Download Triangular Numbers

InterMath | Workshop Support | Write Up Template
Title
Triangular Numbers
Problem Statement
Consider the pattern formed by these dots.
The number used to describe each "triangle" is called a triangular number. Given a
number, can you arrange that many dots into a triangle? If you can, you have
identified a triangular number. The sequence of triangular numbers starts with 1,
3, 6, and so on.
Draw a picture of the next triangle in the sequence and state the next triangular
number.
Without drawing any more triangles, determine the next five triangular numbers and
explain how you found your results.
Problem setup
I am trying to find the pattern for triangular numbers. I need to find the next five
triangular numbers and explain how the sequence works.
Plans to Solve/Investigate the Problem
Prediction: The sequence will be the sum of counting numbers (e.g. 1=1, 1+2=3,
1+2+3=6, 1+2+3+4=10, 1+2+3+4+5=15, 1+2+3+4+5+6=21). Therefore, the sequence is
1, 3, 6, 10, 15, 21.


Create rectangles out of dots and try to see if half of a rectangle can help derive
the triangular numbers.
Try to find a sequence starting with the given numbers: 1, 3, 6, …
Investigation/Exploration of the Problem
1. First I am going to create rectangles (out of dots) in GSP. I will create six
different rectangles with increasing dimensions: 1 x 2, 2 x 3, 3 x 4, and 4 x 5.
2. In creating these rectangles, I used the graph feature in GSP to make the points
equal in distance. Then, I visually cut each number in half.
3. This means, the first array is a 1 x 2 rectangle, followed by a 2 x 3 rectangle,
followed by a 3 x 4 rectangle, and finally a 4 x 5 rectangle. This makes it easy to
see that if we divide them in half (or by two), then we will get the triangular
numbers.
4. Therefore, the 1st triangular number = (1 x 2) / 2, the 2nd triangular number = (2 x
3) / 2, the 3rd triangular number = (3 x 4) / 2, the 4th triangular number = (4 x 5 ) /
2. This means the general pattern for the nth triangular number is equal to
 (n  (n  1))  2 .
5. So, the next triangular number in the sequence 1, 3, 6, is 10.
Triangular Number
Number of dots by column
Total number of dots
1st
1
1
2nd
1+2
3
3rd
1+2+3
6
4th
1+2+3+4
10
5th
1+2+3+4+5
15
.
.
.
Nth
1 + 2 + 3 + ….. + n
(n  (n  1))  2
6. This means that by adding natural consecutive natural numbers, the triangular
numbers can be found. So, the next five triangular numbers are 1, 3, 6, 10, 15, 21,
28, 36.
(4  (4  1))  2  10
(5  (5  1))  2  15
(6  (6  1))  2  21
(7  (7  1))  2  28
(8  (8  1))  2  36
Extensions of the Problem
What is the 100th triangular number? Explain your result.
If you notice, the sum of the first and last number (1+100=101) and the sum of the
second and second to last number (2+99=101), and so on, where every pair has a sum of
101. So, if you had 100 numbers in all to be added, resulting in 50 pairs, the 101 * 50 =
5050. This means that the formula for finding the 100th triangular number is:
 (n  2)  (n  1) , where n  2 is the number of pairs and n  1 is the sum of each pair.
Author & Contact
Lauren Johnson, Middle Grades Cohort at Georgia College and State University
[email protected]
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2