Sprowston Cluster

Maths Calculation Policy ©

July 2015

Contents

1 Schools involved 2 Aims 3 Rationale 4 Overview of approach 5 Addition 6 Addition 7 Addition 8 Addition 9 Addition 10 Addition 11 Addition 12 Subtraction 13 Subtraction 14 Subtraction 15 Subtraction 16 Multiplication 17 Multiplication 18 Multiplication 19 Multiplication 20 Multiplication 21 Multiplication 22 Division 23 Division 24 Division 25 Division 26 Division 27 Division 28 Division 29 Division 30 Division 31 Division 32 Key vocabulary 33 Using representations 34 Use of technology

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School involved

This policy has been developed by the following schools within the cluster

Name of school Name of contact

Cecil Gowing Infant Helen Hoye

Falcon Junior Rachel Harrod Rachel Atkinson

Garrick Green Infant Helen Prophet

Hainford/Frettenham Primary Partnership

David Board

Lodge Lane Infant Catrina Hilditch

Sparhawk Infant and Nursery

Claire Butler Ben Wilson

Spixworth Infant Rebecca Buller

Sprowston Community High

Avril Mallett

Sprowston Infant Andy Palmer

Sprowston Junior Matt Walton

Woodland View Junior Heallen Payne

Policy created – 06/02/2015 Policy reviewed – 24/06/2015

Date of next review – June 2016

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Aims

For each of the four operations, students will develop reliable written and mental methods of calculation which further promote and build upon their conceptual understanding of number;

Students receive a consistent approach to the teaching of calculations to ensure smooth transition between phases;

Fluency, reasoning and problem solving underpins the teaching of mathematics from Reception onwards;

A consistent Concrete-Pictorial-Abstract approach is used throughout;

Students can confidently use a range of strategies to calculate mentally.

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Rationale

This policy was developed as a result of research that was collated from every school in the cluster. Analysis showed that many of the strategies that the students were using for addition, subtraction, multiplication and division were unsuccessful. Findings from cluster data showed that a large percentage of the students in the cluster answering age related questions scored lower than nationally expected. This was in line with data found in both Norfolk and national contexts. One of the research findings from the Norfolk data showed that students from Year 6 were able to answer Year 8 age related questions better than the Year 8 students, an example of this is shown below.

Subtraction example: Year 8 age related question. Schools A to D are Year 6 students. It was noted that students in School B reverted to using vertical calculation methods and made several mistakes whereas schools A,C and D used number lines and were more accurate. These results support our need, as a cluster, to produce a Calculations policy.

82 – 3.8 Number Correct

Number Incorrect

Percentage Correct

Percentage Incorrect

Y8 data 563 422 57% 43%

School A 25 3 89% 11%

School B 13 16 45% 55%

School C 55 6 90% 10%

School D 16 7 70% 30%

In order to improve these results, and to ensure successful progression and smooth transitions, both within and across schools, every Maths Subject Leader from each school within the cluster met to discuss and agree strategies for each of the four operations. This is a cross-phase cluster calculation policy which starts in Reception and continues into Key Stage 3.

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Overview of Approach This policy contains the written and mental calculation strategies that will be taught within the Sprowston Cluster. Strategies were informed by:

A range of research (most notably the Norfolk Calculations Research by Borthwick & Harcourt-Heath, 2006 to present day);

Careful consideration of strategies currently in use;

Advice from Mathematics Advisers for Norfolk Integrated Education Advisory Service.

Underlying our approach is the belief that students should be taught to understand concepts, rather than relying on procedural or instrumental learning. In this way, students will be able to successfully understand, use and apply their mathematical skills, both inside the classroom, and out in the real world. This will be achieved by:

Adopting the Concrete-Pictorial-Abstract (CPA) approach to learning;

Using mathematical representations to support strategies at every stage in every phase;

Ensuring that Fluency, Reasoning and Problem Solving play an integral part in every maths lesson;

Considering how mental methods and written methods can work in partnership (The ‘Teaching Children to Calculate Mentally (DfE, 2010) booklet will provide a framework for teaching children mental calculation strategies).

All calculations will be presented in a variety of ways to enable students to choose a strategy that works for them and this may well be dependent on the numbers themselves, or the context in which they are presented. It is advantageous for calculations to be presented horizontally to mirror the way in which GCSE calculations are set out.

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Addition

Definition Addition is the process of combining 2 or more quantities.

Early Learning Students practise counting in 1s.

They count chorally up to any given number;

They count how many children would like school dinners;

They count given objects to see how many they have;

They sing counting songs. Students use given apparatus or their fingers to find 1 more. Students are introduced to written numbers. Students are introduced to the + symbol. Begin to relate addition to combining two groups of objects.

