A copy of this booklet is available on‐line on the school website

( www.fairfieldinfant.co.uk or www.colneisjunior.co.uk )

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At Fairfield Infant and Colneis Junior Schools, children receive a daily maths lesson. As a basis for planning, the staff use the government curriculum guidance which outlines what is expected for children from Reception to Year 6.

The purpose of this booklet is to outline the various calculation methods that the children are taught as they progress through school, many of which look different to the methods that you may have been taught in your primary school days.

As children progress through the school, they are building a bank of strategies that can be ap-plied when appropriate. Each strategy can be refined or extended to suit the calculation needed. We hope the explanations and examples of strategies will help you to assist your child at home.

Some parts of the booklet refer to the use of ‘Numicon’ and ‘Dienes’ . These are resources used in class to help with counting, demonstrating number bonds, understanding place value and cal-culating.

Also included in the booklet are various ideas and suggestions for maths activities that you can enjoy doing with your child in the world away from school. It is not an exhaustive list and you will doubtless have many more ideas of your own.

Introduction

Numicon equipment

Thousand Unit Hundred Ten

Dienes equipment

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Calculations

A lot of emphasis in Numeracy teaching is placed on using mental calculations and, where pos-sible, using jottings to help support thinking. As children progress through the school and are taught more formal written methods, they are still encouraged to think about mental strategies they could use first and to only use written methods for those calculations they cannot solve in their heads. It is important that children are secure with number bonds (adding numbers to-gether and subtracting them, e.g. 6 + 4 = 10; 10 - 6 = 4; 13 + 7 = 20; 20 - 7 = 13, etc. and can remember them). They also need a good understanding of place value (tens and units, etc.) be-fore embarking on formal written methods.

Each numeracy lesson generally starts with a mental maths session, referred to as the oral and mental starter. The children are introduced to different methods of calculating numbers in their head. Number cards, hundred squares, number lines and other apparatus may also be used during this session. These resources are available for you to print and use on the school web-site : Parents section (www.fairfieldinfant.co.uk and www.colneisjunior.co.uk ).

Mental maths plays an important role in the end of Year 6 assessments. Practice is important and skills are built throughout the school right from the Foundation years in Nursery and Recep-tion.

Discussing the efficiency and suitability of different strategies is an important part of maths les-sons. Explaining strategies and processes orally helps to develop the use of appropriate mathe-matical vocabulary.

When faced with a calculation problem, encourage your child to ask:

• Can I do this in my head? • Could I do this in my head using drawings and / or jottings to help me? • Do I need to use a written method? • Should I use a calculator? (This is not encouraged in the earlier years until we are

sure the children have the essential strategies in place.)

Also help your child to estimate and then check the answer. Encourage them to ask:

• Is the answer sensible?

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The use of number lines

Number lines are a very important tool used in many calculations. Children are introduced to them as soon as their knowledge and understanding of numbers is sufficient for them to be able to find them a useful resource.

Number lines can take many forms and are used in a huge variety of ways to help develop chil-dren’s understanding of numbers. Children make jumps up and down a number line to help them solve a mathematical problem.

Classrooms use number lines of various types, appropriate to the age group, including 0-10; 0-50 and number lines incorporating negative numbers, e.g.

As children progress through the school, they are also taught the value of drawing a blank num-ber line that can accommodate relevant numbers to solve calculations.

E.g. finding change from 50p after spending 36p

The number line is used by counting on from 36p to get to 50p, just the way a shopkeeper would give change.

We also show children how number lines are part of everyday life, e.g. on measuring jugs, clocks, height charts, etc.

0 2 3 4 5 6 7 8 9 10 1

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Addition

Children are taught to understand addition as combining sets and counting on, and encouraged to use the language of ‘more than’, ‘count on’, ‘add’, ‘plus’ and ‘altogether’. Calculations are put into practical contexts so that children see the relevance of the method they are learning. The methods below are a progression from Reception to Year 6. Children should fully understand the previous method before moving on. They should not just jump to the last method if they have not understood the other methods.

