Want to know how to teach place value to KS2 Maths pupils? You’ve found the right place!

This post is part of our interventions bootcamp series: designed to support Year 5 and 6 teachers and SATs booster group leaders achieve age related expectations with pupils who need that extra intervention.

It’s 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.

Tracking back to different stages of understanding is examined, before giving strategies that can be used with whole classes, booster groups or alongside 1 to 1 interventions, such as those provided by Third Space Learning.

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#### Each post follows a similar structure:

- First you’ll diagnose where pupils 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.

#### In this post on place value, we help you solve problems such as:

- The pupil’s mental maths keeps on surprising me – they should know that!
- That answer can’t be right, they aren’t estimating or checking their results.
- Why do they keep making mistakes with their carrying/exchanging?

#### Diagnosis: what are pupils’ key misconceptions with place value?

**From the below pupils, find the best fit for those you are looking at. Then use this to guide where in the post to read for support:**

**Karen **is surprisingly under-confident in Maths. She seems fairly able in most areas but makes seemingly random errors. This is often due to unsecured mental methods or errors relating to manipulating the digits she has in their representation. For example, when calculating 406 + 397, she prefers a written method. When calculating mentally, she doesn’t see that she can exchange 3 ones from 406 to add to 397 to make her calculation 403 + 400. Stage: SECURING.

**Hamza **is bubbly and always tries hard. He can identify digits in large numbers and add or subtract multiples of that digit well as long as they don’t cross the boundary into the next digit. Hamza struggles when being asked to represent a number in multiple different ways or split the number up from its separate digits (e.g 4603 = 4000+600+3). He can order numbers, although occasionally makes “silly errors”. He frequently makes slips in carrying/exchanging when calculating written methods. Stage: EMERGING.

**Stuart **is able to recall to the point where he doesn’t like to write anything down as he prefers to do everything in his head. However, this causes him to make unnecessary errors when crossing boundaries, similarly to Hamza. When he is forced to use written methods, he makes errors as he has not secured using them – particularly with exchanging. He doesn’t check his answers to make sure they are reasonable. Stuart much prefers calculations he can ‘see’ rather than using and applying contexts, where he struggles to know what to do. Stage: DEVELOPING.

**Tommy **struggles with identifying which digit is a hundred, ten and one. He has difficulties when trying to add or subtract multiples of 10 to a number but can add/subtract a single ten including over the hundreds boundary. He can use concrete materials to support him exchanging and calculating written addition and subtraction but makes seemingly random errors that are difficult to pin-point. Using and applying contexts throw Tommy. Stage: PRE-BOOSTER.

*Tommy’s needs are beyond the scope of a booster group or this post. He needs specifically targeted 1:1 support from a trainer professional, such as that provided by Third Space Learning to catch him up to a point where booster intervention could be considered.*

For a more personalised diagnosis of pupils’ misconceptions in place value, download the free Place Value Diagnostic Quiz for years 5 and 6.

#### Misconceptions and strategies for ‘EMERGING’ in place value

##### Step-by-step – establishing a baseline

Start with bundles of straws – this may seem extreme but when pupils like Hamza are making errors with boundaries between ThHTU or struggle to represent numbers in different ways. We need to ensure they have secure foundations to their understanding of place value as it is likely fundamental a lack of understanding in this area is causing their errors and lack of progress.

- Have straws on their own (ones), bundles of ten (tens) and a few of bundles of a hundred (hundreds).
- Ask the pupils to make a number that you have written in front of them (e.g. 465) out of straws. Watch for any hesitation.
- Ask them others but without showing them the number in written form. This will check they truly understand the language, order and relative size abstractly. Pupils that struggle here are below the entry point for this booster.
- Assuming the pupils are able to do this, introduce zero as a place-holder, both written and only spoken, e.g. 503. Ensure the pupils are not representing the zero 3 or 5 tens (mixing their hundreds or ones).

##### Next steps – building understanding

- Initially with straws (even if just tens and ones) and then with base 10 equipment, ask the pupils to make and then write down numbers such as 241, representing each digit (241=200+40+1). This should not be a challenge.
- Then get the pupils to move a ten to their ones column. Ask them to re-write (241=200+30+11). Discuss with the pupils how it is possible to have the same number but have it represented in different ways, even with the tens in the ones column.
- Then build on this, asking the pupils to represent 241 in as many different ways as possible. This will allow you to assess who has a strong understanding and who doesn’t.
- Remove the concrete apparatus and see who can transfer their understanding to another number such as the 503 that has already been met. Using zero as a place holder should not be a barrier and will again establish pupils level of understanding (503=200+300+3).
- If the pupils struggle here, go back to using concrete apparatus and ask them to physically move their hundreds/tens/ones and recount to find the total until they fully understand they have the same number no matter how they represent the number.

##### Final steps – cement the learning

The above activities will have supported the pupils in their understanding of the relative size of each digit and they are then ready to eliminate those mistakes made when ordering numbers.

