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      Using the associative, commutative and distributive laws together

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

      Lesson 17 of 17
      New
      New
      • Year 7

      Using the associative, commutative and distributive laws together

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

      Lesson details

      Key learning points

      1. A calculation may be made easier by applying more than one of the associative, commutative or distributive laws.
      2. Steps taken should progress the calculation steps.
      3. 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

      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.
      Teacher tip

      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 (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      Match each law with an example of the law

      Correct Answer:Commutative law,$$3 + 10 + 8 = 8 + 10 + 3$$
      Correct Answer:Associative law,$$6\times5\times4= (6\times5)\times4$$
      Correct Answer:Distributive law,$$45\times(0.7+0.3)=45\times0.7 + 45\times0.3$$

      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

      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

      Lesson appears in

      UnitMaths / Arithmetic procedures with integers and decimals