Number Shapes Revisited
Start with a large collection of small objects, such as raisins. For each number, investigate which rectangles and other shapes you can make with that many objects.
Rectangles
The sides of rectangles are values that evenly divide the number and multiply together to give the number. Making rectangles is a direct way to experience divisibility.
Unit, prime, composite
1 is a unit, and can only be made with a 1 by 1 rectangle. The numbers, such as 5, that only have flat rectangles, are called primes. Numbers that are not a unit or a prime are called composite because they are composed of primes being multiplied together, such as 12 = 2 × 2 × 3. Numbers, such as 9, are called squares because one of their rectangles is a square – one rectangle for 9 is the 3 by 3 square.
Trapezoidal numbers
There are other shapes that are fun to investigate. For example, which numbers are trapezoidal? These are the numbers that can be represented as stair steps (where each level changes its length by 1)? If you include triangular numbers in this group, the answer is surprising – it is all numbers that are not a power of 2!
Helping your child
Investigations are meant for your child to play with and think about. Let your child explore these looking for interesting patterns and beautiful relationships. Resist the temptation to unveil what is going on and give the answer. If your child seems to have reached a dead end, suggest that they come back to the investigation at a future time to play with it again.
Many investigations benefit from organizing results, and this is a great thing for you to help your child with. Help them make tables, drawings, or whatever may help them see more easily what is going on. And of course, it is perfectly fine to give them gentle nudges from time to time in the right direction. Remember that your child will learn a lot by developing persistence and learning how to look more deeply at things.