Make a record in pictures, words or symbols of addition activities already carried out;

Construct number sentences to go with practical activities;

Use of games, songs and practical activities to begin using vocabulary;

Solve simple word problems using their fingers.

Students will begin to double given equipment. Students use their knowledge of the number system to count along a number line. Students use equipment to find number bonds for numbers up to 10.

Introduce students to the language of addition.

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They continue to combine two groups of increasingly large numbers using concrete material. Students continue to find number bonds. They now find number bonds of numbers up to 20 and beyond. To support this, teachers could model using Numicon, beadstrings and uni-fix.

Mental Calculations

Counting forwards and backwards in ones, twos, fives, tens etc

Reordering

Partitioning: counting on or back

Partitioning: bridging through multiples of 10

Partitioning: compensating (rounding and adjusting)

Partitioning: using near doubles

Partitioning: bridging through 60 to calculate time intervals

Using addition as the inverse of subtraction

Written Methods

Students use number lines. Students use a numbered line to count on in ones. Students use number lines and practical resources to support calculation and teachers demonstrate the use of the number line. 7+ 4

0 1 2 3 4 5 6 7 8 9 10 11 12

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Students draw their own empty number lines.

7 + 4 = 11

Students to practise counting on from any number, to include 2-digit numbers, to 20 and beyond. Number lines will increase in size as needed. Students will also use 100 squares to count on from a given number to find a total. Students need to understand the concept of equality before using the ‘=’ sign. Calculations should be written either side of the equality sign so that the sign is not just interpreted as ‘the answer’. 2 = 1+ 1 2 + 3 = 4 + 1 3 = 3 2 + 2 + 2 = 4 + 2 4 + 2 = 2 + 2 + 2 Missing numbers need to be positioned in all possible places.

3 + 4 = = 3 + 4

3 + = 7 7 = + 4

+ 4 = 7 7 = 3 +

+ = 7 7 = + Students should be encouraged to choose the equipment and the strategy that suits them when adding.

Partition into tens and ones and recombine 12 + 23 = 10 + 2 + 20 + 3 = 30 + 5 = 35

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Count on in 10s and 1s 23 + 12 = 23 + 10 + 2 = 33 + 2 = 35

The Empty Number Line: Partitioning and bridging through 10

The steps in addition often bridge through a multiple of 10.

Students should be able to partition the 7 to relate adding the 2 and then the 5.

8 + 7 = 15

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Add near 10s by adding multiples of 10 and adjusting

Add 9 by adding 10 and adjusting by 1 35 + 9 = 44

Previous methods are extended to include larger numbers. 35 + 28 = 63

Students are taught to estimate an answer first by rounding numbers and using near doubles.

Partition into 10s and 1s

Partition both numbers and recombine;

Count on by partitioning the second number only. 36 + 53 = 53 + 30 + 6 = 83 + 6 = 89

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Add a near multiple of 10 to a 2-digit number Secure mental methods by using a number line to model the method. Students need to be secure adding multiples of 10 to any 2-digit number including those that are not multiples of 10. 48 + 36 = 84

Partition into 10s and 1s and recombine

Either partition both numbers and recombine or partition the second number only. 55 + 37 = 55 + 30 + 7 = 85 + 7 = 92

Add the nearest multiple of 10, and then adjust.

Extend to decimals in the context of money.

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Partition into 100s, 10s and 1s and recombine Either partition both numbers and recombine or partition the second number only. 358 + 73 = 358 + 70 + 3 = 428 + 3 = 431

Add or subtract the nearest multiple of 10 or 100, then adjust 458 + 79 is the same as 458 + 80 - 1

Partition into 100s, 10s, 1s and decimal fractions and recombine

Either partition both numbers and recombine or partition the second number only. 35.8 + 7.3 = 35.8 + 7 + 0.3 = 42.8 + 0.3 = 43.1

Add the nearest multiple of 10, 100 or 1000, then adjust Continue with numbers including extending to adding 0.9, 1.9, 2.9

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Subtraction

Definition Subtraction is the inverse of addition. It can be defined as the process of taking away one number or amount from another, or as the act of finding the difference between two numbers or amounts.

Early Learning

One firework goes off. How many are left? 3 – 1 = 2

How many more green cubes? How many less black cubes?

5 – 3 = 2

2 skittles are knocked down. How many left?

6 – 2 = 4

3 – 1 = 2

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Mental Calculations

Counting forwards and backwards in ones, twos, fives, tens etc;

Reordering;

Partitioning: counting on or back;

Partitioning: bridging through multiples of 10;

Partitioning: compensating (rounding and adjusting);

Partitioning: using near doubles;

Partitioning: bridging through 60 to calculate time intervals;

Using addition as the inverse of subtraction.

