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50 mins

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Properties of Operations and the Distributive Law

Learn how to split one factor into place-value parts and use the distributive law to make multiplication easier. Visualise why the partial products always add to the whole using area models.

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1 Getting Started
2 Watch and Notice
3 Try It Together
4 Show the Split in Your Copy
5 Class Challenge
6 What Did We Notice?
7 What's Next
Pupil practice

1 - Getting Started ~4 mins

Here is a multiplication: 6 × 13. Have a go at it in your head, but don't just rush at the whole thing. Could you break the 13 into a part that's really easy to multiply by 6? What part would you pick, and what would be left over?

Hands up with your idea. There's no answer revealed yet, we're going to find it together.

Teacher Notes

Take three hands-up ideas, not open call-outs. You want someone to suggest splitting 13 into 10 + 3 — but if nobody does, that's fine, this is the genuine problem the lesson opens up.

Do not reveal the answer or the method here. This is a problem-first lesson: the rule must emerge from the cases pupils examine in the next step, not be announced now.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~11 mins

6 × 13 as a rectangle

Let's look at these together. Here is a rectangle 6 high and 13 wide. We've cut the 13 side into 10 and 3, so the rectangle splits into a 6 by 10 piece and a 6 by 3 piece. Read off the two areas: 60 and 18. These two areas — 60 and 18 — are called the partial products. What do you notice when you add them?

4 × 25 split into 20 and 5

Now this one. The 25 side is cut into 20 and 5. We get a 4 by 20 piece and a 4 by 5 piece. Read the two partial products off the picture — 80 and 20 — then think: do they add back to the whole rectangle?

7 × 8 and 7 × 2 tiling one rectangle

Last one. Look at the 7 by 10 rectangle. We've cut the 10 side into 8 and 2, so it splits into a 7 by 8 piece and a 7 by 2 piece, side by side with no gap and no overlap. The two partial products are 56 and 14. What does that tell you about 7 × 8 + 7 × 2 compared with 7 × 10?

So here is what every picture is showing us: when we split one side and add the partial products, we get exactly the same total. That is the distributive law.

Teacher Notes

Do not state the rule before pupils have looked. Walk each example and draw the noticing out of the class. Give the third example the most room — it is the visual proof, so don't let the clock rush you off it.

6 × 13 — point at the two pieces; ask the class to read off 60 and 18, name them as the partial products, and add them. Land on 78.
4 × 25 — 80 and 20, total 100; a tidy one to build confidence.
7 × 10 cut into 8 and 2 — the two pieces 7 × 8 and 7 × 2 tile the 7 × 10 rectangle exactly, so 7 × 8 + 7 × 2 must equal 7 × 10. This is the visual proof; spend the time to draw it out.

Only after the third picture do you name the rule, drawn from what they saw: splitting one side and adding the pieces gives the same total every time.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~12 mins

Now we work through these three together at the board, one at a time. For each one I'll rebuild the area model up here: we split the product the smart way, build the two pieces, and check the partial products add to the whole.

8 × 14 (split the 14)
5 × 23 (split the 23)
9 × 12 (split the 12)

Tip

We'll say the split aloud each time before we check it on the board.

Split and check

Teacher Notes

This round is for talking it through together — rebuild each of the three products in turn on the board (the widget loads with 8 × 14; reset it and re-enter the factors for 5 × 23 and 9 × 12). Pupils take turns at the board and the class agrees or corrects out loud.

Insist the split is read aloud before the check: '8 times 10, plus 8 times 4'. Watch for pupils splitting the wrong factor or forgetting the second partial product. Revoice a strong answer: so the two pieces are 80 and 32, and 80 plus 32 is 112.

+15 XP

4 - Show the Split in Your Copy ~4 mins

COPYBOOK MOMENT

In your maths copy, write three products and beside each show the distributive split. For example: 8 × 14 = 8 × 10 + 8 × 4. Work out both totals and underline them when they match.

8 × 14
6 × 17
7 × 22

Teacher Notes

Walk the room glancing for the split written before the answer and the two equal totals underlined — this is whole-class copybook practice, not marking.

+10 XP

5 - Class Challenge ~12 mins

Now we test the rule on fresh numbers. Split each product, build it on the area model, and check the pieces add to the whole.

Distributive challenge

Teacher Notes

This round is the practice bank — pupils take turns at the board, check each answer, and the class confirms before moving on. Keep the board work brisk rather than over-explaining.

The challenge widget prompts pupils to choose the place-value split for each product; the label states which side to cut and into which tens and units. On the larger-unit items (18, 24, 35) pause and model the tens/units choice aloud before the second partial product, since that is where weaker pupils stall. The 4 × 25 → 100 case is a good one to pause on: ask why the answer is so tidy.

+15 XP

Pupil practice

Module 10 · Algebra: Patterns, Expressions and Equations Mixed

Lesson 106 · Properties of Operations and the Distributive Law

Download Activity Book page (PDF)

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Properties of Operations and the Distributive Law

Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~11 mins

6 × 13 as a rectangle

4 × 25 split into 20 and 5

7 × 8 and 7 × 2 tiling one rectangle

3 - Try It Together ~12 mins

Split and check

4 - Show the Split in Your Copy ~4 mins

5 - Class Challenge ~12 mins

Distributive challenge

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