OBLONG, OBLONG WHERE DO WE BELONG?
(Oblong Numbers) I. Introduction Math investigations engage one’s mind to formulate conjectures from observed numerical phenomenon. Patterns are being observed and these patterns are then translated into mathematical expressions such as the equation. Thus this investigation would lead to mathematical models that need to be proven. However, the formulation of a mathematical model is not an easy task. It requires skills and experience to device such. Then, after the modeling, it requires testing the model by verifying extreme cases and later proves that this model is true to all cases. This investigation involved two cases: (1) relationship exists between the number of columns and the number of dots (oblong numbers) and (2) relationship exists between the number of rows and the number of dots (oblong numbers). As we go further with the investigation, we’d come up with a mathematical model that leads to answer the aforementioned cases. Moreover, we are looking for the possible extent if there are still existing mathematical facts involving these cases that are uncovered and waiting to be explored.
Situation:
OBLONG NUMBERS are numbers which can be represented in a rectangular array having one dimension one unit longer than the other.
[pic]
II. Statement of the Problem This mathematical investigation aimed to ascertain the relationships exist between the number of columns/rows and the number of dots (oblong numbers). Specifically it sought to answer the following questions: 1. What relationship exists between the number of columns (c) and the number of dots (oblong number “O”)? 2. What relationship exists between the number of rows (r) and the number of dots (oblong number “O”)? 3. How many dots (oblong numbers) are there in 25 columns? 100 columns? nth columns? 4. How many dots (oblong numbers) are there in 25 rows? 100 rows? nth rows? III. Data Gathering and Conjectures [pic] Based from the above illustration, relationships exist between number of columns/rows and the number of dots (oblong numbers). Table A shows that there is a relationship exists between the number of columns (c) and the number of dots (O). Moreover, Table B shows that there is a relationship exists between the number of rows (r) and the number of dots (O).

Table A: Relationship between number of columns and number of dots
|Number of columns (c)| | | | |
| |1 |2 |3 |4 |
|No. of dots (O) | | | | |
| |2 |6 |12 |20 |

1st difference: 4 6 8

2nd difference: 2 2
Table B: Relationship between number of rows and number of dots
|Number of rows (r) | | | | |
| |0 |1 |2 |3 |
|No. of dots (O) | | | | |
| |2 |6 |12 |20 |

1st difference: 4 6 8

2nd difference: 2 2 Further investigation was made and found out that between the results of the number of dots there is a difference of two from second difference. Thus, quadratic equation is applicable to use in deriving a mathematical model that leads to answer the problem.
A. Deriving a mathematical model to determine the number of dots in nth column using a quadratic equation
Let x be the number of columns
Let y be the number of dots
If x = 1 then y = 2 If x = 2 then y = 6 If x = 3 then y = 12 y = Ax2 + Bx + C y = Ax2 + Bx + C y = Ax2 + Bx + C
2 = A(1)2 + B(1) + C 6 = A(2)2 + B(2) + C 12 = A(3)2 + B(3) + C
2 = A + B + C (Eq. 1) 6 = 4A + 2B + C (Eq. 2) 12 = 9A + 3B + C (Eq. 3) Using elimination method (Eq. 2 and Eq. 3) we have, 6 = 4A + 2B + C ( 12 = 9A + 3B + C ( 6 = ( 5A – B (Eq. 4)
Using elimination method (Eq. 1 and Eq. 2) we have, 2 = A + B + C ( 6 = 4A + 2B + C ( 4 = ( 3A ( B (Eq. 5)
Using elimination method (Eq. 4 and Eq. 5) we have, ( 6 = ( 5A ( B ( ( 4 = ( 3A ( B ( 2 = ( 2A 1 = A To find the value of B using Eq. 4 if A = 1 through substitution method we have, - 6 = - 5A – B (Eq. 4) - 6 = - 5(1) – B - 6 = - 5 – B - 6 + 5 = - B - 1 = - B ( B = 1 To find the value of C using Eq. 1 if A = 1 and B = 1 through substitution method we have, 2 = A + B + C 2 = 1 + 1 + C 2 = 2 + C 2 – 2 = C 0 = C
A = 1
B = 1
C = 0 Substituting the values of A, B and C to the quadratic equation y = Ax2 + Bx + C, the obtained equation was y = x2 + x or y = x(x + 1). Hence, conjecture 1 was formulated.
Conjecture 1:
For every number of columns, the total number of dots (oblong number) is determined by O = c(c + 1)
|Number of columns (c) |1 |2 |3 |4 |c |
|No. of dots (O) |2 |6 |12 |20 |c(c+1) |

B. Deriving a mathematical model to determine the number of dots in nth row using a quadratic equation
Let x be the number of rows
Let y be the number of dots

