+ - x ÷ + - x ÷ + - x ÷ + - x ÷ + - x ÷ + - + - x ÷ + - x ÷

+ - + - x ÷ + - x ÷ + - x ÷

Introduction

There are many different approaches to teaching and learning Mathematics. Approaches vary in different parts of the world and over time. We would like to share with you the Mathematics strategies your child is learning at BIS.

This Mathematics Booklet contains information about the following:

The progression of methods used to perform calculations Vocabulary connected with addition, subtraction, multiplication and division Ways to help your child develop agility in their mental calculations Mathematics lessons and homework Setting in Mathematics A brief overview of the Mathematics curriculum content in Mile Post 2 Assessment Some ways to help your child develop their Mathematical skills at home

Learning ‘Why’ to explain the ‘How’.

The calculation methods learnt at BIS are in line with the standard expectations of the National Curriculum Mathematics Framework of England and Wales for this age group. They are

progressive and build on the methods learned in Mile Post 1. The aim of this progression is that students fully understand the concept of number, the properties of numbers and the

relationships between numbers. An understanding of multiple calculation methods can assist children to independently choose suitable strategies to solve problems and apply in real life

situations. The methods listed below are sequential and assist with mental arithmetic as well as written calculations.

What are Levels?

There is wide range of levels of mathematical understanding throughout Mile Post 2. Students will work sequentially through the National Curriculum Levels (or stages) at their own rate. These levels do not relate to Year Groups. Please see the Assessment section at the back of this booklet which explains the report grade system and how it relates to the Levels below. Please see your child’s teacher if you would like more details about the level at which your

child is working. Please also take note that while your child is working at a level in their competency in Mthematics in different areas of the subject may vary.

Calculation Methods used in Mile Post 2

(Year 3 and 4)ADDITION

Year 3 and Year 4 Examples

Level 2, Level 3 and Level 4Level 2

Counting on using a number line Steps in addition can be recorded on a number line. The steps often bridge through a multiple of 10.

8 + 7 = 15

48 + 36 = 84

or:

Partitioning

Add the tens and then the ones to form partial sums and then add these partial sums.

Partitioning both numbers into tens and ones mirrors the column method where ones are placed under ones and tens under tens. This also links to mental methods.

Record steps in addition using partitioning:

47 + 76 = 47 + 70 + 6 = 117 + 6 = 123

47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123

In class 3W there are 14 boys and 17 girls. How many children are there altogether?

52 cars and 38 motorbikes passed the school gates. What is the total of number of vehicles?

Year 3 had 56 House Points. The following week they won another 38 House Points. What is the total?

A DVD costs $29 and an electronic game costs $36. What is the total cost?

Level 3

Informal expanded column addition Partitioned numbers are then written under one another:

Expanded column additionWrite the numbers in columns.Adding the tens first:

A farmer has 86 sheep and 27 goats. How many animals does he have?

I have 65 books and my sister has 88. What is the sum total of our books?

ADDITION

Adding the ones first:

Ad

dit

ion

Year 3 and 4 Examples

Level 2, Level 3 and Level 4

Level 4

Compact column addition

In this method, recording is reduced further. Carry digits are recorded below the line (or above the top digit).

Use the words ‘carry ten’ or ‘carry one hundred’, not ‘carry one’.

Column addition is efficient when used with larger whole numbers and decimals. Once learned, the method is quick and reliable. Children should fully understand the place value of the digits when using this method.

It is considered important for children to understand and be able to use the previous stages of addition calculation in order to build their understanding of number, their relationship to each other and place value of digits.

Children’s mental agility and their ability to figure out problem solving strategies independently will be extended by attaining a full understanding of all the numerical calculation methods.

Addition with money using decimals

7 8 ¢ £ 2 . 3 5

+ 6 3 ¢ £ 1 . 4 5

1 1 ¢ - adding the units £ 3 . 8 0

1 3 0 ¢ - adding the tens 1

1 4 1 ¢

= $ 1. 4 1

Using money

Children should know the position of the currency sign for commonly used currencies. Eg.

