Intermath | Workshop Support

Write-up


Title
Triangular Numbers

 

Problem Statement
Consider the pattern formed by these dots.


The number used to describe each "square" is called a square number. Given a number, can you arrange that many dots into a square? If you can, you have identified a square number. The figure above shows the first three square numbers, 1, 4, and 9. How can you make the next square number from parts of the third square number? Generalize your work. What is the sum of the first n square numbers?

Problem setup

How could you quickly identify square numbers?

 

Plans to Solve/Investigate the Problem

Well, this looks like a problem for… dunh dunh dunh!  Excel!

 

Investigation/Exploration of the Problem

I created an Excel spreadsheet.  Of course square numbers must have been named so because they are formed when you square a number.  After numbering several rows, I created two formulas for finding a square (=SUM((A2)*(A2)) in the second column and =SUM((A2^2)) in the third column). 

Square numbers

 

1

1

1

2

4

4

3

9

9

4

16

16

5

25

25

6

36

36

7

49

49

8

64

64

9

81

81

10

100

100

11

121

121

12

144

144

13

169

169

14

196

196

15

225

225

16

256

256

Voila!  Square numbers! 

 

Now for making the next square number from parts of the third square number.  Looking at the visual arrangement, a very simple patter emerges in my mind.  To create another set of circles to make this number square you would add to the previous square number the number of circles in the base row and also add that number minus 1.  So the next square number would be 9+4+3= 16.

 

The sum of the first n square numbers would be (n^2)+(n-1)^2+ (n-2)^2+ (n-3)^2…

 

Extensions of the Problem

Purchasing flooring for a square room.  How much material should you purchase for a square room?

Decorating a square cake.  How many pre-made decorations would you need to balance your cake?

Author & Contact
Lorri Worman
lworman@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
http://www.eduref.org/cgi-bin/printlessons.cgi/Virtual/Lessons/Mathematics/Arithmetic/ATH0020.html

http://www.k-state.edu/smartbooks/Lesson055.html