...New York State Common Core Mathematics Curriculum GEOMETRY • MODULE 1 Table of Contents1 Congruence, Proof, and Constructions Module Overview .................................................................................................................................................. 3 Topic A: Basic Constructions (G-CO.1, G-CO.12, G-CO.13).................................................................................... 7 Lesson 1: Construct an Equilateral Triangle ............................................................................................. 8 Lesson 2: Construct an Equilateral Triangle II ........................................................................................ 16 Lesson 3: Copy and Bisect an Angle........................................................................................................ 21 Lesson 4: Construct a Perpendicular Bisector ........................................................................................ 30 Lesson 5: Points of Concurrencies .......................................................................................................... 37 Topic B: Unknown Angles (G-CO.9) ..................................................................................................................... 43 Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point ....................................................... 44 Lesson 7: Solve for Unknown Angles—Transversals .................................
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...History of Geometry Geometry was thoroughly organized in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'. The books covered not only plane and solid geometry but also much of what is now known as algebra, trigonometry, and advanced arithmetic. Through the ages, the propositions have been rearranged, and many of the proofs are different, but the basic idea presented in the 'Elements' has not changed. In the work facts are not just cataloged but are developed in a fashionable way. Even in 300 BC, geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is how to follow lines of reasoning, how to say precisely what is intended, and especially how to prove basic concepts by following these lines of reasoning. Taking a course in geometry is beneficial for all students, who will find that learning to reason and prove convincingly is necessary for every profession. It is true that not everyone must prove things, but everyone is exposed to proof. Politicians, advertisers, and many other people try to offer convincing arguments. Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction. Geometry provides a simplified universe, where points and lines obey believable rules and where conclusions are easily verified. By first studying how to reason in...
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...understand these different geometric shapes, and it can also be quite fun, as the children will play by themselves eventually. Materials needed- Pictures of a pentagons, trapezoids, triangles, hexagon, rectangle, and a parallelogram, these items are going to be large enough so the children are able to see them. I will also have pictures from magazines as well, so they are able to see these geometric shapes in a real-life manner. National Council of Teacher Mathematics Standards. –This activity meets many of the National Council of Teacher Mathematics Standards, including showing the different characteristics of two and three dimensional geometric shapes as well as developing mathematical arguments about geometric relationships. According to "Geometry Standard" (2013),”They will be able to recognize, and name two dimensional shapes, as well as describing the attributes as well as the parts of two and three dimensional shapes.” They also will be able to show the...
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...Let the coordinates of point A be (x1 y1), the coordinates of B be (x2 y2), and the coordinates for C, which is a right angle, be (x2 y1) It is understood from the image that triangle ABC is a right angle triangle. AB is, in fact, the hyptenouse, and AC & BC are the corresponding sides. The length of the long leg, side AC, is equal to the difference in the x-coordinates of points C and A. Points C and B have the same x-coordinate, so the length of side AC is also equal to the difference in the x-coordinates of points B and A. So, AC = x2 − x1 The length of the short leg, side BC, is equal to the difference in the y-coordinates of points B and C. Points A and C have the same y-coordinate, so the length of side BC is also equal to the difference in the y-coordinates of points B and A. So, BC = y2 − y1 Thus, from pythagorous theorum... a2 + b2 = c2 We substitute... AB2 = AC2 + BC2 We substitute again... d2= (x2-x1)2 + (y2-y1)2 Now, we will take the square root of either side and arrive at the distance forumla... d = √(x2-x1)2 + (y2-y1)2 B. Let the coordinates of point A be (x1 y1 z1), of B be (x2 y2 z1), C be (x3 y3 z1), D be (x2 y2 z2) This derivation is two step process. First, we will find AB and, using it, we will find AD It is understood from the image that triangle ABC is a right angle triangle. AB is, in fact, the hyptenouse, and AC & BC are the corresponding sides Let's suppose that AC is actually along the x-axis while BC is along the y-axis Now triangle...
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...Islamic Architecture and Geometry When studying Islamic architecture and archaeology one can easily become distracted by the beauty and grace of the many different and Iconic Islamic structures. Coming from New York City it is becoming increasingly difficult to learn about the cities past by studying its Architectural history. Everyday older buildings are being knocked down and replaced by newer and more visually appealing skyscrapers. However, this trend has not come to pass in the major Islamic cities of the east. From Damascus to Baghdad or Jerusalem or Samara one can study and see the history that is still currently present within their cities. One of the most fascinating aspects of Islamic architecture and archaeology for me has always been the immense attention to detail in which the Islamic monuments were built with. For example Ludovico Micara talks about the importance of Geometry within the context of Islamic architecture and design. He references the well-known historian of Islamic art Oleg Grabar. Grabar talks about how writing, geometry, architecture and nature go hand in hand within Islam “In viewers well-defined emotions and stances: control and forcefulness of assertion with writing, Order with geometry, boundaries and protection with architecture, life forces with nature and throughout sensory pleasure”, This concept of interweaving architecture and design with geometry and nature has always been the most interesting concept for me when studying Islamic...