30,000 VND

$ 20.00

£12.00

The currency of the child’s own country

Children should know:

$1.00 = 100 cents (¢) $00.45 = 45¢

£1.00 = 100 pence (p) £00.75 = 75p

€1.00 = 100 centimes (c)

- 77 = 86

8 6+ 2 5 6 3 1 I have 285 ml of red paint, 353 ml of yellow paint and 188 ml of blue paint.What is the maximum amount of purple paint can I make?

Entrance prices for a Fair are:1 Child - £ 0.601 Adult - £ 1.30Children must be accompanied by an adult.I have £5.00.What is the maximum number of people who can visit the Fair?Explain your answer.

$ 5.80

$ 6.35

What is the cost of 2 books and a football?

SUBTRACTION

Year 3 and Year 4 Examples

Level 2, Level 3 and Level 4 Level 2

Counting on using a number line

Subtracting the tens in one jump and the units in one jump.

47 – 23 = 24

or: Count back 10, then another 10 then 3

Partitioning on a number lineSteps in subtraction can be recorded on a number line. The steps often bridge through a multiple of 10.15 – 7 = 8

74 – 27 = 47 worked by counting back:

The steps may be recorded in a different order:

or combined:

There are 45 people on a bus. 26 people get off. How many people are left on the bus?

A piece of string is 32 cm long.18 cm is cut off.How much is left?

What number is 34 less than 62?(Use a number line)

Level 3

Partitioning

Subtraction can be recorded using partitioning to write equivalent calculations that can be carried out mentally.

Take away 10 from a two-digit number mentally. The method of recording links to counting back on the number line.

Counting on using a number line and vertical columns

There are 62 sweets in a bag.35 sweets are eaten. How many remain in the bag?

80 - = 14

In a quiz Linh scores 84 points. Jan’s score is 27 less than Linh. What is Jan’s score?

Decrease 91 by 56.

-20-3

24 27 47

SUBTRACTION

Year 3 and Year 4

Level 2, Level 3 and Level 4

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Level 4

Counting on using a number line and vertical columnsWith three-digit numbers the number of steps can again be reduced, provided that children are able to work out answers to calculations such as 178 +  = 200 and 200 +  = 326 mentally.The most compact form of recording remains reasonably efficient.

Expanded column method Partitioning the numbers into tens and units Partitioned numbers are then written under one another:

Example: 74 – 27

60 + 14 7 0 + 4 7 0 + 4 – 2 0 + 7 – 2 0 + 7 4 0 + 7

Example: 741 − 367

Children should be familiar with subtraction by partitioning to help develop quick mental subtraction and apply their understanding to problem solving tasks.

There is 330ml of orange juice in a can. Erin drinks 125ml. How much is left?

An elephant is 482m metres tall.A zebra is 325 cm shorter. How tall is the zebra?

What is 37 less than 83?

A car is travelling at 110 km per hour. It reduces its speed to 45 km per hour.How much slower is it now travelling?

The price of a MP3 player is reduced by $128.00.It was $362.00 before discount. What is its new price?

MULTIPLICATION

Year 3 and Year 4 Examples

Level 2, Level 3 and Level 4

Level 2

Practical, visual, arrays

7 x 3 = ( 5 x 3) + (2 x 3)

5 groups of 5 beads = 5 x 5(2 x 5) + (3 x 5)

Level 3

PartitioningThe tens and ones can be multiplied separately. These are then added to find the total product. Either the tens or the ones can be multiplied first but it is more common to start with the tens. 43 x 6 =

Also record mental multiplication using partitioning:

There are 36 tables in the canteen. Each table has 6 chairs around it.

Calculate the number of chairs.

Level 3 -

4

The grid methodThe tens and ones are partitioned and multiplied separately by the second number. Children need to be familiar with mentally multiplying by multiples of 10.

38 x 7 =

Answer = 266

It is better to place the number with the most digits in the left-hand column of the grid so that it is easier to add the partial products

26 x 0.4 =

X 0.4 20 8.0 6 2.4 10.4 Answer = 10.4

There are 26 biscuits in each packet. We have 8 packets. How many biscuits are there?

327 children are in the school. They each use 4 pencils during the year. What is the total many pencils will be required?

