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)+(n1)^2+ (n2)^2+
(n3)^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 premade 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/cgibin/printlessons.cgi/Virtual/Lessons/Mathematics/Arithmetic/ATH0020.html
http://www.kstate.edu/smartbooks/Lesson055.html
