from the larger one.
What is the difference between 63 m2 and 12 m2 ?
 63 m2 − 12 m2 = 51 m2
Is there an advantage to one of these methods over the others?
 Answers will vary. In this example, either one or two calculations are necessary when decomposing the figure. Consider the two expressions: 18 m2 + 8 m2 + 25 m2 and 63 m2 − 12 m2 .
What do the terms in these expressions represent in this problem?
 The first is a “sum of the parts” expression, and the second is a “whole minus part” expression. More specifically, the first expression shows that the total area is the sum of the areas of three rectangles; the second expression shows that the total area is the area of a large rectangle minus the area of a small one.

Allow some time for discussion before moving on.

Example 1 (10 minutes): Decomposing Polygons into Rectangles
Example 1: Decomposing Polygons into Rectangles
The Intermediate School is producing a play that needs a special stage built. A diagram is shown below (not to scale).
a.

On the first diagram, divide the stage into three rectangles using two horizontal lines. Find the dimensions of these rectangles and calculate the area of each. Then, find the total area of the stage.
Dimensions: by , by , and by

Area: × = , × = , × =

Total: + + =

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Date:

The Area of Polygons Through Composition and Decomposition
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b.

6•5

On the second diagram, divide the stage into three rectangles using two vertical lines. Find the dimensions of these rectangles and calculate the area of each. Then, find the total area of the stage.
Dimensions: by , by , and by

Area: × = , × = , × =

Total:

c.

On the third diagram, divide the stage into three rectangles using one horizontal line and one vertical line.
Find the dimensions of these rectangles and calculate the area of each. Then, find the total area of the stage.
Dimensions: by , m by , and by

Area: × = , × = , × =

Total:

d.

Think of this as a large rectangle with a piece removed.
Dimensions: by and by

i.

What are the dimensions of the large rectangle and the small rectangle?

ii.

Area: × = , × =

iii.

What are the areas of the two rectangles?

What operation is needed to find the area of the original figure?
Subtraction
− =

iv.

What is the difference in area between the two rectangles?

v.

What do you notice about your answers to (a), (b), (c), and (d)?
The area is the same.

vi.

Why do you think this is true?
No matter how we decompose the figure, the total area is the sum of its parts. Even if we take the area around the figure and subtract the part that is not included, the area of the figure remains the same, .
Lesson 5:
Date:

The Area of Polygons Through Composition and Decomposition
11/4/14

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6•5

Scaffolding:
1

As an extension, ask students to manipulate these unsimplified numerical expressions to demonstrate equivalence of areas (MP.2).

2

3

1

For example, using the factors of the area, showing that
2×4+2×4+7×5=
2×9+2×9+3×5 by applying the distributive property and using decomposition of whole numbers. 2
3

Using the products (areas), the equivalence should be made clear: 8 + 8 + 35 = 18 + 18 + 15
= 63 − 12

Example 2 (10 minutes): Decomposing Polygons into Rectangles and Triangles
In this example, a parallelogram is bisected along a diagonal. The resulting triangles are congruent, with the same base and height of the parallelogram. Students should see that the area for a parallelogram is equal to the base times the height, regardless of how much the bases are skewed. Ask how we could find the area using only triangles.
Parallelogram is part of a large solar power collector. The base measures and the height is .
Example 2: Decomposing Polygons into Rectangles and Triangles

a.

Draw a diagonal from to . Find the area of both triangles and .
Student drawing and calculations are shown here.

Lesson 5:
Date:



=

= ( )( )

=

Scaffolding:
Some students will benefit from actually cutting the parallelograms from paper to prove their congruency. There are reproducible copies included. △

=

= ( )( )

=

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12 m2

What is the area of each triangle?




6•5

24 m2

What is the area of the parallelogram?

b.

Draw in the other diagonal, from to . Find the area of both triangles and .

Student drawing and calculations are shown here.



=

= ( )( )

=



=

= ( )( )

=

Example 3 (10 minutes): Decomposing Trapezoids
Drawing one of the diagonals in a trapezoid separates the figure into two non-congruent triangles. Note that the height of these triangles is the same if the two bases of the trapezoid are used as bases of the triangles. If students want to consider the area of the rectangle around the trapezoid, two exterior right triangles will be formed. For isosceles trapezoids, these triangles will be congruent. For scalene trapezoids, two non-congruent triangles will result. A reproducible copy of trapezoids is included at the end of this lesson for use in further investigation. In all cases, the area can be found by averaging the length of the bases and multiplying by the height.


What is the area of the garden plot? Use what you know about decomposing and composing to determine the area. Example 3: Decomposing Trapezoids
The trapezoid below is a scale drawing of a garden plot.

If students need prompting, ask them to draw a diagonal from to .

Lesson 5:
Date:

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Find the area of both triangles and . Then find the area of the trapezoid.


=

= ( )( )

=

Student drawing and calculations are shown here.

6•5



=

= ( )( )

=

= + =

If necessary, further prompt students to draw in the other diagonal, from to .
Find the area of both triangles and . Then find the area of the trapezoid.

Student drawing and calculations are shown here.



=

= ( )( )

=



=

= ( )( )

=

= + =

How else could we find this area?

We could consider the rectangle that surrounds the trapezoid. Find the area of that rectangle, and then subtract the area of both right triangles.

Student drawing and calculations are shown here. =

Area of Rectangle

= × =



=

= ( )( )

= .



=

= ( )( )

= .

= − . − . =

= − (. + . ) =
OR

Lesson 5:
Date:

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6•5

Closing (2 minutes)


How can we find the area of irregularly shaped polygons?




They can be broken into rectangles and triangles; we can then calculate the area of the figure using the formulas we already know.

Which operations did we use today to find the area of our irregular polygons?


Some methods used addition of the area of the parts. Others used subtraction from a surrounding rectangle. Exit Ticket (3 minutes)

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Date:

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Name ___________________________________________________

6•5

Date____________________

Lesson 5: The Area of Polygons Through Composition and
Decomposition
Exit Ticket
1.

Find the missing dimensions of the figure below, and then find the area. The figure is not drawn to scale.

2.

Find the area of the parallelogram below. The figure is not drawn to scale.

Lesson 5:
Date:

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6•5

Exit Ticket Sample Solutions
1.

Find the missing dimensions of the figure below, and then find the area. The figure is not drawn to scale.

Solutions can be any of the below.

2.

Find the area of the parallelogram below. The figure is not drawn to scale.

8 mi.

= × . × .

=

=

= × . × .

=

=

= +

The area of the parallelogram is .

Lesson 5:
Date:

= + =

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6•5

Problem Set Sample Solutions
1.

If = , = , = , and = , find the length of both other sides. Then, find the area of the irregular polygon.

= , = , =

2.

If = . , = . , = . , and = . , find the length of both other sides. Then, find the area of the irregular polygon.

= . , = . , = .
3.

Determine the area of the trapezoid below. The trapezoid is not drawn to scale.

= × ×

=

= × ×

Area of Triangle 1

=

=

Area of Triangle 2

=

= + = + =

Lesson 5:
Date:

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4.

6•5

Determine the area of the isosceles trapezoid below. The image is not drawn to scale. =

Area of Rectangle

= × =

= × . ×

=

Area of Triangles 1 and 2

=

= − − = − − =

5.

Here is a sketch of a wall that needs to be painted:

a.

Whole wall: . × . =

The windows and door will not be painted. Calculate the area of the wall that will be painted.
Window: .× . = There are two identical windows, × =

Door: .× . =

− − =

b.

If a quart of Extra-Thick Gooey Sparkle paint covers , how many quarts must be purchased for the painting job? ÷ =

Therefore, quarts must be purchased.

Lesson 5:
Date:

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6.

6•5

The figure below shows a floor plan of a new apartment. New carpeting has been ordered, which will cover the living room and bedroom but not the kitchen or bathroom. Determine the carpeted area by composing or decomposing in two different ways, and then explain why they are equivalent.

MP.7

Bedroom: . × . =

Answers will vary. Sample student responses are shown.

Living room: . × . =

Sum of bedroom and living room: + =
Alternatively, the whole apartment is . × . =

Subtracting the kitchen and bath ( and ) still gives .

The two areas are equivalent because they both represent the area of the living room and bedroom.

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Date:

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Date:

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A

B

F

F

E

D

C

A

B

F

D

B

E

D

A

F

E

Lesson 5:
Date:

A

C

6•5

C

B

E

D

C

The Area of Polygons Through Composition and Decomposition
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Date:

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The Area of Polygons Through Composition and Decomposition
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Date:

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The Area of Polygons Through Composition and Decomposition
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6•5

Horizontal
V
e r t i c a l

Lesson 5:
Date:

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Lesson 6

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 6: Area in the Real World
Student Outcomes


Students determine the area of composite figures in real-life contextual situations using composition and decomposition of polygons.



Students determine the area of a missing region using composition and decomposition of polygons.

Lesson Notes
Finding area in real-world contexts can be done around the classroom, in a hallway, or in different locations around the school. This lesson will require the teacher to measure and record the dimensions of several objects and calculate the area ahead of time. Choices will be dependent on time available and various students’ needs. Different levels of student autonomy can be taken into account when grouping and deciding which objects will be measured. Further, the measurement units and precision can be adjusted to the students’ ability level.
Floor tile, carpet area, walls, and furniture in the classroom can be used for this lesson. Smaller objects within the classroom may also be used, such as bulletin boards, notebooks, windows, and file cabinets. Exploring the school building for other real-world area problems might lead to a stage in an auditorium or walkway around a school pool. Of course, adhere to school policy regarding supervision of students, and be vigilant about safety. Students should not have to climb to make measurements.
Throughout the lesson, there are opportunities to compare unsimplified numerical expressions. These are important and should be emphasized because they help prepare students for algebra.

Classwork

Gauge students’ ability level regarding which units and level of precision will be used in this lesson. Using metric units
MP.5 for length and height of the classroom wall will most likely require measuring to the nearest 0.1 meter or 0.01 meter and
&
MP.6 will require multiplying decimals to calculate area. Choosing standard units allows precision to be set to the nearest foot, half foot, etc. but could require multiplying fractional lengths.
Scaffolding:

Discussion (5 minutes)
Decide if the whole group will stay in the classroom or if carefully selected groups will be sent out on a measurement mission to somewhere outside the classroom. All students will need to understand which measurement units to use and to what precision they are expected to measure.


Area problems in the real world are all around us. Can you give an example of when you might need to know the area of something?


Area needs to be considered when covering an area with paint, carpet, tile, or wallpaper; wrapping a present; etc.

Lesson 6:
Date:

As noted in the classwork section, there is great flexibility in this lesson, so it can be tailored to the needs of the class and can be easily individualized for both struggling and advanced learners. English language learners might need a minilesson on the concept of wallpaper with accompanying visuals and video, if possible.

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Lesson 6

NYS COMMON CORE MATHEMATICS CURRICULUM



The Problem Set from the last lesson had a wall that was to be painted. What measurement units were used in that problem?
All linear measurements were made in feet. Paint was calculated in quarts.




How precisely were the measurements made?
Measurements were most likely measured to the nearest foot. Paint was rounded up to the next quart.




Could those measurements have been made more precisely?
Yes, measurements could have been made to the nearest inch, half inch, or some other smaller fraction of an inch. Paint can be purchased in pints.




6•5

We can measure the dimensions of objects and use those measurements to calculate the surface area of the object. Our first object will be a wall in this classroom.

Exploratory Challenge 1: Classroom Wall Paint (34 minutes)
Scaffolding:

Exploratory Challenge 1: Classroom Wall Paint
The custodians are considering painting our classroom next summer. In order to know how much paint they must buy, the custodians need to know the total surface area of the walls. Why do you think they need to know this, and how can we find the information?
All classroom walls are different. Taking overall measurements then subtracting windows, doors, or other areas will give a good approximation.
Make a prediction of how many square feet of painted surface there are on one wall in the room.
If the floor has square tiles, these can be used as a guide.

This same context can be worded more simply for ELL students, and beginner-level students would benefit from a quick pantomime of painting a wall. A short video clip might also set the context quickly.

Students make a prediction of how many square feet of painted surface there are on one wall in the room. If the floor has square tiles, these can be used as a guide.
Decide beforehand the information in the first three columns. Measure lengths and widths, and calculate areas. Ask students to explain their predictions.
Estimate the dimensions and the area. Predict the area before you measure. My prediction: ______________ .
a.

Measure and sketch one classroom wall. Include measurements of windows, doors, or anything else that would not be painted.
Student responses will be determined by the teacher’s choice of wall.

Object or item to be measured Measurement units Precision
(measure to the nearest)

door

feet

half foot

Lesson 6:
Date:

Length

Width

.

.

Expression that shows the area

t. × .

Area

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b.

c.

6•5

Work with your partners and your sketch of the wall to determine the area that will need paint. Show your sketch and calculations below; clearly mark your measurements and area calculations.
A gallon of paint covers about . Write an expression that shows the total area.
Evaluate it to find how much paint will be needed to paint the wall.

Answers will vary based on the size of the wall. Fractional answers are to be expected.
d.

How many gallons of paint would need to be purchased to paint the wall?
Answers will vary based on the size of the wall. The answer from part (d) should be an exact quantity because gallons of paint are discrete units. Fractional answers from part (c) must be rounded up to the nearest whole gallon. Exploratory Challenge 2 (Optional—15 minutes)
Assign other walls in the classroom for groups to measure and calculate the area, or send some students to measure and sketch other real-world area problems found around the school. The teacher should measure the objects prior to the lesson using the same units and precision the students will be using. Objects may have to be measured multiple times if the activity has been differentiated using different units or levels of precision.
Exploratory Challenge 2
Object or item to be measured

Measurement units Precision
(measure to the nearest) Door

feet

half foot

Length

Width

.

.

Area

Closing (3 minutes)


What real-life situations require us to use area?




Floor covering, like carpets and tiles, require area measurements. Wallpaper and paint also call for area measurements. Fabric used for clothing and other items also demand that length and width be considered. Wrapping a present, installing turf on a football field, or laying bricks, pavers, or concrete for a deck or patio are other real-world examples.

Sometimes measurements are given in inches and area is calculated in square feet. How many square inches are in a square foot?


There are 144 square inches in a square foot, 12 in. × 12 in. = 144 in2

Exit Ticket (3 minutes)

Lesson 6:
Date:

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Lesson 6

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 6: Area in the Real World
Exit Ticket
Find the area of the deck around this pool. The deck is the white area in the diagram.

Lesson 6:
Date:

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Lesson 6

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Find the area of the deck around this pool. The deck is the white area in the diagram.

=

Area of Walkway and Pool

= ×

=

− =

Area of Pool

=

Area of Walkway

= × =

Problem Set Sample Solutions
1.

Below is a drawing of a wall that is to be covered with either wallpaper or paint. The wall is . high and . long. The window, mirror, and fireplace will not be painted or papered. The window measures . by . The fireplace is . wide and . high, while the mirror above the fireplace is . by .

a.

Total wall area = . × . =

Window area = . × . . =

How many square feet of wallpaper are needed to cover the wall?

Fireplace area = . × . =

Mirror area = . × . =

Net wall area to be covered − ( + + ) =

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Date:

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Lesson 6

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

The wallpaper is sold in rolls that are . wide and . long. Rolls of solid color wallpaper will be used, so patterns do not have to match up.

b.

i.

Area of one roll of wallpaper: . × . . = .

ii.

How many rolls would be needed to cover the wall?

What is the area of one roll of wallpaper?

÷ . ≈ . ; therefore, rolls would need to be purchased.

c.

This week, the rolls of wallpaper are on sale for $. /roll. Find the cost of covering the wall with wallpaper. d.

A gallon of special textured paint covers and is on sale for $. /gallon. The wall needs to be painted twice (the wall needs two coats of paint). Find the cost of using paint to cover the wall.

We need two rolls of wallpaper to cover the wall, which will cost $. × = $. .

Total wall area= . × . =
Window area= . × . . =
Fireplace area= . × . =
Mirror area= . × . =

Net wall area to be covered − ( + + ) =

If the wall needs to be painted twice, we need to paint a total area of × = . One gallon is enough paint for this wall, so the cost will be $. .

2.

A classroom has a length of . and a width of . The flooring is to be replaced by tiles. If each tile has a length of . and a width of ., how many tiles are needed to cover the classroom floor?

Area of the classroom: . × . =
Area of each tile: . × . =

=
=

Allow for students who say that if the tiles are . × ., and they orient them in a way that corresponds to the . × . room, then they will have ten rows of ten tiles giving them tiles. Using this method, the students do not need to calculate the areas and divide. Orienting the tiles the other way, students could say that they will

need tiles as they will need rows of tiles, and since of tiles.

3.

of a tile cannot be purchased, they will need rows

Challenge: Assume that the tiles from Problem 2 are unavailable. Another design is available, but the tiles are square, . on a side. If these are to be installed, how many must be ordered?

Solutions will vary. An even number of tiles fit on the foot width of the room ( tiles), but the length requires

tiles. This accounts for a tile by tile array tiles × tiles = tiles.

The remaining area is . × . . ( ×
Since of the

)

tiles are needed, additional tiles must be cut to form . of these will be used with of tile

Using the same logic as above, some students may correctly say they will need tiles. left over.

Lesson 6:
Date:

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Lesson 6

NYS COMMON CORE MATHEMATICS CURRICULUM

4.

6•5

A rectangular flower bed measures by . It has a path wide around it. Find the area of the path.
Total area: × =

Flower bed area: × = 2

Area of path: − = 2

5.

Tracy wants to cover the missing portion of his deck with soil in order to grow a garden.
a.

Find the missing portion of the deck. Write the expression and evaluate it.

Students will use one of two methods to find the area: finding the dimensions of the garden area (interior rectangle, × ) or finding the total area minus the sum of the four wooden areas, shown below.

× =

× − × − × − × − × = (All linear units are in meters; area is in square meters.)

OR

b.

Find the missing portion of the deck using a different method. Write the expression and evaluate it.
Students should choose whichever method was not used in part (a).

c.

× − × − × − × − ×

Write your two equivalent expressions. ×

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Date:

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NYS COMMON CORE MATHEMATICS CURRICULUM

d.

6.

6•5

One expression shows the dimensions of the garden area (interior rectangle, × ), and one shows finding the total area minus the sum of the four wooden areas.
Explain how each demonstrates a different understanding of the diagram.

The entire large rectangle below has an area of

. If the dimensions of the white rectangle are as shown

below, write and solve an equation to find the area, , of the shaded region.

. × . =

+ =

=

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Date:

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New York State Common Core

6

Mathematics Curriculum

GRADE

GRADE 6 • MODULE 5

Topic B:

Polygons on the Coordinate Plane
6.G.A.3
Focus Standard:

6.G.A.3

Instructional Days:

4

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

Lesson 7: Distance on the Coordinate Plane (P) 1
Lesson 8: Drawing Polygons in the Coordinate Plane (P)
Lesson 9: Determining Perimeter and Area of Polygons on the Coordinate Plane (P)
Lesson 10: Distance, Perimeter, and Area in the Real World (E)

In Lesson 7 of Topic B, students apply prior knowledge from Module 3 by using absolute value to determine the distance between integers on the coordinate plane in order to find side lengths of polygons. Then they move to Lesson 8, where students draw polygons in the coordinate plane when given coordinates for vertices. They find the area enclosed by a polygon by composing and decomposing, using polygons with known area formulas. They name coordinates that define a polygon with specific properties. In Lesson 9, students find the perimeter of rectilinear figures using coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. They continue to find the area enclosed by a polygon on the coordinate plane by composition and decomposition. The topic concludes with Lesson 10, where students apply their knowledge of distance, perimeter, and area to real-life contextual situations.
Students learn more than a key word reading of contexts. They comprehend different problem contexts and apply concepts accordingly.

1

Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

Topic B:
Date:

Polygons on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 7: Distance on the Coordinate Plane
Student Outcomes


Students use absolute value to determine distance between integers on the coordinate plane in order to find side lengths of polygons.

Lesson Notes
Students build on their work in Module 3. More specifically, they build on their work with absolute value from Lessons
11 and 12 as well as on their work with coordinate planes from Lessons 17–19.
Also note that each square unit on the coordinate planes represents 1 unit.

Classwork
Fluency Exercise (5 minutes): Addition of Decimals
Sprint: Refer to the Sprints and the Sprint Delivery Script sections of the Module Overview for directions to administer
Sprints.

Example (15 minutes)
Example
Determine the lengths of the given line segments by determining the distance between the two endpoints.
Line
Segment

MP.8

����

����

����

�����

����

����

����

����

����

����

����

Point

Point

(, )

Distance

(, −)

(, )

(−, −)

(−, )

(−, )

(−, )
(, )
(, )
(, )

(−, −)
(−, −)
(−, −)
(−, −)
(−, −)

Lesson 7:
Date:

(, −)
(, −)

(−, −)
(−, )
(−, )
(−, )
(−, )

Proof

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM



In each pair, either the -coordinates are the same or the -coordinates are the same.

What do you notice about each pair of points?




6•5

How could you calculate the lengths of the segments using only the coordinate of their endpoints? (Please note, it is possible that ELLs may not understand this question and may need modeling to understand. In addition, students may need to be reminded that distances or lengths are positive.)

Either the -values will be the same or the -values will be the same. We will ignore these and focus on the coordinates that are different. We can subtract the absolute values of the endpoints if both points have the same sign. If the signs are different, we will add the absolute values.
Why are the steps different? For example, why are the steps for ���� different than the steps for ���� ?





It may be helpful for students to go back to the image and walk through the steps visually when trying to describe the steps and the difference between the two.


MP.8


When we determine the distance from to , we are really adding together the distance from to the
-axis and the distance from to the -axis. We add the distances together because they are on opposite sides of the -axis. When determining the distance from to , we are taking the distance from to the -axis and to the -axis and finding the difference because they are on the same side of the -axis.

Add a fourth column to the table to show proof of your distances.
Line Segment
����

����

Point
(−, )

Distance

(, −)

(−, −)

(−, )

(−, )

(, )

����

�����

(, )
(, )

����

����

(−, −)

����

����

(−, −)
(−, −)

����

����

(−, −)

����



Point
(, )

(−, −)

(, )

(, −)
(, −)

(−, −)
(−, )
(−, )
(−, )
(−, )

Proof
|| + | − | =
|| − || =

|| + | − | =

|| + | − | =
|| + | − | =

| − | − | − | =

| − | + || =

| − | + || =

| − | + || =
|| − || =

| − | + || =

How would the distances from one point to another change if each square unit on the plane were 2 units in length? Or 3 units in length?


The distance would double if each square unit were worth 2 units. The distance would triple if each square unit were actually equal to 3 units in length.

Lesson 7:
Date:

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exercise (15 minutes)
Exercise
Complete the table using the diagram on the coordinate plane.

����

(−, )

�����

(−, −)

����

(, −)

Line Segment
�����

����

����

����

����

����

����

Point

Distance

(−, )

(−, )

(−, −)

(−, −)

(−, −)

(, −)

| − | − | − | =

| − | − | − | =

|| + | − | =

Point

(−, )

(−, −)
(−, −)
(−, −)
(−, )

(−, −)

(, −)

(, −)

(−, −)
(−, )

(−, −)

|| + || =
Proof

|| + | − | =

|| + | − | =
| − | + || =
| − | + || =

| − | − | − | =
| − | + || =

Extension (3 minutes)
For each problem below, write the coordinates of two points that are units apart with the segment connecting these points having the following characteristics.
Extension

a.

b.

Answers may vary. One possible solution is (, ) and (, ).
The segment is vertical.

The segment intersects the -axis.

Answers may vary. One possible solution is (, −) and (, ).

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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

c.

The segment intersects the -axis.

d.

6•5

The segment is vertical and lies above the -axis.

Answers may vary. One possible solution is (−, ) and (, ).
Answers may vary. One possible solution is (−, ) and (−, ).

Closing (2 minutes)


What did all of the segments used in the lesson have in common?




How could you determine whether the segments were vertical or horizontal given the coordinates of their endpoints? 



They were all either vertical or horizontal.
If the -coordinates were the same for both points, then the segment was vertical. If the -coordinates were the same, then the segment was horizontal.

How did you calculate the length of the segments given the coordinates of the endpoints?


If the coordinates that were not the same had the same sign, we subtracted the absolute values of the coordinates. 

If the coordinates that were not the same had different signs, we added the absolute values of the coordinates. Exit Ticket (5 minutes)

Lesson 7:
Date:

Distance on the Coordinate Plane
11/4/14

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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 7: Distance on the Coordinate Plane
Exit Ticket

Use absolute value to show the lengths of ���� , ���� , ���� , ���� , and ���� .

Line
Segment
����

Point

Point

Distance

Proof

����

����

����

����

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Date:

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Use absolute value to show the lengths of ����, ����, ����, ����, and ����.

Line
Segment

Point

����

(, )

(−, −)

(, −)

����

(, )

����

(, )

����

Distance

(, )

(−, )

����

Point

(, −)

(, −)

(, −)

Proof

| − | + ||
|| − ||

|| + | − |

| − | − | − |
|| + | − |

Problem Set Sample Solutions
1.

Given the pairs of points, determine whether the segment that joins them will be horizontal, vertical, or neither.
a.
b.
c.

2.

(, ) and (−, )

Horizontal

(−, ) and (, −)

Neither

(−, ) and (−, )

Vertical

Complete the table using absolute value to determine the lengths of the line segments.
Line
Segment

Point

����

(−, )

����

(−, )

����

(, −)

����

(, −)

�����

Lesson 7:
Date:

(, )

Point

(, )

Distance

(−, −)
(−, )

(, −)
(, )

Proof

| − | + ||
|| + | − |

|−| − | − |
|| − ||
|| + ||

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

3.

Complete the table using the diagram and absolute value to determine the lengths of the line segments.
Line
Segment
����

(−, )

����

(−, −)

����

����

����

����

4.

6•5

Point

(, )
(, )

(−, )

(−, −)

Point

(, )

Distance

(, )

(−, −)

| − | + ||

|−| − | − |

(−, )

(−, −)
(−, )

Proof

|| − ||

|| + | − |

|| + | − |

|−| + ||

Complete the table using the diagram and absolute value to determine the lengths of the line segments.
Line
Segment
����

(−, )

�����

(, −)

����

����

����

L

����

����

����

Point

(, )
(, )
(, )

(, −)
(, −)
(−, )

Point

(, )

Distance

(, )

| − | + || =

(, −)

|| + || =

(−, )
(−, )

(−, −)
(, )

(−, )

Proof

|| − || =

|| + | − | =
|| + | − | =
|| + | − | =

| − | + || =
|| − || =

5.

