How to teach subtraction for KS2 interventions - Maths Year 5 & Year 6 Bootcamp 
Make the best use of your Key Stage 2 Maths interventions to help KS2 pupils reach age related expectations for SATs with differentiated teaching activities and ideas to tackle all Year 5/Year 6 misconceptions such as partitioning, use of the column method, and how to exchange.
This is the second post in our new KS2 Bootcamp series for Year 5 and Year 6 teachers on common Maths topics and misconceptions such as addition, subtraction and fractions.
They're designed to help Maths coordinators, intervention leads and SATS booster group leaders to support their Year 5 and Year 6 pupils achieve their national curriculum age related expectations, and make good progress in their final years of Key Stage 2.
The differentiated guidance is particularly aimed at those running interventions in school but is also relevant much more widely to your whole class teaching, supporting a mastery approach, to help you ensure each child is getting appropriate support and challenge.
Each post follows a similar structure:
First you’ll diagnose where children are struggling with ‘the nuts and bolts’ of that area of Mathematics from the National Curriculum.
Next you’ll track back to the different stages of understanding and examine what the misconception might be in detail.
Finally we’ll give you strategies that can be used with whole classes, booster groups or alongside 1 to 1 interventions.
Solve addition problems such as:
They just don’t understand subtraction like they do with addition…why?!
Why do they keep making mistakes with their exchanging?
How do I stop them switching around the numbers they are subtracting?
Diagnosis of misconceptions in Subtraction
From the below children, find the best fit for those you are looking at. Then use this to guide where in the post to read for support:
Karah likes a chat and can reason well during whole class discussions. However, she struggles during independent sessions and her mind wanders. She knows some number facts to 10 but her recall is a little sketchy. She enjoys the formal written method of addition and usually calculates correctly with one carry/exchange. In tests, she tends to only use written methods when shown an arithmetic calculation rather than a problem. When tackling written subtraction she frequently switches numbers (e.g. in 471 – 328, for the ones she will calculate 8-1 rather than 1 – 8 and exchange). Stage: EMERGING
Tom loves written addition and he will often choose it as his preferred method. Even when attempting what should be a subtraction word problem! During discrete sessions of written subtraction he sometimes calculates correctly but makes mistakes with exchanging, particularly when having to exchanging over multiple columns such as in 407 – 138. Stage: DEVELOPING
Oliver has extremely quick recall of addition and subtraction facts to 20. He often works things out mentally quicker than his teacher. He does, however, overuse his mental strength when he should use a written method which leads to unnecessary errors. He also sometimes chooses the incorrect type of calculation to solve problems. Stage: SECURING
Aaliyah struggles with remembering number facts to 10, particularly subtraction. She doesn’t yet fully understand what ‘subtraction being the inverse of addition’ actually means. Aaliyah doesn’t offer suggestions in class and often appears to be playing catchup with what is being learnt. She learns through repetitive practice but can’t apply it in context. She is used to having a teaching assistant support children on her table and is over reliant on the support. Stage: PRE-BOOSTER Aaliyah’s needs are beyond the scope of a booster group or this post. She needs specifically targeted 1-to-1 support from a trained professional to catch her up to a point where a booster intervention could be considered. (If you are looking for 1-to-1 support, then read this review I wrote of Third Space Learning's 1-to-1 Maths programme.)
For a more personalised diagnosis of a pupil’s misconceptions in addition, download the free Addition and Subtraction Diagnostic Quiz for Years 5 and 6.
Misconceptions and strategies for EMERGING in Subtraction
For a child like Karah, although she is not fluent with number facts to 20, the focus of the booster should be on developing understanding of how to use the written method of subtraction.
What about their bonds and number facts?
Some aspects of a pupil’s Maths fluency don’t need to be taught through resource intensive boosters. Practice of number facts can be done just as well, if not better, using online maths games or even worksheets. Additional time outside of English and Maths could be given for children not secure on facts to repetitively practise using worksheets or online packages and apps such as Numbergym Bond Builder or Mathsframe.
