Using the associative, commutative and distributive laws together

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Lesson details

Learning outcome

I can use the associative, distributive and commutative laws to flexibly and efficiently solve problems

Key learning points

A calculation may be made easier by applying more than one of the associative, commutative or distributive laws.
Steps taken should progress the calculation steps.
If a more efficient way cannot be spotted, the calculation can still be calculated using other methods.

Keywords

Commutative - An operation is commutative if the values it is operating on can be written in either order without changing the calculation.
Associative - The associative law states that a repeated application of the operation produces the same result regardless how pairs of values are grouped.
Distributive - The distributive law says that multiplying a sum is the same as multiplying each addend and summing the result.

Common misconception

The common multiplier is the number seen to be common eg. 1.2 x 20 + 1.2 x 30 + 1.2 x 10

Students can use the associative law to find more common multipliers eg. 1.2 x 10 x 2 + 1.2 x 10 x3 + 1.2 x 10 = 12 x 2 + 12 x 3 + 12 x 1

Teacher tip

Encourage student to see more factors of multipliers using the associative law. Give the class 2.4 x 30 - 2.4 x 20 and ask to rewrite it e.g 2.4 x 10 x 3 - 2.4 x 10 x 2 etc and decide which calculation is easiest to work out the answer.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

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Prior knowledge starter quiz

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6 Questions

Q1.
Match each law with an example of the law

Correct Answer:Commutative law,$$3 + 10 + 8 = 8 + 10 + 3$$

$$3 + 10 + 8 = 8 + 10 + 3$$

Correct Answer:Associative law,$$6\times5\times4= (6\times5)\times4$$

$$6\times5\times4= (6\times5)\times4$$

Correct Answer:Distributive law,$$45\times(0.7+0.3)=45\times0.7 + 45\times0.3$$

$$45\times(0.7+0.3)=45\times0.7 + 45\times0.3$$

Q2.
Which of the following is an example of the distributive law for this calculation $$39\times99$$?

$$99\times39$$

Correct answer: $$39\times(100 - 1)$$

$$39\times(9 \times 9)$$

Q3.
Which of the following is an example of the distributive law for this calculation: $$22\times42$$

Correct answer: $$22\times(40+2)$$

$$22\times(21\times 2)$$

$$22\times(4+2)$$

Correct answer: $$22\times(10+10+10+10+2)$$

Q4.
Without using a calculator, use the distributive law to work out: $$4.8 \times 6.1 + 4.8 \times 3.9 = $$

Correct Answer: 48

Q5.
Work out the missing number: $$5.67 \times 0.3 + 5.67 \times$$ $$= 5.67$$

Correct Answer: 0.7, 7/10

Q6.
Work out the missing number: $$23 \times 0.8 + 23 \times 1.4 − 23 \times$$ $$= 46$$

Correct Answer: 0.2, 2/10, 1/5

Assessment exit quiz

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6 Questions

Q1.
Select the calculation that is equivalent to $$8\times(10+12)$$.

$$80 + 92$$

Correct answer: $$8\times 10 + 8\times12$$

$$8 \times2$$

Q2.
Select the calculation that is equivalent to $$4\times9 + 4\times11$$.

Correct answer: $$4\times20$$

$$4\times9\times 11$$

$$4\times13$$

Q3.
Select the calculations that are equivalent to $$ 24\times44$$.

Correct answer: $$24\times(40+4)$$

Correct answer: $$6\times4\times11\times4$$

$$2\times(12\times22)$$

Correct answer: $$16\times66$$

$$20\times4 + 40\times4$$

Q4.
Work out the missing number: $$14.3 \times 0.6 + 14.3 \times$$ $$= 14.3$$

Correct Answer: 0.4, 4/10, 2/5

Q5.
Select the calculations that are equivalent to $$70\times4 + 12\times4+9\times8$$.

$$4\times(70+12+9)$$

Correct answer: $$4\times(70+12+18)$$

Correct answer: $$8\times(35+6+9)$$

$$8\times(70+12+9)$$

Q6.
The same number is written in each square: $$(3\times\square+5\times\square)+\square = \triangle \times \square$$. Find the number that should go in the triangle so that the equation is always true.

Correct Answer: 9, nine

To help you plan your 7 maths lesson on: Using the associative, commutative and distributive laws together, download all teaching resources for free and adapt to suit your pupils' needs...

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Using the associative, commutative and distributive laws together

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Match each law with an example of the law

Q2.Which of the following is an example of the distributive law for this calculation $$39\times99$$?

Q3.Which of the following is an example of the distributive law for this calculation: $$22\times42$$

Q4.Without using a calculator, use the distributive law to work out: $$4.8 \times 6.1 + 4.8 \times 3.9 = $$

Q5.Work out the missing number: $$5.67 \times 0.3 + 5.67 \times$$ $$= 5.67$$

Q6.Work out the missing number: $$23 \times 0.8 + 23 \times 1.4 − 23 \times$$ $$= 46$$

Assessment exit quiz

6 Questions

Q1.Select the calculation that is equivalent to $$8\times(10+12)$$.

Q2.Select the calculation that is equivalent to $$4\times9 + 4\times11$$.

Q3.Select the calculations that are equivalent to $$ 24\times44$$.

Q4.Work out the missing number: $$14.3 \times 0.6 + 14.3 \times$$ $$= 14.3$$

Q5.Select the calculations that are equivalent to $$70\times4 + 12\times4+9\times8$$.

Q6.The same number is written in each square: $$(3\times\square+5\times\square)+\square = \triangle \times \square$$. Find the number that should go in the triangle so that the equation is always true.

How to plan a lesson using our resources

Q1.
Match each law with an example of the law

Q2.
Which of the following is an example of the distributive law for this calculation $$39\times99$$?

Q3.
Which of the following is an example of the distributive law for this calculation: $$22\times42$$

Q4.
Without using a calculator, use the distributive law to work out: $$4.8 \times 6.1 + 4.8 \times 3.9 = $$

Q5.
Work out the missing number: $$5.67 \times 0.3 + 5.67 \times$$ $$= 5.67$$

Q6.
Work out the missing number: $$23 \times 0.8 + 23 \times 1.4 − 23 \times$$ $$= 46$$

Q1.
Select the calculation that is equivalent to $$8\times(10+12)$$.

Q2.
Select the calculation that is equivalent to $$4\times9 + 4\times11$$.

Q3.
Select the calculations that are equivalent to $$ 24\times44$$.

Q4.
Work out the missing number: $$14.3 \times 0.6 + 14.3 \times$$ $$= 14.3$$

Q5.
Select the calculations that are equivalent to $$70\times4 + 12\times4+9\times8$$.

Q6.
The same number is written in each square: $$(3\times\square+5\times\square)+\square = \triangle \times \square$$. Find the number that should go in the triangle so that the equation is always true.