Name two points in different quadrants that form a vertical line segment that is units in length.

6.

Name two points in the same quadrant that form a horizontal line segment that is units in length.

Answers will vary. One possible solution is (, ) and (, −).

Answers will vary. One possible solution is (−, −) and (−, −).

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Date:

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Number Correct: ______

Addition of Decimals—Round 1
Directions: Determine the sum of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

4.2 + 3.5

18.

9.2 + 2.8

19.

23.4 + 45.5

20.

45.2 + 53.7

21.

6.8 + 7.5

22.

5.62 + 3.17

23.

23.85 + 21.1

24.

32.45 + 24.77

25.

112.07 + 54.25

26.

64.82 + 42.7

27.

87.5 + 45.21

28.

16.87 + 17.3

29.

27.84 + 34.21

30.

114.8 + 83.71

31.

235.6 + 78.26

32.

78.04 + 8.29

33.

176.23 + 74.7

Lesson 7:
Date:

34.

89.12 + 45.5

416.78 + 46.5

247.12 + 356.78
9 + 8.47

254.78 + 9

85.12 + 78.99
74.54 + 0.97
108 + 1.75

457.23 + 106

841.99 + 178.01
154 + 85.3

246.34 + 525.66
356 + 0.874

243.84 + 75.3

438.21 + 195.7
85.7 + 17.63
0.648 + 3.08

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Addition of Decimals—Round 1 [KEY]
Directions: Determine the sum of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

4.2 + 3.5

.

9.2 + 2.8

23.4 + 45.5

.

45.2 + 53.7

.

6.8 + 7.5

.

5.62 + 3.17

.

23.85 + 21.1

.

32.45 + 24.77

.

112.07 + 54.25

.

64.82 + 42.7

.

87.5 + 45.21

.

16.87 + 17.3

.

27.84 + 34.21

.

114.8 + 83.71

.

235.6 + 78.26

.

78.04 + 8.29

.

176.23 + 74.7

Lesson 7:
Date:

.

18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

89.12 + 45.5

.

9 + 8.47

.

416.78 + 46.5

247.12 + 356.78

.
.

254.78 + 9

.

108 + 1.75

.

85.12 + 78.99
74.54 + 0.97

457.23 + 106

841.99 + 178.01
154 + 85.3

246.34 + 525.66

.
.

.

.

356 + 0.874

.

85.7 + 17.63

.

243.84 + 75.3

438.21 + 195.7
0.648 + 3.08

.
.
.

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

Number Correct: ______
Improvement: ______

Addition of Decimals—Round 2
Directions: Determine the sum of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

2.5 + 3.1

18.

7.4 + 2.5

19.

7.5 + 9.4

20.

23.5 + 31.2

21.

43.4 + 36.2

22.

23.08 + 75.21

23.

41.41 + 27.27

24.

102.4 + 247.3

25.

67.08 + 22.51

26.

32.27 + 45.31

27.

23.9 + 34.6

28.

31.7 + 54.7

29.

62.5 + 23.9

30.

73.8 + 32.6

31.

114.6 + 241.7

32.

327.4 + 238.9

33.

381.6 + 472.5

Lesson 7:
Date:

6•5

34.

24.06 + 31.97
36.92 + 22.19
58.67 + 31.28
43.26 + 32.87

428.74 + 343.58
624.85 + 283.61
568.25 + 257.36
841.66 + 382.62
526 + 85.47

654.19 + 346

654.28 + 547.3
475.84 + 89.3

685.42 + 736.5
635.54 + 582

835.7 + 109.54
627 + 225.7

357.23 + 436.77

Distance on the Coordinate Plane
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Lesson 7

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Addition of Decimals—Round 2 [KEY]
Directions: Determine the sum of the decimals.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

2.5 + 3.1

.

7.4 + 2.5

.

7.5 + 9.4

.

23.5 + 31.2

.

43.4 + 36.2

.

23.08 + 75.21

.

41.41 + 27.27

.

102.4 + 247.3

.

67.08 + 22.51

.

32.27 + 45.31

.

23.9 + 34.6

.

31.7 + 54.7

.

62.5 + 23.9

.

73.8 + 32.6

.

114.6 + 241.7

.

327.4 + 238.9

.

381.6 + 472.5

Lesson 7:
Date:

.

18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

24.06 + 31.97

.

43.26 + 32.87

.

36.92 + 22.19
58.67 + 31.28

.
.

428.74 + 343.58

.

841.66 + 382.62

, .

654.28 + 547.3

, .

624.85 + 283.61
568.25 + 257.36
526 + 85.47

654.19 + 346

475.84 + 89.3

685.42 + 736.5
635.54 + 582

835.7 + 109.54
627 + 225.7

357.23 + 436.77

.
.
.

, .
.

, .
, .
.
.

Distance on the Coordinate Plane
11/4/14

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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 8: Drawing Polygons on the Coordinate Plane
Student Outcomes


Given coordinates for the vertices, students draw polygons in the coordinate plane. Students find the area enclosed by a polygon by composing or decomposing using polygons with known area formulas.



Students name coordinates that define a polygon with specific properties.

Lesson Notes
Helping students understand the contextual pronunciation of the word coordinate may be useful. Compare it to the verb coordinate, which has a slightly different pronunciation and a different stress. In addition, it may be useful to revisit the singular and plural forms of this word vertex (vertices).

Classwork
Examples 1–4 (20 minutes)
Students graph all four examples on the same coordinate plane.
Plot and connect the points (, ), (, ), and (, ). Name the shape, and determine the area of the polygon.

Examples
1.

= ( )( )

= ( )

= .

Right Triangle =

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Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM



6•5

In this example, I subtracted the values of the coordinates. For , I subtracted the absolute value of the -coordinates. For , I subtracted the absolute value of the -coordinates.

How did you determine the length of the base and height?


Plot and connect the points (−, ), (−, ), and (−, ). Then give the best name for the polygon, and determine the area.

2.

The shape is a triangle. =

Area of Square

= ( )

=

( )( )

( )

Area of Triangle 2 =

=

=

=

Area of Triangle 1

=

= ( )( )

= ( )

=

Area of Triangle 3

=

= ( )( )

= ( )

= .

= − − − .

Total Area of Triangle = .


MP.1

How is this example different than the first?


The base and height are not on vertical and horizontal lines. This makes it difficult to determine the measurements and calculate the area.

Lesson 8:
Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM



6•5

What other methods might we try?

Students may not come up with the correct method in discussion and may need to be led to the idea. If this is the case, ask students if the shape can be divided into smaller pieces. Try drawing lines on the figure to show this method will not work. Then draw one of the outside triangles to show a triangle whose area could be determined, and help lead
MP.1 students to determine that the areas of the surrounding triangles can be found.




1
2

1
2

1
2

62 − (1)(6) − (6)(3) − (5)(3)

What expression could we write to represent the area of the triangle?




Answers will vary. We can draw a square around the outside of the shape. Using these vertical and horizontal lines, we can find the area of the triangles that would be formed around the original triangle.
These areas would be subtracted from the area of the square leaving us with the area of the triangle in the center.

The 62 represents the area of the square surrounding the triangle.

Explain what each part of the expression corresponds to in this situation.


MP.2




3.

The (1)(6) represents the area of triangle 1 that needs to be subtracted from the square.
1
2
1

The (6)(3) represents the area of triangle 2 that needs to be subtracted from the square.
2
1

The (5)(3) represents the area of triangle 3 that needs to be subtracted from the square.
2

Plot and connect the following points: (−, −), (−, −), (−, −), and (−, −). Give the best name for the polygon, and determine the area.
This polygon has sides and has no pairs of parallel sides.
Therefore, the best name for this shape is a quadrilateral.
To determine the area I will separate the shape into two triangles. Area of Triangle 1

=

= ( )( )

= ( )

=

MP.1

= ( )( )

= ( )

=

Area of Triangle 2 =

Total Area = +
Total Area =
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Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM



6•5

What method(s) could be used to determine the area of this shape?



MP.1


We could decompose the shape, or break the shape, into two triangles using a horizontal line segment to separate the two pieces.
We could also have used a similar method to Example 2, where we draw a rectangle around the outside of the shape, find the area of the pieces surrounding the quadrilateral, and then subtract these areas from the area of the rectangle.

In this case, which method is more efficient?




MP.2

(6)(3) + (2)(2)

What expression could we write to represent the area of the triangle?




It would be more efficient to only have to find the area of the two triangles, and then add them together. 1
2

1
2

The (6)(3)represents the area of triangle 1 that needs to be added to the rest of the shape.

Explain what each part of the expression corresponds to in this situation.


4.

1
2
1

The (2)(2) represents the area of triangle 2 that needs to be added to the rest of the shape.
2

Plot and connect the following points: (, −), (, −), (, −), (, −), and (, −). Give the best name for the polygon, and determine the area.
This shape is a pentagon.
Area of Shape 1

=

= ( )( )

= ( )

=

= ( )( )

= ( )

=

Area of Shape 2 and Shape 4 =

MP.1

Because there are two of the same triangle, that makes a total of . =

Area of Shape 3

= ( )( ) =

Total Area = + +
Total Area =
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Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM



Do we have a formula that we typically use to calculate the area of a pentagon?

MP.1




Answers will vary. We can break up the shape into triangles and rectangles, find the areas of these pieces, and then add them together to get the total area.
(8)(2) + 2 � (4)(2)� + (4)(4)

What expression could we write to represent the area of the pentagon?




No, we have formulas for different types of triangles and quadrilaterals.

How could we use what we know to determine the area of the pentagon?




This shape has 5 sides. Therefore, the best name is pentagon.

What is the best name for this polygon?




6•5

1
2

1
2

The (8)(2) represents the area of triangle 1 that needs to be added to the rest of the areas.

Explain what each part of the expression corresponds to in this situation.

MP.2





1
2
1

The (4)(2) represents the area of triangles 2 and 4 that needs to be added to the rest of the areas. It
2

is multiplied by 2 because there are two triangles with the same area.

The (4)(4) represents the area of rectangle 3 that needs to also be added to the rest of the areas.

Example 5 (5 minutes)
Two of the coordinates of a rectangle are (, ) and (, ). The rectangle has an area of square units. Give the possible locations of the other two vertices by identifying their coordinates. (Use the coordinate plane to draw and check your answer.)

5.

A

B

One possible location of the other two vertices is (, ) and
(, ). Using these coordinates will result in a distance, or side length, of .

Since the height is , × = .

Another possible location of the other two vertices is (−, ) and (−, ). Using these coordinates will result in a distance, or side length, of .
Since the height is , × = .

What is the length of ���� ?

Allow students a chance to try this question on their own first, and then compare solutions with a partner.



|7| − |2| = 7 − 2 = 5; therefore, = 5 units.

If one side of the rectangle is 5 units, what must be the length of the other side?




Since the area is 30 square units, the other length must be 6 units so that 5 × 6 will make 30.
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Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM



How many different rectangles can be created with segment as one side and the two sides adjacent to segment having a length of 6 units?




6•5

There are two different solutions. I could make a rectangle with two new points at (9, 7) and (9, 2), or
I could make a rectangle with two new points at (−3, 7) and (−3, 2).

How are the -coordinates in the two new points related to the -coordinates in point and point ?


They are 6 units apart.

Exercises 1–4 (10 minutes)
Students will work independently.
Exercises
For Exercises 1 and 2, plot the points, name the shape, and determine the area of the shape. Then write an expression that could be used to determine the area of the figure. Explain how each part of the expression corresponds to the situation. 1.

(, ), (, ), (, ), (, −), (, −), and (, )

This shape is a hexagon.
Area of 1

=

= ( )( )

= ( )

= = = ( )( )

Area of 2

=

= = ( )( )

Area of 3

=

Area of 4

=

= ( )( )

= ( )

= .
Area of 5

=

= ( )( )

= ( )

=

Total Area = + + + . + Total Area = .

Expression

()() + ()() + ()() + ()() + ()()

Each term represents the area of a section of the hexagon. They must be added together to get the total.
The first term is the area of triangle 1 on the left.
The second term is the area of rectangle 2.
The third term is the area of the large rectangle 3.
The fourth term is the area of triangle 4 on the left.
The fifth term is the area of triangle 5 on the right.

Lesson 8:
Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM

(−, ), (−, −), and (−, −)

2.

This shape is a triangle. =

Area of Outside Rectangle

= ( ) ( ) =

1

Area of Triangle 1

=

= ( )( )

= ( )

= .

2

3

()() − ()() − ()() − ()()

Expression

6•5

= ( )( )

= ( )

=

Area of Triangle 2 =

Area of Triangle 3

=

= ( )( )

= ( )

= .

Total Area = − . − −
.
Total Area =

The first term in the expression represents the area of the rectangle that goes around the outside of the triangle.
The next three terms represent the areas that need to be subtracted from the rectangle so that we are only left with the given triangle.
The second term is the area of the top right triangle.
The third term is the area of the bottom right triangle.
The fourth term is the area of the triangle on the left.
3.

A rectangle with vertices located at (−, ) and (, ) has an area of square units. Determine the location of the other two vertices.

4.

Challenge: A triangle with vertices located at (−, −) and (, −) has an area of square units. Determine one possible location of the other vertex.

The other two points could be located at (−, ) and (, ) or (−, ) and (, ).

Answers will vary. Possible solutions include points that are from the base. (−, ) or (, −).

Closing (5 minutes)


What different methods could you use to determine the area of a polygon plotted on the coordinate plane?


In order to find the area of a polygon on a coordinate plane, it is important to have vertical and horizontal lines. Therefore, the polygon can be decomposed to triangles and rectangles or a large rectangle can be drawn around the polygon.

Lesson 8:
Date:

Drawing Polygons on the Coordinate Plane
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NYS COMMON CORE MATHEMATICS CURRICULUM



Lesson 8

6•5

How did the shape of the polygon influence the method you used to determine the area?


If the shape is easily decomposed with horizontal and vertical lines, then this is the method that I would use to calculate the area. If this is not the case, then it would be easier to draw a rectangle around the outside of the shape.

Exit Ticket (5 minutes)

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Date:

Drawing Polygons on the Coordinate Plane
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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 8: Drawing Polygons on the Coordinate Plane
Exit Ticket
Determine the area of both polygons on the coordinate plane, and explain why you chose the methods you used. Then write an expression that could be used to determine the area of the figure. Explain how each part of the expression corresponds to the situation.

Lesson 8:
Date:

Drawing Polygons on the Coordinate Plane
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6•5

Exit Ticket Sample Solutions
Determine the area of both polygons on the coordinate plane, and explain why you chose the methods you used. Then write an expression that could be used to determine the area of the figure. Explain how each part of the expression corresponds to the situation.
Methods to calculate the answer will vary.

#1 Area of shape a

Area of shape b

= ( )( )

= ( )( )

=

=

=

= ( ) =

Total Area = + =

Explanations will vary depending on method chosen.
()() + ()()

Expression

The first term represents the area of the rectangle on the left, which makes up part of the figure.
The second term represents the area of the triangle on the right that completes the figure.

#2 Area of outside rectangle Area of shape c =

= ( )( ) =

=

=

= ( )( )

= ( )

Expression

= ( )

=

Explanations will vary depending on method chosen.

= ( )( ) = .

Total Area = − − . −

Total Area = .

Area of shape d

Area of shape e =

= ( )( )

= ( )

=

()() − ()() − ()() − ()()

The first term in the expression is the area of a rectangle that goes around the triangle.
Each of the other terms represents the triangles that need to be subtracted from the rectangle so that we are left with just the figure in the center.

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Date:

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Lesson 8

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Problem Set Sample Solutions
Plot the points for each shape, determine the area of the polygon, and then write an expression that could be used to determine the area of the figure. Explain how each part of the expression corresponds to the situation.
1.

(, ), (, ), (, ), (, ), and (, −)

( )( )

( )

Area of Triangle 1 =

=

=

=

Area of Triangle 2

=

= ( )( )

= ( )

= .

Area of Triangle 3

=

= ( )( )

= ( )

= .

Pentagon total area = + . +
.
Total Area =

()() + ()() + ()()

Expression

Each term in the expression represents the area of one of the triangular pieces that fits inside the pentagon. They are all added together to form the complete figure.

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Date:

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2.

(−, ), (−, ), and (−, −)

=

Area of Outside Rectangle = ( )( ) =

Area of Top Triangle

=

= ( )( )

= ( )

=

()() − ()() − ()() − ()()

Expression

6•5

Area of Bottom Left Triangle

=

= ( )( )

= ( )

=

= ( )( )

= ( )

= .

Area of Bottom Right Triangle =

Area of center triangle = − − −
.
Area of center triangle = .

The first term in the expression represents the area of the rectangle that would enclose the triangle. Then the three terms after represent the triangles that need to be removed from the rectangle so that the given triangle is the only shape left.
3.

(, ), (, −), and (, −)

= ()()

= =

Area of Triangle on the Left

= ()()

=

Area of Triangle on the Right

=

Total Area = + =

()() + ()()

Expression

Each term in the expression represents the area of a triangle that makes up the total area. The first term is the area of the triangle on the left, and the second term is the area of a triangle on the right.

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Date:

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4.

6•5

Find the area of the triangle in Problem 3 using a different method. Then, compare the expressions that can be used for both solutions in Problems 3 and 4. =

Area of Rectangle

= ( )( ) =

Area of Triangle on Top Left

=

= ( )( )

=

Expression

= ( )( )

=

Area of Triangle on Bottom Left =

= ( )( )

=

Area of Triangle on Right =

Total Area = − − −
Total Area =

()() − ()() − ()() − ()()

The first term in the expression is the area of a rectangle around the outside of the figure. Then we subtracted all of the extra areas with the next three terms.
The two expressions are different because of the way we divided up the figure. In the first expression, we split the shape into two triangles that had to be added together to get the whole. In the second expression, we enclosed the triangle inside a new figure, and then had to subtract the extra area.
5.

Two vertices of a rectangle are (, −) and (, ). If the area of the rectangle is square units, name the possible location of the other two vertices.

6.

A triangle with two vertices located at (, −) and (, ) has an area of square units. Determine one possible location of the other vertex.

(, −) and (, ) or (, −) and (, )

Answers will vary. Possible solutions include points that are from the base. (, −) or (−, −).

Lesson 8:
Date:

Drawing Polygons on the Coordinate Plane
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Lesson 9

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 9: Determining Area and Perimeter of Polygons on the Coordinate Plane
Student Outcomes


Students find the perimeter of irregular figures using coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.



Students find the area enclosed by a polygon on the coordinate plane by composing or decomposing using polygons with known area formulas.

Lesson Notes
The solutions given throughout the lesson only represent some of the correct answers to the problems. Discussion throughout the lesson about other possible solutions should be welcomed.
The formulas = and = ℎ are used intermittently. Both are correct strategies for determining the area of a rectangle and should be accepted.
Please note that in each coordinate plane, each square unit is one unit in length.

Please also note that some of the formulas are solved in a different order depending on the problem. For example, when using the formula for the area of triangles, students could multiply the base and the height and then multiply by
1
2

or they could take of either the base or the height before multiplying by the other. Because multiplication is

1
2

commutative, multiplying in different orders is mathematically sound. Students should be comfortable with using either order and may see opportunities when it is more advantageous to use one order over another.

Classwork
Fluency Exercise (5 minutes): Addition and Subtraction Equations
Sprint: Refer to the Sprints and the Sprint Delivery Script sections in the Module Overview for directions to administer a
Sprint.

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Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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6•5

Example 1 (8 minutes)
Example 1
Jasjeet has made a scale drawing of a vegetable garden she plans to make in her backyard. She needs to determine the perimeter and area to know how much fencing and dirt to purchase. Determine both the perimeter and area.

=

=

=

=

=

=

Perimeter = + + + + +

Perimeter =

The area is determined by making a horizontal cut from (, ) to point . = =

Area of Top

= ( )( )

=

Area of Bottom

= ( )( ) =

Total Area = +
Total Area =



How can we use what we worked on in Lessons 7 and 8 to help us calculate the perimeter and area?


We can determine the lengths of each side first. Then, we will add the lengths together to get the perimeter. 

Next, we can break the shape into two rectangles, find the area of each rectangle using the side lengths, and add the areas together to get the total area of the polygon.

Lesson 9:
Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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6•5

Example 2 (8 minutes)
Example 2
Calculate the area of the polygon using two different methods. Write two expressions to represent the two methods, and compare the structure of the expressions.
Answers will vary. The following are two possible methods. However, students could also break the shape into two triangles and a rectangle or another correct method.
Method One:

Method Two:

Area of Triangle 1 and 4

=

= ( )( )

= ( )

=

Since there are , we have a total area of .

Area of Triangle 2 and 3

=

= ( )( )

= ( )

=

Since there are , we have a total area of .

Total Area = + =

� ()()� + � ()()�

Expressions

=

= ( )( ) =

= ( )( )

=

=

There are triangles of equivalent base and height. ( ) =

Total Area = −

or

()() − � ()()�

The first expression shows terms being added together because I separated the hexagon into smaller pieces and had to add their areas back together.
The second expression shows terms being subtracted because I made a larger outside shape, and then had to take away the extra pieces.

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Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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Allow time for students to share and explain one of their methods.
MP.1
&
MP.3



Scaffolding:

What were the strengths and weaknesses of the methods that you tried?
Responses will vary. Some students may prefer methods that require fewer steps while others may prefer methods that only include rectangles and triangles.



6•5

As ELL students discuss their thinking, it may be useful to provide support for their conversations. Sentence starters may include, “My favorite method is ...” or
“First, I ...”

Exercises 1–2 (16 minutes)
Students work on the practice problems in pairs, so they can discuss different methods for calculating the areas.
Discussions should include explaining the method they chose and why they chose it. Students should also be looking to see if both partners got the same answer.
Consider asking students to write explanations of their thinking in terms of decomposition and composition as they solve each problem.
Exercises 1–2
1.

Determine the area of the following shapes.
a.
=

Area of Rectangle

= ( )( ) =

= ( )( )

= .

Area of Triangle =

triangles with equivalent base and height
(. ) =

Area = −
Area =

Teachers please note that students may also choose to solve by decomposing. Here is another option:

Lesson 9:
Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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6•5

Area of Triangle 1

=

= ( )( )

= ( )( )

b.

=

Another correct solution might start with the following diagram:

Area of Triangles 2 and 4

=

= ( )( )

= ( )

= .

Since triangles 2 and 4 are congruent, the combined area is . =

Area of Rectangle 3

= ( )( ) =

2.

Total Area = + +
Total Area =

Determine the area and perimeter of the following shapes.
a.

Area
Large Square = = ( ) =

Removed Piece = = ( )( ) =

Other correct solutions might start with the following diagrams:

Lesson 9:
Date:

Area = −
Area =

Perimeter = + + + + +
Perimeter =

Determining Area and Perimeter of Polygons on the Coordinate Plane
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b.

6•5

Area
Horizontal Area = = ( )( ) =
Vertical Area = = ( )( ) =

Total Area = +
Total Area =

Perimeter = + + + + + + +
Perimeter =

Other correct solutions might start with the following diagrams:

Closing (4 minutes)


Share with the class some of the discussions made between partners about the methods for determining area of irregular polygons.

Ask questions to review the key ideas:


There appear to be multiple ways to determine the area of a polygon. What do all of these methods have in common? 


The areas cannot overlap.



When you decompose the figure, you cannot leave any parts out.




Answers will vary.

When drawing a rectangle around the outside of the shape, the vertices of the original shape should be touching the perimeter of the newly formed rectangle.

Why did we determine the area and perimeter of some figures and only the area of others?


In problems similar to Exercise 1 (parts (a) and (b)), the sides were not horizontal or vertical, so we were not able to use the methods for determining length like we did in other problems.

Exit Ticket (4 minutes)

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Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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Lesson 9

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 9: Determining Area and Perimeter of Polygons on the
Coordinate Plane
Exit Ticket

Determine the area and perimeter of the figure below. Note that each square unit is 1 unit in length.

Lesson 9:
Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Determine the area and perimeter of the figure below. Note that each square unit is in length.
Area

Area of Large Rectangle = = ( )( ) =
Area of Small Square = = ( ) =

Area of Irregular Shape = − =

Perimeter = + + + + + + +
Perimeter =

Other correct solutions might start with the following diagrams:

Lesson 9:
Date:

Determining Area and Perimeter of Polygons on the Coordinate Plane
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Lesson 9

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Problem Set Sample Solutions
1.

Determine the area of the polygon.
Area of Triangle 1

=

= ( )( )

= ( )

=

Area of Triangle 2

=

= ( )( )

= ( )

=

= ( )( )

= ( )

=

Area of Triangle 3 =

Total Area = + +

Total Area =

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Date:

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NYS COMMON CORE MATHEMATICS CURRICULUM

2.

6•5

Determine the area and perimeter of the polygon.
Area

=

Horizontal Rectangle

= ( )( ) = =

Vertical Rectangle

= ( )( ) = =

Square

= ( )

=

Total Area = + +
Total Area =

Perimeter = + + + + + + +
Perimeter

Perimeter =

Lesson 9:
Date:

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129

Lesson 9

NYS COMMON CORE MATHEMATICS CURRICULUM

3.

Determine the area of the polygon. Then, write an expression that could be used to determine the area.

=

=

Area of Rectangle on Left

Area of Rectangle on Right

= ( )( )

= ( )( )

=

=

Total Area = + + . = .
Expression
4.

6•5

()() + ()() + ()()

= ( )( )

= .

Area of Triangle on Top =

If the length of each square was worth instead of , how would the area in Problem 3 change? How would your expression change to represent this area?
If each length is twice as long, when they are multiplied, × = . Therefore, the area will be four times larger when the side lengths are doubled.

I could multiply my entire expression by to make it times as big. �()() + ()() + ()()�

5.

Determine the area of the polygon. Then, write an expression that represents the area.

Area of Outside Rectangle =

= ( )( ) =

Area of Rectangle on Left =

= ( )( ) =

Total Area = − −
Total Area =

Expression

()() − ()() − ()()

Lesson 9:
Date:

Area of Rectangle on Right =

= ( )( ) =

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NYS COMMON CORE MATHEMATICS CURRICULUM

6.

