Square Pyramids, Difference of Squares, and Amazing Square Patterns

Pyramids:

What a week of squares! I had the children explore the world of squares with nine square arrays of tennis balls: 1x1, 2x2, 3x3 to 9x9. They first built what they called a “regular pyramid.” We discussed that pyramid is categorized by its base; in this case, a square or rectangle. So, a “square pyramid.” We added one more mathematical phrase to that pyramid called “right.” It is “right square pyramid” when the apex (top) is aligned directly above the center of the base. We added the sum of the squares: 1 + 4 + 9 + 16 + 25 + 36 + 49. We used a strategy of adding first 1 + 49, then 4 + 36, then 9 + 16 + 25, to get 50 + 40 + 50, to yield 140. Keeping the same 140 tennis ball pyramid, they shifted each square forward one row which resulted in a pyramid with an apex not aligned above the center of the base, this is called an oblique square pyramid. We were able to create several different kinds of oblique square pyramids; one had two acute triangular faces and two obtuse triangular faces, the other had four right triangular faces.

Difference of Consecutive Squares are Consecutive Odd Numbers:

The kindergarteners (see the pdf. Square Pyramids and Consecutive Odd Square Differences) and first graders (see the pdf. Consecutive Odd Square Differences, Amazing Square Pyramids) worked on counting squares in an oblique square pyramid from 1, 4, 9, 16, 25, 36, 49, and 64. They used several methods of counting by 2s, 3s, 4s, 5s, and so on. The most challenging part for even older children in counting by 2s after an odd number. For example, if you have a 3x3 square of 9 and are counting the additional 7 squares to get to 4x4 or 16, you start with 9, then 11, then 13, then 15, plus one more to 16. Each time they see that the difference of consecutive squares are consecutive odd numbers:

4 — 1     = 3

9 — 4     = 5

16 — 9   = 7

25 — 16 = 9

36 — 25 = 11

49 — 36 = 13

and so on.

Most of the K and 1st graders finished coloring the odd difference of consecutive squares up to 64 so I gave them a second square sheet that went beyond 8x8 to 24x24 or 576 squares. They should continue on this quest during the week trying to identify each square.

Is there a pattern in the number of squares in each range of one hundred. 

Range of Numbers Number of Square #s  List of square #s

0 and 100 =   10 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

101 and 200 = 4 121, 144, 169, 196

201 and 300 = 3 225, 256, 289

301 and 400 =

401 and 500 =

And so on through 1,000. I provided the 1st-7th grades with a list of the first 100 square numbers ending in 100x100=10,000. They can continue this exploration looking at the numbers in the ranges from:

1,001 and 2,000 =

2,001 and 3,000 =

3,001 and 4,000 =

4,001 and 5,000 =

5,001 and 6,000 =

6,001 and 7,000 =

7,001 and 8,000 =

8,001 and 9,000 =

9,001 and       10,000 =

111,111,111^2

We looked at the palindromic patterns when squaring numbers with ones such as:

1^2 =   1

11^2 =  121

111^2 =  12,321

1,111^2 =

11,111^2 =

111,111^2 = 

1,111,111^2 =

11,111,111^2 =

111,111,111^2 = 12,345,678,987,654,321

The challenge was whether they needed the third solution to solve 111,111,111^2. Once they saw that 111^2 =  12,321, they could solve the challenge. The real big challenge is what happens when there are ten ones in the square 1,111,111,111^2. It is really interesting at 1,234,567,900,987,654,321.

3,333,333,333,333^2

We looked at the patterns when squaring numbers with threes such as:

3^2 = 9

33^2 =  1,089

333^2 =  110,889

3,333^2 = 11,108,889

33,333^2 =

333,333^2 = 

3,333,333^2 =

33,333,333^2 =

333,333,333^2 =

3,333,333,333^2 = 

33,333,333,333^2 =

333,333,333,333^2 =

3,333,333,333,333^2 = 11,111,111,111,108,888,888,888,889

This pattern is easier to solve with large numbers of digits because the pattern continues indefinitely.

Difference of Squares

One of the most useful factoring methods in algebra is the difference of squares. The difference between any two square numbers is the product of the sum of the roots and the difference of the roots: a^2 - b^2 = (a + b)(a - b). For example:

5^2 - 2^2 = (5 + 2)(5 - 2)= 7 x 3 = 21= 25 - 4

The 3rd —7th graders tackled this challenge with enthusiasm. I am really trying to see which strategies they employ when multiplying (a + b)(a - b). When solving 12^2 — 7^2, they have to multiply 19 x 5. Many of them used the distributive property 5(10 + 9) to get 50 + 45 = 95. This is a great strategy but there is one better: 5(20-1) to get 100—5 = 95.

See the pdf called Difference of Squares, Consecutive Odd Differences, and Amazing Square Patterns. This also provides a list of the first 100 squares so they can create their own Difference of Square challenges.