Written Methods

Taking Away or Counting Back:

Solve problems using a marked or empty number line to subtract 1 digit numbers from 2 digit numbers by counting backwards e.g. 35 – 6

Solve problems using number tracks to take away by counting back e.g. 9 - 3 = 6

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Students can use 100 squares and Base 10 equipment to support their calculations and then transfer this onto an empty number line. Solve problems involving subtraction of 2 digit numbers from 2 digit numbers e.g.57-25.

Finding the Difference or Counting Up

Solve problems involving subtraction of 2 digit numbers from 3 digit numbers e.g. 215 – 28.

Solve subtraction problems involving decimal numbers e.g. 6.6 – 1.6

Solve problems using a number track to count up from the smallest number e.g. 6 – 4.

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Solve problems using a marked or empty number line to find the difference between 2 digit numbers e.g. 72 – 58.

Solve problems involving subtraction or larger numbers by counting up e.g. 274 – 187.

Solve problems involving decimal numbers e.g. 6.2 – 4.7

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Multiplication

Definition

Multiplication is the product of two or more numbers or repeatedly adding a number or quantity. For example 4 multiplied by 5 (4 x 5) is 5 groups of 4 or 4+4+4+4+4. It is an inverse operation of division.

Early Learning

Students need opportunities to count groups of the same number of objects and add them together. They need a wide variety of experiences, engaging in songs, rhymes and real life contexts. Encourage students to draw pictures and to use equipment such as Numicon, beadstrings and cubes to show their representations. Students recognise that doubling and multiplying by 2 are the same.

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Mental Calculations

Counting forwards and backwards in equal steps e.g. in 2’s, 5’s and 10’s; Repeated addition;

Rapid recall of multiplication facts;

Partitioning;

Secure understanding of place value;

Multiplying and dividing by 10, 100 and 1000;

Doubling and halving;

Using division as the inverse of multiplication.

Students should develop fluency and reasoning by making connections between different methods of mental calculation.

Written Methods

Repeated Addition

It is important to develop a student’s understanding of repeated addition; counting in regular steps of various sizes and to show the links between this and a number sentence. A variety of equipment can be used to help the student develop an understanding of the process and as a visual tool to help develop their understanding.

5 children have 3 cars each. 4 children have collected 10 How many cars are there altogether? cards each. How many cards

have they collected altogether?

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Number line

Number lines can be used to effectively show repeated addition.

Arrays Once the student recognises multiplication as repeated addition, they can then be taught that multiplication can be done in any order (4 x 2 = 2 x 4).

Multiplication can be taught pictorially in the use of arrays. An array consists of a set of objects that have been put into rows and columns. The student can either be given a set of everyday objects, or draw symbols when representing the number sentence or problem.

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The following array, consisting of four columns and three rows could be used to represent the number sentence 3 x 4 = 12 or 4 x 3 = 12.

4x3=12

3x4=12

How many do you have?

How many in each row?

How many in total do you have?

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Empty Arrays / the Grid Method

With a student’s understanding developing, progression will lead to the teaching of the grid method. This will have already been introduced with arrays. Grids should be proportioned and students should be given the freedom to decide how to partition each number. This may lead to the 10s being partitioned into more than one column. Students will start using this method by multiplying a 2-digit by a 1-digit number, progressing to much more challenging calculations.

The method helps the student to see the value of each part of the calculation which enables them to understand the values being multiplied. Variations can be used to add the answers together.

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Multiply numbers with decimals e.g. 42.5 x 3

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Division

Definition

Division is the inverse of multiplication and is a way of determining how many

times one quantity is contained within another. Either sharing or grouping can

divide a quantity. For example:

Sharing

12 ÷ 4

Sharing 12 objects between 4.

Grouping

12 ÷ 4

How many groups of 4 are there in 12?

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Early Learning

Concrete objects and sharing

Students use objects, diagrams and pictorial representations to solve

problems involving sharing. They will understand equal groups and share

items out in play and problem solving.

They will share a set of objects equally. For example, I have 6 beanbags to

share between 2 people. How many does each person get?

Students are able to share objects with a remainder. For example, I have 9

beanbags to share between 2 people. How many does each person get?

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Mental Calculations

Counting forwards and backwards in equal steps e.g. in 2’s 5’s and

10’s;

Repeated addition;

Rapid recall of multiplication facts;

Partitioning;

Secure understanding of place value;

Multiplying and dividing by 10, 100 and 1000;

Doubling and halving;

Using division as the inverse of multiplication.

Students should develop fluency and reasoning by making connections

between different methods of mental calculation.

