Goytre Fawr Primary SchoolGoytre Fawr Primary School Parents Meeting on:Parents Meeting on:

Progression through CalculationsProgression through Calculations66thth March 2013 March 2013

Can I do it in my head?

Do I need

jottings ?

Do I need to use a

calculator?

Shall I use a pencil and

paper method?

DfEE 1999 (14 years ago!)DfEE 1999 (14 years ago!)

‘ ‘Parents who are confident about maths Parents who are confident about maths tend to have children who are also tend to have children who are also confident, and these children are ready to confident, and these children are ready to tackle and assimilate new ideas in a way tackle and assimilate new ideas in a way that is impossible for children who feel that is impossible for children who feel uncertain about, or even fear, maths.’uncertain about, or even fear, maths.’

AimsAims

To look at the ways in which the To look at the ways in which the teaching of mathematics has teaching of mathematics has developed;developed;

To look at how children calculate;To look at how children calculate; Try activities to develop calculation Try activities to develop calculation

strategies;strategies; To look at ways in which parents can To look at ways in which parents can

help their childrenhelp their children

Sound familiar…Sound familiar…

When I was at school I remember the few occasions that the teacher stood at the front of the class and did some maths teaching, it involved teaching rules of how we do things. For example: long multiplication.

We didn't question and often we didn't understand why we were "carrying ten" or "borrowing one". The rest of the time I was taught Maths through a text book and would line up at the teacher’s desk to get my work marked once I had finished.

Activities like this were probably okay for the

children who ‘got’ maths, but not for those who struggled or were uninspired by the work on offer.

How has mathematics changed?How has mathematics changed?

Daily mathematics lesson; Daily mathematics lesson; Emphasis on mental calculations;Emphasis on mental calculations; Interactive whole class and group Interactive whole class and group

teaching;teaching; Enjoyable practical approaches;Enjoyable practical approaches; Mathematics with understandingMathematics with understanding

OverviewOverview

Up to Year 3 the emphasis is on:Up to Year 3 the emphasis is on:o working mentally, working mentally, o calculations recorded in calculations recorded in horizontalhorizontal number number

sentencessentenceso some jottings for more challenging numberssome jottings for more challenging numberso Models and ImagesModels and Images

In Year 3-6 children will be gradually taught moreIn Year 3-6 children will be gradually taught moreformal written methods of calculation butformal written methods of calculation butthey will still use mental methods and jottingsthey will still use mental methods and jottingswhere appropriate.where appropriate.

So - how can we give children the So - how can we give children the best foundations for success with best foundations for success with

written calculations?written calculations?

o We need to encourage children to useWe need to encourage children to use mental calculation strategies formental calculation strategies for smaller/ simpler numbers.smaller/ simpler numbers.

o We need to encourage children toWe need to encourage children to ask the question “Can I do it in my ask the question “Can I do it in my head?” or “Can I do it in my head with head?” or “Can I do it in my head with jottings/ a number line?”jottings/ a number line?”

4 Calculation areas4 Calculation areas

Addition Addition

SubtractionSubtraction

MultiplicationMultiplication

DivisionDivision

A

S

M

D

Bead bars to Empty Number Bead bars to Empty Number LinesLines

Addition- ProgressionAddition- Progression

o Mental calculation supported by:Mental calculation supported by: Modelling of method by teacher Modelling of method by teacher JottingsJottings Number linesNumber lines

o Expanded method using partitioningExpanded method using partitioning

o Compact ‘carrying’ methodCompact ‘carrying’ method

JottingsJottings

Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. They develop ways of recording calculations using pictures, etc.

The first stage in written addition

Count all – a child doing 3 + 5 counts out three counters and then five counters and then finds the

total by counting all the counters.

Counting on from the first number – a child finding 3 + 5 counts on from the first number: ‘four, five, six, seven, eight’.

Counting on from the larger number : 3 + 5 A child chooses the larger number, even when it is not the first number, and counts on from there: 'six, seven, eight'

As well as children using their fingers or bead strings to add on small amounts, a number line will be introduced.It is more efficient to count on from the larger number because you have less to work out. It also shows children that addition can be done in any order ; it doesn’t matter which number you

add first, you get the same answer.

Number linesNumber lines Frequently used in each year group and provide Frequently used in each year group and provide

children with a very visual method of calculationchildren with a very visual method of calculation 15 + 8 = 15 + 8 = 29 + 23 = 29 + 23 =

__________________________________________________________________Can you get from a starting number to 100 in 3 Can you get from a starting number to 100 in 3

hops?hops? 27 + ? = 100 27 + ? = 100 48 + ? = 10048 + ? = 100 2 + ? = 1002 + ? = 100 _________________________________100_________________________________100

Mental Method (using partitioning)Mental Method (using partitioning)  

47 + 76 = (40 + 70) + (7 + 6) = 110 + 13 = 12347 + 76 = (40 + 70) + (7 + 6) = 110 + 13 = 123

Use an expanded layout:Use an expanded layout:

4747+76+76 13 (7+6)13 (7+6)110 (40+70)110 (40+70)123123

  

Standard Method Column Standard Method Column AdditionAddition

The final step, when the children have a The final step, when the children have a sound grasp of place value & of the whole sound grasp of place value & of the whole process…process…

364364

+ 54+ 54

418418

11

Subtraction - ProgressionSubtraction - Progression

Mental calculations supported by:Mental calculations supported by:

Modelling of method by teacherModelling of method by teacher

JottingsJottings Number lineNumber line Expanded methodExpanded method Compact written methodCompact written method

Children develop ways of recording calculations using pictures etc.