What is the product of 8 and 43?

How much wood is required to cut 23 pieces each of which are 0.6 metres in length?

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

DivisionYear 3 and 4 Examples

Level 2, Level 3 and Level 4Level 2

Practical, visualUse practical objects to show children objects can be divided into groups. Eg candy, pencils or counters. Use the words ‘grouping’ and ‘sharing equally.’ Help children to see the connection between multiplication and division.

Sharing equally

6 sweets shared between 2 people, how many do they each get?

Grouping or repeated subtraction

There are 6 sweets, how many people can have 2 sweets each?

Repeated subtraction using a numbered line or bead bar.There are 12 sweets, how many people can have 3 sweets each? - 3 - 3 - 3 - 3

4 3 2 1

Begin to introduce simple remainders. If I have 13 sweets and I share them between 2 of my friends will I have any left over for myself?

Share out 15 pencils between 3 friends.

How many groups of 4 make 20?

(Use a number line)

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Level 3

Division as repeated subtraction on a number line

Ensure that the emphasis is on grouping rather than sharing. Children will continue to use:Repeated subtraction using a number lineThere are 24 sweets, how many people can have 4 sweets each?24 ÷ 4 = 6 -4 - 4 - 4 - 4 - 4 -4

6 5 4 3 2 1 0 4 8 12 16 20 24

There are 13 sweets, how many people can have 4 sweets each?13 ÷ 4 = 3 r 1

- 4 - 4 - 4 3 2 1

1 5 9 13

Use a number line.

There are 28 people altogether. 4 travel in each car. How many cars are needed?

31 children sit in groups of 8. How many groups are there?How many children remaining who are not in a group of 8?

Division

Year 3 and Year 4 Examples

Level 2, Level 3 and Level 4

Level 3

Division by chunks on a number lineChildren will develop their use of repeated subtraction to be able to subtract multiples of the divisor. Initially, these should be multiples of 10s, 5s, 2s and 1s – numbers with which the children are more familiar.

72 ÷ 5 -2 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5

0 2 7 12 17 22 27 32 37 42 47 52 57 62 67 72

Moving onto: -50

0 2 7 12 17 22 72

Children should also be able to find a remainder mentally, for example the remainder when 34 is divided by 6.

Also record mental division using partitioning:

Remainders after division can be recorded similarly.

57 DVDs need to go into packs of 4. How many packs are there?

Each basket holds 6 oranges.How many baskets are needed for 80 oranges.How many oranges are left over?

Each 5-a-side football team has 5 players. There are 85 players in the competition.How many teams are present?

140 ml is divided equally between 6 glasses. How much will be in each glass.How much is left in the bottle?

Level 3 -

4

Division by chunking vertically This method is based on subtracting multiples of the

divisor from the number to be divided, the dividend. For TU ÷ U there is a link to the mental method. As you record the division, ask: ‘How many nines in 90?’

or ‘What is 90 divided by 9?’I have 97 pence. Sweets are 9 pence each. How many can I buy?

97 ÷ 9

10 sweets (with 7p left)

A 2 metre 12 cm piece of wood is divided into 7 pieces. How long is each piece?What length is left over?

Find 1/8 of $192?

There are 238 books. Each shelf holds 9 books. How many shelves will be full?How many books are left over?

Vocabulary Used to Describe Number Operations

During Mile Post 2 children will be introduced to the following vocabulary to describe the number operations. Understanding vocabulary is important to tackle problems.

r2 -5 -51 1

-5 -510

1 1

+ –addition

plusin addition to

totalsum of

increase bymore than

greater thangain

larger thanaltogether

subtracttake away

minusdecrease by

less thanfewer than

reducewhat is missing

a loss ofsmaller than

x ÷multiply by

times greater thanproduct of groups ofmultiple of

double lots of

divideshare

times smaller thanfactor of

mid-point of …. (÷ 2)divisible by

half

An extensive of vocabulary that children will be introduced to and use in all areas of Mathematics in each Year Group is available. Pease contact your child’s Mathematics teacher. A detailed on-line Maths dictionary can be found at: http://www.amathsdictionaryforkids.com/(This dictionary is designed for slightly older children but is also useful for those in Mile Post 2)

