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# Calculation methods and mathematical models Some of the methods and models used in primary maths lessons are a bit different from what you may have used at school. The maths techniques and methods children are taught in schools now are based on giving them a deep understanding of mathematics and helping them to articulate that through explaining, discussing their work with each other and involving them in solving problems that apply to everyday life.

Different schools and different teachers will choose to introduce children to different methods, models and images. A good place to start is your school’s website, where you are likely to find their calculation policy (for schools in England, this is statutory). This will show you the calculation strategies that will be used at your child’s school. Lots of schools run maths information evenings for parents, which will give you an insight into the methods and models used.

We’ve put together some information about some of the methods and models your children may be introduced to in their maths lessons.

### Number lines

A number line is a continuous line representing numbers. It is a visual image which will help children to understand our number system.

It can be used from age 4 upwards, and helps teach addition and subtraction and reinforce numbers and counting.

At 4–5 years old, number lines are used to reinforce the order of numbers up to 20, and to help teach simple addition and subtraction. At 5–6 years old, number lines are used to help children count on or back in 1s or 10s when solving additions and subtractions. At 6–7 years old, children will have had a good deal of experience and practice using number lines and will have a mental image of numbers to 50. Teachers will start to use unstructured lines (i.e. not all the numbers are marked and labelled) to support counting and calculations (this will of course depend on the child’s attainment level). At 7–8 years old, children will use both a structured number line and an ‘empty’ one. The structured number line is an important image to help children develop mental methods for addition and subtraction. ### Number squares

A number square is a grid of sequential numbers. For example, a 1–100 square is very common: Number squares can help with:

• Counting: when children are very young, a number square (e.g. 1–20) can help to reinforce the order of the numbers, and help children see the patterns in numbers in the same column, e.g. 7 and 17.
• Addition and subtraction: a 1–100 square is great for supporting addition and subtraction, as it really helps them to understand what is happening when they add / subtract 1 or 10. E.g. if they were asked to solve 42 + 25, they could put their finger on 42, then move it 2 spaces down the grid (to count on 2 tens), landing on 62, and then 5 spaces to the right (to count on 5 ones), ending up at 67.
• Multiplication: young children need to learn to count in 2s, 5s and 10s, and a 1–100 square is very helpful to support this. The squares in the count (e.g. 5, 10, 15, 20, 25, etc.) can be coloured in, to help children see the patterns. The same thing can be done with slightly older children, when they are learning the other times tables.

### Arrays

An array is objects or shapes arranged in rows and columns. Arrays are very important in helping children to understand how multiplication works. This is a great example of a way to use real objects or pictures to help children see what is actually happening when they multiply. A child can count the objects, but this can be tiresome and inaccurate. If they can count in 2s or know their 2 times table, they can calculate easily.

They learn that in the array the answer is always the same (e.g. 2 × 7 is the same as 7 × 2) even when you rotate the array.

Egg boxes and muffin trays are real-life examples of arrays.

### Grid method

By the age of 10 or 11, children will be expected to use formal written methods for multiplication (such as long multiplication and short multiplication), particularly in tests. The grid method is sometimes used as an interim step between working with arrays, and moving on to formal methods. It involves partitioning numbers into tens and ones before they are multiplied, so it really supports children in understanding what is happening at each step.

Not all schools use the grid method, so check with your child’s teacher if you are unsure.

For example: 7 × 35 Draw a grid with two rows and three columns. Fill in the first row and column like this (partition 35 into 30 and 5): Multiply each pair of numbers (7 × 30 and 7 × 5) and write the products (i.e. the answers to the multiplications) in the blank spaces in the grid. Then add the two products to find the answer. The grid method can also be used for more difficult multiplications, such as 35 × 26. To be able to use the grid method efficiently, children should be confident with their times tables up to 10 × 10 and have a good understanding of place value.

If children are not familiar enough with their times tables to complete the grid in this way, the numbers can be broken down further, e.g. 35 can be broken into 10, 10, 10, 5; 26 can be broken into 10, 10, 6. A multiplication grid up to the 10 times table can be used as a ready reckoner. It is also useful for investigating patterns in numbers.

Methods such as this are important for helping children understand that calculations can be built up to reach a total and set out in different ways. It helps to develop their understanding of more complex calculations and it's also a stage in the progression towards long multiplication.

### Chunking

By the age of 10 or 11, children will be expected to use formal written methods for division (such as long division), particularly in tests. Chunking is sometimes used before long division is introduced, to help children’s conceptual understanding. Although younger children can be taught the long division method, and they can learn to find the right answer, it is better if they actually understand where the answer comes from rather than just following a set of steps – and that’s where chunking can help.