6•5

Describe another method you could use to find the area of the polygon in Problem 5. Then, state how the expression for the area would be different than the expression you wrote.
I could have broken up the large shape into many smaller rectangles. Then I would need to add all the areas of these rectangles together to determine the total area.
My expression showed subtraction because I created a rectangle that was larger than the original polygon, and then
I had to subtract the extra areas. If I break the shape into pieces, I would need to add the terms together instead of subtracting them to get the total area.

7.

Write one of the letters from your name using rectangles on the coordinate plane. Then, determine the area and perimeter. (For help see Exercise 2(b). This irregular polygon looks sort of like a T.)
Answers will vary.

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Date:

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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Number Correct: ______
Directions: Find the value of in each equation.

Addition and Subtraction Equations—Round 1

1.
2.
3.
4.
5.
6.
7.
8.
9.

10.
11.
12.
13.
14.
15.
16.
17.

+ 4 = 11 + 2 = 5

19.

+ 5 = 8

20.

− 7 = 10

21.

− 8 = 1

22.

− 4 = 2

23.

+ 12 = 34

24.

+ 25 = 45

25.

+ 43 = 89

26.

− 20 = 31

27.

− 13 = 34

28.

− 45 = 68

29.

+ 34 = 41

30.

+ 29 = 52

31.

+ 37 = 61

32.

− 43 = 63

33.

− 21 = 40

Lesson 9:
Date:

18.

34.

− 54 = 37
4 + = 9

6 + = 13
2 + = 31

15 = + 11
24 = + 13
32 = + 28
4 = − 7
3 = − 5

12 = − 14

23.6 = − 7.1

14.2 = − 33.8
2.5 = − 41.8

64.9 = + 23.4
72.2 = + 38.7

1.81 = − 15.13

24.68 = − 56.82

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132

Lesson 9

NYS COMMON CORE MATHEMATICS CURRICULUM

Directions: Find the value of in each equation.

6•5

Addition and Subtraction Equations—Round 1 [KEY]

1.
2.
3.
4.
5.
6.
7.
8.
9.

10.
11.
12.
13.
14.
15.
16.
17.

+ 4 = 11

=

+ 2 = 5

=

+ 5 = 8

=

− 7 = 10

=

− 8 = 1

=

− 4 = 2

=

+ 12 = 34

=

+ 25 = 45

=

+ 43 = 89

=

− 20 = 31

=

− 13 = 34

=

− 45 = 68

=

+ 34 = 41

=

+ 29 = 52

=

+ 37 = 61

=

− 43 = 63

=

− 21 = 40

Lesson 9:
Date:

=

18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

− 54 = 37

=

2 + = 31

=

32 = + 28

=

4 + = 9

6 + = 13

15 = + 11
24 = + 13
4 = − 7
3 = − 5

12 = − 14

= = =

= = =

=

23.6 = − 7.1

= .

64.9 = + 23.4

= .

14.2 = − 33.8
2.5 = − 41.8

72.2 = + 38.7

1.81 = − 15.13

24.68 = − 56.82

=

= . = .

= . = .

Determining Area and Perimeter of Polygons on the Coordinate Plane
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133

Lesson 9

NYS COMMON CORE MATHEMATICS CURRICULUM

Directions: Find the value of in each equation.

Number Correct: ______
Improvement: ______

Addition and Subtraction Equations—Round 2

1.
2.
3.
4.
5.
6.
7.
8.
9.

10.
11.
12.
13.
14.
15.
16.
17.

+ 2 = 7

+ 4 = 10

20.

+ 7 = 23

21.

+ 12 = 16

22.

− 5 = 2

23.

− 3 = 8

24.

− 4 = 12

25.

− 14 = 45

26.

+ 23 = 40

27.

+ 13 = 31

28.

+ 23 = 48

29.

+ 38 = 52

30.

− 14 = 27

31.

− 23 = 35

32.

− 17 = 18

33.

− 64 = 1

Lesson 9:
Date:

18.
19.

+ 8 = 15

6•5

34.

6 = + 3

12 = + 7

24 = + 16
13 = + 9
32 = − 3

22 = − 12
34 = − 10
48 = + 29
21 = + 17
52 = + 37
6
4
= +
7
7
2
5
= −
3
3
1
8
= −
4
3
5
7
= −
6
12
7
5
= −
8
12
7
16
+ =
6
3
1
13
+ =
3
15

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134

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NYS COMMON CORE MATHEMATICS CURRICULUM

Directions: Find the value of in each equation.

6•5

Addition and Subtraction Equations—Round 2 [KEY]

1.
2.
3.
4.
5.
6.
7.
8.
9.

10.
11.
12.
13.
14.
15.
16.
17.

+ 2 = 7

=

+ 4 = 10

=

+ 8 = 15

=

+ 7 = 23

=

+ 12 = 16

=

− 5 = 2

=

− 3 = 8

=

− 4 = 12

=

− 14 = 45

=

+ 23 = 40

=

+ 13 = 31

=

+ 23 = 48

=

+ 38 = 52

=

− 14 = 27

=

− 23 = 35

=

− 17 = 18

=

− 64 = 1

Lesson 9:
Date:

=

18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

6 = + 3

=

13 = + 9

=

12 = + 7

24 = + 16

= =

32 = − 3

=

48 = + 29

=

22 = − 12
34 = − 10
21 = + 17
52 = + 37
6
4
= +
7
7
2
5
= −
3
3
1
8
= −
4
3
5
7
= −
6
12
7
5
= −
8
12
7
16
+ =
6
3
1
13
+ =
3
15

= = =

=

=

=

= = = = =

Determining Area and Perimeter of Polygons on the Coordinate Plane
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Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 10: Distance, Perimeter, and Area in the Real World
Student Outcomes


Students determine distance, perimeter, and area in real-world contexts.

Lesson Notes
This lesson is similar to Lesson 6 from this module. The teacher can determine ahead of time whether to do the exploration in the classroom or venture out into hallways or some other location. The measuring tools, units, and degree of precision to be used in this activity should be chosen in a manner that best meets the needs of the students.
For large distances, a long measuring tape or trundle wheel can be used. For very small objects, a millimeter ruler would be more appropriate.
The critical understanding for students is that area involves covering, while perimeter involves surrounding. Since plane objects have both area and perimeter, the distinction between the two concepts must be made.
When choosing objects to be measured, look for composite objects that require more than just measuring length and width. Avoid curved edges, as students will not be able to find area. When possible, choose objects that explicitly lend themselves to both area and perimeter. Such objects could include a frame or mat around a picture, wood trim around the top of a table, piping around a dinner napkin, or baseboard molding along walls. Some of the objects that were chosen for Lesson 6 of this Module can also be used.
It is appropriate for students to use a calculator for this lesson.

Classwork
Opening Exercises 1–2 (6 minutes)

Scaffolding:
There is a great deal of flexibility in this lesson, so it can be tailored to the needs of the class and can be easily individualized for both struggling and advanced learners. Opening Exercises 1–2
1.

Find the area and perimeter of this rectangle:

= = × =

= + + + =

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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136

Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Find the width of this rectangle. The area is . , and the length is . .

2.

= ×

. = . ×

. . ×
=
.
.
. =

Discussion (5 minutes)


How many dimensions does the rectangle have, and what are they?




What units are used to express area?




The line segments around the rectangle have one dimension; line segments only have length.

What units are used to express perimeter?

We do not typically write linear units with an exponent because the exponent is 1.




Peri- means around; -meter means measure. Perimeter is the measure of the distance around an object. Area is the measure of the surface of an object and has two dimensions.

How many dimensions do the line segments around the rectangle have?




Periscope (seeing around), periodontal (surrounding a tooth), pericardium (the sac around the heart), period (a portion of time that is limited and determined by some recurring phenomenon, as by the completion of a revolution of the earth or moon), etc.

How can focusing on the meaning of the word help you remember the difference between area and perimeter? 



Around

Are there any other words that use this prefix?




The number 2, for two dimensions

The term dimensions may be new to ELL students and as such, may need to be taught and rehearsed. Similarly, the word superscript may be new for ELL students and should be taught or reviewed.

What does the prefix peri- mean?




Square units, such as square centimeters as we had in the first Opening
Exercise.

What superscript is used to denote square units?




Two dimensions: length and width

Scaffolding:

Linear units

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Example 1 (6 minutes): Student Desks or Tables
Distribute measuring tools. Explain the units that will be used and level of precision expected.
Scaffolding:

Example 1: Student Desks or Tables
1.
2.

How do you find the perimeter?

4.



How do you find the area of the top of your desk?

3.

Record these on your paper in the appropriate column.

Let’s do an example before starting out on this investigation. Measure the dimensions of the top of your desk (or tabletop, etc.).




Multiply the length by the width.

How do you find the perimeter?




Dimensions will vary.

How do you find the area of the surface?




Consider asking some students to measure to the nearest inch, others to the nearest half inch, and others to the nearest quarter inch, depending on ability. Compare these.

Measure the dimensions of the top of your desk.

Any of three ways: Add the length and width (to find the semi-perimeter), and then double the sum; double the length, double the width, and add those two products; or add the length, length, width, and width. Record these on your paper in the appropriate column.

Exploratory Challenge (17 minutes)
Exploratory Challenge
Estimate and predict the area and perimeter of each object. Then measure each object, and calculate both the area and perimeter of each.
Answers are determined by the teacher when objects are chosen. Consider using examples like decorating a bulletin board: bulletin board trim (perimeter), paper for bulletin board (area).
Object or
Item to be
Measured

Measurement Units

Ex: door

feet

Precision
Area
(Measure to
Prediction
the
(Square Units)
Nearest)
half foot

Desktop

Lesson 10:
Date:

Area (Square Units)
Write the expression and evaluate it.

. × . =

Perimeter
Prediction
(Linear Units)

Perimeter
(Linear Units)

. + .�

= . �

Distance, Perimeter, and Area in the Real World
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Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Optional Challenge (10 minutes)
If desired, send some students to measure other real-world objects found around the school. Set measurement units and precision parameters in advance. The teacher should measure these objects in advance of the activity and calculate the corresponding perimeters and areas. Measuring the school building from the outside could be a whole group activity or could be assigned as an extra credit opportunity.
Optional Challenge
Object or
Item to be
Measured

Measurement
Units

Precision
(Measure to the Nearest)

Ex: door

feet

half foot

Area
(Square Units)

. × . =

Perimeter
(Linear Units)



. + .� = .

Closing (4 minutes)


What are some professions that use area and perimeter regularly?




Can you think of any circumstances where you or someone you know has or might have to calculate perimeter and area?




Surveyors, garment manufacturers, packaging engineers, cabinetmakers, carpenters

Answers will vary. Encourage a large quantity of responses.

Would you like to work in an occupation that requires measuring and calculating as part of the duties?


Answers will vary.

Exit Ticket (7 minutes)

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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139

Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 10: Distance, Perimeter, and Area in the Real World
Exit Ticket
1.

The local school is building a new playground. This plan shows the part of the playground that needs to be framed with wood for the swing set. The unit of measure is feet. Determine the number of feet of wood that will be needed to frame the area.

2.

The school will fill the area with wood mulch for safety. Determine the number of square feet that need to be covered by the mulch.

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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6•5

Exit Ticket Sample Solutions
1.

The local school is building a new playground. This plan shows the part of the playground that needs to be framed with wood for the swing set. The unit of measure is feet. Determine the number of feet of wood that will be needed to frame the area.

Perimeter: . + . + . + . + . + . + . + . = .
2.

The school will fill the area with wood mulch for safety. Determine the number of square feet that need to be covered by the mulch. = = ( . × .) = = = ( . × .) = = + =

Problem Set Sample Solutions
Note: When columns in a table are labeled with units, students need only enter numerical data in the cells of the table and not include the units each time.
1.

How is the length of the side of a square related to its area and perimeter? The diagram below shows the first four squares stacked on top of each other with their upper left-hand corners lined up.

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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141

Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

a.

Complete this chart calculating area and perimeter for each square.
Expression
Showing the
Area

Area in Square
Feet

×

Side Length in
Feet

×

×

× ×

×

×

×

× ×

×

Perimeter in
Feet

×

×

×

Expression
Showing the
Perimeter

×

×

×

×

×

×

×

×

It depends. For side length < , perimeter is greater; however, for side length > , area is greater.

b.

In a square, which numerical value is greater, the area or the perimeter?

c.

When is the numerical value of a square’s area (in square units) equal to its perimeter (in units)?

d.

2.

6•5

Why is this true?

When the side length is exactly .

= is only true when = .

This drawing shows a school pool. The walkway around the pool needs special non-skid strips installed but only at the edge of the pool and the outer edges of the walkway.

+ + + + + + + =

a.

Find the length of non-skid strips that is needed for the job.

b.

The non-skid strips are sold only in rolls of . How many rolls need to be purchased for the job? ÷

= .

Therefore, rolls will need to be purchased.

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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142

Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

3.

6•5

A homeowner called in a painter to paint the walls and ceiling of one bedroom. His bedroom is . long, . wide, and . high. The room has two doors, each . by . and three windows each . by . The doors and windows do not have to be painted. A gallon of paint can cover . A hired painter claims he will need gallons. Show that his estimate is too high.
Area of long walls:

( . × .) =

Area of doors:

( . × .) =

Area of ceiling:

( . × .) =

Area of windows

( . × .) =

Area of short walls:

. × . =

Area to be painted:

( + + ) − ( + ) = ÷ = .

The painter will need a little more than gallons. The painter’s estimate for how much paint is necessary was too high. Gallons of paint needed:

4.

Theresa won a gardening contest and was awarded a roll of deer-proof fencing. The fence is yards long. She and her husband, John, discuss how to best use the fencing to make a rectangular garden. They agree that they should only use whole numbers of feet for the length and width of the garden.
a.

What are all of the possible dimensions of the garden?

Length in Feet

Width in Feet

b.

Which plan yields the maximum area for the garden? Which plan yields the minimum area?

Length in feet

Width in feet

Area in square feet

The . by . garden would have the maximum area ( ), while the . by . garden would have only of garden space.

Lesson 10:
Date:

Distance, Perimeter, and Area in the Real World
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143

Lesson 10

NYS COMMON CORE MATHEMATICS CURRICULUM

5.

6•5

Write and then solve the equation to find the missing value below.

= ×

. = . ×
.
=
.
. =

6.

Challenge: This is a drawing of the flag of the Republic of the Congo. The area of this flag is
a.

Using the area formula, tell how you would determine the value of the base.

b.

.

=

÷ =

÷ . =

. =

. = . +

Using what you found in part (a), determine the missing value of the base.

Lesson 10:
Date:

. =

Distance, Perimeter, and Area in the Real World
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144

Mid-Module Assessment Task

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

1. David is the groundskeeper at Triangle Park, shown below.

300 yd.

50 yd.

a.

David needs to cut the grass four times a month. How many square yards of grass will he cut altogether each month?

b.

During the winter, the triangular park and adjacent square parking lot are flooded with water and allowed to freeze so that people can go ice skating. What is the area of the ice?

300 yd.

Module 5:
Date:

50 yd.

50 yd.

Area, Surface Area, and Volume Problems
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NYS COMMON CORE MATHEMATICS CURRICULUM

Mid-Module Assessment Task

6•5

2. Marika is looking for a new computer table. Part (b) presents a sketch of two computer tables she likes when looking at them from above. All measurements are in feet.
a.

If Marika needs to choose the one with the greater area, which one should she choose? Justify your answer with evidence, using coordinates to determine side lengths.

b.

If Marika needs to choose the one with the greater perimeter, which one should she choose? Justify your answer with evidence, using coordinates to determine side lengths.

Table A

Table B

Module 5:
Date:

Area, Surface Area, and Volume Problems
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NYS COMMON CORE MATHEMATICS CURRICULUM

Mid-Module Assessment Task

6•5

3. Find the area of the triangular region.
6 in.

5 in.
13 in.

4. The grid below shows a birds-eye view of a middle school.

Point
A

B

F

H

E

D

C

G

Coordinates

Segment
����

����

Length (m)

����

����

����

����

����

����

a.

Write the coordinates of each point in the table.

b.

Each space on the grid stands for 10 meters. Find the length of each wall of the school.

c.

Find the area of the entire building. Show your work.

Module 5:
Date:

Area, Surface Area, and Volume Problems
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NYS COMMON CORE MATHEMATICS CURRICULUM

Mid-Module Assessment Task

6•5

A Progression Toward Mastery
Assessment
Task Item

1

a
6.G.A.1

b
6.G.A.1

2

a
6.G.A.3

STEP 1
Missing or incorrect answer and little evidence of reasoning or application of mathematics to solve the problem

STEP 2
Missing or incorrect answer but evidence of some reasoning or application of mathematics to solve the problem

STEP 3
A correct answer with some evidence of reasoning or application of mathematics to solve the problem, or an incorrect answer with substantial evidence of solid reasoning or application of mathematics to solve the problem

STEP 4
A correct answer supported by substantial evidence of solid reasoning or application of mathematics to solve the problem

Student response is incorrect and shows no application of the triangle area formula.

Student uses the triangle area formula but answers incorrectly, perhaps by only calculating the area of the triangle (7,500 yd2).

Student uses the triangle area formula, correctly finds the area of the park, 7,500 yd2, and multiplies that area by 4.
In the final answer, an arithmetic mistake might be made, or the units are either missing or are in yards instead of square yards. Student uses the triangle area formula, correctly finds the area of the park, 7,500 yd2, and multiplies that area by 4.
Student response is correct, both in number and in units
(30,000 yd2).

Student response is incorrect and shows no application of area formulas. Student uses the triangle area formula and/or rectangle area formula but response is incorrect because of arithmetic errors. Units are not correct. Student uses the triangle area formula, and correctly finds the area of the grass, 7,500 yd2 , or correctly finds the area of the parking lot,
2,500 yd2 .

Student uses area formulas and correctly finds the area of the grass, 7,500 yd2 , and parking lot, 2,500 yd2 , and adds them correctly, totaling 10,000 yd2.
Units are correct in the final answer.

Student response is incorrect and shows no application of area formulas. Perimeter calculations may have been made.

Student incorrectly calculates the area of both tables. The student chooses the greater of the two areas calculated, regardless of the mistake. Units are incorrectly identified.

Student correctly calculates the area of one table, either Table A is 39 ft 2 or Table B is
37 ft 2 . The student chooses the greater of the two areas calculated, regardless of the mistake. Units are correctly identified.

Student correctly calculates the area of both tables, Table A is
39 ft 2 and Table B is
37 ft 2 , and concludes
Table A has a larger area.
Units are correctly identified, and coordinates are appropriately used in order to determine side lengths. Module 5:
Date:

Area, Surface Area, and Volume Problems
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NYS COMMON CORE MATHEMATICS CURRICULUM

b
6.G.A.3

3

6.G.A.1

4

a
6.G.A.3

b
6.G.A.3

c
6.G.A.3

Mid-Module Assessment Task

6•5

Student incorrectly calculates the perimeter of both tables. Units are incorrectly identified.
Area calculations may have been made.

Student incorrectly calculates the perimeter of both tables. The student chooses the greater of the two calculated perimeters, regardless of the mistake. Units are incorrectly identified.

Student correctly calculates the perimeter of one table, either Table
A is 32 ft. or Table B is
36 ft., and concludes
Table B has a longer perimeter. Units are correctly identified.

Student correctly calculates the perimeter of both tables, Table A is
32 ft. and Table B is
36 ft., and concludes
Table B has a longer perimeter. Units are correctly identified, and coordinates are appropriately used in order to determine side lengths. Student does not calculate the altitude of the triangle to be 7 in., and the final response is incorrect.

Student correctly calculates the altitude of the triangle to be
7 in., but the final area of the triangle is incorrect.

Student correctly calculates the altitude and area of the triangle, but the units are incorrectly identified.

Student correctly calculates the area of the triangle as 17.5 in2 .

Student correctly identifies fewer than
2 of the 8 points.

Student correctly identifies at least 4 of the 8 points.

Student correctly identifies at least 6 of the 8 points.

Student correctly identifies fewer than 2 of the 8 lengths.

Student correctly identifies at least 4 of the 8 lengths; alternatively, the response ignores the scale factor and finds 6 of the 8 lengths to be one-tenth of the correct answers. Student correctly identifies at least 6 of the 8 lengths; alternatively, the response ignores the scale factor and finds all
8 lengths to be onetenth of the correct answers. Student response is incorrect in both number and units.

Student ignores the scale and incorrectly calculates the area of the building as 83 m2 . Units can be correct, incorrect, or missing.

Student incorrectly calculates the area of the building to be something other than 8300 m2 due to an arithmetic error.
Units are correct.

Module 5:
Date:

Student correctly identifies all 8 points.
Point

Coordinates
(−4, 4)
(6, 4)
(6, −6)
(4, −6)
(4, −2)
(−1, −2)
(−1, −7)
(−4, −7)

Student correctly identifies all 8 lengths correctly. Segment
����

����

����

����

����

����

����

����

Length (m)
100
100
20
40
50
50
30
110

Student correctly calculates the area of the building: 8300 m2 .
Both the number and units are correct.

Area, Surface Area, and Volume Problems
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Mid-Module Assessment Task

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

1. David is the groundskeeper at Triangle Park, scale shown below.

300 yd.

50 yd.

a.

David needs to cut the grass four times a month. How many square yards of grass will he cut altogether each month?

b.

During the winter, the triangular park and adjacent square parking lot are flooded with water and allowed to freeze so that people can go ice skating. What is the area of the ice?

300 yd.

Module 5:
Date:

50 yd.

50 yd.

Area, Surface Area, and Volume Problems
11/5/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Mid-Module Assessment Task

6•5

2. Marika is looking for a new computer table. Below is a sketch of two computer tables she likes when looking at them from above. All measurements are in feet.
a.

If Marika needs to choose the one with the greater area, which one should she choose? Justify your answer with evidence, using coordinates to determine side lengths.

b.

If Marika needs to choose the one with the greater perimeter, which one should she choose? Justify your answer with evidence, using coordinates to determine side lengths.

Table A

Table B

Module 5:
Date:

Area, Surface Area, and Volume Problems
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NYS COMMON CORE MATHEMATICS CURRICULUM

Mid-Module Assessment Task

6•5

3. Find the area of the triangular region.
6 in.

5 in.
13 in.

4. The grid below shows a birds-eye view of a middle school.

A

B

F

E

D
H

C

G

a.

Write the coordinates of each point in the table.

b.

Each space on the grid stands for 10 meters. Find the length of each wall of the school.

c.

Find the area of the entire building. Show your work.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/5/14

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New York State Common Core

6

Mathematics Curriculum

GRADE

GRADE 6 • MODULE 5

Topic C:

Volume of Right Rectangular Prisms
6.G.A.2
Focus Standard:

6.G.A.2

Instructional Days:

4

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Lesson 11: Volume with Fractional Edge Lengths and Unit Cubes (P) 1
Lesson 12: From Unit Cubes to the Formulas for Volume (P)
Lesson 13: The Formulas for Volume (P)
Lesson 14: Volume in the Real World (P)

In Topic C, students extend their understanding of the volume of a right rectangular prism with integer side lengths to right rectangular prisms with fractional side lengths. They apply the known volume formula, = ℎ, to find the volume of these prisms and use correct volume units when writing the answer. In
1 3
5
Lesson 11, students determine the volume of a rectangular prism with edges , , and by packing it with 15
8 8

1 length 8;

8

they then compare that volume to the volume computed by multiplying the side cubes with edge lengths. In Lesson 12, students extend the volume formula for a right rectangular prism to the formula = area of base ⋅ height. Students explore the bases of right rectangular prisms and understand that any face can be the base. They find the area of the base first and then multiply by the height. They determine that two formulas can be used to find the volume of a right rectangular prism. In Lesson 13, students apply both formulas from Lesson 12 to application problems dealing with volume formulas of right rectangular prisms and cubes with fractional edge lengths. The topic concludes with Lesson 14, in which students determine the volume of composite solid figures and apply volume formulas to find missing volumes and missing dimensions in real-world contexts.
1

Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

Topic C:
Date:

Volume of Right Rectangular Prisms
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Lesson 11

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 11: Volume with Fractional Edge Lengths and Unit
Cubes
Student Outcomes


Students extend their understanding of the volume of a right rectangular prism with integer side lengths to right rectangular prisms with fractional side lengths. They apply the formula = ⋅ ⋅ ℎ to find the volume of a right rectangular prism and use the correct volume units when writing the answer.

Lesson Notes
This lesson builds on the work done in Grade 5, Module 5, Topics A and B. Within these topics, students determine the volume of rectangular prisms with side lengths that are whole numbers. Students fill prisms with unit cubes in addition to using the formulas = ℎ and = ⋅ ⋅ ℎ to determine the volume.

Students start their work on volume of prisms with fractional lengths so that they can continue to build an understanding of the units of volume. In addition, they must continue to build the connection between packing and filling. In the following lessons, students move from packing the prisms to using the formula.
The sample activity provided at the end of the lesson will foster an understanding of volume, especially in students not previously exposed to the Common Core standards.

Classwork
Fluency Exercise (5 minutes): Multiplication of Fractions II
Sprint: Refer to the Sprints and the Sprint Delivery Script sections in the Module Overview for directions to administer a Sprint.

Scaffolding:
Use unit cubes to help students visualize the problems in this lesson. One way to do this would be to have students make a conjecture about how many cubes will fill the prism, and then use the cubes to test their ideas. Provide different examples of volume (electronic devices, loudness of voice), and explain that although this is the same word, the context of volume in this lesson refers to threedimensional figures.

Opening Exercise (3 minutes)
Please note that although scaffolding questions are provided, this Opening Exercise is an excellent chance to let students work on their own, persevering and making sense of the problem.
Which prism will hold more . × . × . cubes? How many more cubes will the prism hold?
Opening Exercise

MP.1

.

Lesson 11:
Date:

.

.

.

.

.

Volume with Fractional Edge Lengths and Unit Cubes
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6•5

How many 1 in. × 1 in. × 1 in. cubes will fit across the bottom of the first rectangular prism?

Students discuss their solutions with a partner.






How did you determine this number?






It holds 6 more layers.

The second rectangular prism has 6 more layers than the first, with 40 cubes in each layer.

How many more cubes does the second rectangular prism hold?





Both rectangular prisms hold the same number of cubes in one layer, but the second rectangular prism has more layers.

How many more layers does the second rectangular prism hold?




The second rectangular prism will hold more cubes.

How did you determine this?




I will need 12 layers because the prism is 12 in. tall.

Which rectangular prism will hold more cubes?




40 cubes will fit across the bottom.

