Maths EveningNovember 2015

To help you support your children at home by encouraging the use of methods promoted in school

To consider the progression of addition, subtraction, multiplication and division

To introduce resources used to support children in their mathematical development

Aims

Mathematics involves providing children with the opportunities to:

Count reliably with numbers 1-20 Place numbers in order Say which number is one more or less than a given number Use quantities and objects Add and subtract two single digit numbers and count on and back to

find the answer Solve problems, including doubling, halving and sharing Use everyday language to talk about size, weight, capacity, position

distance, time and money to compare quantities and objects and to solve problems

Recognise, create and describe patterns Explore characteristics of everyday objects and shapes and use

mathematical language to describe them

Early Learning Goals

The national curriculum for mathematics aims to ensure that all pupils:

become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.

reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language

can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

Aims of the National Curriculum

Excellence and enjoyment for all

Progressive and Challenging

Embedded across the curriculum

Inclusive and collaborative

Our Vision

Addition(Stage 1)

In the early stages of Reception children will learn the foundations of number, developing their number recognition skills through lots of fun games and

activities.

Children will then begin to understand addition as the combination of two groups. Children will be encouraged to develop a mental picture of the number system in

their heads and develop pictorial representations to record their calculations;

Bead strings will be used to demonstrate addition to children;

8+2=10

End of Year Objective – Using quantities and objects, children add 2 single-digit numbers and count on to find the answer.

Year 1 (Stage 2)

Children will continue to use pictures to represent addition calculations.

Children will then begin to use readymade number lines to support their own calculations by using a number line or bead string to count on in ones.

8 + 5 = 13

Children move on to using Base 10 equipment to support their developing understanding of addition.  

11 + 5 = 16

11 cubes are lined up (1 ten and 1 unit/one).5 cubes are added to the line of 11 giving a total of 16

End of Year Objective – add one and two-digit numbers to 20, including zero.

Children continue to use the Base 10 equipment, cubes, art straws and Numicon to support their calculations including exchanging 10 units/ones for 1 ten when the total of the units/ones is 10 or more. They will record their own drawings of the Base 10 equipment, using lines for 10 rods and dots for the unit blocks.

34 + 23 = ?The units/ones are added first 4 + 3 = 7The tens are added next30 + 20 = 50answers are put together 50 + 7 = 57

28 + 36 = ?The units/ones are added first

8 + 6 = 14 with ten units/ones exchanged for 1 ten.A ring is put around the units/ones not exchanged – this is the units part of the answer. The tens are then added, including the exchanged ten, to complete the sum.

Year 2 (Stage 2)

End of Year Objective – add numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and ones; a two-digit numbers and tens; two two-digit numbers; adding three one-digit numbers.

Continue to use Base 10 for practical work and recording.

65 + 27

Year 3 (Stage 4)

End of Year Objective – Add numbers with up to three digits using formal written methods of column addition.

Year 4,5,6 (Stage 4)

This is the final stage of the method, and should be continued to be used for all written addition calculations.

The example would be ‘said’ as follows:5 + 8 = 13, put 3 down and carry the 10

20 + 40 + 10 that was carried over = 70 (7 written in the tens column)

600 + 0 = 600 (6 written in the hundreds column)Chil

Year 4,5,6 (Stage 4)

End of Year Objective Y4 – Add numbers with up to 4-digits and decimals with one decimal place using formal written method of column addition.

End of Year Objective Y5 – Add whole numbers with more than 4-digits and decimals with two decimal places using formal written method of column addition.

End of Year Objective Y6– Add whole numbers and decimals using formal written method of column addition.

Children will be expected to use this method for adding numbers with more than 3 digits, numbers involving decimals and adding any number of amounts together.

Vocabulary

+

ADDPLUS

SUM

MOREALTOGETHER

TOTAL

INCREASE

Subtraction (Stage 1)In Reception children will subtract two numbers by taking one away from the other and counting how many are left.

Touch counting and dragging.

7 – 3 = 4

They begin to recognise and use Numicon equipment to help ‘take away’.

When ready, they begin to record subtraction number sentences such as 8 – 5 = 3.

End Of Year Objective - Using quantities and objects, children subtract two single-digit numbers and count on or back to find the answer.