Not all schools use the chunking method, so check with your child’s teacher if you are unsure.

Imagine you were given this problem:

You are holding a party for 25 children. You have 830 sweets to share equally between the party bags. How many sweets should you put in each bag? Will any be left over? That might seem pretty tough (without using a calculator). With chunking, you can just start dividing the sweets between the party bags in ‘chunks’, making sure you don’t exceed the total.

You could do it like this: ### Compensation

Compensation is one of several efficient written methods for addition of larger numbers, where one of the numbers is close to a ‘friendly’ number (i.e. multiple of 10 / 100 / 1000). It involves adding too much and then taking off the extra that you have added.

At 8–9 years old:

For example: 744 + 86
Round 86 to the nearest 100: 86 rounds up to 100
744 + 100 = 844
We have added 14 too many (100 – 86 = 14) so we must take it away
844 – 14 = 830
744 + 86 = 830

At 9–10 years old

For example: 654 + 296
Round 296 to the nearest 100: 296 rounds up to 300
654 + 300 = 954
Take away the extra 4 (300 – 296 = 4)
954 – 4 = 950

At 10–11 years old

For example: 5377 + 2974
Round 2974 to the nearest 1000: 2974 rounds up to 3000
5377 + 3000 = 8377
Take away the extra 26 (3000 – 2974 = 26)
8377 – 26 = 8351

### Tens frames

Tens frames are two-by-five grids which teachers sometimes use alongside coloured counters to help children to form mental pictures of numbers up to 10. E.g. this arrangement could be used to demonstrate the number 7. Children can see that 7 is 5 plus 2 more. Tens frames are very useful for supporting simple addition and subtraction. E.g. if children were asked to solve 7 + __ = 10, they could use red counters to represent the 7, then yellow counters to represent what they need to add to make 10. The arrangement above could be used to show four different calculations:
7 + 3 = 10
3 + 7 = 10
10 – 7 = 3
10 – 3 = 7

It helps children to really see and understand the relationship between these four number facts.

Two tens frames can be used alongside each other to help with understanding place value in 2-digit numbers. E.g. children can see that 17 is made up of 10 and then 7 more.

### Part-whole model

Part-whole models can help children to see numbers as being made up of two or more parts. E.g. a teacher might draw this simple model to show one way to partition the number 10 into two parts: The circle on the right is the ‘whole’, and the two circles on the left are the ‘parts’ that add together to make the whole. The teacher might use dots or circles in some or all of the circles to represent the numbers, or they might write the numerals.

This model can support children with solving simple addition and subtraction problems. E.g.

You have 10 grapes. Your friend gives you 5 more. How many do you have in total? Children might draw a part-whole model with two parts. They might draw 10 dots in the first circle and 5 in the second. They could then add up the total number of dots and write the answer in the ‘whole’ circle. As with the tens frame, the purpose of the part-whole model is to help children to ‘see’ the numbers and the relationships between them. This helps with their conceptual understanding, which will give them a firm foundation to work from when they go on to tackle more challenging problems.

### Bar model

Bar modelling is a great way to help children to visualise an abstract maths problem, enabling them to easily see the relationships between numbers and operations.

The bar model shows the relationship between addition and subtraction – both can be seen within one representation. It really helps children to see the link between the two operations.

For example, the following can be used to show four different calculations: 2 + 8 = 10, 8 + 2 = 10, 10 – 2 = 8, 10 – 8 = 2. The top bar represents the ‘whole’, and the bottom bar represents the ‘parts’ that make up the whole. (The bottom bar can be divided into more than two parts.)

For younger children, it might take them a while to understand the structure behind the bar model, and they will need to start off by using physical objects before they progress to using a version of the bar model as shown above.

For example:
If you have 8 toy cars, and you give 3 to your sister, how many cars do you have left?

Stage 1 might be to use real toy cars. Line up 8 cars in a row. Then line up 3 cars below the first row, with the cars in each row lining up with each other. The child could find the answer to the problem by counting how many cars in the first row don’t have a ‘partner’ in the second row (5).

Stage 2 might be to draw it, using squares to represent the cars. Stage 3 might be to draw this as ‘bars’, but with the divisions shown. And the final stage would be to draw the bars, with no divisions. Multiplication and division
The bar model can also be used for multiplication and division problems.

Multiplication: There are 6 eggs in a box. How many eggs are in 3 boxes? Division: I have a piece of wood 60cm long. If I cut it into 4 equal pieces, how long is each piece? Like with addition and subtraction, the bar model helps children to see the relationship between multiplication and division.

Reviewed: July 2020

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