Sharing with closed arrays

Students will sort a set of objects by sharing them equally. For example,

There are 12 cubes and 4 children. How many cubes does each person get?

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Grouping with closed arrays

Students will sort a set of objects by grouping them equally. For example,

There are 12 cubes. How many groups of 4 can be made?

Students use a blank number line to group from zero in equal jumps of the

divisor to answer problems. For example, How many groups of 4 in 24?

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Blank number lines are used to demonstrate division as repeated subtraction.

For example, There are 24 bulbs and 4 flower pots. How many bulbs are

planted in each pot?

Open Arrays

Many students experience difficulty with this aspect of maths so it is extremely important that the inverse links between division and multiplication are made and explored.

The open array is a tool which allows students to organise their thinking in to rows and columns.

Open arrays show partitioning for division using known multiplication facts. A good place to begin with for the known facts is by encouraging the use of ‘Coin Facts’

Coin Facts are based around the 10p, 5p, 2p and 1p coins.

For example:

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Once known facts have been established these can then be applied to the open array.

122 ÷ 3 = ?

The representations on the array could also be physically shown on the grid using counters, dots or with Numicon blocks.

Encourage the use of a number line to add the groups of numbers in jumps that the student is secure with.

For example,

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Challenge students by:

● Including remainders to be expressed as fractions and as decimals;

● Asking them to find errors within calculations and explain what has

happened.

Grouping

76832

32 320

448

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Grouping with a remainder

30

10

2

10

70

777 35 = 22 7/35 = 22 1/5 = 22.2

? 7

350

350

35

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Understanding of the inverse relationship of multiplication and division;

Understanding of the distributive law of division.

Division should be taught alongside the teaching of fractions, decimals,

percentages and ratios in order to develop the conceptual understanding of

these related areas of mathematics.

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Key Vocabulary

Addition

Add Addition Plus More than Altogether Sum Total Increase Count up

Subtraction

Count back Count up Decrease Fewer Find the difference Difference between Subtract How many left? How many more? Less Less than Minus Reduce Take away Take away from

Multiplication

Multiply Multiplies

Times Groups of Lots of Repeated addition Arrays Multiplied by Product

Division

Grouping Sharing

Divisor Remainder Repeated subtraction Dividend Share equally Divided by

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Using Representations in Maths Representations are a useful tool in mathematics to support learners in developing conceptual understanding through: communicating, reasoning, solving problems, making connections between ideas and learning new concepts. (Principles and Standards of School Mathematics (NCTM, 2000) on

https://learner.org/courses)

With conceptual understanding we:

Know more than just facts;

Know why a mathematical idea is important;

Learn new ideas by connecting them to the ones we already know;

Can remember or retain ideas. Often older and higher-attaining students view practical resources as a tool for those who find mathematics difficult – we should actively challenge this perception and ensure learners of all ages and stages have the opportunity to deepen their levels of understanding through the use of representations; allowing them to become mathematicians and not just a follower of mathematical processes. Practical approaches to work can often mean the knowledge and understanding is more likely to be retained after the session has ended. The following image shows some of the wide range of materials that teachers can draw upon when planning to use representations in their mathematics lessons.

For further research and ideas about the use and value of representations read: 'Developing the use of Visual Representations in the Primary Classroom: A Nuffield Foundation Project. (Barnby, Bolden, Rain and Thompson) which can be accessed via www.nationalstemcentre.org.uk

“How we understand something is we see how it is related or connected to other things we know.’ J.Hiebert, Signposts for Teaching Mathematics through Problem Solving 2003

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Use of Technology

Technology should play an important role in developing the student’s practical calculation skills. Technology can be used to promote the thinking and reasoning skills needed for problem solving within mathematics as well as increasing the student’s understanding of arithmetic operations and numerical relationships. The use of technology should be linked to the appropriate age and ability of the stage the student is working at. It should be carefully planned to advance learning within the classroom and it should not be used as a replacement for basic understanding. Technology should not replace the need for students to develop efficient and accurate methods for both mental and written calculations as well as performing sensible estimations. Technology resources that may be used to support the teaching and learning of calculations could include:-

Calculators;

Computers;

Tablets;

Interactive whiteboards;

Promethean Slates;

Bee-Bots.

Useful apps and websites include:- Addition & Multiplication (Unripe Grape) Maths Party

Mathoku Lite Math Academy Free

www.stemnet.org.uk www.mathsframe.co.uk www.ncetm.org.uk www.nrich.maths.org www.youcubed.stanford.edu

Acknowledgements Sprowston Cluster would like to thank everyone involved in the creation of this policy. Whilst every effort has been made to acknowledge source material throughout the document, if there have been any oversights please contact Rachel Harrod at Falcon Junior School and corrections/amendments will be made.

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