Children move on to using Base 10 and Numicon equipment alongside a number track to support their developing understanding of subtraction. They will also continue to use cubes and counters.13 - 4 = ?13 cubes are lined up.4 cubes are removed from the end of the line leaving 9 left.It is important that children keep track of how many have been removed. 

Touch count and remove the number to be taken away.Touch count to find the number that remains.

 

Year 1 (Stage 2)

End of Year Objective – Subtract one-digit and two-digit numbers to 20, including zero.

Children continue to use the Base 10 equipment to support their calculations. Eg 54 – 23 

which leaves They will record their own drawings of the Base 10 equipment, using lines for 10 rods and dots for the unit blocks. 39 - 17 = ?39 is drawn17 is crossed outA ring is drawn around what is left to give the answer giving 22

Children will begin to use empty number linesEnd of Year Objective - Subtract numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and ones; a two-digit number and tens; two two-digit numbers.

47-23=

Year 2 (Stage 3)

24 25 26 27 37 47

-3 -20

End of Year Objective - Subtract numbers using concrete objects, pictorial representations, and mentally, including: a two-digit number and ones; a two-digit number and tens; two two-digit numbers.

Children move from using the Base 10 method to expanded vertical method, using base 10 notation and arrow cards.

89-57 

Arrow cards help with partitioning

Children learn to subtract the least significant digits first (start with the numbers on the right and work from right to left).

The answer to each individual subtraction is written underneath before these answers are recombined.

Year 3 (Stage 4a)

Exchange needed

71-46

It is clear that there are not enough units to subtract 6 from 1, so one of the tens from the 70 is exchanged for 10 ones.

Year 3 (Stage 4b)

The initial number 71 is rearranged as 60 and 11 to make the calculation easier

This would be recorded by the children as:

End of Year Objective - Subtract numbers with up to three digits, using formal written method of columnar subtraction.

Move from subtraction through partitioning

Year 4 (Stage 5)

to decomposition

End of Year Objective - Subtract numbers with up to 4 digits and decimals with one decimal place using the formal written method of columnar subtraction where appropriate.

Using decomposition

Year 5 (Stage 5)

 2

  13  

 

3

. 4 12

-1

. 7 6

  1 . 6 6

 6

  6  

 7

10 7 12

-3

2 2 6

  3 8 4 6

(4-digits) (decimals to two places)

End of Year Objective - Subtract whole numbers with more than 4 digits and decimals with two decimal places, including formal written methods (columnar subtraction).

When subtracting decimals with different numbers of decimal places, children should be taught and encouraged to make them the same through identification that 2 tenths is the same as 20 hundredths, therefore, 0.2 is the same value as 0.20

Year 6 (Stage 5)

 3  

6   11

 

 4 11

7 . 210

-  3

4 . 7 1

  3 8 2 . 4 9

End of Year Objective - Subtract whole numbers and decimals using formal written methods (columnar subtraction).

Vocabulary

-

DIFFERENCE

MINUS

TAKE AWAYLESS

SUBTRACT

DECREASE

Multiplication(Introduction to Stage 1)

Children are encouraged to develop a mental image of the size of numbers. They learn to think about equal groups or sets of objects in practical, real life situations. They begin to record these situations using pictures.

A child’s jotting showing fingers on each hand as a double.

A child’s jotting showing double three as three cookies on each plate.

Introduction to arrays using practical activities such as placing items in an egg box.

End of Year Objective - Children solve problems, including doubling.

Year 1(Stage 1)

Children will use practical equipment and jottings.They may use the equipment to make groups of objects. Children should see everyday versions of arrays, e.g. egg boxes, baking trays, ice cube trays, wrapping paper etc.

End of Year Objective - Solve one-step problems involving multiplication by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.

Year 2(Stage 2)

Children understand that multiplication is repeated addition and that it can be done by counting in equal steps/groups using practical apparatus.

Children can then be introduced to images of rectangular arrays; and use these to show that counting up in equal groups can be a quicker way of finding a total. Numicon can be used for exploring arrays.

Children also understand that3 x 5 is the same as 5 x 3

End of Year Objectives - Calculate mathematical statements for multiplication (using repeated addition) and write them using the multiplication (x) and equals (=) signs.

Year 3 (Stage 3)

Children continue to use arrays and create their own to represent multiplication calculations. Use of numicon to explore arrays.