Step-by-step exchanging using straws
Go back to the beginning of exchanging by using straws to demonstrate and let children practice the process (most misconceptions in subtraction come from a lack of understanding of the exchanging process).
For example, in 53 - 12, have straws in 5 bundles of ten with elastic bands to group and 3 on their own.
It is vital that children understand that unlike addition, where they are building both numbers, subtraction is the process of removing an amount from another. Therefore, children only build the amount they ‘have’ (the minuend number) in preparation to remove an amount (the subtrahend number).
In the example, children should remove 2 of the 3 individual straws. They should then remove 1 of the bundles of ten (taking 12 away in total).
Give children a short amount of time to do the same with a few calculations just with tens and ones. Straws can then be revisted with children who are not fully understanding exchanging with base 10 equipment.
Step-by-step base 10 method
Now swap out straws for base 10, making sure children understand that the long ‘tens’ are the same as a bundle of straws.
At this stage, get the children to layout their base ten on a HTU place value frame (which can be drawn on whiteboards).
Quickly after this, re-introduce the formal written addition layout whilst using the base 10 on place value grids alongside it.
Try by working with the calculation 251 - 125.
When children need to exchange, as with this example, time should be taken. Deep understanding at this stage will enable faster progress later on. Once it has been established that there are not enough ones to take 5 of them away, children should ‘swap’ (exchange) one of the tens to 10 ones. It is recommended that children put the ten to be swapped in an ‘exchange’ area as it is common for children to think they are taking this ten away. They can then get 10 ones and also put those in the exchange area. Then the ten can be removed. It is important to emphasis to the children that the amount now in the exchange area is still ten and the number they have is still 251, it has just been made into 200 + 40 + 11.
The children should then remove 2 tens from the remaining 4 tens, followed by removing 1 hundred from the 2 hundreds.
If issues persist, as mentioned previously, return to using straws to model and practice the exchanging process.
When children can confidently exchange once in a calculation using base ten, without error, they are ready for the next stage of the booster – ‘Developing’.
Misconceptions and strategies for DEVELOPING in Subtraction
Children like Tom, who like to use written methods but still make frequent mistakes should be working towards eliminating those errors by tackling misconceptions and multiple exchanges. They can then begin applying their increased fluency in word problem contexts.
Step by step from base 10 to worded problems
Begin by using base 10 equipment on place value charts.
Give calculations that involve two exchanges such as 435 − 269. Get the children to look at the calculation before they start. Nudge them into noticing the double exchange. This will support their understanding as they go through the calculation. Children go through the exchanging process for the 3 − 9 ones.
Next, when faced with the 2 remaining tens, discuss what is different when subtracting 6 tens from the 2 remain tens. IE. Nothing. The same process is followed.
Children then complete the calculation, subtracting from the remaining hundreds.
Introduce the written method alongside future calculations with base 10 equipment or place value counters on place value charts. When children are confident with double exchanges, move to step 3.
Still with concrete resources alongside the formal method, introduce children to exchanges across digits with 0 in. For example, 305 − 237. By now the children should be proficient at instantly noticing they will need to exchange because of 5 − 7 and 0 tens − 3 tens. Allow them to begin calculating 5 − 7. Watch closely when they realise they can’t exchange from the tens as there are none.
Allow them time to think and discuss as the more they can solve this themselves using their prior knowledge, the better they will do when calculating these situations independently. Using a mix of yours and children’s explanations, model how we must exchange the hundred with 10 tens and then we can exchange one of those tens with 10 ones. Our calculation then becomes 2 hundreds, 9 tens, fifteen ones. We can now subtract 7 ones away. The rest of the calculation should be straightforward as long as the children are correctly following the written method alongside the concrete resources:
It is important the children begin to be more mindful, paying attention to potential additional exchanges once the calculation is in process. For example, in the calculation 6,149 – 1,162 there appears to be only 1 exchange (4 tens – 6 tens) but after exchanging a hundred so there are enough tens to subtraction, there would be 0 hundreds left to subtract 1 hundred from.