How many layers will you need?


MP.1

There are 6 inches in the height; therefore, 6 layers of cubes will fit inside.

How many 1 in. × 1 in. × 1 in. cubes will fit across the bottom of the second rectangular prism?




Answers will vary. Students may determine how many cubes will fill the bottom layer of the prism and then decide how many layers are needed. Students who are English language learners may need a model of what “layers” means in this context.

How many layers of 1 in. × 1 in. × 1 in. cubes will fit inside the rectangular prism?




40 cubes will fit across the bottom.

6 × 40 = 240 more cubes.

We can also use the formula = ∙ ∙ ℎ.

What other ways can you determine the volume of a rectangular prism?


Example 1 (5 minutes)
Example 1

. ×

. ×

. cubes. How many dice of this size can fit in the box?

A box with the same dimensions as the prism in the Opening Exercise will be used to ship miniature dice whose side lengths have been cut in half. The dice are

Scaffolding:

.

.

Lesson 11:
Date:

Students may need a considerable amount of time to make sense of cubes with fractional side lengths.

.

An additional exercise has been included at the end of this lesson to use if needed.

Volume with Fractional Edge Lengths and Unit Cubes
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How can you use this information to determine the number of box? 





20 × 8 × 12 = 1920 of the smaller cubes.

1
2

in. ×

1
2

in. ×

I can multiply the number of cubes in the length, width, and height.

1
2

in. cubes it will take to fill the

How many of these smaller cubes will fit into the 1 in. × 1 in. × 1 in. cube?




Two cubes would fit across a 1-inch length. So, I would need to double the lengths to get the number of cubes. Twenty cubes will fit across the 10-inch length, 8 cubes will fit across the 4-inch width, and 12 cubes will fit across the 6-inch height.

How many cubes could we fit across the length? The width? The height?




6•5

Two will fit across the length, two across the width, and two for the height. 2 × 2 × 2 = 8. Eight smaller cubes will fit in the larger cube.

How does the number of cubes in this example compare to the number of cubes that would be needed in the
Opening Exercise?



=

new old 1920
240

=

8
1

If I fill the same box with cubes that are half the length, I will need 8 times as many.



How is the volume of the box related to the number of cubes that will fit in it?



What is the volume of 1 cube?

1
8







The volume of the box is of the number of cubes that will fit in it. =

=

1

2
1
8

in. × in3 1
2

in. ×

1
2

in.

1920 × = 240

What is the product of the number of cubes and the volume of the cubes? What does this product represent?



1
8

The product represents the volume of the original box.

Example 2 (5 minutes)
Example 2
A

. cube is used to fill the prism.

How many

. cubes will it take to fill the prism?

What is the volume of the prism?

How is the number of cubes related to the volume?

.

Lesson 11:
Date:

.

.

Volume with Fractional Edge Lengths and Unit Cubes
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How would you determine, or find, the number of cubes that fill the prism?
One method would be to determine the number of cubes that will fit across the length, width, and height. Then, I would multiply.





6 will fit across the length, 4 across the width, and 15 across the height.
6 × 4 × 15 = 360 cubes

How are the number of cubes and the volume related?
360 cubes ×

in3 =

The volume of one cube is = ℎ

1

64

360
64

1
4

1

in. ×

4
40

in3 = 5

64

1
4

in. =

1

64

in3 .

in3 = 5 in3
5
8

What other method can be used to determine the volume? = �1 in.� (1 in.) �3 in.�



=



in. ×

The volume is equal to the number of cubes times the volume of one cube.





6•5

=

3

1
2

in. ×

2
45
8

1
1

in. ×

15
4

in3 = 5 in3
5
8

3
4

in.

in., in., or

in. because there would be

Would any other size cubes fit perfectly inside the prism with no space left over?
We would not be able to use cubes with side lengths of



1
2

1
3

spaces left over. However, we could use a cube with a side length of over. 2

3
1
8

in. without having spaces left

Exercises (20 minutes)
Students will work in pairs.
Exercises
1.

Use the prism to answer the following questions.
a.

=

= � � � � � �

= × ×

Calculate the volume.

=

b.

or

If you have to fill the prism with cubes whose side lengths are less than , what size would be best?

The best choice would be a cube with side lengths of

c.

.

× × = cubes

How many of the cubes would fit in the prism?

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Date:

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157

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NYS COMMON CORE MATHEMATICS CURRICULUM

d.

Use the relationship between the number of cubes and the volume to prove that your volume calculation is correct. The volume of one cube would be

×

×

=

Since there are cubes, the volume would be ×
2.

b.

.

=

= � � � � � �

= × ×

= or

.

= or .

Calculate the volume of the following rectangular prisms.
a.

3.

6•5

=

= � .� � .� � .�

= . × . × .

, = or

.

.

. These smaller boxes are then packed into a shipping box with dimensions of

A toy company is packaging its toys to be shipped. Some of the very small toys are placed inside a cube-shaped box

. × . × .

with side lengths of

× × = toys

a.

How many small toys can be packed into the larger box for shipping?

b.

Use the number of toys that can be shipped in the box to help determine the volume of the box.
One small box would have a volume of = =

. ×

. ×

. =

.

Now, I will multiply the number of cubes by the volume of the cube. ×

Lesson 11:
Date:

Volume with Fractional Edge Lengths and Unit Cubes
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158

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NYS COMMON CORE MATHEMATICS CURRICULUM

4.

6•5

A rectangular prism with a volume of cubic units is filled with cubes. First, it is filled with cubes with side lengths

unit. Then, it is filled with cubes with side lengths of unit.

a.
How many more of the cubes with -unit side lengths than cubes with -unit side lengths will be needed to

of

fill the prism?

There are cubes with -unit side lengths in cubic unit because the volume of one cube is

Since we have cubic units, we would have × = total cubes with -unit side lengths.

cubic units.

There are cubes with -unit side lengths in cubic unit because the volume of one cube is

cubic units.

Since we have cubic units, we would have × = total cubes with -unit side lengths. − = more cubes

b.

Why does it take more cubes with

unit side lengths to fill the prism?

< . The side length is shorter for the cube with a -unit side length, so it takes more to fill the rectangular

prism.
5.

Calculate the volume of the rectangular prism. Show two different methods for determining the volume. =

= � � � � � �

= � � � � � �

=

=

Method 1:

Method 2:
Fill the rectangular prism with cubes that are
The volume of the cubes is

.

×

×

.

We would have cubes across the length, cubes across the width, and cubes across the height. × × = cubes total

cubes ×

=

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Date:

Volume with Fractional Edge Lengths and Unit Cubes
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6•5

Closing (2 minutes)


When you want to find the volume of a rectangular prism that has sides with fractional lengths, what are some methods you can use?



One method to find the volume of a right rectangular prism that has fractional side lengths is to use the volume formula = ℎ.

Another method to find the volume is to determine how many cubes of fractional side lengths are inside the right rectangular prism, and then find the volume of the cube. To determine the volume of the right rectangular prism, find the product of these two numbers.

Exit Ticket (5 minutes)

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Date:

Volume with Fractional Edge Lengths and Unit Cubes
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160

6•5

Lesson 11

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

Date

Lesson 11: Volume with Fractional Edge Lengths and Unit Cubes
Exit Ticket
Calculate the volume of the rectangular prism using two different methods. Label your solutions Method 1 and
Method 2.

2

1

Lesson 11:
Date:

3 cm 8

1 cm 4

5 cm 8

Volume with Fractional Edge Lengths and Unit Cubes
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161

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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Calculate the volume of the rectangular prism using two different methods. Label your solutions Method 1 and
Method 2. =

Method 1:

= � � � � � �

= × ×

=

Method 2:
Fill shape with

cubes.

× × = cubes

Each cube has a volume of = ×

×

×

= =

=

.

Problem Set Sample Solutions
1.

Answer the following questions using this rectangular prism:

a.

=

What is the volume of the prism?

.� � .�

= � .� � .� � .�

=

=

.

.

.

= ( .) �

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Date:

Volume with Fractional Edge Lengths and Unit Cubes
11/5/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

Linda fills the rectangular prism with cubes that have side lengths of . How many cubes does she need to
She would need across by wide and high.

fill the rectangular prism?

Number of cubes = × ×

Number of cubes = , cubes with . side lengths
c.

× =

How is the number of cubes related to the volume?
The number of cubes needed is times larger than the volume.

d.

e.

Because the cubes are not each ., the volume is different from the number of cubes. However, I could multiply the number of cubes by the volume of one cube and still get the original volume.
Why is the number of cubes needed different from the volume?

Should Linda try to fill this rectangular prism with cubes that are . long on each side? Why or why not?

Because some of the lengths are and some are , it would be difficult to use side lengths of to fill the

prism.
2.

Calculate the volume of the following prisms.
a.

b.

.

Lesson 11:
Date:

.

.

=

� � �

= ( ) � � � �

,

=

= = ( ) �

=

= � . � � . � � . �

. � = � . � � . � �

=

=

Volume with Fractional Edge Lengths and Unit Cubes
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163

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NYS COMMON CORE MATHEMATICS CURRICULUM

3.

A rectangular prism with a volume of cubic units is filled with cubes. First, it is filled with cubes with

unit side

unit side lengths.

How many more of the cubes with unit side lengths than cubes with unit side lengths will be needed to fill

lengths. Then, it is filled with cubes with
a.

6•5

the prism?

There are cubes with -unit side lengths in cubic unit because the volume of one cube is

cubic units.

Since we have cubic units, we would have × = total cubes with -unit side lengths.

There are cubes with -unit side lengths in cubic unit because the volume of one cube is

cubic units.

Since we have cubic units, we would have × = total cubes with -unit side lengths. − = more cubes

b.

Finally, the prism is filled with cubes whose side lengths are fill the prism?

unit. How many unit cubes would it take to

cubic units.

Since there are cubic units, we would have × = total cubes with side lengths of unit.

There are cubes with -unit side lengths in cubic unit because the volume of one cube is
4.

lengths of . These boxes are then packed into a shipping box with dimensions of . × . × .

A toy company is packaging its toys to be shipped. Some of the toys are placed inside a cube-shaped box with side × × = toys

a.

How many toys can be packed into the larger box for shipping?

b.

Use the number of toys that can be shipped in the box to help determine the volume of the box.

One small box would have a volume of . × . × . = .

Now, I will multiply the number of cubes by the volume of the cube. × =

5.

A rectangular prism has a volume of . cubic meters. The height of the box is . meters, and the length is
. meters.
a.

Write an equation that relates the volume to the length, width, and height. Let represent the width, in meters. b.

Solve the equation.

. = (. )(. )

. = . = .

The width is . .

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Date:

Volume with Fractional Edge Lengths and Unit Cubes
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6•5

Additional Exercise from Scaffolding Box

dimensional) squares with side lengths of 1 unit, unit, and unit, which leads to an understanding of three-dimensional
1

1

This is a sample activity that fosters understanding of a cube with fractional edge lengths. It begins with three (two2

cubes that have edge lengths of 1 unit, unit, and unit.
1
2



3

3

How many squares with -unit side lengths will fit in a square with 1 unit side lengths?
1
2





1

Four squares with -unit side lengths will fit in the square with 1-unit side lengths.
1
2

1
2

The area of a square with -unit side lengths is of the area of a square with 1-unit side lengths, so it

What does this mean about the area of a square with -unit side lengths?


1
2

1
4

has an area of square units.

Lesson 11:
Date:

1
4

Volume with Fractional Edge Lengths and Unit Cubes
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How many squares with side lengths of units will fit in a square with side lengths of 1 unit?
1
3





Nine squares with side lengths of unit will fit in a square with side lengths of 1 unit.
1
3

1
3

The area of a square with -unit side lengths is of the area of a square with 1-unit side lengths, so it

What does this mean about the area of a square with -unit side lengths?


1
3

1
9

has an area of square units.


1
9

Let’s look at what we have seen so far:
Side Length (units)

How many fit into a unit square?

1
2

4

1

1

1
3

Sample questions to pose:


6•5

9
1
4

Make a prediction about how many squares with -unit side lengths will fit into a unit square; then, draw a
16 squares

picture to justify your prediction.


Lesson 11:
Date:

Volume with Fractional Edge Lengths and Unit Cubes
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6•5

How could you determine the number of -unit side length squares that would cover a figure with an area of 15
1
2

1
3

4 squares of -unit side lengths fit in each 1 square unit. So, if there are 15 square units, there will be

square units? How many -unit side length squares would cover the same figure?




1
2

15 × 4 = 60 squares of -unit side lengths.
1
2

9 squares of -unit side lengths fit in each 1 square unit. So, if there are 15 square units, there will be
1
3

15 × 9 = 135 squares of -unit side lengths.
1
3



Now let’s see what happens when we consider cubes of 1-, -, and -unit side lengths.



How many cubes with -unit side lengths will fit in a cube with 1-unit side lengths?

1
2

1
3

1
2



Eight of the cubes with -unit side lengths will fit into the cube with a 1-unit side lengths.
1
2

Lesson 11:
Date:

Volume with Fractional Edge Lengths and Unit Cubes
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1
2

The volume of a cube with -unit side lengths is of the volume of a cube with 1-unit side lengths, so it

What does this mean about the volume of a cube with -unit side lengths?


1
2

1
8

1
3




8

27 of the cubes with -unit side lengths will fit into the cube with 1-unit side lengths.
1
3

1
3

of the volume of a square with 1-unit side lengths, so

What does this mean about the volume of a cube with -unit side lengths?


1

1
3

The volume of a cube with -unit side lengths is it has a volume of



1

How many cubes with -unit side lengths will fit in a cube with 1-unit side lengths? has a volume of cubic units.



27

cubic units.

1

27

Let’s look at what we have seen so far:
Side Length (units)

How many fit into a unit cube?

1
2

8

1
1
3

Sample questions to pose:


6•5

1

1

27

4

Make a prediction about how many cubes with -unit side lengths will fit into a unit cube, and then draw a
64 cubes

picture to justify your prediction.


Lesson 11:
Date:

Volume with Fractional Edge Lengths and Unit Cubes
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6•5

How could you determine the number of -unit side length cubes that would fill a figure with a volume of 15
1

1
3

2

8 cubes of -unit side lengths fit in each 1 cubic unit. So, if there are 15 cubic units, there will be 120

cubic units? How many -unit side length cubes would fill the same figure?



1
2

cubes because 15 × 8 = 120.

27 cubes of -unit side lengths fit in each 1 cubic unit. So, if there are 15 cubic units, there will be 405
1
3

cubes because 15 × 27 = 405.

Understanding Volume
Volume



Volume is the amount of space inside a three-dimensional figure.



It is measured in cubic units.



It is the number of cubic units needed to fill the inside of the figure.

Cubic Units







Cubic units measure the same on all sides. A cubic centimeter is one centimeter on all sides; a cubic inch is one inch on all sides, etc.
Cubic units can be shortened using the exponent 3.

6 cubic centimeter = 6 cm3

Different cubic units can be used to measure the volume of space figures—cubic inches, cubic yards, cubic centimeters, etc.

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Date:

Volume with Fractional Edge Lengths and Unit Cubes
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169

Lesson 11

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Number Correct: ______

Multiplication of Fractions II—Round 1
1 5
×
2 8
3 3
×
4 5
1 7
×
4 8
3 2
×
9 5
5 3
×
8 7
3 4
×
7 9
2 3
×
5 8
4 5
×
9 9
2 5
×
3 7
2 3
×
7 10
3 9
×
4 10
3 2
×
5 9

Directions: Determine the product of the fractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.

2 5
×
10 6
5 7
×
8 10
3 7
×
5 9

Lesson 11:
Date:

28.
29.
30.

2 3
×
9 8
3 8
×
8 9
3 7
×
4 9
3 10
×
5 13
2 7
1 ×
7 8
1
5
3 ×3
2
6
7
1
1 ×5
8
5
4
2
5 ×3
5
9
2
3
7 ×2
5
8
2
3
4 ×2
3
10
3
1
3 ×6
5
4
7
1
2 ×5
9
3
3
1
4 ×3
8
5
1
2
3 ×5
3
5
2
2 ×7
3

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6•5

Multiplication of Fractions II—Round 1 [KEY]
1 5
×
2 8
3 3
×
4 5
1 7
×
4 8
3 2
×
9 5
5 3
×
8 7
3 4
×
7 9
2 3
×
5 8
4 5
×
9 9
2 5
×
3 7
2 3
×
7 10
3 9
×
4 10
3 2
×
5 9

Directions: Determine the product of the fractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

=

=

=

=

=

=

2 5
×
10 6
5 7
×
8 10
3 7
×
5 9

Lesson 11:
Date:

16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.

2 3
×
9 8
3 8
×
8 9
3 7
×
4 9
3 10
×
5 13
2 7
1 ×
7 8
1
5
3 ×3
2
6
7
1
1 ×5
8
5
4
2
5 ×3
5
9
2
3
7 ×2
5
8
2
3
4 ×2
3
10
3
1
3 ×6
5
4
7
1
2 ×5
9
3
3
1
4 ×3
8
5
1
2
3 ×5
3
5
2
2 ×7
3

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

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Lesson 11

NYS COMMON CORE MATHEMATICS CURRICULUM

Number Correct: ______
Improvement: ______

Multiplication of Fractions II—Round 2
2 5
×
3 7
1 3
×
4 5
2 2
×
3 5

Directions: Determine the product of the fractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

16.
17.
18.

5 5
×
9 8
5 3
×
8 7
3 7
×
4 8
2 3
×
5 8
3 3
×
4 4
7 3
×
8 10
4 1
×
9 2
6 3
×
11 8
5 9
×
6 10
3 2
×
4 9
4 5
×
11 8
2 9
×
3 10

Lesson 11:
Date:

6•5

19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.

3 2
×
11 9
3 10
×
5 21
4 3
×
9 10
3 4
×
8 5
6
2
×
11 15
2 3
1 ×
3 5
1 3
2 ×
6 4
2
2
1 ×3
5
3
2
1
4 ×1
3
4
1
4
3 ×2
2
5
3
3×5
4
2
1
1 ×3
3
4
3
2 ×3
5

5
1
1 ×3
7
2
1
9
3 ×1
3
10

Volume with Fractional Edge Lengths and Unit Cubes
11/5/14

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172

Lesson 11

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Multiplication of Fractions II—Round 2 [KEY]
2 5
×
3 7
1 3
×
4 5
2 2
×
3 5

Directions: Determine the product of the fractions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.

15.

19.

22.

=

25.

=

28.

17.

20.

23.

=

26.

21.

=

24.

=

2 9
×
3 10

18.

=

5 5
×
9 8
5 3
×
8 7
3 7
×
4 8
2 3
×
5 8
3 3
×
4 4
7 3
×
8 10
4 1
×
9 2
6 3
×
11 8
5 9
×
6 10
3 2
×
4 9
4 5
×
11 8

Lesson 11:
Date:

16.

27.

=

29.

30.

3 2
×
11 9
3 10
×
5 21
4 3
×
9 10
3 4
×
8 5
6
2
×
11 15
2 3
1 ×
3 5
1 3
2 ×
6 4
2
2
1 ×3
5
3
2
1
4 ×1
3
4
1
4
3 ×2
2
5
3
3×5
4
2
1
1 ×3
3
4
3
2 ×3
5
5
1
1 ×3
7
2

1
9
3 ×1
3
10

=

=

=

=

=

=

=
=

=

=
=

=
=

=

=

=

=

=

=

Volume with Fractional Edge Lengths and Unit Cubes
11/5/14

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173

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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 12: From Unit Cubes to the Formulas for Volume
Student Outcomes


Students extend the volume formula for a right rectangular prism to the formula = Area of base ∙ height. They understand that any face can be the base.

Lesson Notes
This lesson is a continuation of the ideas in Lesson 11 and the lessons in Grade 5, Module 5, Topics A and B.
The word face, though referenced in the last lesson, should be taught to students who may not know this meaning of it.
A student-friendly definition and illustration can be posted on the wall (along with definitions of edge(s) and vertex/ vertices). Here is a link to a useful illustration: http://www.11plusforparents.co.uk/Maths/shape8.html.

Classwork
Example 1 (10 minutes)


Look at the rectangular prisms in the first example. Write a numerical expression for the volume of each rectangular prism.




What do these expressions have in common?





MP.8

Answers provided below.

We could use area of the base times the height. = ; = (15 in. ) �1

1
1
in. �; and = 22 in2
2
2

How would we use the area of the base to determine the volumes? (Think about the unit cubes we have been using. The area of the base would be the first layer of unit cubes. What would the height represent?)




You may want to use unit cubes to help students visualize the layers in this problem.

What is the area of the base of each of the rectangular prisms?




Scaffolding:

If we know volume for a rectangular prism as length times width times height, what is another formula for volume that we could use based on these examples?




They represent the area of the bases of the rectangular prisms.

Rewrite each of the numerical expressions to show what they have in common.




They have the same dimensions for the lengths and widths.

What do these dimensions represent?


MP.7

Answers provided below.

We would multiply the area of the base times the height. The height would represent how many layers of cubes it would take to fill up the rectangular prism. Sample answers are below.

How do the volumes of the first and second rectangular prisms compare? The first and third?


The volume of the second prism is twice that of the first because the height is doubled. The volume of the third prism is three times that of the first because the height is tripled.

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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6•5

Example 1

. � ( . )

. � ( . )

. � ( . )

a.

Write a numerical expression for the volume of each of the rectangular prisms above.

( . ) �

b.

What do all of these expressions have in common? What do they represent?

c.

Rewrite the numerical expressions to show what they have in common.



d.

If we know volume for a rectangular prism as length times width times height, what is another formula for volume that we could use based on these examples?

e.

What is the area of the base for all of the rectangular prisms?

f.

Determine the volume of each rectangular prism using either method.

( . ) �

All of the expressions have ( . ) � . �. This is the area of the base.

� ( . )



� ( . )



� ( . )

We could use ( )(), or area of the base times height.
( . ) �
( . ) �
( . ) �
( . ) �

g.

( . ) �

. � =

. � ( . ) =

. � ( . ) =

. � ( . ) =

×

or or or







� ( . ) =

� ( . ) =

� ( . ) =

How do the volumes of the first and second rectangular prisms compare? The volumes of the first and third? =

= ×

The volume of the second prism is twice that of the first because the height is doubled. The volume of the third prism is three times as much as the first because the height is triple the first prism’s height.

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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NYS COMMON CORE MATHEMATICS CURRICULUM



What do you think would happen to the volume if we turn this prism on its side so that a different face is the base? (Have students calculate the area of the base times the height for this new prism. To help students visualize what is happening with this rotation, you could use a textbook or a stack of index cards and discuss how this prism is similar and/or different to the rectangular prisms in part (a).)


Answers will vary. Some students may see that the volume will be the same no matter which face is the base.
2

1
2

Volume = Area of the base × height
Volume = �4

Volume = 67

1

1 3 in 2

1 in. 2

3 in.

The volumes in both solutions are the same.

What other expressions could we use to determine the volume of the prism?


MP.7


1 2 in � (15 in. )
2

How does this volume compare with the volume you calculated using the other face as the base?




15 in.

Area of the base = (3 in. ) �1 in. �

Area of the base = 4.5 in



6•5

Answers will vary. Some possible variations are included below.
1
15 in. × 1 in. × 3 in.
2
1
15 in. × 3 in. × 1 in.
2
1
3 in. × 15 in. × 1 in.
2
1
45 in2 × 1 in.
2

We notice that 3 in. × 15 in. × 1 in. and 45 in2 × 1 in. are equivalent, and both represent the volume.
1
2

2

The first expression (3 in. × 15 in. × 1 in.) shows that the volume is the product of three edge lengths.

How do they communicate different information?


1

1
2

The second (45 in2 × 1 in.) shows that the volume is the product of the area of the base and the
1
2

height.

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
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6•5

Example 2 (5 minutes)
Example 2

The base of a rectangular prism has an area of = ×

= � � � . �

= � � � . �

=

rectangular prism.



. The height of the prism is . Determine the volume of the

Do we need to know the length and the width to find the volume of the rectangular prism?


The length and width are needed to calculate the area of the base, and we already know the area of the base. Therefore, we do not need the length and width. The length and width are used to calculate the area, and we are already given the area.

Exercises (20 minutes)
The cards are printed out and used as stations or hung on the classroom walls so that students can move from question to question. Copies of the questions can be found at the end of the lesson. Multiple copies of each question can be printed so that a small number of students visit each question at a time. Students should spend about three minutes at each station, where they will show their work by first writing a numerical expression, and then using the expression to calculate the volume of the rectangular prism described. They will use the rest of the time to discuss the answers, and the teacher can answer any questions students have about the lesson.
Card
a.

Draw a sketch of the figure. Then, calculate the volume. Area of the base = 4 ft 2

Rectangular Prism
Height = 2 ft.
1
2

b.

3
8

Draw a sketch of the figure. Write the length, width, and height in feet. Then, calculate the volume. Length is 2 times as long as the height.

Rectangular Prism
3
4

1
2

Height = 3 ft.

Width is as long as the height.

Lesson 12:
Date:

Sample Response = Area of base × height

3 2
1
ft � �2 ft. �
8
2
35 2 5 = � ft � � ft. �
8
2
175 3 = ft 16
1 15
Length = 3 ft. × 2 = ft. 2
2
3
9
Width = 3 ft. × = ft.
4
4 = �4

= ℎ

15
9
ft. � � ft. � (3 ft. )
2
4
405 3 = ft 8 = �

From Unit Cubes to the Formulas for Volume
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177

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

c.

Write two different expressions to represent the volume, and explain what each one represents.

MP.7

d.

Calculate the volume.

.

=

e.

Answers will vary. Some possible solutions include
2
1
1
14
1
�4 m� � m� �1 m� and � m2 � �1 m� .
3
3
8
9
8
The first expression shows the volume as a product of the three edge lengths. The second expression,
�4

2
1
m� � m�, shows the volume as a product of the
3
3

base area times the height.

= Area of base × height

4
3
= � ft 2 � � ft. �
3
10
12 3 ft =
30
2 = ft 3
5

= Area of base × height = �13

Calculate the volume.

.

=

=

1 2
1
in � �1 in. �
2
3

108 3 in 6

= 18 in3

Lesson 12:
Date:

6•5

From Unit Cubes to the Formulas for Volume
11/5/14

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178

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

f.

Challenge:

Determine the volume of a rectangular prism whose length and width are in a ratio of 3: 1. The width and height are in a ratio of 2: 3. The length of the rectangular prism is 5 ft.

