An operation is commutative if the values it is operating on can be written in either order without changing the calculation.

The associative law states that a repeated application of the operation produces the same result regardless how pairs of values are grouped.

The distributive law says that multiplying a sum is the same as multiplying each addend and summing the result.

Using the associative, commutative and distributive laws together

Maths

Pupils use their understanding of place value and fluency with decimals from this unit in order to extend this understanding to fractions.  In the future unit, pupils will be asked to compare and order both fractions and decimals and will assess when conversion is required.

The prior unit on factors, multiples, squares and cubes helped pupils build on their understanding that any positive integer can be expressed as a product of its factors. This skill is used here to explore multiplying and dividing with decimals.

In this unit pupils develop their ability and efficacy with the four operations including more complicated and challenging calculations involving fractions, decimals and negative numbers. This leads into an exploration of the priority order of operations.

Arithmetic procedures with integers and decimals

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

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

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

Work out the missing number:  $$14.3 \times 0.6 + 14.3 \times$$ {{ }} $$= 14.3$$

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

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.

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

Match each law with an example of the law

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

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

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

Work out the missing number: $$5.67 \times 0.3 + 5.67 \times$$ {{ }} $$= 5.67$$

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

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