(Stage 4)Children will continue to use arrays to lead into the grid method of multiplication.14 x 6

End of Year Objective - Write and calculate mathematical statements for multiplication using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, progressing to formal written methods.

Year 3 and Year 4(Stage 5)

In this stage, the array is removed and children use the grid method. This is an important step in retaining children’s understanding of multiplication.Y3: 2-digit numbers x by 1-digit numbers; Y4: Solve problems involving money and measures.

End of Year Objectives – Multiply two-digit and three-digit numbers by a one-digit number using formal written layout.

Year 5(Stage 6)

x 2000 600 90 3

20 40000 12000 1800 60

4

8000 2400 360 12

40000

+ 8000+ 12000

+ 2400+ 1800+ 360+ 60 + 12

64632

End of Year Objectives - Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers.

Year 5 (Stage 6 contd)

Introduce formal ‘expanded vertical method’ from grid method

(Stage 7) The expanded method can then be taken into the compact vertical method.

Year 5 (stage 8)

.

When ready the children can be introduced to long multiplication.

Year 6

End of Year Objectives - Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication.

Multiply one-digit numbers with up to two decimal places by whole numbers.

T U . t h

4 . 9 2 x 3

0 . 0 6 (0.02 x 3)

2 . 7 0 (0.9 x 3)

+ 1 2 . 0 0 (4 x 3)

1 4 . 7 6

T U . t h

4 . 9 2 x 3

1 4 . 7 6

Vocabulary

Multiplication PRODUCT

GROUPS OFARRAY

LOTS OF

MULTIPLYTIMES TABLE

REPEATED ADDITION

DIVISION(Stage 1)

End of Year Objective – Children solve problems, including halving and sharing.

Children are encouraged to develop a mental image of the number system in their heads to use for calculation. They should experience practical calculation opportunities involving equal groups and equal sharing.

Use pictures and jottings. A child’s jotting showing halving six spots between two sides of a ladybird.

A child’s jotting showing how they shared the apples at snack time between two groups.

Year 1(Stage 2)

End of Year Objective - Solve one-step problems involving division by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher.

Practical contexts for sharing and grouping

Sharing Grouping

Equal sharing (6 shared equally between 2)6 football stickers are shared equally between 2 people, how many do they each get? Children may solve this by using a ‘one for you, one for me’ strategy until all of the stickers have been given out.

Equal grouping (How many groups of 2 are there in 6?)

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

Year 2(Stage 3)

End of Year Objective - Calculate mathematical statements for division within the multiplication tables and write them using the division (÷) and equals (=) signs.

Children continue to use practical equipment including numicon to represent division calculations as grouping (repeated subtraction) and use jottings to support their calculation.

12 ÷ 3 = ? Children begin to read this calculation as,‘How many groups of 3 are there in 12?’

Introduction to remainders 13 ÷ 4 = 3 remainder 1

Year 3(Stage 4)

Division by grouping continued.

Children learn if a remainder should be rounded up or down e.g. 62 ÷ 8 = 7 remainder 6I have 62p. Sweets are 8p each. How many can I buy?Answer: 7 (the remaining 6p is not enough for another sweet)Apples are packed into boxes of 8. There are 62 apples. How many boxes do I need?Answer: 8 (the remaining 6 apples still need to be placed into a box)

(Stage 5)Use repeated subtraction on a number line.Begin to use vertical number line to show repeated subtraction.Subtract ‘chunks’ of the divisor.

End of Year Objectives Y3 - Write and calculate mathematical statements for division using the multiplication tables that they know, including for two-digit numbers divided by one-digit numbers, progressing to formal written methods

Year 4 (Stage 6)

End of Year Objectives - Divide numbers up to 3 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context.

The 'chunking' method, following on from repeated subtraction.

Year 5(Stage 6 contd)

End of Year Objective - Divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context. Solve money and measures problems.

Continue with more efficient methods of chunking the divisor.

Year 6(Stage 6 contd)

End of Year Objective - Divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context.

Show the remainder as a fraction and a decimal.

Year 6 contd

4) 752

Stage 7 – Introduce short division quick method

Vocabulary

DivisionDIVIDED

BY

DIVISIBLE BY

HALVE

REPEATED SUBTRACTION

GROUPSHARE

DIVIDE

INTO