6•5

Length = 5 ft.

5 ft. 3
5
3 5
Height = ft. ×
= ft.
3
2
2
Width = 5 ft. ÷ 3 = = ℎ

5
5
= (5 ft. ) � ft. � � ft. �
2
3
125 3 = ft 6

Extension (3 minutes)
A company is creating a rectangular prism that must have a volume of . The company also knows that the area of
Extension

the base must be . How can you use what you learned today about volume to determine the height of the

rectangular prism?

I know that the volume can be calculated by multiplying the area of the base times the height. So, if I needed the height instead, I would do the opposite. I would divide the volume by the area of the base to determine the height. = ×

= � �

÷ =

. =

How is the formula = ∙ ∙ h related to the formula = Area of the base ∙ height?

Closing (2 minutes)




When we multiply the length and width of the rectangular prism, we are actually finding the area of the base. Therefore, the two formulas both determine the volume of the rectangular prism.

Exit Ticket (5 minutes)

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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179

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 12: From Unit Cubes to the Formulas for Volume
Exit Ticket
1.

Determine the volume of the rectangular prism in two different ways.
3
ft.
4

3 ft. 4

2.

3 ft. 8

The area of the base of a rectangular prism is 12 cm2 , and the height is 3 rectangular prism.

Lesson 12:
Date:

1 cm. Determine the volume of the
3

From Unit Cubes to the Formulas for Volume
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180

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
1.

= ∙ ∙

= � =

2.

= ∙

Determine the volume of the rectangular prism in two different ways.

. � � . � � . �

= �

=

.

� � . �

The area of the base of a rectangular prism is , and the height is = ∙

= ( ) � �

=

=

rectangular prism.

.

.

. Determine the volume of the

Problem Set Sample Solutions
1.

=

Determine the volume of the rectangular prism. = �

=

2.

� � � � �

The area of the base of a rectangular prism is = ×

rectangular prism. = � = �

=

� � . �

, and the height is

. Determine the volume of the

� � . �

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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181

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

3.

6•5

The length of a rectangular prism is times as long as the width. The height is of the width. The width is . =

Determine the volume. = × = ×

= = � =

4.

a.

=

=

� ( ) � �

.

.

.

Write numerical expressions to represent the volume in two different ways, and explain what each reveals.


. � � . � ( . ) represents the product of three edge lengths. � � ( ) represents the

product of the base area times the height. Answers will vary.

b.

5.

. � � . � ( . ) = or � � ( . ) =

Determine the volume of the rectangular prism.


An aquarium in the shape of a rectangular prism has the following dimensions: length = , width = and height =
a.

.

Write numerical expressions to represent the volume in two different ways, and explain what each reveals.
( ) �

� � � represents the product of the three edge lengths.

( ) �
b.

,

� represents the base area times the height.

� =

Determine the volume of the rectangular prism.
( ) �

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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182

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6.

6•5

The area of the base in this rectangular prism is fixed at . This means that for the varying heights, there will be various volumes.
a.

Complete the table of values to determine the various heights and volumes.
Height in Centimeters

Volume in Cubic
Centimeters

b.

Write an equation to represent the relationship in the table. Be sure to define the variables used in the equation. Let be the height of the rectangular prism in centimeters.

Let be the volume of the rectangular prism in cubic centimeters. =

c.

The unit rate is .

What is the unit rate for this proportional relationship? What does it mean in this situation?

For every centimeter of height, the volume increases by because the area of the base is . In order to determine the volume, multiply the height by .

The volume of a rectangular prism is . . The height is . .
a.

Let represent the area of the base of the rectangular prism. Write an equation that relates the volume, the area of the base, and the height.

b.

7.

Solve the equation for .

. = .

. .
=
.
.
= .

The area of the base is . .

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
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183

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Station A
Make a sketch of the figure. Then, calculate the volume.
Rectangular prism:

Area of the base =
Height =

Lesson 12:
Date:

.

From Unit Cubes to the Formulas for Volume
11/5/14

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184

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Station B
Make a sketch of the figure. Write the length, the width, and height in feet. Then, calculate the volume.
Rectangular prism:

Length is times the height.

Width is as long as the height.
Height = .

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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185

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Station C
Write two different expressions to represent the volume, and explain what each expression represents.

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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186

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Station D
Calculate the volume.

.

=

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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187

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Station E
Calculate the volume.

=

Lesson 12:
Date:

.

From Unit Cubes to the Formulas for Volume
11/5/14

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188

Lesson 12

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Station F
Challenge:
width are in a ratio of :. The width and height are in a ratio of
Determine the volume of a rectangular prism whose length and
:. The length of the rectangular prism is .

Lesson 12:
Date:

From Unit Cubes to the Formulas for Volume
11/5/14

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189

Lesson 13

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 13: The Formulas for Volume
Student Outcomes


Students develop, understand, and apply formulas for finding the volume of right rectangular prisms and cubes. Lesson Notes
This lesson is a continuation of the two previous lessons, Lessons 11 and 12, in this module and Grade 5, Module 5,
Topics A and B.

Classwork
Fluency Exercise (5 minutes): Multiplication and Division Equations with Fractions
RWBE: Refer to the Rapid White Board Exchanges section in the Module Overview for directions to administer an RWBE.

Example 1 (3 minutes)
Scaffolding:
Example 1

Determine the volume of a cube with side lengths of .

=

� � � � �

= × ×

=

= �

MP.1

Provide a visual of a cube for students to label. If needed, begin with less complex numbers for the edge lengths.

Students work through the first problem on their own, and then discuss.





Answers will vary. Sample response: I chose to use the = ℎ formula to solve.

Which method for determining the volume did you choose?

9 cm

9 cm

9 cm

= (9 cm)(9 cm)(9 cm) = 729 cm3

Why did you choose this method?


Explanations will vary according to the method chosen. Sample response: Because I know the length, width, and height of the prism, I used = ℎ instead of the other formulas.

Lesson 13:
Date:

The Formulas for Volume
11/5/14

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Lesson 13

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Example 2 (3 minutes)
Example 2
Determine the volume of a rectangular prism with a base area of

and a height of .

= ×

= � � � . �

=



What makes this problem different than the first example?
This example gives the area of the base instead of just giving the length and width.




I could try fitting cubes with fractional lengths. However, I could not use the = ℎ formula because I do not know the length and width of the base.

Would it be possible to use another method or formula to determine the volume of the prism in this example?


Exercises (27 minutes)
In the exercises, students will explore how changes in the lengths of the sides affect the volume. Students can use any method to determine the volume as long as they can explain their solution. Students work in pairs or small groups.
Please note that the relationships between the volumes will be more easily determined if the fractions are left in their original form when solving. If time allows, this could be an interesting discussion point, either between partners, groups, or as a whole class when discussing the results of their work.
Exercises
1.

Use the rectangular prism to answer the next set of questions.

.

a.

=

= × = � � � . �

=

Determine the volume of the prism.

Lesson 13:
Date:

Scaffolding:
 The wording half as long may confuse some students. Explain that half as long means that the original length was multiplied by one-half or divided by 2. A similar explanation can be used for one-third as long and one-fourth as long.
 Explain to students that the word doubled refers to twice as many or multiplied by two.

The Formulas for Volume
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Lesson 13

NYS COMMON CORE MATHEMATICS CURRICULUM

b.

c.

6•5

× = � . × � =
.

Determine the volume of the prism if the height of the prism is doubled. = ×

= � � � .�

= or

Compare the volume of the rectangular prism in part (a) with the volume of the prism in part (b). What do you notice?
When the height of the rectangular prism is doubled, the volume is also doubled.

d.

Complete and use the table below to determine the relationship between the height and volume.

Height in Feet

MP.2

Volume in Cubic Feet

What happened to the volume when the height was tripled?
The volume tripled.

What happened to the volume when the height was quadrupled?
The volume quadrupled.
What conclusions can you make when the base area stays constant and only the height changes?
Answers will vary but should include the idea of a proportional relationship. Each time the height is multiplied by a number, the original volume will be multiplied by the same amount.

a.

If represents the area of the base and represents the height, write an expression that represents the volume. b.

If we double the height, write an expression for the new height.

c.

Write an expression that represents the volume with the doubled height.

2.

MP.7

Lesson 13:
Date:

The Formulas for Volume
11/5/14

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Lesson 13

NYS COMMON CORE MATHEMATICS CURRICULUM

d.

MP.7

3.

6•5

Write an equivalent expression using the commutative and associative properties to show the volume is twice the original volume.
()

Use the cube to answer the following questions.
a.

Determine the volume of the cube.

=

= ()()() =

=

= � � � � � �

=

b.

Determine the volume of a cube whose side lengths are half as long as the side lengths of the original cube.

c.

Determine the volume if the side lengths are one-fourth as long as the original cube’s side lengths.

d.

Determine the volume if the side lengths are one-sixth as long as the original cube’s side length.

=

= � � � � � �

=

=

= � � � � � �

=

OR

=

e.

Explain the relationship between the side lengths and the volumes of the cubes.

figure will be � � of the original volume. For example, if the sides are as long, then the volume will be

� � = as much.

If each of the sides are changed by the same fractional amount, , of the original, then the volume of the new

Lesson 13:
Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

4.

6•5

Check to see if the relationship you found in Exercise 3 is the same for rectangular prisms. .

.

.
a.

=

Determine the volume of the rectangular prism. = ( . )( . )( . ) =

b.

=

= � .� � .� � .�

=

Determine the volume if all of the sides are half as long as the original lengths.

=
c.

OR

=

= � .� � . � � . �

=

Determine the volume if all of the sides are one-third as long as the original lengths.

=
OR

d.

Is the relationship between the side lengths and the volume the same as the one that occurred in Exercise 3?
Explain your answer.
Yes, the relationship that was found in the problem with the cubes still holds true with this rectangular prism.
When I found the volume of a prism with side lengths that were one-third the original, the volume was

the original.

� � =

MP.2
&
MP.7

a.

If represents a side length of the cube, create an expression that shows the volume of the cube.

b.

5.

If we divide the side lengths by three, create an expression for the new edge length.

or

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Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

c.

� � or � �

d.

Write an equivalent expression to show that the volume is

6•5

Write an expression that represents the volume of the cube with one-third the side length.

� �

� � � � � �

� � � �

MP.2
&
MP.7

of the original volume.

Closing (2 minutes)


How did you determine which method to use when solving the exercises?






If I were given the length, width, and height, I have many options for determining the volume. I could use = ℎ. I could also determine the area of the base first and then use = Area of the base × height. I could also use a unit cube and determine how many cubes would fit inside. If I was given the area of the base and the height, I could use the formula = Area of the base × height, or I could also use a unit cube and determine how many cubes would fit inside.

What relationships did you notice between the volume and changes in the length, width, or height?


Answers will vary. Students may mention that if the length, width, or height is changed by a certain factor, the volume will be affected by that same factor.



They may also mention that if all three dimensions are changed by the same factor, the volume will be � � as large as the original.

1
2

change by that factor cubed. For example, if all the sides are as long as the original, the volume will
1 3
2

Exit Ticket (5 minutes)

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Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 13: The Formulas for Volume
Exit Ticket
1.

prism with a rectangular base with an area of 23
Determine the volume of the sample box.

2.

1 2
1
in . The height of the prism is 1 in.
3
4

A new company wants to mail out samples of its hair products. The company has a sample box that is a rectangular

A different sample box has a height that is twice as long as the original box described in Problem 1. What is the volume of this sample box? How does the volume of this sample box compare to the volume of the sample box in
Problem 1?

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Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
1.

prism with a rectangular base with an area of

= ×

= � � � . �

= × .

=

the sample box.

=
2.

. The height of the prism is . Determine the volume of

A new company wants to mail out samples of its hair products. The company has a sample box that is a rectangular

OR

A different sample box has a height that is twice as long as the original box described in Problem 1. What is the volume of this sample box? How does the volume of this sample box compare to the volume of the sample box in
Problem 1? = ×

= � � � . �

� � . � = �

=

=

OR

By doubling the height, we have also doubled the volume.

Problem Set Sample Solutions
1.

Determine the volume of the rectangular prism. = ×

= � � � �

=

OR

=

Lesson 13:
Date:

=

The Formulas for Volume
11/5/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

2.

6•5

Determine the volume of the rectangular prism in Problem 1 if the height is quadrupled (multiplied by four). Then, determine the relationship between the volumes in Problem 1 and this prism. = ×

= � � � �

=

=

OR

When the height was quadrupled, the volume was also quadrupled.
3.

The area of the base of a rectangular prism can be represented by , and the height is represented by . =

a.

Write an equation that represents the volume of the prism.

b.

If the area of the base is doubled, write an equation that represents the volume of the prism.

c.

If the height of the prism is doubled, write an equation that represents the volume of the prism.

d.

Compare the volume in parts (b) and (c). What do you notice about the volumes?

=

= =

The expressions in part (b) and part (c) are equal to each other.
e.

= =

Write an expression for the volume of the prism if both the height and the area of the base are doubled.

.

4.

Determine the volume of a cube with a side length of

5.

Use the information in Problem 4 to answer the following:

=

= � .� � .� � .�

=
. ×
. ×
.

=

a.

=

= � .� � .� � .�

= . × . × .

=

Determine the volume of the cube in Problem 4 if all of the side lengths are cut in half.

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Date:

The Formulas for Volume
11/5/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

How could you determine the volume of the cube with the side lengths cut in half using the volume in
Problem 4?
Because each side is half as long, I know that the volume will be

×

×

× =

=

the volume of the cube in Problem 4. This is

because the length, the width, and the height were all cut in half.

6.

Use the rectangular prism to answer the following questions.

a.

Complete the table.
Length

=

=

=

=

= =

=

Volume

is one-third of . Therefore, when the length is one-third as long, the volume is also one-third as much.

b.

How did the volume change when the length was one-third as long?

c.

How did the volume change when the length was tripled?

d.

What conclusion can you make about the relationship between the volume and the length?

is three times as much as . Therefore, when the length is three times as long, the volume is also three times as much.

When the length changes but the width and height stay the same, the change in the volume is proportional to the change in the length.

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Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

7.

Lesson 13

6•5

The sum of the volumes of two rectangular prisms, Box A and Box B, are . . Box A has a volume of
. .
a.

Let represent the volume of Box B in cubic centimeters. Write an equation that could be used to determine the volume of Box B.

b.

Solve the equation to determine the volume of Box B.

c.

d.

. = . + = .

If the area of the base of Box B is . , write an equation that could be used to determine the height of
Box B. Let represent the height of Box B in centimeters.
. = (. ) = .

Solve the equation to determine the height of Box B.

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Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 13

6•5

Multiplication and Division Equations with Fractions
Progression of Exercises
1.

2.

3.

4.

5.

6.

7.

8.

9.

5 = 35 =

3 = 135 =

12 = 156 =

3

= 24

=

7

= 42

= 4

13

= 18

=

2
3

= 6

=

3
5

= 9

=

3
4

= 10

=

=

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Date:

The Formulas for Volume
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NYS COMMON CORE MATHEMATICS CURRICULUM

10.

11.

12.

13.

14.

15.

5
8

Lesson 13

6•5

= 9

=

=

=

=

=

=

3
7

ℎ = 13

4

3

=

=

3
5

2
7

=

=

=

=

=

2
5
3
4

=

=

3
7
5
8

Lesson 13:
Date:

The Formulas for Volume
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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 14: Volume in the Real World
Student Outcomes


Students understand that volume is additive and apply volume formulas to determine the volume of composite solid figures in real-world contexts.



Students apply volume formulas to find missing volumes and missing dimensions.

Lesson Notes
This lesson is a continuation of the three previous lessons, Lessons 11–13, in this module and Grade 5, Module 5, Topics
A and B.

Classwork
Example 1 (6 minutes)
Example 1
a.

The area of the base of a sandbox is the sandbox.

. The volume of the sandbox is . Determine the height of

MP.1 Students make sense of this problem on their own before discussing.


What information are we given in this problem?




How can we use the information to determine the height?




We have been given the area of the base and the volume.
We know that the area of the base times the height will give the volume.
Since we already have the volume, we can do the opposite and divide to get the height.

Notice that the number for the volume is less than the number for the area.
What does that tell us about the height?


If the product of the area of the base and the height is less than the area, we know that the height must be less than 1.
Lesson 14:
Date:

Note to Teacher:
In these examples, it might be easier for students to use common denominators when dividing and working with dimensions. Students can use the invert and multiply rule, but it may cause more work and make it harder to see the relationships. Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM



6•5

Volume = Area of the base × height

Calculate the height.


= ℎ

1
1 3 ft = �9 ft 2 � ℎ
2
8
19 2
57 3 ft = � ft � ℎ
2
8
19
2 2
57 3 2 ft ×
=
×� ft � ℎ
19
2
19
8
114
3 ft. = ℎ or ft. = ℎ
152
4
7

MP.1

b.

height of



What new information have we been given in this problem?




. Determine the volume of the sand.

The sandbox was filled with sand, but after kids played, some of the sand spilled out. Now, the sand is at a

This means that the sandbox is not totally filled. Therefore, the volume of sand used is not the same as the volume of the sandbox.

How will we determine the volume of the sand?


1

ft. instead of the height of the sandbox.

To determine the volume of the sand, I will use the area of the base of the sand box, but I will use the height of

2

Volume = Area of the base × height

1 2 1 ft × ft.
2
2
19 2 1
Volume = ft × ft.
2
2
19 3
Volume = ft 4
3
Volume = 4 ft 3
4
Volume = 9

The volume of the sand is 4 ft 3 .
3
4

Lesson 14:
Date:

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Example 2 (6 minutes)
Example 2
A special order sandbox has been created for children to use as an archeological digging area at the zoo. Determine the volume of the sandbox.



Describe this three-dimensional figure.





I could think of it as a piece on the left and a piece on the right.



MP.7

This figure looks like two rectangular prisms that have been placed together to form one large prism.

Or, I could think of it as a piece in front and a piece behind.

How can we determine the volume of this figure?




Does it matter which way we divide the shape when we calculate the volume?




We can find the volume of each piece and then add the volumes together to get the volume of the entire figure.
Answers will vary.

At this point, you can divide the class in half and have each half determine the volume using one of the described methods.


Volume of prism on the left = ℎ.
1
3 = 2 m × 2 m × m
5
4
1
11 m×2m× m =
5
4

If the shape is divided into a figure on the left and a figure on the right, we would have the following:

=

22 3 m 20

Lesson 14:
Date:

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Volume of the prism on the right = ℎ.
1
1
1
= 2 m × 4 m × m
3
5
4
13
1
9 m× m = m ×
3
5
4
117 3 m =
60
39 3 m =
20

Total volume = volume of left + volume of right
22 3 39 3 m + m 20
20
61 3
1
Total volume = m =3 m3 20
20

Total volume =



Volume of the back piece = ℎ
1
= 5 m × 2 m × m
5
= 2 m3

If the shape is divided into a figure with a piece in front and piece behind, we have the following:

Volume of the front piece = ℎ
1
1
1
= 2 m × 2 m × m
3
5
4
7
1
9 = m × m × m
3
5
4
3
1 3
63 3 m =1 m3 = 1 m =
60
20
60



Total volume = volume of back + volume of front
1
m3
Total volume = 2 m3 + 1
20
1 3 m Total volume = 3
20
What do you notice about the volumes determined in each method?


The volume calculated with each method is the same. It does not matter how we break up the shape.
We will still get the same volume.

Lesson 14:
Date:

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exercises (20 minutes)
Students work in pairs. When working with composite figures, have one student solve using one method and the other solve the problem another way so they can compare answers.
Exercises
1.
a.

The volume of the rectangular prism is given. Determine the missing measurement using a one-step equation. =

= � �

× = � � � �

=

=

The height is .
b.

The volume of the box is =

=

� �� � =

=

=

The area is
2.

=

=

= ?

. Determine the area of the base using a one-step equation.

� �

� � � �

.

Marissa’s fish tank needs to be filled with more water.
a.

=

Determine how much water the tank can hold.

� � � � �

=

= �

b.

=

Determine how much water is already in the tank.

� � � � �

=

= �

Lesson 14:
Date:

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

c.

6•5

How much more water is needed to fill the tank?

− = − =

Height of empty part of tank: =

=

� � � �


=

= �

3.

Determine the volume of the composite figures.
a.

=

= ( )( ) � �

( )( ) � =


=

= +

Lesson 14:
Date:

=

� � � � �

= �
� � � �


= =

= �

=

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

b.

.

.

.

.

.

=

= � .� � .� � . �

= � =

.� � .� � .�

=

= � .� � .� � .�

=

=

=

=

.

= = �

=

.� � .� � .�

=

=

=

=

+ +

+ +

.� � .� � .� + � .� � .� � .� + � .� � .� � .�

Another possible solution: = �

6•5

+ +

+ +

Closing (5 minutes)
Students take time to share their solutions with the class. Discuss the differences between the types of problems and how working with volume and the many formulas or methods for solving can help in determining how to get to a solution. Exit Ticket (8 minutes)

Lesson 14:
Date:

Volume in the Real World
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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 14: Volume in the Real World
Exit Ticket
1.

Determine the volume of the water that would be needed to fill the rest of the tank.

3 m 4

2.

Determine the volume of the composite figure.

1
1 m
4

5 ft. 8

1 ft. 4

Lesson 14:
Date:

1 m 2

1 ft. 6

1 m 2

1 ft. 4

1 ft. 3

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
1.

=

= � � � � � �

=

Determine the volume of the water that would be needed to fill the rest of the tank.

=

= � � � � � �

=

=

=

2.

− =

=

= � .� � .� � .�

=

.

Determine the volume of the composite figure.

=

= � .� � .� � .�

=

=

+ =

.

.

.

.

Problem Set Sample Solutions
1.

The volume of a rectangular prism is = ÷ = =

÷ .

, and the height of the prism is

. Determine the area of the base.

÷
.

= ÷ . =

Lesson 14:
Date:

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

2.

The volume of a rectangular prism is

= ÷

prism.

= =

=
3.

÷

. Determine the height of the rectangular

÷

. OR
.

Determine the volume of the space in the tank that still needs to be filled with water if the water is

=

= ( .) � =

= = =
4.

. The area of the base is

6•5

.� ( .)

.



. deep.

.

.

=

= ( .) � =

.� � .�



Determine the volume of the composite figure.

= = � = =

� � � � �

+ = OR

Lesson 14:
Date:

=

= � =

� � � � �

Volume in the Real World
11/5/14

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Lesson 14

NYS COMMON CORE MATHEMATICS CURRICULUM

5.

.

.� � .� + ( .) � .� � .�

Determine the volume of the composite figure. = ( .) � = ( .) � =

=

.� � .� + ( .) � .� � .�

= OR

.

.

.

+

6.

6•5

.

.

× × � + � × × �

a.

Write an equation to represent the volume of the composite figure.

b.

Use your equation to calculate the volume of the composite figure.

= �

× × � + � × × �

= � × × � + � × × �

= +

= +

=

=

= �

Lesson 14:
Date:

Volume in the Real World
11/5/14

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New York State Common Core

6

Mathematics Curriculum

GRADE

GRADE 6 • MODULE 5

Topic D:

Nets and Surface Area
6.G.A.2, 6.G.A.4
Focus Standard:

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.A.4

Instructional Days:

6.G.A.2

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. 5

Lesson 15: Representing Three-Dimensional Figures Using Nets (M) 1
Lesson 16: Constructing Nets (E)
Lesson 17: From Nets to Surface Area (P)
Lesson 18: Determining Surface Area of Three-Dimensional Figures (P)
Lesson 19: Surface Area and Volume in the Real World (P)
Lesson 19a: Addendum Lesson for Modeling―Applying Surface Area and Volume to Aquariums
(Optional) (M)

Topic D begins with students constructing three-dimensional figures through the use of nets in Lesson 15.
They determine which nets make specific solid figures and also determine if nets can or cannot make a solid figure. Students use physical models and manipulatives to do actual constructions of three-dimensional figures with the nets. Then, in Lesson 16, students move to constructing nets of three-dimensional objects using the measurements of a solid’s edges. Using this information, students will move from nets to determining the surface area of three-dimensional figures in Lesson 17. In Lesson 18, students determine that a right rectangular prism has six faces: top and bottom, front and back, and two sides. They determine
1

Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

Topic D:
Date:

Nets and Surface Area
11/5/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Topic D

6•5

that surface area is obtained by adding the areas of all the faces and develop the formula = 2 + 2ℎ + 2ℎ. They develop and apply the formula for the surface area of a cube as = 6 2 .
For example:

Topic D concludes with Lesson 19, in which students determine the surface area of three-dimensional figures in real-world contexts. To develop skills related to application, students are exposed to contexts that involve both surface area and volume. Students are required to make sense of each context and apply concepts appropriately. Topic D:
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Nets and Surface Area
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Lesson 15

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6•5

Lesson 15: Representing Three-Dimensional Figures Using
Nets
Student Outcomes


Students construct three-dimensional figures through the use of nets. They determine which nets make specific solid figures and determine if nets can or cannot make a solid figure.

Lesson Notes
Using geometric nets is a topic that has layers of sequential understanding as students progress through the years. For
Grade 6, specifically in this lesson, the working description of a net is this: If the surface of a three-dimensional solid can be cut along enough edges so that the faces can be placed in one plane to form a connected figure, then the resulting system of faces is called a net of the solid.
A more student-friendly description used for this lesson is the following: Nets are two-dimensional figures that can be folded to create three-dimensional solids.
Solid figures and the nets that represent them are necessary for this lesson. These three-dimensional figures include a cube, a right rectangular prism, a triangular prism, a tetrahedron, a triangular pyramid (equilateral base and isosceles triangular sides), and a square pyramid.

There are reproducible copies of these nets included with this lesson. The nets of the cube and right rectangular prism are sized to wrap around solid figures made from wooden or plastic cubes with 2 cm-edges. Assemble these two solids prior to the lesson in enough quantities to allow students to work in pairs. If possible, the nets should be reproduced on card stock and pre-cut and pre-folded before the lesson. One folded and taped example of each should also be assembled before the lesson.
The triangular prism has a length of 6 cm and has isosceles right triangular bases with identical legs that are 2 cm in length. Two of these triangular prisms can be arranged to form a rectangular prism.
The rectangular prism measures 4 cm × 6 cm × 8 cm, and its net will wrap around a Unifix cube solid that has dimensions of 2 × 3 × 4 cubes.

