KS3 · Maths

Associative, commutative and distributive laws

8 × 14 + 8 × 6 becomes 8 × 20 or 16 × 10: two routes, one answer, 160. The laws give you permission, and flag the traps.

Maths · Number

Which law lets you do that?

Pick a rewrite, then pick where it belongs. You'll see the reason straight away — even when you get it wrong.

Still to sort

Commutative law (0)

The numbers swap places.

Where the line is: Commutative swaps the order of the numbers. Associative moves the brackets that show the grouping.

Associative law (0)

The brackets move to show a different grouping.

Where the line is: If the brackets move, it's the associative law. If two numbers change places, it's the commutative law.

Distributive law (0)

Multiplying a sum means multiplying each part, then adding.

Where the line is: Distributive needs a multiplication outside a bracket that holds an addition.

Not true — the law doesn't hold (0)

The two sides give different answers.

Where the line is: A rewrite can look just like a law and still be false. Subtraction and division are not commutative or associative.

9 of 9 still to sort.

Every line below is a rewrite: the left side has been changed into the right side. Some changes are allowed by a law. Some are traps that look allowed.

Maths · Distributive law

Run the distributive law backwards

You've seen the law expand 10 × (12 + 3) into 10 × 12 + 10 × 3. Now run it backwards. This grid holds 8 × 14 + 8 × 6, with 14 and 6 across the top and each product inside its cell. Each cell is its row times its column — so what number belongs down the side?

Grid: each cell is its row times its column
146
11248

Type x² as x^2 if you cannot type ². Spaces do not matter.

Two routes to the same answer

Problem

You know 8 × 14 + 8 × 6 = 8 × (14 + 6) = 160. Find a second route to 160, then use the same idea on 1.2 × 20 + 1.2 × 30 + 1.2 × 10.

Maths · Common factors with decimals

Your turn: take out the common factor

Work out 4.5 × 12 + 1.3 × 8 + 0.9 × 4 by finding a common factor.

  1. Look at the whole numbers being multiplied: 12, 8 and 4.The decimals 4.5, 1.3 and 0.9 don't share a useful factor, so start with the whole numbers.
  2. missing step
Which line is step 2?

Commit first

Don't calculate yet. Look at the shape of each side and decide.

Are 17² − 7 × 17 and 2 × 17 × (3.2 + 1.8) equal?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Rewrite a calculation into an easier one that must give the same answer — and know when you're not allowed to.

What you need to know

  • Commutative law: you can write the numbers in either order without changing the calculation. Addition and multiplication are commutative: 3 + 4 = 4 + 3 and 2 × 6 = 6 × 2.
  • Associative law: when you repeat the same operation, the grouping of pairs doesn't change the result, and brackets show the grouping. Addition and multiplication are associative: (2 + 3) + 4 = 2 + (3 + 4).
  • Distributive law: multiplying a sum is the same as multiplying each part and adding the results: 10 × (12 + 3) = 10 × 12 + 10 × 3 = 150.
  • Subtraction and division are not commutative or associative: 10 − 2 is not 2 − 10, and 24 ÷ (8 ÷ 4) is not (24 ÷ 8) ÷ 4.
  • Run the distributive law backwards to take out a common factor: 8 × 14 + 8 × 6 = 8 × (14 + 6) = 160.

The big picture

The commutative, associative and distributive laws let you rewrite a calculation into an easier one that gives the same answer. Addition and multiplication are both commutative and associative; subtraction and division are neither. Naming the law you used tells you whether the rewrite is allowed.

Key points

1The laws let you rewrite a calculation so it's easier to work out, as long as the rewrite is allowed.
2Commutative is about order, associative is about grouping with brackets, distributive is about multiplying a sum.
3Only addition and multiplication are commutative and associative.
4A common factor plus the distributive law can make a calculation easier, or show that two calculations are equal.
5There are endless ways to rewrite a calculation. Choose steps that progress it, and if you can't spot a neat one, calculate another way.

Worked example

Problem

Work out 6 × 17 + 6 × 3 without a calculator, and name the law you use.

⚠ Watch out

Treating subtraction and division like addition and multiplication. 10 − 2 and 2 − 10 look like a harmless swap, but they're 8 and −8. And 24 ÷ (8 ÷ 4) is 12 while (24 ÷ 8) ÷ 4 is 0.75.

🧠

Memory hook

C is for Change places, A is for Adjust the brackets, D is for Deal the multiplication out to every part. And − and ÷ fail the C and A tests.

✓

Check yourself

Without a calculator, use a common factor to work out 9 × 15 + 9 × 5. Then say why 9 − 4 = 4 − 9 is false.

Flashcards

(14)
What does the commutative law say?
The numbers can be written in either order without changing the calculation.
What does the associative law say?
When an operation is repeated, it doesn't matter how pairs of values are grouped. Brackets show the grouping.
What does the distributive law say?
Multiplying a sum is the same as multiplying each part and adding the results.
Which two operations are both commutative and associative?
Addition and multiplication.
Show that subtraction isn't commutative.
10 − 2 is not the same as 2 − 10.
Show that division isn't commutative.
8 ÷ 4 is not the same as 4 ÷ 8.
Show that subtraction isn't associative.
10 − (2 − 1) is not the same as (10 − 2) − 1.
Show that division isn't associative.
24 ÷ (8 ÷ 4) is not the same as (24 ÷ 8) ÷ 4.
Which law lets you write 10 × (12 + 3) as 10 × 12 + 10 × 3?
The distributive law. Both sides make 150.
What is a common factor move, in one line?
If one number multiplies every term, take it out of the sum and write it in front of brackets.
Why bother rewriting a calculation using the laws?
To find the most efficient way to calculate, and to solve problems flexibly.
What should every rewriting step do?
Progress the calculation. If you can't spot a more efficient way, you can still calculate it another way.
How do you find a common factor when decimals are involved?
Look at the whole numbers being multiplied. Their common factor is the one to take out.
How can the associative law help find a bigger common multiplier?
Split numbers into factors and group them, so 1.2 × 10 becomes 12 and 12 shows up in every term.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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