OuiSi · Math

Arrays in pictures

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Grades 3–5 Teaching 35 min Plus 3 to set out, 3 to clear Class any size Set any Prep squared paper, and a few numbers of your own ready

set out 3′ Count, then group 8′ Make a rectangle 12′ The numbers that refuse 10′ Close 5′ clear 3′

Goal: Children find a quantity in a photograph, try to arrange it into a rectangle, and meet the numbers that refuse — discovering factors and primes by running into them.
Materials: One OuiSi Photo-Card set, about six cards per pair. Paper, squared if you have it.

Grade 3: Numbers to 24, rectangles and factor pairs. Steps 1 and 2, then the close — about 25 min.
Grade 4: Add the stubborn numbers and name them — the full 35 min.
Grade 5: Prove a number has no other rectangle, and say how you know — the full 35 min.

1 · Count, then group

8 min

Pairs, about six cards face up.

Find a card where you can count something. Write the number down.

Then before any arithmetic.

How did you count them? One by one, or in groups?

Children who counted in groups are most of the way to a row. Grouping is not yet an array — the array needs equal rows and equal columns — but it is the move the next step builds on, so name it.

2 · Make a rectangle

12 min

Draw your number as a rectangle. Rows and columns, no gaps, nothing left over.

Once a pair has one.

Can you make another rectangle with the same number? How will you know when you have found them all?

Twelve gives three — one by twelve, two by six, three by four. Twenty-four gives four. Some numbers are generous and some are mean, which is the right instinct, and the word for it comes later.

Collect the factor pairs on the board under each number as they come.

Three rows of four. Say it as a multiplication.

3 · The numbers that refuse

10 min

Find a card with a number that will only make one rectangle. One long line and nothing else.

Seven, eleven, thirteen. Expect pairs to insist they just have not found it yet, and let them hunt — the conviction that there must be another one is what makes the finding matter. Do not leave it to the deck, though. If nobody has landed on one after five minutes, hand them seven and ask the same question.

When a pair is sure.

How do you know there is not another one? Convince me.

Then give the words, after the thing has happened.

The numbers that make lots of rectangles have lots of factors. The stubborn ones are prime — greater than one, with exactly two factors: one and itself.

That definition is why one is not prime, which somebody usually asks. A good question, not a detour.

4 · Close

5 min

Pick a number from the board. How many equal groups could you make from it, with nothing left over?

A useful way to feel what factors are for. Resist saying a prime shares badly — seven shares perfectly among seven.

Watch for

Rectangles with gaps. The array has to be full. A gappy rectangle is a different and much weaker idea, so hold the no-gaps rule.
Counting the same rectangle twice. Three rows of four and four rows of three are one factor pair turned sideways. Say so in step two, before any number goes on the board — the counts only mean anything once the room has agreed on this.
Numbers too large. Over thirty and counting eats the lesson. Steer toward cards in the ten to twenty-four range, and keep the smaller primes for handing out in step three.
Giving the word prime too early. The hunting is the lesson. Name it only once a pair has convinced themselves.

Adapt

Grade 3 — stop after step two. Factor pairs alone are a full lesson.
Grade 5 — ask for every factor pair of a number and a reason the list is complete.
Children who think faster than they write — let them build arrays with counters before drawing them.
Multilingual learners — rows and columns are the only vocabulary load. Put a sketch of each on the board.
Number anxiety — give them twelve. It is generous, it works three ways, and success comes fast.
When the deck will not supply it — have 7, 12, 15, 18 and 24 ready to hand out. The photograph starts the lesson, it does not have to carry it.
Extension — find two cards whose numbers multiply to something that makes a square.

Curriculum links. Counting a quantity and recording it · Building rectangular arrays and reading them as multiplication · Finding every factor pair of a number · Identifying primes by the arrays they refuse to make · Arguing that a search is complete.
Standard codes vary by state and district — these are the skills exercised, for you to map to your own.

For grades K–2 see Number bonds, which does the same counting with physical counters and splitting.
Screen-free. No codes and no devices. Works with any OuiSi Photo-Card set.