The tetrahedron has an edge length of 6 cm. The triangular pyramid has a base edge length of 6 cm and isosceles sides with a height of 4 cm.
The square pyramid has a base length 6 cm and triangular faces that have a height of 4 cm.

Also included is a reproducible sheet that contains 20 unique arrangements of six squares. Eleven of these can be folded to a cube, while nine cannot. These should also be prepared before the lesson, as indicated above. Make enough sets of nets to accommodate the number of groups of students.
Prior to the lesson, cut a large cereal box into its net which will be used for the Opening Exercise. Tape the top flaps thoroughly so this net will last through several lessons. If possible, get two identical boxes and cut two different nets like the graphic patterns of the cube nets below. Add a third uncut box to serve as a right rectangular solid model.

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Representing Three-Dimensional Figures Using Nets
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Classwork
Mathematical Modeling Exercise (10 minutes)
Display the net of the cereal box with the unprinted side out, perhaps using magnets on a whiteboard. Display the nets below as well (images or physical nets).



What can you say about this cardboard (the cereal box)?




How do you think it was made?




Accept all correct answers, such as it is irregularly shaped; it has three sets of identical rectangles; all vertices are right angles; it has fold lines; it looks like it can be folded into a 3D shape (box), etc.
Accept all plausible answers, including the correct one.

Similarities: There are 6 sections in each; they can be folded to make a 3D shape; etc.

Compare the cereal box net to these others that are made of squares.



Differences: One is made of rectangles; others are made of squares; there is a size difference; etc.

Turn over the cereal box to demonstrate how it was cut. Reassemble it to resemble the intact box. Then, direct attention to the six-square arrangements.


What do you think the six-square shapes will fold up into?




Cubes

If that were true, how many faces would it have?


Six

Fold each into a cube.


Consider this six-square arrangement:



Do you think it will fold into a cube?

Encourage a short discussion, inviting all views. As students make claims, ask for supporting evidence of their position.
Use the cut-out version to demonstrate that this arrangement will not fold into a cube. Then, define the term net.


Today we will work with some two-dimensional figures that can be folded to create three-dimensional solids.
These are called geometric nets, or nets.

Ask students if they are able to visualize folding the nets without touching them. Expect a wide variety of spatial visualization abilities necessary to do this. Those that cannot readily see the outcome of folding will need additional time to handle and actually fold the models.

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Date:

Representing Three-Dimensional Figures Using Nets
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Use the previously cut out six-square arrangements. Each pair or triad of students will need a set of 20 with which to experiment. These are sized to wrap around a cube with side lengths of 4 cm, which can be made from eight Unifix cubes. Each group needs one of these cubes.

Exercise (10 minutes): Cube



There are some six-square arrangements on your student page. Sort each of the six-square arrangements into one of two piles, those that are nets of a cube (can be folded into a cube) and those that are not.
Exercise: Cube
1.

Nets are two-dimensional figures that can be folded up into three-dimensional solids. Some of the drawings below are nets of a cube. Others are not cube nets; they can be folded, but not into a cube.

a.

Experiment with the larger cut out patterns provided. Shade in each of the figures above that will fold into a cube. b.

Write the letters of the figures that can be folded up into a cube.

MP.1

A, B, C, E, G, I, L, M, O, P, and T
c.

Write the letters of the figures that cannot be folded up into a cube.
D, F, H, J, K, N, Q, R, and S

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Representing Three-Dimensional Figures Using Nets
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Example 1 (10 minutes): Other Solid Figures
Provide student pairs with a set of nets for each of the following: right rectangular prism, triangular prism, tetrahedron, triangular pyramid (equilateral base and isosceles triangular sides), and square pyramid.
PRISM: A prism is a solid geometric figure whose two bases are parallel to identical polygons and whose sides are parallelograms.
PYRAMID:

A pyramid is a solid geometric figure formed by connecting a polygonal base and a point and forming triangular lateral faces. (Note: The point is sometimes referred to as the apex.)
Display one of each solid figure. Assemble them so the grid lines are hidden (inside).
Allow time to explore the nets folding around the solids.


Why are the faces of the pyramid triangles?




Why are the faces of the prism parallelograms?




 English language learners may hear similarities to the words ladder or literal, neither of which are related nor make sense in this context.

The length of the base edges will match one set of sides of the parallelogram. The shape of the base polygon will determine the number of lateral faces the prism has.
No, there are six sides on the prism, plus two bases, for a total of eight faces.

What is the relationship between the number of sides on the polygonal base and the number of faces on the prism? 



 All students may benefit from a working definition of the word lateral. In this lesson, the word side can be used (as opposed to the word base).

If the bases are hexagons, does this mean the prism must have six faces?




The two bases are identical polygons on parallel planes. The lateral faces are created by connecting each vertex of one base with the corresponding vertex of the other base, thus forming parallelograms.

 Assembled nets of each solid figure should be made available to students who might have difficulty making sharp, precise folds. How are these parallelograms related to the shape and size of the base?




The base of the triangle matches the edge of the base of the pyramid.
The top vertex of the lateral face is at the apex of the pyramid. Further, each face has two vertices that are the endpoints of one edge of the pyramid’s base, and the third vertex is the apex of the pyramid.

Scaffolding:

The total number of faces will be two more than the number of sides on the polygonal bases.

What additional information do you know about a prism if its base is a regular polygon?


All the lateral faces of the prism will be identical.

Example 2 (8 minutes): Tracing Nets
If time allows, or as an extension, ask students to trace the faces of various solid objects (i.e., wooden or plastic geosolids, paperback books, packs of sticky notes, or boxes of playing cards). After tracing a face, the object should be carefully rolled so one edge of the solid matches one side of the polygon that has just been traced. If this is difficult for students because they lose track of which face is which as they are rolling, the faces can be numbered or colored differently to make this easier. These drawings should be labeled “Net of a [Name of Solid].” Challenge students to make as many different nets of each solid as they can.

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Date:

Representing Three-Dimensional Figures Using Nets
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Closing (3 minutes)


What kind of information can be obtained from a net of a prism about the solid it creates?




When looking at a net of a pyramid, how can you determine which faces are the bases?




If the net is a pyramid, there will be multiple, identical triangles that will form the lateral faces of the pyramid, while the remaining face will be the base (and will identify the type of pyramid it is).
Examples are triangular, square, pentagonal, and hexagonal pyramids.

How do the nets of a prism differ from the nets of a pyramid?




We can identify the shape of the bases and the number and shape of the lateral faces (sides). The surface area can be more easily obtained since we can see all faces at once.

If the pyramid is not a triangular pyramid, the base will be the only polygon that is not a triangle. All other faces will be triangles. Pyramids have one base and triangular lateral faces, while prisms have two identical bases, which could be any type of polygon, and lateral faces that are parallelograms.

Constructing solid figures from their nets helps us see the “suit” that fits around it. We can use this in our next lesson to find the surface area of these solid figures as we wrap them.

Lesson Summary
Nets are two-dimensional figures that can be folded to create three-dimensional solids.
A prism is a solid geometric figure whose two bases are parallel to identical polygons and whose sides are parallelograms. A pyramid is a solid geometric figure formed by connecting a polygonal base and a point and forming triangular lateral faces. (Note: The point is sometimes referred to as the apex.)

Exit Ticket (4 minutes)

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Date:

Representing Three-Dimensional Figures Using Nets
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Name

6•5

Date

Lesson 15: Representing Three-Dimensional Figures Using Nets
Exit Ticket
1.

What is a net? Describe it in your own words.

2.

Which of the following will fold to make a cube? Explain how you know.

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Representing Three-Dimensional Figures Using Nets
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Exit Ticket Sample Solutions
1.

What is a net? Describe it in your own words.
Answers will vary but should capture the essence of the definition used in this lesson. A net is a two-dimensional figure that can be folded to create a three-dimensional solid.

2.

Which of the following will fold to make a cube? Explain how you know.
Evidence for claims will vary.

Problem Set Sample Solutions
1.

Match the following nets to the picture of its solid. Then, write the name of the solid.
a.

d.

Right triangular prism
b.

e.

Rectangular pyramid
c.

f.

Rectangular prism

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Sketch a net that will fold into a cube.
Here are the possible nets for a cube.

Sketches will vary but will match one of the shaded ones from earlier in the lesson.

3.

Below are the nets for a variety of prisms and pyramids. Classify the solids as prisms or pyramids, and identify the shape of the base(s). Then, write the name of the solid.
a.

b.

Prism, the bases are pentagons.
Pentagonal Prism
c.

Pyramid, the base is a rectangle.
Rectangular Pyramid
d.

Pyramid, the base is a triangle.
Triangular Pyramid
e.

Prism, the bases are triangles.
Triangular Prism
f.

Pyramid, the base is a hexagon.
Hexagonal Pyramid

Prism, the bases are rectangles.
Rectangular Prism

Below are graphics needed for this lesson. The graphics should be printed at 100% scale to preserve the intended size of figures for accurate measurements. Adjust your copier or printer settings to actual size, and set page scale to none.
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A

B

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C

D
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E

F

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H

G

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I
J

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K

L

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M

N

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S

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Lesson 16: Constructing Nets
Student Outcomes


Students construct nets of three-dimensional objects using the measurements of a solid’s edges.

Lesson Notes
In the previous lesson, a cereal box was cut down to one of its nets. On the unprinted side, the fold lines should be highlighted with a thick marker to make all six faces easily seen. These rectangles should be labeled Front, Back, Top,
Bottom, Left Side, and Right Side. Measure each rectangle to the nearest inch, and record the dimensions on each.
During this lesson, students are given the length, width, and height of a right rectangular solid. They cut out six rectangles (three pairs), arrange them into a net, tape them, and fold them up to check the arrangement to ensure the net makes the solid. Triangular pieces are also used in constructing the nets of pyramids and triangular prisms.
When students construct the nets of rectangular prisms, if no two dimensions, length, width, or height, are equal, then no two adjacent rectangular faces will be identical.
The nets that were used in Lesson 15 should be available so that students have the general pattern layout of the nets.
Two-centimeter graph paper works well with this lesson. Prior to the lesson, cut enough polygons for Example 1.
Cutting all the nets used in this lesson will save time as well but removes the opportunity for students to do the work.

Classwork
Opening (2 minutes)
Display the cereal box net from the previous lesson. Fold and unfold it so students will recall the outcome of the lesson.


How has this net changed since the previous lesson?




What can you say about the vertices where 3 faces come together?




Some students will need more opportunities than others to manipulate the nets in this lesson. What can you say about the angles between the faces when it is folded up?




They are 90 degrees, or right angles.

What can you say about the angles in each rectangle?




It now has labels and dimensions.

Scaffolding:

The two faces also form a right angle.
Again, they form right angles.

This refolded box is an example of a right rectangular prism. It is named for the angles formed at each vertex.

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Opening Exercise (3 minutes)
Opening Exercise
Sketch the faces in the area below. Label the dimensions.

Display this graphic using a document camera or other device.

MP.1



How could you create a net for this solid? Discuss this with a partner.

Allow a short time for discussion with a partner about this before having a whole-class discussion.

Example 1 (10 minutes): Right Rectangular Prism


How can we use the dimensions of a rectangular solid to figure out the dimensions of the polygons that make up its net?




8 cm × 3 cm

What are the dimensions of the bottom?




8 cm × 3 cm

What are the dimensions of the top of this prism?




6

How many faces does the rectangular prism have?




The length, width, and height measurements of the solid will be paired to become the length and width of the rectangles.

3 cm × 5 cm

What are the dimensions of the right side?


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3 cm × 5 cm



What are the dimensions of the left side?



What are the dimensions of the front?





8 cm × 5 cm

What are the dimensions of the back?

The 6 faces of this rectangular solid are all rectangles that make up the net. Are there any faces that are identical to any others?




8 cm × 5 cm



There are three different rectangles, but two copies of each will be needed to make the solid. The top is identical to the bottom, the left and right sides are identical, and the front and back faces are also identical. Make sure each student can visualize the rectangles depicted on the graphic of the solid and can make three different pairs of rectangle dimensions (length × width, length × height, and width × height).

Display the previously cut six rectangles from this example on either an interactive whiteboard or on a magnetic surface.
Discuss the arrangement of these rectangles. Identical sides must match.
Working in pairs, ask students to rearrange the rectangles into the shape below and use tape to attach them. Having a second copy of these already taped will save time during the lesson.

Scaffolding:
 Some students will benefit from using precut rectangles and triangles.
Using cardstock or lamination will make more durable polygons.
 Other students benefit from tracing the faces of actual solids onto paper and then cutting and arranging them.

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6•5

If this is truly a net of the solid, it will fold up into a box. In mathematical language, it is known as a right rectangular prism.

Students should fold the net into the solid to prove that it was indeed a net. Be prepared for questions about other arrangements of these rectangles that are also nets of the right rectangular prism. There are many possible arrangements. Exploratory Challenge 1 (9 minutes): Rectangular Prisms
Students will make nets from given measurements. Rectangles should be cut from graph paper and taped. Ask students to have their rectangle arrangements checked before taping. After taping, it can be folded to check its fidelity.
Exploratory Challenge 1: Rectangular Prisms
a.

Use the measurements from the solid figures to cut and arrange the faces into a net.
One possible configuration of rectangles is shown here:

b.

A juice box measures inches high, inches long, and inches wide. Cut and arrange all faces into a net.
One possible configuration of faces is shown here:

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

NYS COMMON CORE MATHEMATICS CURRICULUM

c.

6•5

Challenge: Write a numerical expression for the total area of the net for part (b). Explain each term in your expression. Possible answer: ( . × .) + ( . × .) + ( . × .). There are two sides that have dimensions . by ., two sides that are . by ., and two sides that are . by .

Exploratory Challenge 2 (7 minutes): Triangular Prisms
Cutting these prior to the lesson will save time during the lesson.
Exploratory Challenge 2: Triangular Prisms
Use the measurements from the triangular prism to cut and arrange the faces into a net.

One possible configuration of rectangles and triangles is shown here:

Exploratory Challenge 3 (9 minutes): Pyramids
Exploratory Challenge 3: Pyramids
Pyramids are named for the shape of the base.
a.

Use the measurements from this square pyramid to cut and arrange the faces into a net. Test your net to be sure it folds into a square pyramid.

One possible configuration of rectangles and triangles is shown here:

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

A triangular pyramid that has equilateral triangles for faces is called a tetrahedron. Use the measurements from this tetrahedron to cut and arrange the faces into a net.

One possible configuration of triangles is shown here:

Closing (2 minutes)


What are the most important considerations when making nets of solid figures?




After all faces are made into polygons (either real or drawings), what can you say about the arrangement of those polygons?




All faces are rectangles. Opposite faces are identical rectangles. If the base is a square, the lateral faces are identical rectangles. If the prism is a cube, all of the faces are identical.

Describe the similarities between the nets of pyramids.




Edges must match like on the solid.

Describe the similarities between the nets of right rectangular prisms.




Each face must be taken into account.

All of the faces that are not the base are triangles. The number of these faces is equal to the number of sides the base contains. If the base is a regular polygon, the faces are identical triangles. If all of the faces of a triangular pyramid are identical, then the solid is a tetrahedron.

How can you test your net to be sure that it is really a true net of the solid?


Make a physical model and fold it up.

Exit Ticket (3 minutes)

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 16: Constructing Nets
Exit Ticket

Sketch and label a net of this pizza box. It has a square top that measures 16 inches on a side, and the height is 2 inches.
Treat the box as a prism, without counting the interior flaps that a pizza box usually has.

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Sketch and label a net of this pizza box. It has a square top that measures inches on a side, and the height is inches.
Treat the box as a prism, without counting the interior flaps that a pizza box usually has.
One possible configuration of faces is shown here:

Problem Set Sample Solutions
1.

A cereal box that measures inches high, inches long, and inches wide

Sketch and label the net of the following solid figures, and label the edge lengths.
a.

One possible configuration of faces is shown here:

7 in.

2 in.

13 in.

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Date:

Constructing Nets
11/6/14

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Lesson 16

NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

A cubic gift box that measures on each edge

One possible configuration of faces is shown here:

c.

Challenge: Write a numerical expression for the total area of the net in part (b). Tell what each of the terms in your expression means.
( × ) or

( × ) + ( × ) + ( × ) + ( × ) + ( × ) + ( × )

There are faces in the cube, and each has dimensions by .

2.

This tent is shaped like a triangular prism. It has equilateral bases that measure feet on each side. The tent is feet long. Sketch the net of the tent, and label the edge lengths.
Possible net:

Lesson 16:
Date:

Constructing Nets
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

3.

Lesson 16

6•5

The base of a table is shaped like a square pyramid. The pyramid has equilateral faces that measure inches on each side. The base is inches long. Sketch the net of the table base, and label the edge lengths.
Possible net:

4.

The roof of a shed is in the shape of a triangular prism. It has equilateral bases that measure feet on each side.
The length of the roof is feet. Sketch the net of the roof, and label the edge lengths.
Possible net:

Lesson 16:
Date:

Constructing Nets
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16

6•5

Rectangles for Opening Exercise

Lesson 16:
Date:

Constructing Nets
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 16

6•5

Rectangles for Exercise 1, part (a)

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Date:

Constructing Nets
11/6/14

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Lesson 16

6•5

Rectangles for Exercise 1, part (b)

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

6•5

Polygons for Exercise 2

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

6•5

Polygons for Exercise 3, part (a)

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 16

6•5

Triangles for Exercise 3, part (b)

Lesson 16:
Date:

Constructing Nets
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 17: From Nets to Surface Area
Student Outcomes


Students use nets to determine the surface area of three-dimensional figures.

Classwork
Fluency Exercise (5 minutes): Addition and Subtraction Equations
Sprint: Refer to the Sprints and the Sprint Delivery Script sections of the Module Overview for directions to administer a
Sprint.

Opening Exercise (4 minutes)
Students work independently to calculate the area of the shapes below.
Opening Exercise
a.

Write a numerical equation for the area of the figure below. Explain and identify different parts of the figure.
i.

ii.

b.

= ( )( ) =

represents the base of the figure because + = , and represents the altitude of the figure because it forms a right angle with the base.

How would you write an equation that shows the area of a triangle with base and height ? =

Write a numerical equation for the area of the figure below. Explain and identify different parts of the figure.
i.

.

.

= ( . )( . ) =

. represents the base of the rectangle, and . represents the height of the rectangle.

ii.

How would you write an equation that shows the area of a rectangle with base and height ? =

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Discussion (5 minutes)
English language learners may not recognize the word surface; take this time to explain what surface area means.
Demonstrate that surface is the upper or outer part of something, like the top of a desk. Therefore, surface area is the area of all the faces, including the bases of a three-dimensional figure.
Use the diagram below to discuss nets and surface area.


Examine the net on the left and the three-dimensional figure on the right. What do you notice about the two diagrams? 

The two diagrams represent the same rectangular prism.

Surface Area
= Area of back + Area of side
+ Area of side + Area of bottom
+ Area of front + Area of top
Surface Area
= 8 cm2 + 4 cm2 + 4 cm2
+ 8 cm2 + 8 cm2 + 8 cm2
= 40 cm2



Examine the second rectangular prism in the center column. The one shaded face is the back of the figure, which matches the face labeled back on the net. What do you notice about those two faces?


The faces are identical and will have the same area.

Continue the discussion by talking about one rectangular prism pictured at a time, connecting the newly shaded face with the identical face on the net.


Will the surface area of the net be the same as the surface area of the rectangular prism? Why or why not?


The surface area for the net and the rectangular prism will be the same because all the matching faces are identical, which means their areas are also the same.

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Example 1 (4 minutes)
Lead students through the problem.
Example 1
Use the net to calculate the surface area of the figure.



When you are calculating the area of a figure, what are you finding?




Surface area is similar to area, but surface area is used to describe three-dimensional figures. What do you think is meant by the surface area of a solid?




The surface area of a three-dimensional figure is the area of each face added together.

What type of figure does the net create? How do you know?

If the boxes on the grid paper represent a 1 cm × 1 cm box, label the dimensions of the net.




The area of a figure is the amount of space inside a two-dimensional figure.

It creates a rectangular prism because there are six rectangular faces.

1 cm

1 cm

Lesson 17:
Date:

2 cm

2 cm

1 cm

1 cm

2 cm

2 cm

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM





The surface area of a figure is the sum of the areas of all faces. Calculate the area of each face, and record this value inside the corresponding rectangle.

In order to calculate the surface area, we will have to find the sum of the areas we calculated since they represent the area of each face. There are two faces that have an area of 4 cm2 and four faces that have an area of 2 cm2 . How can we use these areas to write a numerical expression to show how to calculate the surface area of the net?


MP.2
&
MP.7


(1 cm × 2 cm) + (1 cm × 2 cm) + (1 cm × 2 cm) + (1 cm × 2 cm) + (2 cm × 2 cm) +
(2 cm × 2 cm).
The numerical expression to calculate the surface area of the net would be

4(1 cm × 2 cm) + 2(2 cm × 2 cm)

Write the expression more compactly, and explain what each part represents on the net.





6•5

The expression means there are 4 rectangles that have dimensions 1 cm × 2 cm on the net and 2 rectangles that have dimensions 2 cm × 2 cm on the net.
The surface area of the net is 16 cm2 .

What is the surface area of the net?


Example 2 (4 minutes)
Lead students through the problem.
Example 2
Use the net to write an expression for surface area.

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM



What type of figure does the net create? How do you know?

If the boxes on the grid paper represent a 1 ft. × 1 ft. square, label the dimensions of the net.




It creates a square pyramid because one face is a square and the other four faces are triangles.

2 ft



2 ft

3 ft

2 ft

3 ft

2 ft

5

How many faces does the rectangular pyramid have?

Knowing the figure has 5 faces, use the knowledge you gained in Example 1 to calculate the surface area of the rectangular pyramid.




6•5



Area of Base: 3 ft. × 3 ft. = 9 ft 2

1
× 3 ft. × 2 ft. = 3 ft 2
2

Surface Area: 9 ft 2 + 3 ft 2 + 3 ft 2 + 3 ft 2 + 3 ft 2 = 21 ft 2
Area of Triangles:

Exercises (13 minutes)
Students work individually to calculate the surface area of the figures below.
Exercises

Name the solid the net would create, and then write an expression for the surface area. Use the expression to determine the surface area. Assume that each box on the grid paper represents a × square. Explain how the expression represents the figure.
1.

Name of Shape: Rectangular Pyramid, but more specifically a
Square Pyramid

Surface Area: × + � × × � =

MP.1

+ (

)

=

The figure is made up of a square base that is × and four triangles with a base of and a height of .

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

2.
Surface Area: ( × ) + ( × ) =
( ) + ( ) =
Name of Shape: Rectangular Prism

The figure has square faces that are × and rectangular faces that are × .

3.

Name of Shape: Rectangular Pyramid

Surface Area: × + � × × � +

� × × � = + ( ) + ( ) = =

MP.1

The figure has rectangular base that is × , triangular faces that have a base of and a height of , and other triangular faces with a base of and a height of .

4.
Surface Area: ( × ) + ( × ) +
( × ) = ( ) + ( ) + ( )
Name of Shape: Rectangular Prism

The figure has rectangular faces that are × , rectangular faces that are × , and the final faces that are × .

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Date:

From Nets to Surface Area
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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Closing (5 minutes)


Why is a net helpful when calculating the surface area of pyramids and prisms?




Answers will vary. The nets are helpful when calculating surface area because it is easier to find the areas of all the faces.

What type of pyramids and/or prisms requires the fewest calculations when finding surface area?


Regular pyramids or prisms require the fewest calculations because the lateral faces are identical, so the faces have equal areas.

Exit Ticket (5 minutes)

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 17: From Nets to Surface Area
Exit Ticket
Name the shape, and then calculate the surface area of the figure. Assume each box on the grid paper represents a
1 in. × 1 in. square.

1.23

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exit Ticket Sample Solutions
Name the shape, and then calculate the surface area of the figure. Assume each box on the grid paper represents a . × . square.
Area of Base: . × . =

Name of Shape: Rectangular Pyramid

Area of Triangles:

× . × . = ,

× . × . =

Surface Area: + + + + =

Problem Set Sample Solutions
Name the shape, and write an expression for surface area. Calculate the surface area of the figure. Assume each box on the grid paper represents a . × . square.
1.

Surface Area: ( . × .) + ( . × .) + ( . × .) +
( . × .) + ( . × .) + ( . × .)
Name of Shape: Rectangular Prism

( . × .) + ( . × .) + ( . × .) + + =

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

2.

Name of Shape: Rectangular Pyramid

Surface Area: ( . × .) + � × . × .� +

� × . × .� + � × . × .� + � × . × .�

. × . + � × . × .� + � × . × .�

+ + =

Explain the error in each problem below. Assume each box on the grid paper represents a × square.
3.

Name of Shape: Rectangular Pyramid, but more specifically a Square Pyramid

Area of Base: × =

Area of Triangles: × =

Surface Area: + + + + =

The error in the solution is the area of the triangles. In order to calculate the correct area of the triangles, you must use the correct formula =

. Therefore, the

area of each triangle would be and not .

4.
Name of Shape: Rectangular Prism or, more specifically, a Cube
Area of Faces: × =

Surface Area: + + + + =

The surface area is incorrect because the student did not find the sum of all faces. The solution shown above only calculates the sum of faces. Therefore, the correct surface area should be + + + + + = and not .

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

5.

6•5

Sofia and Ella are both writing expressions to calculate the surface area of a rectangular prism. However, they wrote different expressions.
a.

Examine the expressions below, and determine if they represent the same value. Explain why or why not.
( × ) + ( × ) + ( × ) + ( × ) + ( × ) + ( × )
Sofia’s Expression:

( × ) + ( × ) + ( × )
Ella’s Expression:

Sofia and Ella’s expressions are the same, but Ella used the distributive property to make her expression more compact than Sofia’s.
b.

What fact about the surface area of a rectangular prism does Ella’s expression show that Sofia’s does not?
A rectangular prism is composed of three pairs of sides with identical areas.

Lesson 17:
Date:

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Number Correct: ______

Addition and Subtraction Equations—Round 1
Directions: Find the value of in each equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

+ 4 = 11 + 2 = 5

19.

+ 5 = 8

20.

− 7 = 10

21.

− 8 = 1

22.

− 4 = 2

23.

+ 12 = 34

24.

+ 25 = 45

25.

+ 43 = 89

26.

− 20 = 31

27.

− 13 = 34

28.

− 45 = 68

29.