2 + 3 =

At a party, I eat two cakes and my friend eats three. How many cakes did we eat altogether?

Children could draw a picture to help them work out the answer or use practical equipment (i.e. a counter to represent each cake, Numicon to represent each group) to model the problem.

When counting objects, children are encour-aged to move each object as they count it to reinforce their understanding of 1:1 correspon-dence and reduce the chance of miscounting.

6 + 5 =

Six people are on the bus. Five more people get on at the next stop. How many people are on the bus now?

or

I I I I I I I I I

Children could draw dots or tally marks to represent objects.

Dominoes and dice are also useful resources when practising these skills.

5 + 3 =

What is the total of the numbers on these two dice?

3 + 5 = 8

5 + 3 = 8

Children can count along a number line, mak-ing ‘jumps’ to reach the answer. They can also see that the addition can be done in any order, developing awareness that it is more efficient to put the larger number first.

They are also taught to use a 100 square, counting on in jumps of 1s, 2s, 5s, 10s.

12 + 9 =

12 birds are sitting on the grass. 9 more fly in to join them. How many are there altogether?

Numbers greater than 10 can be worked by holding the larger number in their head and counting on, using fingers. (Children are en-couraged to touch the side of their head as they ‘put the number in their head’.)

They are also introduced to the concept of using an empty number line (not numbered) to count on from the bigger number.

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47 + 25 =

My sunflower is 47cm tall. My friend’s is 25cm taller. How tall is my friend’s sunflower?

Drawing an empty number line helps children to record the steps they have taken in a calcula-tion. Start on 47, then + 20, then + 5. This is more efficient than counting on in ones.

Empty number lines can be used with numbers of any size.

87 + 64 =

One shelf measures 87cm and another meas-ures 64cm. What is their total length in centi-metres and metres?

80 + 60 = 140 7 + 4 = 11 151cm (or 1.51m)

By partitioning (splitting) both numbers into tens and units, each part can be added separately and then the answers combined to give the to-tal. This is the method most frequently used un-til at least Year 5.

487 + 546 =

There are 487 boys and 546 girls in a school. How many children are there altogether?

546 546 487 or 487 900 13 120 120 13 900

1033 1033

Children are taught written methods for those calculations they cannot do in their heads. Ex-panded methods build on mental methods and make the value of the digits clear to the chil-dren. The language used is very important, i.e. 500 + 400, 40 + 80, 6 + 7 and then 900 + 120 + 13 OR starting with the units, tens and then hundreds.

Children are taught the importance of placing digits with the same value underneath each other in clear columns.

3692 + 1435 =

Records show that 3692 people visited the ex-hibition in the morning and 1435 in the after-noon. What was the total number of tickets sold that day?

3692 1435 5127 1 1

Column addition starts with adding units and carrying between the columns. It can be used for adding larger numbers, adding more than two numbers and adding decimals once the children understand the importance of the value of each column and the relationship between them.

16.54 + 3.6 =

I need two pieces of fabric, one 16.54m and one 3.6m. How much fabric do I need to buy altogether?

16.54 3.6 20.14 1 1

When using column addition for adding deci-mals, children are also reminded of the impor-tance of lining up the decimal points in order to maintain accurate place value throughout their calculation.

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Subtraction

Children are taught to understand subtraction as taking away (counting back) and finding the difference (counting on / up). They are encouraged to use the language of ‘less than’, ‘count back’, ‘minus’ and ‘left’ (as in ‘How many are left?’). Calculations are put into practical contexts so that the child sees the relevance of the method they are learning.

5 - 2 =

I had five balloons. Two burst. How many do I have left?

(Take away)

Drawing a picture helps children to visualise the problem.

The use of practical equipment such as pence, cubes, counters and Dienes materials, also helps to model the problem for the children.

8 - 3 =

We baked eight biscuits. I ate three. How many were left?

I I I I I I I I (Take away)

Lisa has eight felt tip pens and Tim has three. How many more does Lisa have?