If x = 0 then y = 2 If x = 1 then y = 6 If x = 2 then y = 12 y = Ax2 + Bx + C y = Ax2 + Bx + C y = Ax2 + Bx + C
2 = A(0)2 + B(0) + C 6 = A(1)2 + B(1) + C 12 = A(2)2 + B(2) + C
2 = C (Eq. 1) 6 = A + B + C (Eq. 2) 12 = 4A + 2B + C (Eq. 3)
Using elimination method (Eq. 2 and Eq. 3) we have, 6 = A + B + C ( 12 = 4A + 2B + C - 6 = - 3A – B (Eq. 4)
Using substitution method (Eq. 1 and Eq. 2) we have, C = 2 (Eq. 1) 6 = A + B + C (Eq. 2) 6 = A + B + 2
6 – 2 = A + B 4 = A + B (Eq. 5)
To find the value of A using Eq. 4 and Eq. 5 through substitution method we have, 4 = A + B (Eq. 5) ( B = 4 – A - 6 = - 3A – B (Eq. 4) - 6 = - 3A – (4 – A) - 6 = - 3A – 4 + A - 6 = - 2A – 4 - 6 + 4 = - 2A - 2 = - 2A 1 = A
To find the value of B using Eq. 5 through substitution method we have, 4 = 1 + B 4 = 1 + B 4 – 1 = B ( B = 3
A = 1
B = 3
C = 2 Substituting the values of A, B and C to the quadratic equation y = Ax2 + Bx + C, the obtained equation was y = x2 + 3x + 2. Hence, conjecture 2 was formulated.
Conjecture 2: For every number of rows, the total number of dots (oblong number) is determined by O = r2 + 3r + 2
|Number of rows (r) |0 |1 |2 |3 |r |
|No. of dots (O) |2 |6 |12 |20 |r2 + 3r + 2 |

IV. Testing/Verifying Conjectures
Conjecture 1:
For every number of columns, the total number of dots (oblong number) is determined by O = c(c + 1)

|No. of Columns (c) |Verifying Conjecture 1 |No. of Dots (O) |
| |O = c(c + 1) | |
|1 |O = 1(1 + 1) = 1(2) = 2 |2 |
|2 |O = 2(2 + 1) = 2(3) = 6 |6 |
|3 |O = 3(3 + 1) = 3(4) = 12 |12 |
|4 |O = 4(4 + 1) = 4(5) = 20 |20 |
|5 |O = 5(5 + 1) = 5(6) = 30 |30 |
|nth column |O = n(n + 1) |n + 1 |

Using the above data as presented, conjecture 1 is therefore verified to be true.
Conjecture 2: For every number of rows, the total number of dots (oblong number) is determined by O = r2 + 3r + 2
|No. of Rows (r) |Verifying Conjecture 1 |No. of Dots (O) |
| |O = r2 + 3r + 2 | |
|0 |O = 02 + 3(0) + 2 = 0 + 0 + 2 = 2 |2 |
|1 |O = 12 + 3(1) + 2 = 1 + 3 + 2 = 6 |6 |
|2 |O = 22 + 3(2) + 2 = 4 + 6 + 2 = 12 |12 |
|3 |O = 32 + 3(3) + 2 = 9 + 9 + 2 = 20 |20 |
|4 |O = 42 + 3(4) + 2 = 16 + 12 + 2 = 30 |30 |
|nth row |O = n2 + 3(n) + 2 |n2 + 3n + 2 |

V. Proving Conjectures/Explaining/Justifying
1. How many dots (oblong numbers) are there in 25 columns? 100 columns? nth columns?

(a) To determine the number of dots (oblong number) in 25 columns we have, O = c(c + 1) O = 25(25 + 1) O = 25(26) O = 650 Hence, there are 650 oblong numbers in 25 columns. (b) To determine the number of dots (oblong number) in 100 columns we have, O = c(c + 1) O = 100(100 + 1) O = 100(101) O = 10, 100 Hence, there are 10, 100 oblong numbers in 100 columns.

(c) To determine the number of dots (oblong number) in nth column we have, O = c(c + 1) O = n(n + 1) Hence, there are n(n + 1) oblong numbers in nth columns.
Proof:
a) If c = 25 then O = 650. Using substitution law we have, O = c(c + 1) 650 = 25(25 + 1) 650 = 25(26) 650 = 650 b) If c = 100 then O = 10, 100. Using substitution law we have, O = c(c + 1) 10, 100 = 100(100 + 1) 10, 100 = 100(101) 10, 100 = 10, 100 c) If c = n then O = n + 1. Using substitution law we have, O = c(c + 1) n + 1 = n(n + 1)
2. How many dots (oblong numbers) are there in 25 rows? 100 rows? nth rows? (a) To determine the number of dots (oblong number) in 25th row we have, O = r2 + 3r + 2 O = (25)2 + 3(25) + 2 O = 625 + 75 + 2 O = 702 Hence, there are 702 oblong numbers in 25 rows. (b) To determine the number of dots (oblong number) in 100 rows we have, O = r2 + 3r + 2 O = (100)2 + 3(100) + 2 O = 1, 000 + 300 + 2 O = 1, 302 Hence, there are 1, 302 oblong numbers in 100 rows.

(c) To determine the number of dots (oblong number) in nth rows we have, O = r2 + 3r + 2 O = n2 + 3n + 2 Hence, there are n2 + 3n + 2 oblong numbers in nth rows.
Proof:
a) If r = 25 then O = 702. Using substitution law we have, O = r2 + 3r + 2 702 = (25)2 + 3(25) + 2 702 = 625 + 75 + 2 702 = 702 b) If r = 100 then O = 1, 302. Using substitution law we have, O = r2 + 3r + 2 1, 302 = (100)2 + 3(100) + 2 1, 302 = 1, 000 + 300 + 2 1, 302 = 1, 302 c) If r = n then O = n2 + 3n + 2. Using substitution law we have, O = r2 + 3r + 2 n2 + 3n + 2 = n2 + 3n + 2

VI. Summary To determine the number of dots (oblong numbers) in terms of number of columns by using O = c(c + 1) where c stands for the number of columns and O as the number of dots (oblong numbers). Moreover, to determine the number of dots (oblong numbers) in terms of rows by using O = r2 + 3r + 2 where r stands for the number of rows and O as the number of dots (oblong number). VII. Possible Extensions It is recommended that further investigation may be conducted along this line by determining the number of rectangles formed in nth rows with the same number of dots (oblong numbers).