+ 34 = 41

30.

+ 29 = 52

31.

+ 37 = 61

32.

− 43 = 63

33.

− 21 = 40

Lesson 17:
Date:

18.

34.

− 54 = 37
4 + = 9

6 + = 13
2 + = 31

15 = + 11
24 = + 13
32 = + 28
4 = − 7
3 = − 5

12 = − 14

23.6 = − 7.1

14.2 = − 33.8
2.5 = − 41.8

64.9 = + 23.4
72.2 = + 38.7

1.81 = − 15.13

24.68 = − 56.82

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Addition and Subtraction Equations—Round 1 [KEY]
Directions: Find the value of in each equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

+ 4 = 11

=

+ 2 = 5

=

+ 5 = 8

=

− 7 = 10

=

− 8 = 1

=

− 4 = 2

=

+ 12 = 34

=

+ 25 = 45

=

+ 43 = 89

=

− 20 = 31

=

− 13 = 34

=

− 45 = 68

=

+ 34 = 41

=

+ 29 = 52

=

+ 37 = 61

=

− 43 = 63

=

− 21 = 40

Lesson 17:
Date:

=

18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

− 54 = 37

=

2 + = 31

=

32 = + 28

=

4 + = 9

6 + = 13

15 = + 11
24 = + 13
4 = − 7
3 = − 5

12 = − 14

= = =

= = =

=

23.6 = − 7.1

= .

64.9 = + 23.4

= .

14.2 = − 33.8
2.5 = − 41.8

72.2 = + 38.7

1.81 = − 15.13

24.68 = − 56.82

=

= . = .

= . = .

From Nets to Surface Area
11/6/14

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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

Number Correct: ______
Improvement: ______

Addition and Subtraction Equations—Round 2
Directions: Find the value of in each equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

+ 2 = 7

+ 4 = 10

20.

+ 7 = 23

21.

+ 12 = 16

22.

− 5 = 2

23.

− 3 = 8

24.

− 4 = 12

25.

− 14 = 45

26.

+ 23 = 40

27.

+ 13 = 31

28.

+ 23 = 48

29.

+ 38 = 52

30.

− 14 = 27

31.

− 23 = 35

32.

− 17 = 18

33.

− 64 = 1

Lesson 17:
Date:

18.
19.

+ 8 = 15

6•5

34.

6 = + 3

12 = + 7

24 = + 16
13 = + 9
32 = − 3

22 = − 12
34 = − 10
48 = + 29
21 = + 17
52 = + 37
6
4
= +
7
7
2
5
= −
3
3
1
8
= −
4
3
5
7
= −
6
12
7
5
= −
8
12
7
16
+ =
6
3
1
13
+ =
3
15

From Nets to Surface Area
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Lesson 17

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Addition and Subtraction Equations—Round 2 [KEY]
Directions: Find the value of in each equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

+ 2 = 7

=

+ 4 = 10

=

+ 8 = 15

=

+ 7 = 23

=

+ 12 = 16

=

− 5 = 2

=

− 3 = 8

=

− 4 = 12

=

− 14 = 45

=

+ 23 = 40

=

+ 13 = 31

=

+ 23 = 48

=

+ 38 = 52

=

− 14 = 27

=

− 23 = 35

=

− 17 = 18

=

− 64 = 1

Lesson 17:
Date:

=

18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.

6 = + 3

=

13 = + 9

=

12 = + 7

24 = + 16

= =

32 = − 3

=

48 = + 29

=

22 = − 12
34 = − 10
21 = + 17
52 = + 37
6
4
= +
7
7
2
5
= −
3
3
1
8
= −
4
3
5
7
= −
6
12
7
5
= −
8
12
7
16
+ =
6
3
1
13
+ =
3
15

= = =

=

=

=

= = = = =

From Nets to Surface Area
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Lesson 18

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 18: Determining Surface Area of Three-Dimensional
Figures
Student Outcomes




Students determine that a right rectangular prism has six faces: top and bottom, front and back, and two sides. They determine that surface area is obtained by adding the areas of all the faces and develop the formula = 2 + 2ℎ + 2ℎ.

Students develop and apply the formula for the surface area of a cube as = 6 2 .

Lesson Notes
In order to complete this lesson, each student will need a ruler and the shape template that is attached to the lesson. To save time, teachers should have the shape template cut out for students.

Classwork
Opening Exercise (5 minutes)
In order to complete the Opening Exercise, each student needs a copy of the shape template that is already cut out.
Opening Exercise
a.

What three-dimensional figure will the net create?
Rectangular Prism

b.

Measure (in inches) and label each side of the figure.

.

. . .

Lesson 18:
Date:

. . . . .

. .

.

.

Determining Surface Area of Three-Dimensional Figures
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c.

6•5

Calculate the area of each face, and record this value inside the corresponding rectangle.

d.

How did we compute the surface area of solid figures in previous lessons?
To determine surface area, we found the area of each of the faces and then added those areas.

e.

( . × .) + ( . × .) + ( . × .) + ( . × .) + ( . × .) + ( . × .)
Write an expression to show how we can calculate the surface area of the figure above.
( . × .) + ( . × .) + ( . × .)

f.

What does each part of the expression represent?
Each part of the expression represents an area of one face of the given figure. We were able to write a more compacted form because there are three pairs of two faces that are identical.

g.

( . × .) + ( . × .) + ( . × .) + ( . × .) + ( . × .) + ( . × .)
What is the surface area of the figure?

( . × .) + ( . × .) + ( . × .)

Example 1 (8 minutes)


Fold the net used in the Opening Exercise to make a rectangular prism. Have the two faces with the largest area be the bases of the prism.



Fill in the second row of the table below.
Example 1
Fold the net used in the Opening Exercise to make a rectangular prism. Have the two faces with the largest area be the bases of the prism. Fill in the second row of the table below.
Area of Top (base)

Lesson 18:
Date:

Area of Bottom (base)

Area of Front

Area of Back

Area of Left
Side

Area of Right
Side

Determining Surface Area of Three-Dimensional Figures
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What do you notice about the areas of the faces?




length × height width × height

How do we calculate the area of the right and left faces of the prism?




length × width

How do we calculate the area of the front and back faces of the prism?




The faces that have the same area are across from each other. The bottom and top have the same area, the front and the back have the same area, and the two sides have the same area.

How do we calculate the area of the two bases of the prism?




Pairs of faces have equal areas.

What is the relationship between the faces having equal area?




Using the name of the dimensions, fill in the third row of the table.
Area of Top
(base)
. × .

×



6•5

Area of Bottom
(base)
. × .

×

Area of Front

Area of Back

Area of Right
Side
. × .

Area of Left Side

. × . ×

. × . ×

. × .

×

×

Examine the rectangular prism below. Complete the table.
Examine the rectangular prism below. Complete the table.

Area of Top
(base)
× ×



MP.8

× ×
Area of Front

× ×
Area of Back

× ×

= × + × + × ℎ + × ℎ + × ℎ + × ℎ

Since we use the same expression to calculate the area of pairs of faces, we can use the distributive property to write an equivalent expression for the surface area of the figure that uses half as many terms.

Lesson 18:
Date:

Area of Left Side

When comparing the methods to finding surface area of the two rectangular prisms, can you develop a general formula?




Area of Bottom
(base)
× ×

Area of Right
Side
× ×

Scaffolding:

Students may benefit from a poster or handout highlighting the length, width, and height of a three-dimensional figure.
This poster may also include that = length, = width, and ℎ = height.

Determining Surface Area of Three-Dimensional Figures
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We have determined that there are two × dimensions. Let’s record that as 2 times times , or simply
2( × ). How can we use this knowledge to alter other parts of the formula?
We also have two × ℎ, so we can write that as 2( × ℎ), and we can write the two × ℎ as
2( × ℎ).



MP.8


= 2( × ) + 2( × ℎ) + 2( × ℎ)

Writing each pair in a simpler way, what is the formula to calculate the surface area of a rectangular prism?




6•5

Knowing the formula to calculate surface area makes it possible to calculate the surface area without a net.

Example 2 (5 minutes)
Work with students to calculate the surface area of the given rectangular prism.
Example 2



The length is 20 cm, the width is 5 cm, and the height is 9 cm.

What are the dimensions of the rectangular prism?




We will use substitution in order to calculate the area. Substitute the given dimensions into the surface area formula. = 2(20 cm)(5 cm) + 2(20 cm)(9 cm) + 2(5 cm)(9 cm)




= 200 cm2 + 360 cm2 + 90 cm2

Solve the equation. Remember to use order of operations. = 650 cm2




Exercises 1–3 (17 minutes)
Students work individually to answer the following questions.
Exercises 1–3
1.

Calculate the surface area of each of the rectangular prisms below.
a.

.

.

Lesson 18:
Date:

.

= ( . )( . ) + ( . )( . )
+ ( . )( . )

= + + =

Determining Surface Area of Three-Dimensional Figures
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b.

= ( )( ) + ( )( ) + ( )( ) = + +

=

c.

6•5

.

.

.

= ( . )( . ) + ( . )( . ) + ( . )( . ) = + +

=

d.

.

.

= ( )(. ) + ( )(. ) + (. )(. )

= . + . + . = .
2.

= ( )( ) + ( )( ) + ( )( )

Calculate the surface area of the cube.

Lesson 18:
Date:

= + + =

Determining Surface Area of Three-Dimensional Figures
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6•5

All the edges of a cube have the same length. Tony claims that the formula = , where is the length of each side of the cube, can be used to calculate the surface area of a cube.

3.

a.

Tony’s formula is correct because = ( ) = , which is the same surface area when we use the surface area formula for rectangular prisms.

b.

Why does this formula work for cubes?

c.

Becca does not want to try to remember two formulas for surface area, so she is only going to remember the formula for a cube. Is this a good idea? Why or why not?

MP.3

Use the dimensions from the cube in Problem 2 to determine if Tony’s formula is correct.

Each face is a square, and to find the area of a square, you multiply the side lengths together. However, since the side lengths are the same, you can just square the side length. Also, a cube has identical faces, so after calculating the area of one face, we can just multiply this area by to determine the total surface area of the cube. Becca’s idea is not a good idea. The surface area formula for cubes will only work for cubes because rectangular prisms do not have identical faces. Therefore, Becca also needs to know the surface area formula for rectangular prisms.

Use two different ways to calculate the surface area of a cube with side lengths of 8 cm.

Closing (5 minutes)








= 2(8 cm × 8 cm) + 2(8 cm × 8 cm) + 2(8 cm × 8 cm) = 128 cm2 + 128 cm2 + 128 cm2

= 384 cm2

= 6 2

= 6(8 cm)2 = 384 cm2

If you had to calculate the surface area of 20 different sized-cubes, which method would you prefer to use, and why?


Answers may vary, but most likely students will chose the formula for surface area of a cube because it is a shorter formula, so it would take less time.

Lesson Summary

Surface Area Formula for a Rectangular Prism: = + +
Surface Area Formula for a Cube: =

Exit Ticket (5 minutes)

Lesson 18:
Date:

Determining Surface Area of Three-Dimensional Figures
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Name

6•5

Date

Lesson 18: Determining Surface Area of Three-Dimensional
Figures
Exit Ticket
Calculate the surface area of each figure below. Figures are not drawn to scale.
1.
10 ft.

2 ft.

12 ft.

2.

8 cm

8 cm

8 cm

Lesson 18:
Date:

Determining Surface Area of Three-Dimensional Figures
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6•5

Exit Ticket Sample Solutions
Calculate the surface area of each figure below. Figures are not drawn to scale. = + +

1.

= ( . )( . ) + ( . )( . ) + ( . )( . ) = + +

=

=

2.

= ( )

= ( ) =

Problem Set Sample Solutions
Calculate the surface area of each figure below. Figures are not drawn to scale.
1.

2.

.

.

.

.

= ( . )( . ) + ( . )( . ) + ( . )( . )

= + + =

.

.

= (. )(. ) + (. )(. ) + (. )(. ) = . + . + .

= .

Lesson 18:
Date:

Determining Surface Area of Three-Dimensional Figures
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3.

.

.

.

= �

= �

= � =

4.
.

6•5

. �

. �



=

.

.

= (. )(. ) + (. )(. ) + (. )(. )

= . + . + . = .

5.

Write a numerical expression to show how to calculate the surface area of the rectangular prism. Explain each part of the expression.
( . × .) + ( . × .) + ( . × .)

The first part of the expression shows the area of the top and bottom of the rectangular prism. The second part of the expression shows the area of the front and back of the rectangular prism. The third part of the expression shows the area of the two sides of the rectangular prism.
The surface area of the figure is .

.

6.

.

.

length = . , width = . , and height = . .

When Louie was calculating the surface area for Problem 4, he identified the following: length = . , width = . , and height = . .

However, when Rocko was calculating the surface area for the same problem, he identified the following:
Would Louie and Rocko get the same answer? Why or why not?

Louie and Rocko would get the same answer because they are still finding the correct area of all six faces of the rectangular prism.

Lesson 18:
Date:

Determining Surface Area of Three-Dimensional Figures
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7.

6•5

Examine the figure below.

a.

What is the most specific name of the three-dimensional shape?
Cube

b.

( × ) + ( × ) + ( × ) + ( × ) + ( × ) + ( × )

Write two different expressions for the surface area. × ( )

c.

The two expressions are equivalent because the first expression shows × , which is equivalent to
( ) . Also, the represents the number of times the product × is added together.
Explain how these two expressions are equivalent.

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Date:

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Lesson 18:
Date:

6•5

Determining Surface Area of Three-Dimensional Figures
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290

Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 19: Surface Area and Volume in the Real World
Student Outcomes


Students determine the surface area of three-dimensional figures in real-world contexts.



Students choose appropriate formulas to solve real-life volume and surface area problems.

Classwork
Fluency Exercise (5 minutes): Area of Shapes
RWBE: Refer to the Rapid White Board Exchange section in the Module Overview for directions to administer an RWBE.

Opening Exercise (4 minutes)
Opening Exercise
A box needs to be painted. How many square inches will need to be painted to cover every surface? .

MP.1

.

= ( . )( . ) + ( . )( . ) + ( . )( . )

= + +

.

=

A juice box is . tall, . wide, and . long. How much juice fits inside the juice box? = . × . × . =

How did you decide how to solve each problem?
I chose to use surface area to solve the first problem because you would need to know how much area the paint would need to cover. I chose to use volume to solve the second problem because you would need to know how much space is inside the juice box to determine how much juice it can hold.

If students struggle deciding whether to calculate volume or surface area, use the Venn diagram below to help them make the correct decision.

Lesson 19:
Date:

Surface Area and Volume in the Real World
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6•5

Discussion (5 minutes)
Students need to be able to recognize the difference between volume and surface area. As a class, complete the Venn diagram below so students have a reference when completing the application problems.
Discussion

Volume


Measures space inside



Includes only space needed to fill inside



Is measured in cubic units

Surface Area
• A way to measure space figures •

Measures outside surface



Includes all faces



Is measured in square units



Can be measured using a net Example 1 (5 minutes)
Work through the word problem below with students. Students should be leading the discussion in order for them to be prepared to complete the exercises.
Example 1
Vincent put logs in the shape of a rectangular prism. He built this rectangular prism of logs outside his house. However, it is supposed to snow, and Vincent wants to buy a cover so the logs will stay dry. If the pile of logs creates a rectangular prism with these measurements: long, wide, and high,

what is the minimum amount of material needed to make a cover for the wood pile?







We need to find the size of the cover for the logs, so we need to calculate the surface area. In order to find the surface area, we need to know the dimensions of the pile of logs.

Why do we need to find the surface area and not the volume?




Scaffolding:

Where do we start?

We want to know the size of the cover Vincent wants to buy. If we calculated volume, we would not have the information Vincent needs when he goes shopping for a cover.

The length is 33 cm, the width is 12 cm, and the height is 48 cm.

What are the dimensions of the pile of logs?


Lesson 19:
Date:

 Add to the poster or handout made in the previous lesson showing that long represents length, wide represents width, and high represents height.  Later, students will have to recognize that deep also represents height.
Therefore, this vocabulary word should also be added to the poster.

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Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM



How do we calculate the surface area to determine the size of the cover?

= 2(33 cm)(12 cm) + 2(33 cm)(48 cm) + 2(12 cm)(48 cm) = 792 cm2 + 3168 cm2 + 1152 cm2 = 5112 cm2

We can use the surface area formula for a rectangular prism.





What is different about this problem from other surface area problems of rectangular prisms you have encountered? How does this change the answer?
If Vincent just wants to cover the wood to keep it dry, he does not need to cover the bottom of the pile of logs. Therefore, the cover can be smaller.




6•5

We know the area of the bottom of the pile of logs has the dimensions 33 cm and 12 cm. We can calculate the area and subtract this area from the total surface area.

How can we change our answer to find the exact size of the cover Vincent needs?


The area of the bottom of the pile of firewood is 396 cm2 ; therefore, the total surface area of the cover would need to be 5112 cm2 − 396 cm2 = 4716 cm2 .



Exercises 1–6 (17 minutes)
Students complete the volume and surface area problems in small groups.
Exercises 1–6
Use your knowledge of volume and surface area to answer each problem.
1.

Quincy Place wants to add a pool to the neighborhood. When determining the budget, Quincy Place determined that it would also be able to install a baby pool that requires less than cubic feet of water. Quincy Place has three different models of a baby pool to choose from.
Choice One: × ×

Choice Two: × ×

Choice Three: × ×

Choice One Volume: . × . × . =

Which of these choices is best for the baby pool? Why are the others not good choices?

Choice Two Volume: . × . × . =

Choice Three Volume: . × . × . =

MP.1

Choice Two is within the budget because it holds less than of water. The other two choices do not work because they require too much water, and Quincy Place will not be able to afford the amount of water it takes to fill the baby pool.
2.

A packaging firm has been hired to create a box for baby blocks. The firm was hired because it could save money by creating a box using the least amount of material. The packaging firm knows that the volume of the box must be .
a.

What are possible dimensions for the box if the volume must be exactly ?

Choice 1: × ×

Choice 2: × ×

Choice 3: × ×

Choice 4: × ×
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NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

Which set of dimensions should the packaging firm choose in order to use the least amount of material?
Explain.
Choice 1: = ( )( ) + ( )( ) + ( )( ) =
Choice 2: = ( )( ) + ( )( ) + ( )( ) =

Choice 3: = ( )( ) + ( )( ) + ( )( ) =
Choice 4: = ( )( ) + ( )( ) + ( )( ) =

The packaging firm should choose Choice 4 because it requires the least amount of material. In order to find the amount of material needed to create a box, the packaging firm would have to calculate the surface area of each box. The box with the smallest surface area requires the least amount of material.
3.

A gift has the dimensions of × × . You have wrapping paper with dimensions of × . Do you have enough wrapping paper to wrap the gift? Why or why not?
Surface Area of the Present: = ( )( ) + ( )( ) + ( )( ) = + + =
Area of Wrapping Paper: = × = ,

I do have enough paper to wrap the present because the present requires , of paper, and I have , of wrapping paper.

MP.1

4.

Tony bought a flat rate box from the post office to send a gift to his mother for Mother’s Day. The dimensions of the medium size box are × × . . What is the volume of the largest gift he can send to his mother?

Volume of the Box: . × . × . . =

Tony would have of space to fill with a gift for his mother.

5.

A cereal company wants to change the shape of its cereal box in order to attract the attention of shoppers. The original cereal box has dimensions of × × . The new box the cereal company is thinking of would have dimensions of × × .
a.

Volume of Original Box: = . × . × . =
Which box holds more cereal?

Volume of New Box: = . × . × . =

The new box holds more cereal because it has a larger volume.
b.

Surface Area of Original Box: = ( . )( . ) + ( . )( . ) + ( . )( . ) = + + =
Which box requires more material to make?

Surface Area of New Box: = ( . )( . ) + ( . )( . ) + ( . )( . ) = + + =

The new box requires more material than the original box because the new box has a larger surface area.

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Date:

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Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Cinema theaters created a new popcorn box in the shape of a rectangular prism. The new popcorn box has a length of inches, a width of . inches, and a height of . inches but does not include a lid.

6.

Scaffolding:

. .

a.

MP.1

.

. .

English language learners may not be familiar with the term lid. Provide an illustration or demonstration. Surface Area of the Box: = ( . )(. . ) + ( . )(. . ) + (. . )(. . ) =
How much material is needed to create the box?

+ + . = .

Area of Lid: . × . . =

The box does not have a lid, so we have to subtract the area of the lid from the surface area.
Total Surface Area: . − = .

. of material is needed to create the new popcorn box.

b.

Volume of the Box: = . × . . × . . = .
How much popcorn does the box hold?

Closing (4 minutes)


Is it possible for two containers having the same volume to have different surface areas? Explain.




Yes, it is possible to have two containers to have the same volume but different surface areas. This was the case in Exercise 2. All four boxes would hold the same amount of baby blocks (same volume), but required a different amount of material (surface area) to create the box.

If you want to create an open box with dimensions 3 inches × 4 inches × 5 inches, which face should be the base if you want to minimize the amount of material you use?


The face with dimensions 4 inches × 5 inches should be the base because that face would have the largest area.

If students have a hard time understanding an open box, use a shoe box to demonstrate the difference between a closed box and an open box.

Exit Ticket (5 minutes)

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Date:

Surface Area and Volume in the Real World
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Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

Name

6•5

Date

Lesson 19: Surface Area and Volume in the Real World
Exit Ticket
Kelly has a rectangular fish aquarium with an open top that measures 18 inches long, 8 inches wide, and 12 inches tall.
Solve the word problem below.
a.

What is the maximum amount of water in cubic inches the aquarium can hold?

b.

If Kelly wanted to put a protective covering on the four glass walls of the aquarium, how big does the cover have to be?

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Surface Area and Volume in the Real World
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6•5

Exit Ticket Sample Solutions
Kelly has a rectangular fish aquarium that measures inches long, inches wide, and inches tall.
Solve the word problem below.

a.

Volume of the Aquarium: = . × . × . =
What is the maximum amount of water the aquarium can hold?

The maximum amount of water the aquarium can hold is , .
b.

If Kelly wanted to put a protective covering on the four glass walls of the aquarium, how big does the cover have to be?

Surface Area of the Aquarium: = ( . )( . ) + ( . )( . ) + ( . )( . ) = + + =

We only need to cover the four glass walls, so we can subtract the area of both the top and bottom of the aquarium. Area of Top: = . × . =

Area of Bottom: = . × . =

Surface Area of the Four Walls: = − − =

Kelly would need to cover the four walls of the aquarium.

Problem Set Sample Solutions
Dante built a wooden, cubic toy box for his son. Each side of the box measures feet.

Solve each problem below.
1.

a.

Surface Area of the Box: = ( ) = ( ) =
How many square feet of wood did he use to build the box?

Dante would need of wood to build the box.
b.

Volume of the Box: = . × . × . =
How many cubic feet of toys will the box hold?

The toy box would hold of toys.

2.

A company that manufactures gift boxes wants to know how many different sized boxes having a volume of cubic centimeters it can make if the dimensions must be whole centimeters.
a.

Choice One: × ×

List all the possible whole number dimensions for the box.
Choice Two: × ×

Choice Three: × ×
Choice Four: × ×

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Surface Area and Volume in the Real World
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Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

Choice One: = ( )( ) + ( )( ) + ( )( ) = + + =

Which possibility requires the least amount of material to make?

Choice Two: = ( )( ) + ( )( ) + ( )( ) = + + =

Choice Three: = ( )( ) + ( )( ) + ( )( ) = + + =

Choice Four: = ( )( ) + ( )( ) + ( )( ) = + + = Choice Four requires the least amount of material because it has the smallest surface area.
c.

3.

I would recommend the company use the box with dimensions of × × (Choice Four) because it requires the least amount of material to make; so, it would cost the company the least amount of money to make. Which box would you recommend the company use? Why?

Volume of the Rice Box: =
4.

.

A rectangular box of rice is shown below. How many cubic inches of rice can fit inside?

.

.

. × . × . = =

The Mars Cereal Company has two different cereal boxes for Mars Cereal. The large box is inches wide, inches high, and inches deep. The small box is inches wide, inches high, and . inches deep.
a.

Surface Area of the Large Box: = ( . )( . ) + ( . )( . ) + ( . )( . ) = + + =
How much more cardboard is needed to make the large box than the small box?

Surface Area of the Small Box: = ( . )( . ) + ( . )(. . ) + ( . )(. . ) = + + =
Difference: − =

The large box requires more material than the small box.
b.

Volume of the Large Box: = . × . × . =

How much more cereal does the large box hold than the small box?

Volume of the Small Box: = . × . × . . =
Difference: − =

The large box holds more cereal than the small box.

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Date:

Surface Area and Volume in the Real World
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5.

6•5

A swimming pool is meters long, meters wide, and meters deep. The water-resistant paint needed for the pool costs $ per square meter. How much will it cost to paint the pool?
You will have to point faces.

a.

How many faces of the pool do you have to paint?

b.

How much paint (in square meters) do you need to paint the pool?

= ( × ) + ( × ) + ( × ) = + + =

Area of Top of Pool: × =

Total Paint Needed: − =

c.

2 × $ = $

How much will it cost to paint the pool?
It will cost $ to paint the pool.

6.

Sam is in charge of filling a rectangular hole with cement. The hole is feet long, feet wide, and feet deep. How much cement will Sam need? = . × . × . =

Sam will need of cement to fill the hole.

7.

The volume of Box D subtracted from the volume of Box C is . cubic centimeters. Box D has a volume of
. cubic centimeters.
a.

Let be the volume of Box C in cubic centimeters. Write an equation that could be used to determine the volume of Box C.

b.

Solve the equation to determine the volume of Box C.

− . = .

− . + . = . + . = .

c.

The volume of Box C is one-tenth the volume of another box, Box E. Let represent the volume of Box E in cubic centimeters. Write an equation that could be used to determine the volume of Box E, using the result from part (b).
. =

d.

= ÷

Solve the equation to determine the volume of Box E.
. ÷

. =

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Date:

Surface Area and Volume in the Real World
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Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Area of Shapes

1.

8 ft.

10 ft.
2.
5m

=

12 m
22 in.

3.

4.

13 m

21 cm

22 in.

=

= ,

49 cm

Lesson 19:
Date:

=

Surface Area and Volume in the Real World
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

5.
10 ft.

8 ft.

6 ft.

8 km

6.
13 km
5 km

12 km

=

12 ft.