Mental Strategies That Will Help Your Child Successfully Complete Written Calculations

AdditionTo add successfully, children need to be able to:

recall all addition pairs to 9 + 9 and complements in 10;

add mentally a series of one-digit numbers, such as 5 + 8 + 4;

Games to speed up agility of mental arithmetic operations

How many multiplication tables can you say in one minute? Year 3 learn multiplication tables: x 2, x 3, x 4, x 5, x 6 and x 10 Year 4 learn multiplication tables: x 7, x 8 and x 9 and consolidate previous knowledge

Use flash cards of multiplication table questions, 6 x 3, 7 x 4 etc. How many can you answer in a minute?

Match up pairs of cards with equivalent answers eg 3 x 6 and 2 x 9 will be a pair.

Bingo using numbers that are the answers to multiplication tables questions.

Websites – eg Tutpup – Mental Arithmetic games against other children around the world. http://www.tutpup.com/

Practise multiplying and dividing numbers by 10, 100, 1000.

Practise multiplying and dividing by multiples of 10, 100, 1000.

Using practical everyday situations speed up agility of mental arithmetic operations

What will our bill come to? How much change should we receive? How many minutes until lunch time? Share these sweets out equally between all the whole family. We are travelling at 60 km per hour. How far are we likely to go in the next 15 minutes? Planning a party or a meal? Involve your child in working out quantities of food required.

add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value;

partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways.

SubtractionTo subtract successfully, children need to be able to:

recall all addition and subtraction facts to 20;

subtract multiples of 10 (such as 160 – 70) using the related subtraction fact,16 – 7, and their knowledge of place value;

partition two-digit and three-digit numbers into multiples of one hundred, ten and one in different ways (e.g. partition 74 into 70 + 4 or 60 + 14).

Multiplication

To multiply successfully, children need to be able to: understand multiplication is repeated addition

add two or more numbers mentally;

recall all multiplication facts to 10 × 10

partition number into multiples of one hundred, ten and one

add two or more numbers mentally;

add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value;

Division

To divide successfully children need to be able to:

subtract numbers

understand division as repeated subtraction

estimate how many times one number divides into another

know how to find a remainder working mentally – for example, find the remainder when 48 is divided by 5

understand and use multiplication and division as inverse operations.

understand and use the vocabulary of division

partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways

recall multiplication and division facts to 10 × 10

multiply a two-digit number by a single-digit number mentally to check division answer

It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for addition, subtraction, multiplication and division.

Organisation of Mathematics Learning

Mathematics Lesson Time

5 x one hour lessons per weekWeekly homework will usually be set. (See below)

Lesson Content

Our Mathematics Curriculum content is based on the ‘English National Curriculum Primary Mathematics Framework’. Students will re-visit topics in Mathematics two or three times during the year. Each time, their knowledge will be revisited, and their understanding and application ability extended.

Each term lessons will cover the following strands of Mathematics:

Number Shape and space Measure Pattern of number and the relationship numbers have with each other Handling data

During each unit students will participate in activities aimed to develop:-

mental arithmetic strategies to help promote speed, agility and understanding understanding and use of new concepts reinforcement of previous ideas understanding and use of mathematical vocabulary knowledge and application of mathematical facts problem solving activities apply Mathematics to real-life practical tasks.

During Mathematics lessons children will learn to work independently, in pairs, in small groups and as part of the whole class. Participating fully in group tasks through exploring and developing ideas is an important part of learning as is gaining the confidence to make mathematical decisions independently.

Children will use their Mathematics skills with in other subject areas such as IPC, Science, Music and PE.

Homework

The aim of Mathematics homework is to reinforce, consolidate and apply concepts learnt in class. Homework tasks may consist of problem solving tasks, practical activities, reinforcement questions, mental agility exercises. Students will sometimes complete these tasks on-line on websites such as ‘My Maths’, sometimes home learning will be of a practical nature and sometimes guidance will be given on a worksheet. Homework may also be posted on ‘Studywiz’. Children are expected to record their homework tasks in their Homework Diary. Your child can be assists with their home learning by having a quiet and comfortable place to work with adequate light and sufficient time allowed. At times your child may need access to a computer to complete parts of their homework.