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...The History of Geometry Geometry, from the ancient Greek “geo” meaning Earth and “metron” meaning measurement, arose as the field of knowledge dealing with spatial relationships. Geometry was revolutionized by Euclid, who introduced mathematical rigor still in use today. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. Greek Geometry The early history of Greek geometry is unclear, because no original sources of information remain and all of our knowledge is from secondary sources written many years after the early period. For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their...
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...Making Geometry Fun with Origami Lucila Cardenas Vega University of Texas at Brownsville Introduction Teachers must have an understanding of students’ mathematical thinking in order to create meaningful learning opportunities. This becomes more relevant when teaching subjects that not all students have an interest for, such as, geometry. Since geometry is the study of shapes and configurations, it is important to understand how a student thinks about the different properties in geometry including, symmetry, congruence, lines and angles. Students remember a lesson better and the information becomes more significant when learning is accessed through hands on activities. (Pearl, 2008). Origami is the art of transforming a flat sheet of material into a finished sculpture through folding and sculpting techniques. The use of origami can be thought of as art; however, there are so many other benefits of incorporating origami in geometry lessons. According to experts, origami teaches students how to follow directions, encourages cooperation among students, improves motor skills and it helps develop multi-cultural awareness (Weirhem, 2005). Origami activities used in geometry lessons reinforces vocabulary words, facilitates the identification of shapes and simplifies congruency and symmetry (Pearl, 2008). In origami, students take a flat piece of paper and create a figure that is three dimensional. The use of origami in geometry is not new. Friedrich Froebel, the founder...
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...The word geometry is Greek for geos - meaning earth andmetron - meaning measure. Geometry was extremely important to ancient societies and was used for surveying, astronomy, navigation, and building. Geometry, as we know it is actually known as Euclidean geometry which was written well over 2000 years ago in Ancient Greece by Euclid, Pythagoras, Thales, Plato and Aristotle just to mention a few. The most fascinating and accurate geometry text was written by Euclid, and was called Elements. Euclid's text has been used for over 2000 years! Geometry is the study of angles and triangles, perimeter, area and volume. It differs from algebra in that one develops a logical structure where mathematical relationships are proved and applied. In part 1, you will learn about the basic terms associated with Geometry. Terms (Undefined) 1. Point Points show position. A point is shown by one capital letter. In the example below, A, B, and C are all points. Notice that points are on the line. 2. Line A line is infinite and straight. If you look at the picture above, is a line, is also a line and is a line. A line is identified when you name two points on the line and draw a line over the letters. A line is a set of continuous points that extend indefintely in either of its direction. Lines are also named with lowercase letters or a single loswer case letter. For instance, I could name one of the lines above simply by indicating an e. Terms (Defined) 1. Line Segment A line...
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...Geometry Notes First Class Point has zero dimensions Postulates/Axioms: statement we accept as being true without a proof Theorems/Corollaries: statement that must be proven before we accept it as being true Corollaries are spin off of another theorem Postulates * P1-1 Given any two points, there is a unique distance between them * P1-2 any segment has exactly one midpoint Theorem * T1-1 “Midpoint Theorem” If m is the midpoint of segment AB, then: 2AM=AB, AM=1/2AB and 2MB=AB, MB=1/2 AB Vocab: Collinear: points that are on the same straight line Noncollinear Points: Points that are not on same straight line Obliquely intercept: not at 90 degree angle Three positions for two lines in space 1. Skew lines: never intercept and not parallel. 2. Parallel Lines: never intercept 3. Intercepting lines: cross each other Planes Planes go on forever, never end, like lines. Line can be: * in plane * Intercept plan * Parallel with plane Two relationships for Planes 1) Parallel 2) Intercept Space contains all points Line with 4 colinear points Can name it line L, or take two points on the line and same it that way. AB, AC, AD, BC, BD, with arrows on top. Subsets of line. * Rays, has only one arrow on top cause starts at point, * Rays going in diff direction on line called opposite rays, same starting points, but opposite directions * Can’t change lettering around with rays cause first letter is...
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...Geometry Manipulative Handout Joan L. Holyfield Math/157 March 8, 2015 Instructor: James Paga Geometry Manipulative Handout The geometry manipulative that I chose to present is how to make a Geo City. With making a Geo City, it teaches students in grades 3-5. This math exercise encompasses learning and identifying cubes, pyramids, cones, rectangular and other shapes in different sizes which are appropriate for this age group. This activity can be done in a group setting, with each child having a share in identifying different geometrical shapes, and putting them in a real-world setting. By using these different geometric shapes to construct buildings and cars, the children will learn about spatial visualization, location and coordinate points, and transformation. The Geo City will help to reinforce the relationships with two and three-dimensional shapes that have already been learned in previous classes. The students will also be able to explore the effects of rotating, reflecting, and transforming shapes. The Geo city also lends itself to real world application by simulating a city block that they may find in their neighborhood. Then they can begin to understand how the geometric shapes are a part of their everyday lives. Constructing the Geo City also helps the children with their mapping and graph skills, because they will be using both to first, decide how to design the city (using graph paper or geo dot paper), then by deciding where to put the houses...