1 2 3 4 5 (Find the difference)

Using a line or dot to represent each object is quicker than drawing a picture.

To find the difference the children match the sets one to one and then count on from 3 (the lower number) to ‘find the difference’, i.e. 5

9 - 5 =

I had nine pence. I spent five pence. How much do I have left?

The children are first introduced to a numbered number line for either jumping on or back. They are taught to count the number of jumps along the line not the numbers, e.g. when tak-ing away they start on 9 and jump back from 9.

It is used for counting back or jumping back. (Take away)

It can also be used for counting on. (Find the difference)

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50 - 36 =

I spent 36p in a shop. How much change did I get from 50p?

36p + ____p = 50p

(Counting on )

Counting on using an empty number line is par-ticularly useful in calculating change.

84 - 27 =

I cut 27cm off a ribbon measuring 84cm. How much is left?

( Counting back)

Children can count back using an empty number line. This is a good way to record the steps they have taken and shows their understanding of how numbers can be partitioned (split) to make a calculation easier,

e.g. start on 84, then -20, then -7 and 57cm is left.

834 - 378

The library owns 834 books. 378 are out on loan. How many are left on the shelves?

i.e. 378 + _____ = 834

400 34 434 20 22 2 456 456

Children can count up from the smaller number to the bigger using an empty number line. It is easiest to count up to a multiple of 10 or 100,

e.g. start on 378 then + 2, + 20, + 400, + 34

Alternatively, as they grow in confidence and understanding they can use fewer jumps (preferably just two). They count up to the next multiple of 100 and then add on the remainder to find the difference,

e.g. start on 378 then + 22, + 434

The jumps can be added together mentally or recorded horizontally starting with the largest number first (e.g. 400 + 34 + 20 + 2 = 456).

Alternatively it can be written as a column addi-tion, making sure that the digits of the same value are always underneath each other.

9 - 5 =

I had nine pence. I spent five pence. How much do I have left?

Once the children understand how to use a numbered number line, they are taught how to use an empty number line (unnumbered)

So, given the same problem they would either count the number of jumps back for take away or count on the number of jumps to find the dif-ference between the two numbers.

Using this method they just need to draw a line and start from one of the numbers in the calcula-tion.

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2006 - 1998 =

Sarah was born in 2006 and Mark in 1998. How much older is Mark than Sarah?

Using an empty number line and the counting on method is particularly helpful when numbers are actually quite close to each other but cross a tens, hundreds or thousands barrier and so look harder than they actually are.

753 - 86 =

753 cars are waiting to load onto the car ferry. 86 drive on. How many are still waiting?

753 = 700 + 50 + 3 86 = 80 + 6 = 700 + 40 + 13 7413 80 + 6 8 6 = 600 + 140 + 13 61413 80 + 6 8 6 600 + 60 + 7 6 6 7

Children progress onto subtraction using ‘decomposition’ where there are fewer units, tens, and/or hundreds, etc. in the larger num-ber.

The use of the expanded written method illus-trated here helps them to understand the proc-ess and they then move onto the more compact standard written method for ‘decomposition’.

Starting with the units, 3 - 6 we can’t do, so we carry over a ten to make 13 units leaving 4 tens. 13 - 6 equals 7 units.

Moving onto the tens column, 4 tens subtract 8 tens we can’t do, so we carry over a hundred to make 14 tens leaving 6 hundreds. Now 14 tens subtract 8 tens equals 6 tens.

Finally, the hundreds column, 6 hundreds sub-tract nothing or zero equals 6 hundreds.

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Multiplication

Times tables

A good knowledge and quick recall of times tables is essential to children’s mathematical pro-gress. The children are taught up to 12 x 12. The target is for all children to know their tables by the end of Year 4. It is very important that children practise their times tables regularly at home.

When learning their tables, children are taught to look for patterns such as odd and even an-swers, or patterns made by adding together the separate digits in the answers.