Mathematics Sets

Year 3 and 4 students work in Mathematics Sets. The sets are known as ‘Core Group’, ‘Extension Group’ and ‘Support Group’. These groups may be differentiated further depending upon need. In some Year Groups there may be five Mathematics Sets all working at different levels.Students are placed in sets which the teachers consider will best suit every child’s learning needs to maximise progress. These decisions are based on the teachers’ assessment of class work contributions; formal assessment tests and the student’s ability to independently apply their knowledge in problem solving situations. In addition, the speed at which a child grasps new concepts, the amount of reinforcement needed to retain their understanding, their confidence in using knowledge and their preferred learning style is also considered when placing children in Mathematics sets.Mathematics Sets are always flexible and are reviewed regularly. Students will move groups at any point during the year when it is considered that the learning needs of a child will be best served in another Set. In this instance the situation will be discussed with the student and parents will be informed of the child’s new Mathematics teacher. The aim is to give each child a solid foundation on which future learning can be understood.

Mathematics in Action at BIS

Mile Post 2 Areas of Study in Mathematics

Below is a list of the general areas of Mathematics that will be explored by the end of the Mile Post. These topics are not learnt in the order listed and each area of study will be re-visited during the year. These objectives are in line with the English National Curriculum Mathematics Framework for the Core Group in each Year Group.

Number Mental and written methods for addition Mental and written methods for subtraction Ordering positive and negative numbers on a number line Rounding numbers Partitioning numbers Multiplication tables to 10 x 10 and division facts Doubling and halving of two-digit numbers Written multiplication methods for two-digit numbers by a one-digit number Written division methods for dividing a two-digit number by a one-digit number Sums and differences of multiples of 10, 100, 1000 Equivalence of fractions Mixed numbers Visual recognition of fractions of shapes and quantities Interpreting the language of ratio and proportion Multiply and divide by 10, 100, 100 Factors and multiples Ways to describe numbers (odd, even factor of ....... , multiples of ......)

Pattern and the Relationship between Numbers Patterns, relationships and properties of numbers and shapes Recognise simple sequences and use the relationship found to generate the

sequence Explain simple sequences Find missing numbers in an equation. (eg 13 + ? = 27)

Shape and Space Properties of polygons Identifying symmetrical shapes and lines of symmetry Visualising 3D and 2D shapes Nets of common solids Names and properties of 2D shapes Classify 2D shapes by the number of sides Use vocabulary to describe position, direction and movement

Measure

Read from scales Compare the impact of different scales Use practical equipment to measure length, weight and liquids Use standard metric units to estimate, measure and record measurements Length – convert units of measurements Read from partly numbered scales Read 12-hour clock time Use time intervals and pm and am notation Area and perimeters of rectangles Measure angles in degrees Recognise acute, obtuse and right angles Draw angles up to 180° Recognise the eight Compass points Use a thermometer to measure temperature Horizontal, vertical and diagonal positions Identify position of a grid using coordinates

Handling Data Construct tables, charts and diagrams, Construct tally charts, pictograms and bar charts Interpret graphs and charts Collection, organising, presenting and interpreting data to answer related questions Identify further questions

Using and Applying

Solve one-step and two-step word problems involving numbers, money, time or measures

Explain methods and reasoning, orally and on paper, using words, diagrams and symbols

Ask questions that will promote mathematical inquiry Use a calculator to help solve problems Explain diagrams and graphs Use charts, graphs and maps to find information Use ICT to handle data Interpret and explain solutions Practical measuring of length, weight, capacity of liquids, money and time

ReferenceMore detailed information regarding the curriculum content and vocabulary for each level can be found on ‘Studywiz’.

Full details of information relating to the’ National Curriculum of England Primary Mathematics Framework for planning the teaching and learning of Mathematics’ can be found at:http://nationalstrategies.standards.dcsf.gov.uk/primary/primaryframework/mathematics

Assessment

Assessment data is used to decide on the next steps for learning, set targets and monitor progress.