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...Analytic Geometry is a branch of mathematics in which problems are solved using the principles of Geometry and the processes of Algebra. He is regarded as the founder of Analytic geometry by introducing coordinates system in 1637. René Descartes The Cartesian Coordinate System * also known as Rectangular Coordinate System or xy-Coordinate System. * It is made up of two mutually perpendicular number lines with the same unit of length and intersecting at their origin. The origin of its number line is its zero point. * The number lines are called the coordinate axes. * The horizontal line is called the x-axis and the vertical line is called the y-axis. * The coordinate axes divide the whole plane into four regions called quadrants. * The plane on which these axis are constructed is called the Coordinate Plane or xy-plane. * The distance of any point P from the y-axis is called x-coordinate or abscissa of the point P. * The distance of any point P from the x-axis is called the y-coordinate or ordinate of the point P. * The pair of real numbers (x,y) is called the coordinate pair of point P. * The symbol P(x,y) is used to indicate the point P on the plane with abscissa x and ordinate y. * The signs of the coordinates determine the quadrant where the point lies. * QI: (+,+) QIII: (-,-) QII: (-,+) QIV: (+,-) Exercise 1.1 Indicate the quadrant or the axis on which the point lies. 1. A(3,-2) 6. F(5,0) 2. B(-1,5)...
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...Task 5 Part A 1. I first used a CD case to draw a line. 2. I placed points at the end of the line and labeled them A and B respectively. 3. I then took my compass and placed it on point A. 4. I stretched the compass out a little past halfway of the middle of the line segment. 5. Next, I made and arc with the compass. 6. Then I placed the compass on point B. 7. Keeping the compass the same length as used on point A, I made an arc and marked where the arcs intersected. 8. Using the CD case, I drew a straight line connecting the points where the arcs met at the top and the bottom. This line crossed through my line segment AB. 9. The line segment created by connecting the intersection of the arcs, was determined to be the midpoint of my original line segment AB. 10. I then repeated steps 1-9 for segments BC, CD, and AD. Part B Diagram A F(a,2b) (0.2b) A . B(2a,2b) (0,b)E . . G (2a,b) . C(0,0) H(a,0) ...
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...Alternatives to Euclidean Geometry Student name: Institution: Alternatives to Euclidean Geometry According to Johnson (2013) Euclidean Geometry , commonly known as high school geometry, is a mathematical study of geometry based on undefined terms such as points, lines and or planes; definitions and other theories of a mathematician known as Euclid (330 B.C.) While a number of Euclid’s research findings had been earlier stated by Greek Mathematicians, Euclid has received a lot of recognition for developing the very first comprehensive deductive systems. Euclid’s approach to mathematical geometry involved providing all the theorems from a finite number of axioms (postulates). Euclidean Geometry is basically a study of flat surfaces. Geometrical concepts can easily be illustrated by drawings on a chalkboard or a piece of paper. A number of concepts are known in a flat surface. These concepts include, the shortest distance between points, which is known to be one unique straight line, the angle sum of a triangle, which adds up to 180 degrees and the concept of perpendicular to any line.( Johnson, 2013, p.45) In his text, Mr. Euclid detailed his fifth axiom, the famous parallel axiom, in this manner: If a straight line traversing any two straight lines forms interior angles on one side less than two right angles, the two straight lines, if indefinitely extrapolated, will meet on that same side where the angles smaller than the two right angles. In today’s mathematics, the parallel...
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...Geometry Charades Objective: Learn basic geometrical terms playing games of charades. Materials: Charades cards with geometrical terms on them, pictures of each geometrical term so that the students can learn what each of them are before the game begins. A prize is needed for the student that wins the most amount of points at the end of the game. Directions: Each student will get a turn in the order of seating positions. The student who is up will draw a Charades card from the pile and act out whatever geometrical term is on the card. The student who is acting out the term has two minutes to get the crowd to guess what they are acting out. Students can raise their hands and guess what the person is trying to act out. The student that guesses the term correctly first wins a point. The student that acts out the term and gets someone in the crowd to guess what he or she is doing wins a point as well. I will keep account of each student’s point balances. Each student will get a turn regardless of who guesses the right answer. Cards: These are the following cards that students can choose: Parallel lines, line, line segment, intersecting lines, ray, right angle, acute angle, obtuse angle Accompanying Lesson Plan: Students will learn the geometry terms definitions, and what each term looks like in a picture before the game is played. The students will use their creativity to create the geometry term that they draw with their bodies in the game of...
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...Coordinate systems in geometry are systems which use numbers known as coordinates to determine the positions of points in space (Wolfram). The two coordinate systems we will be using in this problem are the Cartesian coordinate system and the polar coordinate system. The Cartesian coordinate system specifies points in a plane using pairs of numerical coordinates, which in this case are the x and y values of point A, point B, and point C listed in the diagram (Wolfram). For example, point C on the diagram is denoted by the x-coordinate 1 and the y-coordinate 2. These coordinates are linear and perpendicular. The x-axis is the horizontal axis while the y-axis is the vertical axis (Wolfram). In a two-dimensional Cartesian coordinate system, the x and y coordinates are...
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