Children are also taught to recognise the reversible effect so that they know 6 x 2 = 12 means that they also know that 2 x 6 = 12, that 12 ÷ 6 = 2 and 12 ÷ 2 = 6 ( the ‘fact family’).

To help children with their multiplication, one of the ways we use it is to find all the factors that are used to make a number. For example, the factors of 18 are 1, 2, 3, 6, 9, and 18 because:

1 x 18 2 x 9 3 x 6

18 x 1 9 x 2 6 x 3

all equal 18

Multiplication methods

Children are taught to understand multiplication as repeated addition. It can also describe an array. They are introduced to the concept by daily practice in counting in 2s, 5s and 10s and looking for patterns in a hundred square before progressing to more formal recording. Calcula-tions are put into practical contexts so that the child sees the relevance of the method they are learning.

Each child has two eyes. How many eyes do four children have?

2 + 2 + 2 + 2

The children are introduced first to repeated ad-dition.

Drawing a picture is a helpful way to visualise a problem.

5 x 3 =

There are five cakes in a pack. How many cakes are in three packs?

5 + 5 + 5

Dots or tally marks are often drawn in groups to illustrate the problem.

This shows three lots of five. The children can clearly see the repeated addition

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4 x 3 =

A chew costs four pence. How much do three chews cost?

or

Drawing an array (3 rows of 4 or 3 columns of 4) gives children the image of the answer.

It also helps develop their understanding that 4 x 3 has the same value as 3 x 4.

6 x 4 =

There are four cats. Each cat has six kittens. How many kittens are there altogether?

Children can count on in equal steps recording each jump on an empty number line.

This shows four jumps of 6 so the answer is 24 kittens.

13 x 7 =

There are 13 biscuits in a packet. How many biscuits in seven packets?

13 x 7 = 10 x 7 = 70 = 3 x 7 = 21 = 91

When numbers get bigger, it is inefficient to do lots of smaller jumps. 13 can be partitioned (split) into 10 and 3 and the calculation can be worked out on a number line or horizontally.

124 x 6 =

124 books were sold. Each book cost six pounds. How much money was taken?

600 120 24 744

Once children demonstrate a good understand-ing of place value they move onto using the grid method for larger numbers.

E.g. 124 is partitioned (split) into hundreds, tens and units. Each part is then multiplied by six.

The answers are then added together mentally or set out vertically as shown .

Children are taught the importance of placing digits with the same value underneath each other in clear columns whenever they add verti-cally.

x 100 20 4

6 600 120 24

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72 x 34 =

A cat is 72cm long. A tiger is 34 times longer. How long is the tiger?

2100 + 60 = 2160 280 + 8 = 288 2448

The grid method also works for ‘long multiplica-tion’. The numbers are partitioned (split up) and each part is multiplied separately and then each answer is added together.

The grid method can be used for numbers of any size.

The children are encouraged to apply their un-derstanding of times tables and pattern in num-ber, e.g.

7 x 4 = 28 so

70 x 4 = 280

3.6 x 4 =

I need four pieces of wood 3.6 metres long. How much wood do I need to buy?

(i.e.12 + 2.4 = 14.4 metres)

The grid method can also be used for multiply-ing decimal numbers by partitioning the number as shown. Each part is multiplied separately.

As before, the children are encouraged to apply their understanding of times tables and pattern in number, e.g.

6 x 4 = 24 so

0.6 x 4 = 2.4 and

0.06 x 4 = 0.24 etc ….

7.63 x 8 =

The method can also be applied to multiplying numbers to two or more decimal places.

When using column addition for adding deci-mals, children are also reminded of the impor-tance of lining up the decimal points in order to maintain accurate place value throughout their calculation.

x 70 2

30 2100 60

4 280 8

x 4

3 12

0.6 2.4

14.4

x 8

3 56

0.6 4.8

0.03 0.24

61.04

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Division

Children are introduced to division as sharing, grouping and repeated subtraction. Multiplication and division are interlinked. Calculations are put into practical contexts so that the child sees the relevance of the method they are learning.

