The Shape of Numbers

The shape of numbers: odd, even, composite, prime, square, rectangular, and beyond.

Objective

The students will learn another way to think of numbers, linking numbers to non-counting concepts. The object is to enrich their thinking about numbers, to break the students away from seeing numbers as existing only in arithmetic contexts.

Even and Odd

A number is even if it can be lined up in two rows of equal length:

2 0
  0
4 00
  00
6 000
  000
8 0000
  0000
10 00000
   00000

A number is odd if it cannot be lined up in two rows of equal length:

1 0
3 00
  0
5 000
  00
7 0000
  000

A number is square if it can make a square:

1 0 
4 0 0
  0 0 
9 0 0 0
  0 0 0
  0 0 0

A number is rectangular if it can make a rectangle:

2 0 0 
6 0 0 0
  0 0 0 
8 0 0 0 0
  0 0 0 0 
10 0 0 0 0 0
   0 0 0 0 0

A number is triangular if it can make a triangle:

3
  0
 0 0 
6
  0
 0 0
0 0 0 
10
   0
  0 0
 0 0 0
0 0 0 0

A number is composite if it can be made into a square or rectangle with more than one row:

4 0 0
  0 0
6 0 0 0
  0 0 0
8 0 0 0 0
  0 0 0 0
9 0 0 0
  0 0 0
  0 0 0
10 0 0 0 0 0
   0 0 0 0 0
12 0 0 0 or 0 0 0 0 or 0 0 0 0 0 0
   0 0 0    0 0 0 0    0 0 0 0 0 0
   0 0 0    0 0 0 0
   0 0 0

A number is prime if it cannot be made into a square or rectangle with more than one row. Prime numbers can only be a single row, a single row is a line. All numbers that are not composite are prime.

1 0 (prime because it is only one row)
2 0 0 (prime because it is a one row rectangle, or only a one column rectangle if turned)
3 0 0  0 0 0 ( prime: cannot be made into a multi-row rectangle) 
  0 
5 0 0 0     0 0 0 0 0 
  0 0 
7 0 0 0 0 0 0 0
11 0 0 0 0 0 0 0 0 0 0 0

As a group finish filling in the blanks in the following chart. Use drawings to help you decide.  Due to implementations of CSS in browsers the gridlines will not appear in the empty cells.

Even Odd Square Rectangle Triangle Composite Prime
1 no yes yes no no no yes
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What is the smallest odd composite number that is not a square? _____

Is any number listed both a square number and a rectangular number? _____

If there is a square number and a rectangular number, what is it? _____

Is any number listed both a square number and a prime number? _____

If there is a square number and a prime number, what is it? _____

Is any number listed both a triangular number and a square number? _____

If there is a triangular number and a square number, what is it? _____

Is any number listed triangular, square, and rectangular? _____

If there is a triangular, square and rectangular number, what is it? _____

The Shape of Numbers II

List the line (prime) numbers up to 42: ______________________________

List the triangle numbers up to 42: ______________________________

List the square numbers up to 42: ______________________________

List the rectangle numbers up to 42: ______________________________

As a group agree on a shape that is not listed above. Shape: _____

What is the first number of that shape? _____

What is the second number of that shape? _____

What is the third number of that shape? _____

What is the fourth number of that shape? _____

What is the fifth number of that shape? _____

HW: What is the smallest square that is the sum of two other squares? _____

At the end of the class have each group present its shape and the first three numbers of its shape.