The Shape of Numbers
By Michael Naylor

Thousands of years ago, mathematicians thought about numbers in terms of geometry. Numbers had shapes, and by thinking about the shapes, mathematicians could do a lot of things with numbers that we now recognize as algebra.

Pythagoras was a Greek teacher who lived around 500 BC. He felt everything in the universe was made out of numbers. The Pythagoreans, as his followers were called, wrote numbers as dots in the sand (see below) that allowed them to see all kinds of relationships between numbers.

Number shapes
(Grades K-5)
Show your students these dot diagrams and ask them what shapes they see.

• 3 and 6 are triangles. Ask your students to draw the next two triangular numbers (10 and 15). Write all triangular numbers on the board, both the numbers and their pictures, and ask your students what patterns they see. Each triangle is the sum of consecutive counting numbers, that is, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, and 15 = 1 + 2 + 3 + 4 + 5. Is that easy to see in the dot pictures?

• 4 is a square. 9 is the next square number after 4. Draw a picture of 9 as a square set of dots. What are the next two square numbers? (16 and 25).

Give your students round counters and have each student choose a number from 5 to 12. Ask them what shapes they can make with their number, and have them copy their design onto paper. You can bring out some interesting ideas and practice math facts by examining the student-created designs.

Dot patterns
(Grades 4-8)
Ask your students to find the following sums:

1 =
1 + 3 =
1 + 3 + 5 =
1 + 3 + 5 + 7 =
1 + 3 + 5 + 7 + 9 =

What do they notice about the sums? (The sums are perfect squares – 1, 4, 9, 16 and 25.) Ask them if they think this pattern will continue forever. Why or why not?

Pythagoras figured out a clever way to understand this pattern. He made a picture with dots in the sand that looked like this:

Show your students this pattern and ask them if this helps to explain the curious pattern in the sums. Notice that the dots are connected to make the numbers 1, 3, 5, 7 and 9. These can all be arranged to make a square: 25. If another layer of dots is added, there will be 11 more dots and the shape will be the next greatest square, or 36. A simple picture can prove a tricky idea!

Pythagorean triples
(Grades 6-8)
The Pythagoreans discovered that in a right triangle, if you square the lengths of all three sides, the squares of the shorter sides add up exactly to the square of the hypotenuse. This is called the Pythagorean Theorem. If all three sides of a right triangle have lengths that are whole numbers, the the lengths are called a Pythagorean Triple. The most famous Pythagorean triple is 3-4-5, since 32 + 42 = 52. Another famous triple is 5-12-13.

After your students have learned the Pythagorean theorem, here is an excellent algebra activity with a historical context. Copy these columns of Pythagorean Triples on the board, or make handouts.

Point out that each column is a Pythagorean triple. You may want to verify a couple of the columns with your students to be sure they understand. (One column for example, is 7-24-25. 72 = 49, 242 = 576, 252 = 625 and 49 + 576 = 625.) Challenge your students to extend the rows. How many patterns can they find?

Some patterns in the list:

• The top line consists of the odd numbers.

• The differences in the numbers in the second and third line increase by four each time; the difference between the first two terms is 4, the next difference is 8, then 12 and so on.

• The numbers in the third line are one more than the numbers in the last line.

• The sum of the bottom two numbers in a column make a square number – in fact, it's the square of the top number.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.