Using the associative, commutative and distributive laws together | Maths

Starter quiz

Match each law with an example of the law
- Commutative law
  
  ⇔
  
  $3 + 10 + 8 = 8 + 10 + 3$ ✓
- Associative law
  
  ⇔
  
  $6\times5\times4= (6\times5)\times4$ ✓
- Distributive law
  
  ⇔
  
  $45\times(0.7+0.3)=45\times0.7 + 45\times0.3$ ✓
Which of the following is an example of the distributive law for this calculation $39\times99$ ?
- $99\times39$
- $39\times(100 - 1)$ ✓
- $39\times(9 \times 9)$
Which of the following is an example of the distributive law for this calculation: $22\times42$
- $22\times(40+2)$ ✓
- $22\times(21\times 2)$
- $22\times(4+2)$
- $22\times(10+10+10+10+2)$ ✓
Without using a calculator, use the distributive law to work out: $4.8 \times 6.1 + 4.8 \times 3.9 =$ ______
- '48' ✓
Work out the missing number: $5.67 \times 0.3 + 5.67 \times$ ______ $= 5.67$
- '0.7' ✓
Work out the missing number: $23 \times 0.8 + 23 \times 1.4 − 23 \times$ ______ $= 46$
- '0.2' ✓

Select the calculation that is equivalent to $8\times(10+12)$ .
- $80 + 92$
- $8\times 10 + 8\times12$ ✓
- $8 \times2$
Select the calculation that is equivalent to $4\times9 + 4\times11$ .
- $4\times20$ ✓
- $4\times9\times 11$
- $4\times13$
Select the calculations that are equivalent to $24\times44$ .
- $24\times(40+4)$ ✓
- $6\times4\times11\times4$ ✓
- $2\times(12\times22)$
- $16\times66$ ✓
- $20\times4 + 40\times4$
Work out the missing number: $14.3 \times 0.6 + 14.3 \times$ ______ $= 14.3$
- '0.4' ✓
Select the calculations that are equivalent to $70\times4 + 12\times4+9\times8$ .
- $4\times(70+12+9)$
- $4\times(70+12+18)$ ✓
- $8\times(35+6+9)$ ✓
- $8\times(70+12+9)$
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.
- '9' ✓

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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.

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

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.