6 ÷ 2 =

Six sweets are shared between two children. How many sweets does each child get?

(sharing

between

two)

There are six sweets. How many children can have two each?

(grouping

in 2’s)

Drawing pictures make it easy for the child to visualise the problem and often makes it easier to solve. Practical equipment, e.g. counters, cubes, is also used to model and solve the problem.

12 ÷ 4 =

12 apples are shared equally between four baskets. How many apples are in each basket?

(sharing

between

four)

(grouping

in 4’s)

Dots or tally marks can either be shared out one at a time or split up into groups. This then clearly shows how many groups or how many in each group.

15 ÷ 3 =

How many threes in 15?

To work out how many threes there are, chil-dren can use their fingers to count up in groups of three.

They can also draw these as jumps along a number line (repeated subtraction). This shows you need five jumps back so there are five groups of three.

I I I

I I I I I I

I I I

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84 ÷ 6 =

Each ladybird has 6 legs. How many ladybirds are there if there are 84 legs?

It would take a long time to jump back in sixes from 84, so children can jump back in bigger ‘chunks’. A jump of 10 lots of 6 takes you back to 24. Then you need another 4 lots of 6 to reach zero. Altogether that is 14 sixes so there are 14 ladybirds.

184 ÷ 7 = 26 r2

184 chairs are needed for a concert. They are arranged in rows of seven. How many rows of chairs are needed?

184 - 140 (20 x 7) 44 - 42 (6 x 7) 2

This method is known as ‘chunking’. In this ex-ample, you are taking away chunks of seven.

First subtract 140 (20 lots of 7) and you are left with 44.

Then subtract 42 (6 lots of 7) to leave 2.

Altogether that is 26 sevens with a remainder of 2. So 26 rows are needed with either a small row of 2 chairs or two of the rows will need to have 8 chairs.

Children are encouraged to look for the biggest chunk they can do easily using their mental skills ( x 10; x 100 or doubling, i.e. x 20; x 200, etc.)

347 ÷ 24 = 14 r11

24 apples can fit into a box. How many boxes are needed for 347 apples?

347 - 240 (10 x 24) 01107 - 96 (4 x 24) 11

(10 + 4 = 14 and 11 remaining)

The chunking method works equally well when dividing by a two-digit number. This time you are taking away chunks of 24.

First subtract 240 (10 lots of 24) and you are left with 107.

Then subtract 96 (4 lots of 24) and you are left with 11.

The answer to the problem is that 14 boxes are needed and 11 apples left over. (Depending on how the question is worded, your answer may need to say that you need an additional box for the remaining 11 apples.)

49.6 ÷ 8 = 6.2

49.6 - 48 (6 x 8) 1.6 - 1.6 (0.2 x 8)

0 (6 + 0.2 = 6.2)

The chunking method also works for dividing decimals by a whole number.

First subtract 48 (6 lots of 8) and you are left with 1.6

Then subtract 1.6 (0.2 x 8).

The children are encouraged to use their un-derstanding of multiplying decimals to calculate the answer. (See multiplying decimals on p.12)

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Counting ideas

Practising number facts

♦ Practise chanting the number names. Encourage your child to join in with you. When they are confident, try starting from different numbers, e.g. 4, 5, 6….. Also try counting back-wards. To be able to count backwards is just as important as being able to count forwards. As children understand larger numbers you can start from a two or three-digit number.

♦ Sing number rhymes together. There are lots of commercial CDs available.

♦ Give your child the opportunity to count objects (coins, pasta, shapes, buttons, etc). Encourage them to move each item as they count it. Once confident, also practise counting things you cannot touch, e.g. window panes, jumps, claps, oranges in a bag, etc.

♦ Look for numerals in the environment , e.g. car number plates, house numbers.

♦ Make mistakes when chanting, counting or ordering numbers. Can your child spot what you have done wrong?

♦ Choose a number of the week, e.g. 5: practise counting up to 5, from 5, in 5s, collect groups of 5 items.

