# Use a given additive or multiplicative calculation to derive or complete a related calculation, using arithmetic properties, inverse relationships, and place-value understanding (5)

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## Description

In previous year groups in key stage 2 pupils have learnt about and used the commutative and associative properties of addition (3AS–3), and the commutative, associative and distributive properties of multiplication (4MD–2 and 4MD–3). Pupils have also implicitly used the compensation property of addition, for example, when partitioning two-digit numbers in different ways in year 2:       70 + 2 = 72       60 + 12 = 72. In year 6, pupils should learn the compensation property of addition.

Language focus “If one addend is increased and the other is decreased by the same amount, the sum stays the same.”

Pupils should be able to use the compensation property of addition to complete equations such as 25 + 35 = 27.5 +?, and to help them solve calculations such as 25.7 + 32.5. Similarly, pupils may have implicitly used the compensation property of multiplication, for example, when recognising connections between multiplication table facts: 5 × 8 = 10 × 4 In year 6, pupils should learn the compensation property of multiplication.

Language focus “If I multiply one factor by a number, I must divide the other factor by the same number for the product to stay the same.”

Pupils should be able to use the compensation property of multiplication to complete equations such as 0.3 ×  320 = 3 × ?, and to help them solve calculations such as 0.3 ×  320. Pupils have extensive experience about the effect on the product of  scaling one factor from 3NF–3, 4F–3 and 5NF–2, where they learnt to scale known number facts by 10, 100, one-tenth and one-hundredth. Now they can generalise.

Language focus “If I multiply one factor by a number, and keep the other factor the same, I must multiply the product by the same number.”

Pupils should practise combining their knowledge of arithmetic properties and relationships to solve problems such as the examples here and in the Example assessment questions below. Pupils should learn to write a series of written equations to justify their solutions. Being able to work fluently with related equations in this way will prepare pupils for manipulating algebraic equations in key stage 3 and writing proofs. Pupils can already apply place-value understanding to known multiplication facts to scale one factor, for example, 3 × 4 = 12 , so 3 × 40 = 120. Now they should extend this to scaling both factors, for example, 3 × 4 = 12, so , 30 × 40 = 120.