10 and 4 less

With subtraction calculations we are careful not to over use the vocabulary of ‘taking away’ because some subtraction calculations involve ‘finding the difference’. We use the word ‘subtract’.Counting on – finding a differenceMax has 5 cubes. Milly has 2 cubes. How many more cubes does Max have?

How many less/fewer cubes does Milly have?What is the difference between?

SubtractionSubtractionUsing a Number line for SubtractionUsing a Number line for Subtraction

Step 1: Counting Back Step 1: Counting Back 78 – 1278 – 12 -10-10

-2-2

66 68 7866 68 78

Step 2:Counting on to find the DifferenceStep 2:Counting on to find the Difference 74 – 5774 – 57 +10 +3 +4 +10 +3 +4

57 67 70 7457 67 70 74

1

From this the children will begin to exchange.The calculation should be read as e.g. take 6 from 1.  71 = this leads to… - 46

Step 1 70 + 1 71 - 40 + 6 - 46  25 Step 2 60

70 + 11 - 40 + 6

20 + 5 = 25Children should know that units line up under units, tens under tens, and so on.

Expanded method to compact …

6

Partitioning and decomposition89 = 80 + 9

- 57 50 + 730 + 2 = 32

Initially, the children will be taught using examples that do not need the children to exchange (what you might know as borrowing).

Multiplication - ProgressionMultiplication - Progressiono Mental calculation supported by:Mental calculation supported by: JottingsJottings Number linesNumber lines Modelling of method by teacherModelling of method by teacher

o Understanding of multiplication as:Understanding of multiplication as: an arrayan array repeated additionrepeated addition scalingscaling

o Grid methodGrid method

Children will experience equal groups of objects and will begin to count in 2s, 10s and 5s. They will work on practical problem solving activities involving equal sets or groups.

MultiplicationMultiplication

Show on bead bar or on a number line: 3 x 5 = 5 + 5 + 5

Children need to learn through practise that multiplication can be done in any order. For

example: 3 x 5 = 15 and 5x3 = 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

 Children should be able to model a multiplication calculation using an array.

Using Arrays

3 x 5 = 15

5 x 3 = 15

Grid method of multiplicationGrid method of multiplication

XX 7070 22

3030 21002100 6060

88 560560 1616

38 x 7238 x 72

21602160+ 576+ 576 27362736

Division - ProgressionDivision - Progression

o Mental calculations supported by: Jottings Number lines Modelling of method by teacher

o Understanding division as sharing and grouping.

o Chunking

Children will understand equal groups and share items out in play and problem solving. They will count in 2s and 10s and later in 5s.

Early Division

Grouping or repeated subtractionThere are 6 sweets, how many people can have 2 sweets each?

3 3 3 3

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

Repeated subtraction using a number line or bead bar12 ÷ 3 = 4

Number line divisionNumber line division

72 72 ÷ 6 = 12÷ 6 = 12

0 60 720 60 72

(10 x 6)(10 x 6) (2 x 6) (2 x 6)

ChunkingChunking

Division is repeated subtraction.Division is repeated subtraction.

Chunking uses knowledge of times tables toChunking uses knowledge of times tables to

take away ‘chunks’ from a number.take away ‘chunks’ from a number.

427 427 ÷ 7 = 61÷ 7 = 61

427 427

- 210 (30)- 210 (30)

217217

- 210 (30)- 210 (30)

77

- 7 (1)- 7 (1)

00

Step 3: Standard MethodStep 3: Standard Method

6 1 3 2

221 1

Children would be encouraged to check division calculations by using the inverse operation eg 22 x 6 = 132 (20 x 6 = 120 2 x 6 = 12 120 + 12 = 132

How to help your child with How to help your child with mathematicsmathematics

Parents can help by …Parents can help by … Counting with your childCounting with your child Playing number gamesPlaying number games Involve your child in shopping activitiesInvolve your child in shopping activities Involve your child when taking measurements or weighing itemsInvolve your child when taking measurements or weighing items Take note of numbers in real life e.g. telephone numbers, bus Take note of numbers in real life e.g. telephone numbers, bus

numbers, lottery numbers etcnumbers, lottery numbers etc Give your child opportunities to use money to shop, check change Give your child opportunities to use money to shop, check change

etcetc Talking about the mathematics in football e.g.. How many points Talking about the mathematics in football e.g.. How many points

does your favourite team need to catch the next team in the does your favourite team need to catch the next team in the division?division?

When helping your child calculate, use the method that they have When helping your child calculate, use the method that they have been taught at school.been taught at school.

Thanks for attending tonight’s Thanks for attending tonight’s Mathematics EveningMathematics Evening