Building Rectangles from Tiles

We are learning to express a whole number as a product of its factors by finding every rectangle that number of tiles can make.

Before you startAbout 60 minutes · what success looks like · notes for the helper

About 60 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 build or draw every rectangle for a number of tiles and record each one as a factor pair, like 4 × 9
I can list all the factors of a number in order, testing systematically so no factor pair is missed
I can explain why a number like 13 makes only one rectangle while 12 makes several

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 40 small squares of the same size — square tiles, crackers, sticky notes, or squares cut from cardboard — or some grid paper and a pencil if you'd rather draw. The whole lesson takes about 60 minutes, so two sittings suit most children: the teaching, the guided problems and the check in one, and the chosen task in the next.
What understanding sounds like: Listen for a system: 'one row of 24, two rows of 12, three rows of 8…' tested in order, and for your child noticing when the pairs start repeating. Hearing 'I've already got 4 × 6, so 6 × 4 is the same one turned' is a good sign.
A likely mix-up, and something that helps: A common slip is counting a rectangle twice: 4 × 6 written down, then the same tiles turned and written again as 6 × 4. Another is deciding that every odd number makes only one rectangle. You might turn the tiles round together and ask, 'Is this a new rectangle, or the same one turned?', or set out 15 tiles and wonder aloud whether 3 rows could work.
If it comes easily: If it comes easily, you might ask which number up to 50 your child thinks has the most factors, and let them test a few guesses before the last lesson of the unit answers it.
If it's a struggle: If it's a struggle, one way is to stay with real tiles and one number, 12, and build each rectangle before writing it down. Saying 'rows of, in each row' aloud for each one keeps the bar where it is and makes the pattern easier to see. Factor, factor pair and product carry this lesson; for a child learning English, it can help to say each word with a rectangle of tiles in front of you, and to put a home-language word beside it. If your child shuts down, building rectangles with no writing at all is still the lesson's thinking, and the recording can wait.

Step 1 of 6 · Spark

Lesson 1 of 5 · Factors, Multiples and Number Patterns

The patio puzzle

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.

A landscaper has 24 square pavers and wants to lay a rectangular patio. Every paver has to be used, with no gaps and no bits sticking out. One row of 24 makes a very long, thin path. Two rows of 12 is wider. How many DIFFERENT rectangular patios could she make? And here's the strange part: if she had 23 pavers instead, just one fewer, she could only make one. Why would one paver make such a difference?

24 pavers: 1 row of 24? 2 rows of 12? How many more? And why does 23 only make one?