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How to use previous SATs questions to help pupils consider links and patterns between numbers

It’s vital that we consider how we teach problem-solving strategies, says maths specialist Beth Smith

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As a Maths SLE, I have the opportunity to work with classes across a number of primary schools. One of the most common requests I have is how to help children to answer problem-solving questions.

In the EEF’s research into improving mathematics in KS2 and KS3 (2018), one of its eight recommendations is to teach pupils strategies for solving problems.

It then provides the following suggestions:

• If pupils lack a well-rehearsed and readily available method to solve a problem, they need to draw on problem-solving strategies to make sense of the unfamiliar situation.
• Teach them to use and compare different approaches.
• Show them how to interrogate and use their existing knowledge to solve problems.
• Use worked examples to enable pupils to analyse the use of different strategies.
• Require pupils to monitor, reflect on, and communicate their problem solving.

Let’s consider some of these suggestions in relation to questions taken from recent SATs and White Rose Maths assessments.

Firstly, bar modelling can be a brilliant tool to help children make sense of the information they are given. Take this question from the 2018 SATs:

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Amina is making designs with two different shapes. She gives each shape a value.

Calculate the value of each shape.

Encourage children to use a bar model to draw out what they know.

Notice the way the different colours of shapes are grouped together, helping children to see what is the same and what is different about the bars.

Once the starting bar model has been drawn, children can start to consider what else they can find out.

The first step in this model is to consider what the difference between the designs is and therefore what the value of the larger shape is.

The annotated bar model below shows how this can be modelled:

An important aspect of teaching problem-solving is to then provide children with another problem with a similar structure to practise their skills.

Providing the same problem with different numbers takes the problem-solving elements away and moves the question more towards fluency.

However, another problem with a similar structure but a different context allows children to refine their thinking.

Take this question from the Y6 White Rose Maths spring assessment:

The mass of a box containing 6 tins of beans is 7.5kg. When 2 tins of beans are removed, the mass of the box is 5.1kg. What is the mass of one tin of beans?

The situation is completely different. However, when placed into a bar model children can use similar skills to solve the problem.

Applying knowledge

Next, let’s consider how we can use problem-solving questions to support children to apply their knowledge and skills.

Many questions require children to calculate both mentally and through written methods, practising the skills they have acquired. In the question below, children add and subtract to find change in a money problem:

John buys one toy car (£1.49) and one pack of stickers (£1.64). How much change does John get?

Most commonly, children will use the following method:

£1.49 + £1.64 = £3.13
£10 - £3.13 = £6.87

Some pupils will be able to attempt this mentally. However, others will rely on written methods and this can bring the possibility of errors, especially when calculating £10.00 - £3.13.

Children have to complete multiple exchanges, which may lead to errors.

When subtracting, we can apply the idea of constant difference. If we add or subtract the same amount to both numbers in the subtraction, the difference will remain the same.

Subtracting one penny off both amounts (£10 and £3.13) leads to a much easier subtraction without any exchanging.

£1.49 + £1.64 = £3.13
£9.99 - £3.12 = £6.87

Consider how this method can be applied to the following questions:

Ken is playing a game. He has 4,289 points. Then he scores another 355 points. Ken’s target is 6,000 points. How many more points does Ken need to reach his target? (2019 KS2 SATs)

Morgan is running a 10 kilometre race. So far, she has run 1,340 metres. How far does she have left to run? (White Rose Maths Y6 summer)

Finally, let’s consider how we can use and compare different approaches.

The below question from the 2019 SATs test is, on the surface, a problem requiring a number of steps involving multiplication and addition:

Layla makes jewellery to sell at a school fair. Each bracelet has 53 beads. She makes 68 bracelets. Each necklace has 105 beads. She makes 34 necklaces. How many beads does Layla use altogether?

Many children would approach this problem by using the numbers they are given and calculating in three steps:

The second method, below, shows a link between the calculations.

If we notice the relationship between 34 and 68 in the question, we can use that to help us with our calculations.

Remember, when multiplying, if we half one number and double the other number, the product remains the same:

A third method uses the idea from method two but takes it a step further, using the idea that 105 x 34 + 106 x 34 is equal to 211 x 34.

Interestingly, this is the simplest multiplication to complete, with the least exchanges:

Number sense

Comparing the methods highlights the need for number sense.

Instead of diving straight into written methods, children should look at the numbers they are using and consider if they can see any links or patterns.

Sometimes there won’t be any there, but when the links are there, it can support with calculating more efficiently.

Consider how you could use this idea when answering the question below, taken from the 2018 SATs:

Ken buys 3 large boxes and 2 small boxes of chocolates. Each large box has 48 chocolates. Each small box has 24 chocolates. How many chocolates did Ken buy altogether?

In conclusion, it’s vital that we consider how we teach problem-solving strategies.

Highlighting different methods can lead to mathematical discussion and the chance to unpick the structure of a problem.

Bar modelling can represent the problem and support children with what operations they need to use to solve the problem, giving them the starting point they need.

The first step, as with anything, is to give it a go.

Hopefully, problem-solving will become a little less daunting and a little more engaging and enjoyable.

Beth Smith is senior primary maths specialist for White Rose Maths.