Tips and tricks for adding and subtracting numbers

In Years 1 and 2, children are expected to learn and master the following skills in relation to addition and subtraction:

• Understanding the concept of addition as combining two or more numbers to find a total.
• Developing fluency in adding and subtracting within 20.
• Practising mental strategies, such as counting on or counting back, to solve addition and subtraction problems.
• Introducing the use of concrete objects, pictorial representations and number lines to support addition and subtraction.

When children are at this early stage of developing their skills, you can help them by getting them to use physical objects. These can be cubes, counters or number lines; these help them to visualise the process of adding or taking away different quantities. Try to relate these mathematical concepts to everyday life, e.g., we have 5 sweets and we add 2 more – how many have we got altogether? Try to emphasise the mathematical words such as ‘add’ and ‘altogether’.

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You can also reinforce counting strategies such as counting on (e.g., 5 + 3 = start at 5, count 3 more) and counting back (e.g., 9 - 2 = start at 9, count back 2) to enhance mental arithmetic skills. It can also be helpful to say to children ‘Put 5 in your head and now let’s count on 3 using our fingers” to help them with counting on.

As children progress through Key Stage 1, they will begin to learn some more written methods for helping them solve addition and subtraction problems. Some of the written methods typically introduced during KS1 include:

• Number Line method: In early KS1, children are often introduced to the number line method, which involves representing numbers on a horizontal line and using jumps to visualise addition and subtraction. They can count forward for addition and count backward for subtraction, marking the jumps on the number line.
• Part-whole model: The part-whole model is another visual representation used in KS1. Children partition numbers into smaller parts or groups to understand the composition and relationship between numbers in addition and subtraction. This model helps them see the connection between the whole number, its parts and the operations involved.

• Bar model: As children progress in KS1, they may be introduced to the bar model. This is a relatively new written method which has become a popular approach since the introduction of Mastery approaches in maths. This method involves using rectangular bars or segments to represent quantities, with the length of each bar or segment proportional to the value being represented. Children can use this model to solve addition and subtraction problems by visually comparing the lengths of the bars or segments. For example, this bar shows how 6+2 equals 8. It can also represent how 6 – 2 = 9:

Towards the end of KS1, children may start to explore an expanded column method for addition and subtraction. This method involves breaking down numbers into place value units (ones, tens, hundreds) and arranging them in columns to perform calculations. It helps children understand the concept of regrouping or 'carrying' in addition and 'borrowing' in subtraction. Here is a simple example:

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As children move up into Key Stage 2 (Years 3 and 4), they will develop their skills in addition and subtraction in some of these ways:

• Reinforcing addition and subtraction skills with two- and three-digit numbers.
• Developing mental calculation strategies for addition and subtraction, including counting on or back in steps, using known number facts and applying inverse operations.
• Introducing formal written methods for addition and subtraction, such as column addition and subtraction with regrouping (exchanging).
• Applying addition and subtraction skills to solve real-life problems and mathematical puzzles.
• Developing reasoning skills to explain the strategies and methods used in addition and subtraction.

Useful skills to support your child at this age includes practising their number bonds. Number bonds are pairs of numbers that add up to a given sum. Practising number bond activities helps children develop fluency in addition and subtraction. For example, for the number 10, number bonds could be 7 + 3, 6 + 4, or 2 + 8.

It is also useful to learn and memorise fact families (related addition and subtraction facts) to promote a deeper understanding of the relationship between operations. For instance, if 5 + 3 = 8, then 3 + 5 = 8, and 8 - 3 = 5.

In terms of formal written methods, in Years 3 and 4, children consolidate their understanding of the expanded column method for addition and subtraction. As they become more proficient, they may progress to using the compact column method. This method is an abbreviated version of the expanded column method, where children align the digits vertically without explicitly writing the place value labels. It streamlines the process and emphasises efficient calculation. Here is an example of the compact method used to add two numbers together. This is probably very similar to how we remember being taught at school:

As they move up into their final years of primary school (Years 5 and 6), children will continue to work on more complex methods and problems involving addition and subtraction. Some of the skills they will learn include:

• Extending addition and subtraction skills to larger numbers, including four- and five-digit numbers.
• Strengthening mental calculation strategies for addition and subtraction, including rounding, compensation and using inverse operations.
• Further developing formal written methods for addition and subtraction, including long addition and subtraction with regrouping.
• Applying addition and subtraction skills to solve multi-step word problems, including those involving money, time and measures.
• Developing reasoning and justification skills by explaining the steps, choices and strategies used in addition and subtraction.

In Years 5 and 6, children continue to refine their skills in addition and subtraction and further expand their repertoire of written methods. Here are the written methods typically taught during this stage:

• Compact column method: The compact column method introduced in lower KS2 is reinforced and applied to more complex calculations involving larger numbers. Children should become more proficient in aligning digits and carrying/borrowing as needed.
• Expanded column method with decimals: In Years 5 and 6, children work with decimal numbers and extend their understanding of the expanded column method to include decimal addition and subtraction. They align the digits, including the decimal point, and perform calculations in a similar manner to whole numbers.
• Grid or lattice method: The grid or lattice method is a visual representation that supports multiplication and can also be adapted for addition. In this method, children create a grid or lattice with rows and columns and multiply (or add) numbers within the grid structure. This method allows for a systematic and organised approach to calculations. Here is an example of the grid method used for addition:

This can be a great way to help children remember addition facts.

• Checking strategies: As children progress, they are encouraged to develop checking strategies for addition and subtraction. They may use inverse operations, estimation or mental calculations to verify the accuracy of their results. Checking strategies promote mathematical reasoning and help students identify and correct errors.
• Mental calculation strategies: While written methods are important, children are also encouraged to further develop their mental calculation strategies for addition and subtraction. This includes using known number facts, applying efficient mental strategies and assessing the reasonableness of their answers. These are invaluable skills especially when they are faced with the Year 6 Arithmetic Paper during SATS.