# Manipulate the additive relationship: Understand the inverse relationship between addition and subtraction, and how both relate to the part–part–whole structure (1)

## Description

Pupils will begin year 3 with an understanding of some of the individual concepts covered in this criterion, and will already be familiar with using partitioning diagrams and bar models. However, pupils need to leave year 3 with a coherent understanding of the additive relationship, and how addition and subtraction equations relate to the various additive structures. Pupils must understand that the simplest addition and subtraction equations describe the relationship between 3 numbers, where one is a sum of the other two. They should understand that both addition and subtraction equations can be used to describe the same additive relationship. They should practise writing the full set of 8 equations that are represented by a given partitioning diagram or bar model.

Pupils should learn and use the correct names for the terms in addition and subtraction equations. addend + addend = sum     minuend – subtrahend = difference. Pupils understanding should go beyond the fact that addition and subtraction are inverse operations. They need understand how the terms in addition and subtraction equations are related to each other, and to the parts and whole within an additive relationship, and use this understanding to manipulate equations. With experience of the commutative property of addition, pupils can now learn that, because of the relationship between addition and subtraction, the commutative property has a related property for subtraction.

Language focus “If we swap the values of the subtrahend and difference, the minuend remains the same”

Both of the following equations are therefore correct: 37 – 25 = 12  37 – 12 = 25. Pupils should use their understanding of the additive relationship and how it is related to parts and a whole, the inverse relationship between addition and subtraction, and the commutative property, to manipulate equations. They must recognise that if 2 of the 3 numbers in a given additive relationship are known, the unknown number can always be determined: addition is used to find an unknown whole, while subtraction is used to find an unknown part, irrespective of how the problem is presented. For example,  34 + ? = 56 has an unknown part so is solved using subtraction, even though the problem iswritten as addition. Pupils need to be able to solve:

• missing-subtrahend problems ( minuend – ? = difference)
• missing-minuend problems (? – subtrahend = difference)

Pupils have been solving missing-number problems since year 1. However, with smaller numbers, they were able to rely on missing-number facts (known number bonds) or counting on (for example, counting on 3 fingers to get from 12 to 15 for 12 + ? = 15). Now that pupils are using numbers with up to three digits, they will need to rearrange missing-number equations to solve them, using formal written methods where necessary. Teachers should not assume that pupils will be able to do this automatically: pupils will need to spend time learning the properties of the additive relationship, and practising rearranging equations. They should be able to identify the type of each problem in terms of whether a part or the whole is unknown, and can sketch partitioning diagrams or bar models to help them do this.

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