♦ It’s important that children learn number bonds to 10, e.g. 4 + 6 = 10 and number bonds to 20, e.g. 14 + 6 = 20 by heart. As they move through the school they also need to regularly practise times tables.

♦ Play ‘ping pong’ to practise components with a child. You say a number and they reply with how much more is needed to make 10, 20, 100 or 1000. Encourage your child to answer quickly without counting or using their fingers, e.g. to make 100 you say 40 and they reply 60.

♦ Throw two dice. Ask your child to find the total of the numbers (+), the difference between them (-) or the product (x).

♦ Use a set of playing cards (without the picture cards). Turn over two cards and ask your child to add or multiply the numbers. If they answer correctly, they keep the cards. How many cards can they collect in two minutes?

♦ Play ‘24’ with the whole pack of play cards. You need 4 players: each puts a card down and the first one to make 24 using any or all of the four operations (+, -, x, ÷) and using all or some of the cards, wins that round and keeps all the cards. For example, you put down a Jack, 2 hearts, 7 spades and 2 clubs. You could say ‘2 clubs x Jack, + 2 hearts’.

♦ Give your child an answer. Ask them to write as many number sentences as they can with this answer. You could ask for addition sentences or any type of calculation.

♦ Give your child a number fact, e.g. 5 + 8 = 13. Ask them what else they can find out from this fact (e.g. 8 + 5 = 13, 13 - 8 = 5, 50 + 80 = 130, 130 - 50 = 80, etc)

♦ Play ‘Bingo’. Each player chooses five answers (e.g. numbers to 10 or 20 to practice simple addition and subtraction, multiples of 5 to practise the five times table, etc). Ask a question and if the answer is on a player’s card, they cross it off. The winner is the first player to cross off all their answers.

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♦ Lay the table for a specific number of people ensuring everyone has the right number of cut-lery, etc.

♦ Play simple card games, e.g. ‘Snap’, ‘Rummy’, that involve quickly recognising numbers and/or shapes and adding up scores.

♦ Play board games, e.g. ‘Snakes and ladders’ that encourage your child to add up the numbers on a dice and accurately count backwards and forwards on the board.

♦ Encourage your child to notice door numbers: Can they work out how many doors down the road is the house you plan to visit? Can they see the odd / even pattern?

♦ Go shopping with your child to buy two or three items. Ask them to work out the total amount spent and how much change you will get.

♦ Buy items with a percentage extra free. Help your child to calculate how much of the product is free.

♦ Practise telling the time with your child. Use both digital and analogue clocks. Ask your child to be a ‘timekeeper’, e.g. ‘Tell me when it is half past four because we are going swimming.’

♦ Use a timer / stop watch to time how long it takes to do everyday tasks, e.g. how long does it take to get dressed? Encourage your child to estimate first.

♦ Using a clock or their watch, can your child work out how many minutes before you need to leave for school, get ready for bed, etc?

♦ Plan an outing during the holidays. Ask your child to think about what time you will need to set off and how much money you will need to take, e.g. for ice creams.

♦ Use a bus or train timetable. Ask your child to work out how long a journey between two places should take. Go on the journey. Do you arrive earlier or later than expected? By how much?

♦ Use a TV guide. Ask your child to work out the length of their favourite programme. Can they calculate how long they spend watching TV each day / week?

♦ Let your child help with the cooking. Encourage them to follow a recipe and use the scales to weigh out the ingredients. Help them to measure any liquid ingredients accurately. Talk about what each division on a scale represents. If baking, what time will you need to get the cake out of the oven?

♦ Help your child to scale a recipe up or down to feed different numbers of people.

♦ Practise measuring the lengths and heights of objects in metric measurements. Help your child use different rulers or tape measures carefully and accurately. Encourage them to estimate before measuring. Compare measurements in metric and imperial.

Real Life Maths

Getting children involved in real situations where they are using their mathematical skills is both motivating and stimulating. Below are a few suggestions, but you will no doubt have many good ideas of your own.