Student progress is continually assessed and is based on: -

Contributions to small group and class discussions Contributions to practical work Written class work Homework assignments End of Unit written assessments Problem solving exercises Investigations Standardised Assessment Tests (SATs) and Goal On-line (completed on the computer)

Student achievement is recorded as National Curriculum Level or on the child’s report as an A, B or C grade.

Report Grades and Teacher Assessment Level Correlation

This chart shows the expected National Curriculum Levels in Mathematics at the end of each year group and the corresponding British International School’s Report Grades.

EnglishNational curriculum levels

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

Year 7

65a A

5b B

5c A B

4a B B

4b A B C

4c A B B C

3a B B C C3b A B B D

3c A B B C D

2a B B C D2b A B B E

2c B B C E

1a B C E1b B

1c C

A = ‘above expected level of achievement’B = ‘at expected level of achievement’ C = ‘below expected level of achievement’

(The exact expected level at the end of each year group is indicated by shaded area or a thick black line.)

Practical ways to help your child at home

Games can help children develop their Mathematical skills. eg Snakes and Ladders or other board games involving counting on

and counting back to support mental agility of addition and subtraction.

Card games that require mental agility. Checkers for spatial awareness and logic.

Practise and Experience of Practical Maths:

Counting on in 2’s, 3’s, 4’s, 5’s, 6’s, 10’s and 100’s. Number bonds to 10 and 20 (e.g. 18+2, 3+17 etc). Doubles and halves of numbers to 30 (Double 15 and Half of 30). Addition and subtraction facts to 20 (Also in worded questions such as: ‘There were 19 sweets, I ate 15 how many are there left?’ x 2, x 3, x 4, x 5, x 6 and x 10 times tables (Yr 3). x 7, x 8, x 9 multiplication tables (Y4). Match pairs of equivalent cards. Throw two or three dice and add up numbers as quickly as possible. Talk about fractions when cutting up a pizza or a cake. Involve children in shopping. Involve children in measuring items at home. eg ‘How wide is that rug? Will it fit over

here?’ Point out the time at different times of the day eg, lunch, school. Ask questions such as: ‘What time will it be in …. ?’ ‘How long is it until ….’ Can children tell the time? Link to TV programmes and programming the video/DVD.

‘Can you tell me the time please?’ Use a mirror to see whether shapes are symmetrical Look for right angles (square corners) around the house. Identify ten right angles in

each room. Play shape bingo. At home or on a journey, how many circles, squares, etc. can they

spot? Give them different point values. When cooking encourage to children estimate different measures. What does 10

grams/10 ml/ 1 kilogram looks/feels like? Ask you child to help measure liquids or weigh items when cooking.

Other Ways to help with Homework exercises:

Encourage your child to use a number line to help understand and complete calculations.

Use beads, counters or beans to represent numbers. Eg sharing out beans when dividing.

Use a multiplication square for multiplication and division facts.

Struggling with a written problem?

Ask your child to draw their interpretation of the problem or use real objects so they are able to see the situation visually.

Help your child to break the problem down into steps. Replace the numbers with simpler alternatives to help your child gain confidence

and understand the process, then try again with the given numbers.

Encourage estimation.

Use multiplication squares and calculators to help with the arithmetic calculations so that your child has a clearer understanding of the processes involved in the problem. Work on the calculations separately.

Useful Websites to Help Develop Skills in Mathematics

Maths dictionary:

http://www.amathsdictionaryforkids.com/

Practising skills:

http://www.bbc.co.uk/schools/ks2bitesize/maths/

http://www.woodlands-junior.kent.sch.uk/maths/

http://www.mathplayground.com/games.html

http://www.ixl.com/ (click on ‘2nd Grade’ for Year 3 work and ‘3rd Grade’ for Year 4 work.

http://www.tutpup.com/ (Play mental arithmetic games against students around the world. This website is free but you will need to join. Ask your teacher for your class code. They will set one up for you. You will need to enter the class code when you join.)

My Maths - http://www.mymaths.co.uk/Use the Library to choose your own exercises or games at your own level. (Ask you teacher for advice about which level you should select.)

21.05.11SK