- Initially give pupils ThHTU numbers to order such as 4301, 4103, 4031, 1413, 1431. Let the pupils try and order, smallest to largest, whilst you watch without explaining how.
- Identify who is able to move through the digits (starting with thousands) until finding a digit that is smallest (here they would identify in the tens that 1413 is smaller than 1431). Allow them to continue even if they are making errors. Watch for pupils who think 4103 is the largest as it is likely the ‘see’ the 3 ones as larger than the 1 ones in 4301 and 4031.
- Where the pupils are still making errors at this size of number, use base 10 equipment to support them making each number to compare them. Introduce the idea that they can check each digit in turn starting with the digit worth the greatest value (thousands in the example).
- Continue until the pupils are secure. Then skip straight to large numbers including millions. Ask the pupils what they will have to do differently here. The answer should be very very little.

If the large number throws the pupils, they should just be shown to compare one digit at a time as they did with smaller numbers. If some pupils start to struggle without the base 10, any coloured counters could be used to represent each digit (e.g. millions = pink, hundred thousands = green etc). The pupils can then build using these and compare to consolidate their understanding.

When the pupils can confidently order large numbers and represent numbers in a range of ways, they are ready to move to the next stage of the booster – ‘DEVELOPING’.

#### Misconceptions and strategies for ‘DEVELOPING’ in place value

Begin with pupils like Hamza and Stuart, where they struggle with manipulating across boundaries mentally: for example 5369+1265. Guide the pupils through being mindful, looking for which mental method is most appropriate, as well as potential exchanging. The example most closely linked to place value understanding is partitioning.

The pupils should easily add 1000 to 5000 and 300 to 200. They should already have realised there would be exchanging in the tens because 6 tens + 6 tens = 10 or more tens. Additionally, they should have seen there would also be exchanging in the ones. The focus here should be on the pupils realising they will have to adjust in the hundreds due to the exchanging in the ones and tens.

##### Building on understanding

Explore this abstractly at first as the goal is for the pupils to access this understanding mentally. The mental process to model is:

- 5 + 1 thousands = 6 thousands
- 3 + 2 hundreds = 5 hundreds
- 6 + 6 tens = 12 tens
- Exchange 10 tens for 1 hundred making 6 hundreds and 2 tens
- 9+ 5 ones = 14 ones
- Exchange 10 ones for 1 ten making 3 tens and 4 ones
- 6 thousands, 5 hundreds, 3 tens, 4 ones
- 6634

It is likely that this is the level of complexity the pupils would struggle with due to the (double) exchanging. Some may prefer to do this with a written method.

Let the stage the child is at (Spring term Year 6 compared to Autumn Year 5) guide you as to whether to pursue developing understanding of place value in the context of mental methods or allow them to refine and become increasingly fluent with the written methods.

Where you decide to continue with mental methods, bring out base 10 and, for bigger numbers, place value counters/coloured counters. Use this equipment to model the exchanging process.

Tackling written methods could be done before, during or after the mental methods depending on the needs of the pupils in the booster. For more details on mental or written methods using concrete resources, see our Bootcamp Addition and Bootcamp Subtraction posts.

Once the pupils are more confident with written and mental calculations that require them to have good foundations in place value, they can be given simple using and applying contexts. Include in this mixes sized numbers to check the depth of their place value understanding. For example: Using all these 7 digits once, make a 3 digit and 4 digit number. What is the largest and smallest totals you could make?

To develop pupils like Stuart’s ability in using and applying, they need to be able to pick the right methods. Much of the time, it is this choice that deters or confuses them. Our blog post on ‘How to teach the bar model method in KS1 & KS2 Maths‘ might be a good place to start for addressing this issue. Once pupils are comfortable at this stage, it’s time to move on to the next step – ‘SECURING’.

#### Misconceptions and strategies for ‘SECURING’ in place value

Pupils like Karen are very common – fairly able in written methods but lacking secure enough foundations with recall and place value to continue to develop their mental strategies.

##### Developing fluency and mindfulness

A fluent written method is great and can enable these pupils to achieve. Developing their mental strategies so they are increasingly mindful (looking at the numbers in order to choose which strategy is best) will support both mental and written methods as it will enable them to make quick estimates as well as use the inverse to check their answers.

Using base ten and place value counters can demonstrate to pupils when rounding and adjusting should be used by showing them how close a number is to the next or previous hundred, for example:

- For 4396 + 441, pupils can see that 96 is near 100. They can represent the addition by moving 4 ones from 441 to 4396 to make 4400+437.
- When modelling this, watch for pupils who add the 4 hundreds in 437 to the 4 thousands in 4400. Their understanding of place value is hazy so go back to previous booster steps. Also watch and encourage pupils to estimate first.
- Taking our example, “4396 + 441… well that’s basically 4400 + 400, so my answer should be about 4800.” In other words, their mental strategies, particularly rounding, should be aiding their estimates.

In conclusion, where pupils make occasional errors, they either need to work on their understanding of place value that underpins the method they are using or they need to work on developing a different strategy. It depends on what will help that child with how much time you have left with them. But you can never go wrong with developing their security in place value.

**If your pupils have gaps that still need plugging in the run up to SATs, don’t panic! Read our blog on Year 6 Maths SATs preparation.**

**I****t has everything you need: free practice SATs papers, testing best practice, and guidance on making sure your pupils feel comfortable, straight from a primary headteacher.**