From this point, the children need repetitive practice in different contexts whether straightforward calculations, simple worded problems, or measure contexts. When calculating these the children should focus on sharpening their written method layout and method. This will aid avoiding errors. The teacher should continuously remind children during this stage of the links between their exchanging marks in the written method and process undertaken with concrete resources. The children should become less and less dependent on using concrete resources during this stage.
If children struggle to see the addition word problems and your children are familiar with bar models, get them to speedily draw one in order to visualise the problem. Once children like Tom are able to calculate with a good level of fluency and accuracy, they are ready to secure their understanding of subtraction.
Misconceptions and Strategies for SECURING in Subtraction
The kind of booster that a child like Oliver needs, is less step by step and rather just support to enable him to make good choices regarding when to use the written method of addition and apply this to different contexts such as word problems and missing numbers
A simple and key strategy at this stage is to present calculations horizontally rather than in the vertical written method form. Many children will see this as an instruction to calculate mentally, particularly if they are strong in mental arithmetic. Where this occurs, encourage the children to look at the columns (potentially re-writing vertically) to look for exchanging.
Train the children to be mindful as has occurred in ‘Developing’ and look for the amount of exchanges to guide them on whether they can calculate mentally or with the written method. Nudge them when they ‘they are at their capacity’ when calculating mentally to switch to a written method rather than persist, likely making unnecessary errors.
Where children refuse to switch to a written method, despite making errors mentally, get them to calculate mentally and then do the written method. It is likely that with more complex and large calculations, they will make more errors with the mental method. Once they see this, they will be more likely to accept the need to sometimes use the written method.
Introducing finding the difference calculations
From Year 1 onwards children will be given problems with subtraction ‘finding the difference’ rather than ‘taking away’ but children often struggle to see these problems as subtraction and even more so as finding the difference. If this is the case for children like Oliver, give them a range of finding the difference problems they may face. E.g. involving bar charts, worded problems using language such as ‘How many more’ or ‘How many left’ and anything else that leads to comparisons between amounts. At this late stage it is more time efficient to quickly recap what finding the difference actually means and give plenty of examples. A suggested recap is to draw this part part whole model:
Using a bar model or a number line, give the children a simple finding the difference calculation such as ‘In this survey, how much more popular was Coco-Pops compared to Rice Crispies?’ Explain that whenever you are comparing, you will be finding the difference and when you do that, you do a subtraction calculation.
Introduce a range of contexts, units and sizes
The children should be given a range of contexts for their calculations such as the use of distance, mass and capacity where unit conversions can also be brought in to add a layer of complexity. Encourage children to make jottings to the side when they are converting.
If the introduction of different units throws the children, ensure you are giving them bare practice calculations such as 7245m - 3686m at the same level they have been practicing before introducing word problems or unit conversions.
Money introduces decimals and is a further stage. For some children, they may need to use plastic or real money on a place value frame. If so, use the strategies discussed earlier but in a money context to enable their understanding.
To aid understanding that the decimal place is fixed, get them used to drawing the decimal place in the formal written method answer box before they have begun answering it. This is the same in a money context or with any larger decimals:
You could also try the free Place Value Concertina resource on these children to clarify the position of the decimal point - it’s a resource that proves very effective with all age groups.
Next steps for Subtraction - a free resource pack
Hopefully this has given you some useful points to work on with your KS2 pupils. To support your boosters, we’ve put together a KS2 Subtraction resource pack of some key Subtraction lessons, activities and practice questions that we use in our own one-to-one interventions.
And if your pupils need further support in any topic from the KS2 Mathematics national curriculum, book a demo or our Year 3, 4, 5 Catch-up or Year 6 SATs Foundation to find out more about the 1-to-1 interventions Third Space Learning offer. We’ve just started signing up schools for our September programmes.
Third Space Learning's KS2 Bootcamp series of blog posts on common Maths topics and misconceptions to teach for intervention include: