Finding the Structure of our Number System: Prime and Composite Numbers

I am currently of the belief that our curriculum is weak on number theory. If we want to truly develop the mysterious faculty called “number sense”, we should probably spend more time playing with numbers.

Also, prime numbers are just…interesting. I think kids of all ages would find them interesting. They are the indivisible building blocks of other numbers. They are the most interesting (non)pattern out there.

I regularly retweet the Prime Numbers Twitter account. This is the most recent one. They are going up, prime by prime. It’s fascinating.

What does our curriculum have to say about primes?

Let’s target grade 6, and try and find them in the Overall Expectations for number in the grade.

We have this:

read, represent, compare, and order whole numbers to 1 000 000, decimal numbers to thousandths, proper and improper fractions, and mixed numbers

Clearly, that doesn’t get us there. In this case we need the specific expectation. It’s this one:

identify composite numbers and prime numbers, and explain the relationship

between them (i.e., any composite number can be factored into prime factors)

(e.g., 42=2x3x7).

So yes, kids need to know about prime numbers. Let’s look at a possible lesson path to get there.

Learning Goal: we are learning that some numbers can’t be broken down any further.

Success Criteria:

• I can factor two digit numbers
• I can use a hundreds chart to find prime numbers
• I can explain what makes a number prime or composite.

Prior knowledge needed: you may want to review the word “factors”.

One thing that works for young kids is investigating 24, the smallest number with quite a few factors.

Possible Ways to Activate Thinking

How many rectangles can you make with an area of 24? (make sure the concept of area is present in their minds)

For this, I would deploy snap cubes, or square tiles. I would want to talk about “thinness” vs. “squareness”. There exists a skinny 1 x 24 rectangle, for example.

Other possible way to activate thinking:

How is 17 different from 18?

or:

How is 23 different from 24?

Depending on my mood, I might even throw out a prime like 41, and ask,

Why is this number interesting?

It is also fully possible to stage this as an investigation based on Dan Finkel’s Prime Climb game.

Many of our colleagues us the visual from the game as a provocation at professional sessions.

Developing Thinking

I am partial to using a very classic method to help kids find primes: the Sieve of Eratosthenes.

The hundreds square is a very useful tool here.

I like to stage this as a guided activity.

• cross off 1
• leave 2, 3, 5, and 7 alone
• mark each multiple of 2 with one colour
• mark each multiple of 3 with another colour
• mark each multiple of 5 with another colour
• make each multiple of 7 with another colour

Above is a hundreds square just showing primes. It is fascinating to try and discern a pattern. What would kids say?

Just ask them: what patterns did you notice? Which numbers do you find most interesting?

12, 24, 36, 48, and 60 are all highly composite numbers. You may want to pause and get kids to draw out all possible rectangles for each.

At this point, you could try getting them to extend by asking:

what’s the next prime number after 100?

I think I might dramatically pull out a 1 x 97 rod of snap cubes to show the essential “thinness” of primes (when you use the area model).

Consolidating Thinking

The National Library of Virtual Manipulatives has a niece “sieve”. Show that.

At this point, you want to work toward a good definition of prime number.

Make a list on your whiteboard.

Refer back to the way we activated thinking. Why is 23 different from 24?

Get kids’ thoughts on that.

Pull out any interesting patterns you find in the hundreds chart. Get kids to consider the prime list- is it a pattern?

Ask them to predict what bigger primes would look like. What numbers can they end in?

Exit ticket: why is 1 neither prime nor composite?

Possible future assessment questions:

• is 101 prime? Prove it is or isn’t.
• is 121 prime? Prove it is or isn’t.
• show me all possible rectangular arrangements of 36.