12 km

8 km

13 km

=

5 km

7.

=

8.

8 cm

4 cm

4 cm

8 cm

8 cm

8 cm

8 cm
Lesson 19:
Date:

4 cm

4 cm

8 cm

=

Surface Area and Volume in the Real World
11/6/14

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Lesson 19

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

9.

48 m

36 m

=

16 m
10.

= ,

82 ft.

18 ft.

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Date:

Surface Area and Volume in the Real World
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Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Lesson 19a: Applying Surface Area and Volume to
Aquariums
Student Outcomes


Students apply the formulas for surface area and volume to determine missing dimensions of aquariums and water level.

Lesson Notes
The purpose of this lesson is to demonstrate an abridged version of the modeling cycle in preparation for shortened modeling cycles in Grades 7 and 8 and, finally, in preparation for the complete modeling cycle in Grade 9. The modeling cycle is described and detailed in the New York State P–12 Common Core Standards for Mathematics, pages 61 and 62.
Although the modeling cycle is addressed in detail in high school, the goal of instruction in Grades 6–8 is to prepare students for this kind of thinking. The graphic below is a brief summation of the modeling cycle in which students:


Identify variables in a situation and select those that represent essential figures.



Formulate a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations between variables.



Analyze and perform operations on these relationships to draw conclusions.



Interpret results of the mathematics in terms of the original situation.



Validate conclusions by comparing them with the situation, and then either improve the model or determine if it is acceptable.



Report on the conclusions and the reasoning behind them.

This lesson affords students the opportunity to apply their knowledge of surface area and volume in the real-life context of aquariums. Students will also utilize their knowledge of rates and ratios, as well as apply arithmetic operations and their knowledge of expressions and equations from Module 4 to determine missing aquarium dimensions. Below is an outline of the CCSS addressed in this lesson.

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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Module

Other Related Modules

G6-M5: Area, Surface Area, and
Volume Problems

Standards

M1: Ratios and Rates
M2: Arithmetic Operations
Including Dividing by a Fraction
M4: Expressions and Equations

6.EE.A.2c, 6.EE.B.5, 6.EE.B.6,
6.EE.B.7, 6.G.A.2, 6.G.A.4

6•5

6.RP.A.1, 6.RP.A.2, 6.RP.A.3a,
6.RP.A.3b
6.NS.B.2, 6.NS.B.3, 6.NS.C.5
6.EE.A.2c, 6.EE.B.7, 6.EE.B.8

Students will model with mathematics, demonstrating CCSS Mathematical Practice 4 throughout this lesson. They will use proportional reasoning to plan, approximate, and execute problem solving and calculations in this contextual platform. The activities in this lesson are based on the standard dimensions of a 10-gallon aquarium. Because real-life materials may not be accessible in all classrooms, problems are presented in two ways. Students will either use proportional reasoning to determine a course of action to calculate volume, surface area, and missing dimensions, and/or students will experience a hands-on, tangible experience through optional exercises that are offered for those classrooms that have access to real-life materials. Teacher preparation will include finding aquariums with the dimensions noted in the lesson or adjusting the measurements throughout the lesson to match the aquariums actually used in the lesson.
Teachers will need to prepare stations with liter measuring tools, gallon measuring tools, water, aquariums, and rulers.
The exercises found in this teacher lesson are reproduced for the students in their student materials.

Classwork
Opening Exercise (2 minutes)
Most standard tanks and aquariums have a length of 20 inches, a width of 10 inches, and a height of 12 inches.
Display the following figure.



Using the formula for volume, determine the volume of this aquarium in cubic inches.
Opening Exercise = × ×

Determine the volume of this aquarium. = . × . × . =

.

.

.

Mathematical Modeling Exercise (10 minutes): Using Ratios and Unit Rate to Determine Volume


Below is a table of values that indicates the relationship between gallons of water and cubic inches.



Use the table below to determine how many cubic inches are in one gallon of water, or more specifically, the unit rate of gallons/cubic inches.

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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6•5

Mathematical Modeling Exercise: Using Ratios and Unit Rate to Determine Volume
For his environmental science project, Jamie is creating habitats for various wildlife including fish, aquatic turtles, and aquatic frogs. For each of these habitats, he will use a standard aquarium with length, width, and height dimensions measured in inches, identical to the aquarium mentioned in the Opening Exercise. To begin his project, Jamie will need to determine the volume, or cubic inches, of water that will fill the aquarium.
Use the table below to determine the unit rate of gallons/cubic inches.
Gallons

Cubic Inches

,

There are for every of water. So, the unit rate is .




Since we determined that for every gallon of water, there are 231 cubic inches, determine how many cubic inches are in the 10 gallons of water that Jamie needs for the fish.
How can we determine how many cubic inches are in 10 gallons of water?




We could use a tape diagram or a double number line, or we could find equivalent ratios.

Using either of these representations, determine the volume of the aquarium.
Determine the volume of the aquarium.

Answers will vary depending on student choice. An example of a tape diagram is below.

,



We determined the volume of this tank is 2,310 in3 . This is not the same volume we calculated earlier in the opening exercise. Why do you think the volumes are different?




Generally, it is suggested that the highest level of water in this tank should be approximately 11.55 inches.
Calculate the volume of the aquarium using this new dimension.




= × × ℎ; = 20 in. × 10 in. × 11.55 in. ; = 2,310 in3

What do you notice about this volume?




Answers will vary but should include discussion that there needs to be room for a lid; also, the water level cannot go all the way to the top so that there is room for heaters, filters, and fish, etc., without the water spilling over.

This volume is the same as the volume we determined when we found the volume using ratio and unit rates. Let’s use the dimensions 20 in. × 10 in. × 11.55 in. for our exploration.
Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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6•5

We have determined that the volume for the 10-gallon aquarium with dimensions 20 in. × 10 in. ×
11.55 in. is 2,310 in3 .

Optional Exercise 1





Suppose Jamie needs to fill the aquarium to the top in order to prepare the tank for fish. According to our calculations, if Jamie pours 10 gallons of water into the tank, the height of the water will be approximately
11.55 in.

Let’s test it. Begin pouring water into the aquarium 1 gallon at a time. Be sure to keep track of the number of gallons. Use a tally system.



Tally the Number of Gallons



Measure the height of the water with your ruler.



10

Number of Gallons

What did you find about our height estimation?

Our estimation was correct. The height is approximately 11.55 in.



Exercise 1 (10 minutes)




Next, suppose Jamie needs to prepare another aquarium for aquatic frogs. He contacted the local pet store, and the employee recommended that Jamie only partially fill the tank in order for the frogs to have room to jump from the water to a lily pad or designated resting place. The employee suggested that the tank hold
7 gallons of water. Considering that the length and the width of the tank remain the same (20 in. × 10 in.), use what you know about volume to determine the height of the water that is appropriate for the frogs in the tank. To determine the missing dimension of height, we need the volume formula = ∙ ∙ ℎ.
Exercise 1

a.

Determine the volume of the tank when filled with of water.

∙ =

The volume for of water is , .

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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6•5

Work with your group to determine the height of the water when Jamie places of water in the aquarium. b.

= . ( . ) =

. . =

The tank should have a water height of . inches.

Let’s test it. Begin by pouring water into the aquarium 1 gallon at a time.

Optional Exercise 2



Be sure to keep track of the number of gallons poured. Use a tally system.

Or, have students mark the height of the water using a wax marker or a dry erase marker on the outside of the tank after each gallon is poured in. Then, students measure the intervals (distance between the marks). Students will notice that the intervals are equal.


Test the height at 7 gallons, and record the height measurement.


Tally the Number of Gallons



Number of Gallons

Our estimation was correct. The height is about 8 inches.

What did you find about our estimation?


Exercise 2 (5 minutes)




According to the local pet store, turtles need very little water in an aquarium. The suggested amount of gallons of water in the aquarium for a turtle is 3 gallons. Determine the height of the water in another aquarium of the same size that is housing a turtle when the amount of water Jamie pours into the tank is
3 gallons.
Describe how you would estimate the height level?

First, determine the volume of the water. Then, to determine the missing dimension of height, we need the volume formula = ∙ ∙ ℎ.



Exercise 2
a.

Use the table from Example 1 to determine the volume of the aquarium when Jamie pours of water into the tank.
The volume of the tank is × = .

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

b.

6•5

= . ( . )

Use the volume formula to determine the missing height dimension. =

. . =

The tank should have a water height of . .

Let’s test it. Begin by pouring water into the aquarium 1 gallon at a time.

Optional Exercise 3




Test the height at 3 gallons, and record the height measurement.

Be sure to keep track of the number of gallons poured. Use a tally system.


Tally the Number of Gallons



1
2

Our estimation was correct. The height is about 3 inches.

What did you find about our estimation?


Number of Gallons

Exercise 3 (5 minutes)


Let’s say that when Jamie sets up these aquariums of the same size at home, he does not have any tools that measure gallons. What he does have at home is a few leftover one-liter soft drink bottles. How could Jamie calculate the volume of the aquarium?


Answers will vary but should include that gallons need to be converted to liters.



Using the table of values, determine the unit rate for liters to gallons.



What is the unit rate?




The unit rate is 3.785.

Answers will vary. For every gallon of water, there are 3.785 liters of water.

What does this mean?

If this conversion is accurate, determine the number of liters Jamie will need to fill a 10-gallon tank.






3.785





Answers will vary. For every inch, there are 2.54 centimeters.

liters

gallon

× 10 gallons = 37.85 liters

It is not advantageous to combine liters and inches. Liters and centimeters are both in the metric system of measurement. The ratio of the number of centimeters to the number of inches is 2.54: 1. What does this mean? The unit rate is 2.54.



What is the unit rate?



Use the conversion to determine the length, the width, and the height of the aquariums in centimeters.



Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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308

Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM

6•5

Exercise 3
a.

Using the table of values below, determine the unit rate of liters to gallon.

.

Gallons

The unit rate is . .

Liters

.

.

b.

Using this conversion, determine the number of liters you will need to fill the -gallon tank.

c.

The ratio of the number of centimeters to the number of inches is . : . What is the unit rate?

d.

.

× = .

.

Using this information, complete the table to convert the heights of the water in inches to the heights of the water in centimeters Jamie will need for his project at home.
Height in Inches

×

Convert to Centimeters

.

.

.

.

.

.

.

Height in Centimeters
.

× .

.

× .

.

× .

.

Exercise 4 (5 minutes)




Jamie had the tanks he used at home shipped from the manufacturer. Typically, the manufacturer sends aquariums already assembled; however, they use plastic film to cover the glass in order to protect it during shipping. Determine the amount of plastic film the manufacturer uses to cover the aquarium faces. Draw a sketch of the aquarium to assist in your calculations. Remember that the actual height of the aquarium is 12 inches.
Exercise 4
a.

Determine the amount of plastic film the manufacturer uses to cover the aquarium faces. Draw a sketch of the aquarium to assist in your calculations. Remember that the actual height of the aquarium is . = ( ) + ( ) + ()

= ( ∙ . ∙ .) + ( ∙ . ∙ .) + ( ∙ . ∙ .)

= + +

=
Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM



6•5

We do not include the measurement of the top of the aquarium since it is open without glass. It does not need to be covered with film.
b.

We do not include the measurement of the top of the aquarium since it is open without glass and does not need to be covered with film. Determine the area of the top of the aquarium, and find the amount of film the manufacturer will use to cover only the sides, front, back, and bottom. = ∙

= .∙ .

=

= − =

( . ∙ .)

Side

Side
( . ∙ . )

( . ∙ .)
c.

( . ∙ .)

( . ∙ . )

+ + = or ∙ =

Since Jamie will need three aquariums, determine the total surface area of the three aquariums.

An internet company that sells aquariums charges $300 per aquarium. Jamie is considering building the aquariums at home and buying the parts from a different company that sells glass for $0.11 per square inch. Which option, buying the aquariums already built from the first company or buying the glass and building at home, is a better deal? Closing/Challenge Exercises (5 minutes)
1.

Sample Solution:

2760 in2 ∙ 0.11

dollars in2 = 303.6 dollars or $303.60. It would be a better deal for Jamie to purchase the aquariums

from the company that ships the aquariums because for one aquarium $303.60 > $300. For three aquariums, the comparison is $910.80 > $900.

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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310

Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM

2.

6•5

If Jamie wanted to increase the length of the aquarium by 20%, how would that affect the surface area? How would it affect the volume of water the tank could contain?

Since the length is 20 inches, 20 in.∙ 0.20 = 4 additional inches. The new length would be 20 in. + 4 in. = 24 in. = 2( ) + 2(ℎ) + 2(ℎ)
Sample Solution:

= 2(24 in. ∙ 12 in.) + 2(24 in. ∙ 10 in.) + 2(10 in. ∙ 12 in.)

= 576 in2 + 480 in2 + 240 in2 = 1296 in2

The surface area without the top is 1,296 in2 − 288 in2 , or 1,008 in2 .

The new surface area of 1,008 in2 is 88 in2 more than the original surface area of 920 in2 .

= ∙ ∙ ℎ; = 24 in. ∙ 12 in. ∙ 10 in. = 2,880 in3 , which is 480 in3 more than the original volume of 2,400 in3 .

Exit Ticket (3 minutes)

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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Lesson 19a

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Name ___________________________________________________

6•5

Date____________________

Lesson 19a: Applying Surface Area and Volume to Aquariums
Exit Ticket
What did you learn today? Describe at least one situation in real life that would draw on the skills you used today.

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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6•5

Exit Ticket Sample Solution
What did you learn today? Describe at least one situation in real life that would draw on the skills you used today.
Answers will vary.

Problem Set Sample Solutions
This Problem Set is a culmination of skills learned in this module. Note that the figures are not drawn to scale.
1.

Calculate the area of the figure below.

. .
2.

= ( . )( . )

.

=

Calculate the area of the figure below.

.

= (. )(. )

= . =

.

.
3.

=

.

Calculate the area of the figure below.

.

.

. .

Lesson 19a:
Date:

Area of top rectangle: =

= ( . )( . ) =

Area of bottom rectangle: =

= ( . )( . ) =

= + =

Applying Surface Area and Volume to Aquariums
11/6/14

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313

Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM

4.

Complete the table using the diagram on the coordinate plane.

����

(−, )

����

(, −)

Line Segment

Point

����

(, )

��
��

(, )

Distance

(, )

(−, −)

| − | + || =

|| + | − | =

(−, −)

(−, )

(−, −)

(−, −)

��
��

���

Point

(, −)



5.

6•5

(−, ), (, ), (, −)

(−, )

(−, )
(−, )

Proof

|| − || =

|| + | − | =

| − | + || =

|| + | − | =
|| − || =

Plot the points below, and draw the shape. Then, determine the area of the polygon.

=

= ( )( ) =

= ( )( )

=

=

= ( )( )

=

=

= − − − =

Lesson 19a:
Date:

= ( )( )

=

=

Applying Surface Area and Volume to Aquariums
11/6/14

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314

Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM

6.

Determine the volume of the figure.

7.

6•5

=

= � � � � � �

=

=

Give at least three more expressions that could be used to determine the volume of the figure in Problem 6.

� � �

� � � � � �

� � �


Answers will vary. Some examples include the following:


8.

Determine the volume of the irregular figure.

Volume of the back Rectangular Prism: =

= � . � � . � � . �

=

=

= � . � � . � � . �

=

Volume of the front Rectangular Prism:

=

Lesson 19a:
Date:

+ = =

Applying Surface Area and Volume to Aquariums
11/6/14

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315

Lesson 19a

NYS COMMON CORE MATHEMATICS CURRICULUM

9.

6•5

Draw and label a net for the following figure. Then, use the net to determine the surface area of the figure.

= + + + + +
=

10. Determine the surface area of the figure in Problem 9 using the formula = + + . Then, compare your answer to the solution in Problem 9. = + +

= ( )( ) + ( )( ) + ( )( ) = + + =

The answer in Problem 10 is the same as in Problem 9. The formula finds the areas of each face and adds them together, like we did with the net.

11. A parallelogram has a base of . and an area of . . Tania wrote the equation . = . to represent this situation.
a.

b.

Explain what represents in the equation.

represents the height of the parallelogram.

Solve the equation for .

. .
=
.
.
= .

12. Triangle A has an area equal to one-third the area of Triangle B. Triangle A has an area of

= . Explain what represents in the equation.

a.

Gerard wrote the equation

b.

square meters.

Determine the area of Triangle B.

represents the area of Triangle B in square meters.

∙ = ∙

=

Lesson 19a:
Date:

Applying Surface Area and Volume to Aquariums
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

Name

6•5

Date

1. The juice box pictured below is 4 inches high, 3 inches long, and 2 inches wide.

a.

In the grid above, the distance between grid lines represents one inch. Use the grid paper to sketch the net of the juice box.

b.

Find the surface area of the juice box. Show your work.

c.

Find the volume of the juice box. Show your work.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

2. The Cubic Crystal Company has a new Crystal Cube they want to sell. The packaging manager insists that the cubes be arranged to form a rectangular prism and that the package be designed to hold the Crystal
Cubes exactly, with no leftover packaging. Each Crystal Cube measures 1 in. × 1 in. × 1 in. There are 24
Crystal Cubes to be sold in a box.
a.

What are the dimensions of the possible box designs in inches?

Height

b.

Length

Which Crystal Cube box design will use the least amount of cardboard for packaging? Justify your answer as completely as you can.
Height

c.

Width

Width

Length

Surface Area

Another type of cube is the Mini Crystal Cube, which has an edge length of volume in cubic inches of one Mini Crystal Cube? Show your work.

Module 5:
Date:

3
4

inch. What is the

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

3. Which of these nets can be folded to form a cube?
A

B

C

D

4. Which box below has the larger surface area?
3 in.

1 in.

10 in.

5.

a.

5 in.

3 in.

2 in.

Draw a polygon in the coordinate plane using the given coordinates. (4, −4)
(6, −2)

b.

(8, −6)

Calculate the area of the polygon.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

6. Eaglecrest Elementary School is creating a vegetable garden at the school.
8 ft.

6 ft.

25 ft.

a.

What is the area of the garden?

b.

After more discussion, Eaglecrest decided to change the location of the garden so that the vegetables can get more sunlight. Below is the new garden.

28 ft.
7 ft.

In which garden can the students of Eaglecrest plant more vegetables? Explain your reasoning.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

A Progression Toward Mastery
Assessment
Task Item

1

a
6.G.A.4

b
6.G.A.4

STEP 1
Missing or incorrect answer and little evidence of reasoning or application of mathematics to solve the problem

STEP 2
Missing or incorrect answer but evidence of some reasoning or application of mathematics to solve the problem

Student sketch does not contain 6 rectangles. Student sketch contains
6 rectangles but not 3 different sizes (two each of 2 × 3, 2 × 4, and 3 ×
4); they are arranged in a way that will not fold into a rectangular solid.

Student response does not include the use of a formula and is incorrect (52 in2 ).

Module 5:
Date:

Student uses a formula other than = 2( · + · ℎ + · ℎ), or equivalent, to make the calculation. Alternatively, the volume may have been calculated. STEP 3
A correct answer with some evidence of reasoning or application of mathematics to solve the problem, or an incorrect answer with substantial evidence of solid reasoning or application of mathematics to solve the problem
Student sketch contains
6 rectangles of 3 different sizes (two each of 2 × 3, 2 × 4, and 3 ×
4); however, they are arranged in a way that will not fold into a rectangular solid.

Student uses the formula = 2( · + · ℎ + · ℎ), or equivalent, to make the calculation, but an arithmetic error results in an incorrect final answer.
Alternatively, the correct number is calculated, and the units (in2 ) are incorrect. STEP 4
A correct answer supported by substantial evidence of solid reasoning or application of mathematics to solve the problem

Student sketch is one of many nets of a 2 × 3 × 4 rectangular solid. Here is one example:

Critical performance indicators: the net must have 6 rectangles of 3 different sizes (two each of 2 × 3, 2 × 4, and 3 ×
4), similar rectangles must not be adjacent to one another, and the net must fold to a 2 × 3 × 4 rectangular solid.

Student uses the formula = 2( · + · ℎ + · ℎ), or equivalent, to make the calculation, and the surface area of the box is correctly found
(52 in2 ). Both number and units are correct.

Area, Surface Area, and Volume Problems
11/6/14

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End-of-Module Assessment Task

NYS COMMON CORE MATHEMATICS CURRICULUM

c
6.G.A.2

2

a
6.G.A.2

Student response does not include the use of a formula and is incorrect (24 in3 ).

Student uses a formula other than = · · ℎ, or equivalent, to make the calculation. Alternatively, the surface area may have been calculated. Student response includes none or only one of the six possible configurations of the box. Student response includes at least two of the six possible configurations of the box.

Student uses the formula = · · ℎ, or equivalent, to make the calculation, but an arithmetic error results in an incorrect final answer.
Alternatively, the correct number is calculated, and the units (in3 ) are incorrect. Student uses the formula = · · ℎ, or equivalent, to make the calculation, and the volume of the box is correctly found (24 in3 ).
Both number and units are correct.

Student response includes at least four of the six possible configurations of the box.

Student response includes all six possible configurations of the box
(all measurements in inches): 1 × 1 × 24, 1 ×
2 × 12, 1 × 3 × 8, 1 ×
4 × 6, 2 × 2 × 6, and 2 ×
3 × 4.
1 in.
1 in.
1 in.
1 in.
2 in.
2 in.

1 in.
2 in.
3 in.
4 in.
2 in.
3 in.

L

b
6.G.A.4

Student response does not include a calculation for the surface area of any of the box designs.

Student response includes calculations for at least two of the six possible configurations of the box. The smallest number of these calculations is chosen as the box needing the least amount of cardboard.

6•5

Student response includes calculations for at least four of the six possible configurations of the box. The smallest number of these calculations is chosen as the box needing the least amount of cardboard.

24 in.
12 in.
8 in.
6 in.
6 in.
4 in.

W

H

Student calculates the surface area of all six boxes correctly:
1 in.
1 in.
1 in.
1 in.
2 in.
2 in.

L

1 in.
2 in.
3 in.
4 in.
2 in.
3 in.

W

24 in.
12 in.
8 in.
6 in.
6 in.
4 in.

H

98 in2
76 in2
70 in2
68 in2
56 in2
52 in2

SA

Student concludes that the minimum surface area is found to be on the 2 in.× 3 in.× 4 in. box. That box needs the least amount of cardboard. c
6.G.A.2

Student response does not include a length, width, and height of a Crystal
Cube.

Module 5:
Date:

Student response includes length, width, and height dimensions other than
3
3
3
in. × in. × in.
4

4

4

Student response
3
3 includes in. × in. ×
4

4

in. but is calculated incorrectly. 3
4

Student correctly calculates the volume of a single Crystal Cube:
3
3
3
in. × in. × in. =
4
27
64

in3 .

4

4

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

3

6.G.A.4

Student response does not include choice D.

Student response includes choice D and two or three other
(incorrect) choices.

Student response includes choice D and one other (incorrect) choice. Student response is choice D only.

4

6.G.A.4

Student is not able to calculate the surface area of either rectangular prism.

Student is able to calculate the surface area, but calculations may have mathematical errors. Student calculates the surface area of one prism correctly but one incorrectly. OR
Student calculates both surface areas correctly but does not answer the question. Student finds the surface area of the prisms to be
86 in2 and 62 in2. The student also states that the prism with dimensions 10 in. × 1 in. × 3 in. has the larger surface area.

5

a

Student does not plot any of the points correctly. Student plots the points backwards. For example, student may have plotted the points
(−4, 4), (−2, 6), and
(−6, 8).

Student plots two of the three points correctly.

Student plots all three points correctly.

Student counts the squares inside the shape by estimating the parts of squares that are part of the area.

Student uses the area of rectangles and/or triangles to calculate the area of the shape but does so incorrectly.

Student does not calculate the area.

Student calculates the area incorrectly, perhaps using the wrong dimensions. Student calculates the area correctly but does not label the answer.

Student uses the area of rectangles and/or triangles to calculate the area of the shape and correctly calculates 6 square units as the area.

Student does not calculate the area.

Student calculates the area of the new garden but does not divide by 2.

Student calculates the area of both shapes correctly but does not answer the question.

6.G.A.3

b

Student does not calculate the area.

6.G.A.3

6

a
6.G.A.1
b
6.G.A.1

Module 5:
Date:

Student calculates the area correctly and labels accurately 150 ft 2 .

Student calculates the area of both shapes correctly and explains that the original garden has a larger area because
150 ft 2 is larger than
98 ft 2 ; therefore, students can plant more vegetables in the original garden. Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

Name

6•5

Date

1. The juice box pictured below is 4 inches high, 3 inches long, and 2 inches wide.

a.

In the grid above, the distance between grid lines represents one inch. Use the grid paper to sketch the net of the juice box.

b.

Find the surface area of the juice box. Show your work.

c.

Find the volume of the juice box. Show your work.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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324
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

2. The Cubic Crystal Company has a new Crystal Cube they want to sell. The packaging manager insists that the cubes be arranged to form a rectangular prism and that the package be designed to hold the Crystal
Cubes exactly, with no leftover packaging. Each Crystal Cube measures 1 in. × 1 in. × 1 in. There are 24
Crystal Cubes to be sold in a box.
a.

What are the dimensions of the possible box designs in inches?

b.

Which Crystal Cube box design will use the least amount of cardboard for packaging? Justify your answer as completely as you can.

c.

3

Another type of cube is the Mini Crystal Cube, which has an edge length of inch. What is the
4
volume in cubic inches of one Mini Crystal Cube? Show your work.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

3. Which of these nets can be folded to form a cube?
A

B

C

D

4. Which box below has the larger surface area?
3 in.

1 in.

10 in.

5.

5 in.

3 in.

2 in.

a. Draw a polygon in the coordinate plane using the given coordinates.
(4, −4)
(6, −2)

(8, −6)

b. Calculate the area of the polygon.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

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326
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUM

End-of-Module Assessment Task

6•5

6. Eaglecrest Elementary School is creating a vegetable garden at the school.
8 ft.

6 ft.

25 ft.

a.

What is the area of the garden?

b.

After more discussion, Eaglecrest decided to change the location of the garden so that the vegetables can get more sunlight. Below is the new garden.
28 ft.
7 ft.
In which garden can the students of Eaglecrest plant more vegetables? Explain your reasoning.

Module 5:
Date:

Area, Surface Area, and Volume Problems
11/6/14

© 2014 Common Core, Inc. Some rights reserved. commoncore.org

327
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