Building Rectangles From Tiles

We are learning to find every factor pair of a number by building rectangles, and to use the number of factors to decide whether a number is prime, composite or square.

Before you startAbout 55 minutes · what success looks like · notes for the helper

About 55 minutes. It's fine to stop after the Quick Check and come back another day: the lesson opens where you left it, at the Quick Check.

You'll know you've got it when

I can find every factor pair of a number by testing rows in order, without missing or repeating a pair
I can explain why a prime number has exactly two factors and makes only one rectangle, and why a composite number makes more than one
I can recognise a square number and explain why 1 is neither prime nor composite

For the helperhow to sit beside this lesson, and what this step is for

This step: Spark

Start with the question at the top and chat about it. There is no wrong answer here: it gets your learner thinking about what they already know.

Getting ready: About 36 small square things to arrange — squares cut from a cereal box, coins, buttons or spare tiles — and some paper (grid paper if you have it) for sketching the rectangles. The other tasks need only paper and pencils. The whole lesson takes about 55 minutes, so many families stop after the check and do the task the next day.
What understanding sounds like: Listen for reasons about the number of factors: '13 only has 1 and 13, so it's prime', or 'I found three rectangles, so 12 is composite'. Testing rows in order — 1 row, 2 rows, 3 rows — so that none are missed is a strong sign, and so is 'it makes a square, so it's a square number' or '1 only has one factor, so it's neither'.
A likely mix-up, and something that helps: Many children think 1 is prime, or that every odd number is prime, although 9, 15 and 21 are not. You might ask, 'How many factors does it have? A prime has exactly two,' or 'Can those tiles make more than one rectangle?'
If it comes easily: A stretch worth offering: which numbers under 50 have an odd number of factors, and why does every one of them turn out to be a square number?
If it's a struggle: If it's a struggle, one way is to build each rectangle with real objects before anything is written down, and to stay with numbers up to 20 for now. If your child has had enough, stopping after one number they built and explained is a real finish for today. The words factor, prime and composite carry this lesson, so a child still learning English may find it helps to say each one beside its tiles first.

Step 1 of 6 · Spark

Lesson 1 of 5 · What Numbers Are Made Of: Primes, Squares and Balanced Equations

Twelve tiles, thirteen tiles

For the helper

Start with the question at the top and chat about it. There is no wrong answer here: it gets your learner thinking about what they already know.

You have 12 square tiles. Push them together into a rectangle, with no gaps and no tiles left over. Now make a different rectangle. And another. How many can you find? Now add one more tile and try again with 13. Something strange happens: 13 tiles refuse to make anything except one long row. Why would one extra tile change everything?

12 tiles make three different rectangles. 13 tiles make only one. What is going on?