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New National Curriculum introduced September
2014, for all year groups (except Years 2 and 6 – they are from September 2015).
Expectations have been increased, and some of what was in the Year 4 curriculum is now in Year
3.
All multiplication tables up to 12 x 12 need to be learnt by the end of Year 4.
At Longfield, we expect that by the end of Year 3, children will know their 2x, 3x, 4x, 5x, 8x, 10x and 11x tables.

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What do you understand by:
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Multiplication?
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Division?
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What vocabulary can you think of that applies to each of them?
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What equipment, pictures or images can you think of that might help children when starting multiplication and division?

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X repeated addition eg 5 x 3 is the same as
(equals) 3 + 3 + 3 + 3 + 3 times lots of groups of multiplied by multiply times tables double

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Repeated subtraction eg 20 ÷5 = 20 – 5 – 5 – 5 - 5
Divide
Divided by
Share
Share equally
Groups
Lots
Halve

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At Longfield, we also ask for the 11x table to be known by the end of Year 3.
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By the end of Year 2, children should already know the 2x, 5x and 10x tables, and should already be well on the way to knowing the 3x and 4x tables.
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What is meant by multiplication and division facts?

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Children need to know :
Each multiplication table from
0 x 3 to 12 x 3 0 x 4 to 12 x 4
0 x 8 to 12 x 8 0 x 11 to 12 x 11
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They need to be able to count in 3s, 4s, 8x and
11s, forwards and backwards – at least to the end of that table.
They need to know facts such as 3 more than
18, 8 less than 40.
They also need to be able to answer questions quickly such as ‘What is 10 times 11?’ ‘Twelve lots of 8?’ ‘How many groups of 4 in 28?’
How many 3s in 27? These are multiplication and division facts .

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Write the answers:
8 times 3?
How many 4s in 28?
What are nine groups of eleven?
Sixteen is how many groups of four?
What is 8 multiplied by 5?
Divide 121 by 11.
Seven eights are ... ?
How many 3s in 27?

Practising objectives on Numeracy Passports will help. If they have completed Australasia by the end of Year 3 then they will be well on the way to meeting the National
Curriculum requirement of knowing all tables up to 12 x
12 by the end of Year 4. All Numeracy Passports and parent helpsheets are on the Longfield School website.
Knowing tables and their multiplication and division facts is really important as they are a vital part of being able to problem-solve and reason.

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To meet this objective, children need to:
Understand the place value of numbers
Partition numbers into hundreds, tens and ones
In Year 3, children work with numbers up to 1000.
They need to understand the place value of each digit in a number, as hundreds, tens and units/ones.
So 876 = 800 + 70 + 6 H T U
8 7 6

Partition these numbers into hundreds, tens and ones, as an addition, and using place value headings
70 + 2 T U
7 2
426
504
370
400 + 20 + 6 H T U
4 2 6
500 + 4 H T U
5 0 4
300 + 70 H T U
3 7 0

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Addition and multiplication are commutative operations.
This means that for these operations the numbers can be added or multiplied in any order and the answer will still be the same. (You can think of it like the word ‘commuter’ – like people the numbers can go back and forth or change place and still be the same!)
So 8 + 4 = 12 is the same as 4 + 8 = 12
And 8 x 5 gives the same answer as 5 x 8.
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But subtraction and division are not commutative.
8 – 4 (= 4) is not the same as 4 – 8. (=-4)
And 40 ÷ 5 (= 8) is not the same as 5 ÷ 40 (= 0.125)
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Children need to be taught this – they don’t automatically know or recognise it, and often won’t notice or realise that they have written the calculation the wrong way round.

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Know multiplication facts, including that you can do a multiplication calculation in any order, for example: 2 x 4 = 8 = 4 x 2
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So children need to be able to balance calculations like the one above.
Try these!
5 x 3 = 15 = ? x ?
5 x 3 = 15 = 3 x 5
11 x 7 = ? = 7 x 11 11 x 7 = 77 = 7 x 11
If children are more confident with a particular times table, such as the 5x, then knowing that a calculation can be turned around might make it easier for them to work out.

 Know division facts, including that you cannot do a division calculation in any order, for example:
 6 ÷ 2 = 3 which is not the same as 2 ÷ 6 (= 0.333..)
If children are unsure about this, then they need to continue to work practically with concrete objects.
For example they could be asked to share 6 pencils between 2 children, then 2 pencils between 6 children.
Hopefully they will understand the difference once they try it out!

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One of the things children learn in Year 2 is about inverse operations. They learn to work out addition and subtraction facts such as
4 + 6 = 10 so 6 + 4 = 10 and 10 – 4 = 6 and 10 – 6 = 4
In Year 3 they are introduced to a similar idea but with multiplication and division.
If they know that
8 x 4 is 32, then they also know that 4 x 8 = 32
32 ÷ 8 = 4 and 32 ÷ 4 = 8

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Give all multiplication and division facts for these statements
40 ÷ 8 = 5
10 x 5 = 50 40 ÷ 5 = 8
50 ÷ 10 = 5 8 x 5 = 40
50 ÷ 5 = 10 5 x 8 = 40

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If children know that 2 x 4 = 8, then using knowledge of place value, they will be able to multiply by 10 to find the correct answer.
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Along with knowing number bonds for all numbers up to 20, knowing times tables and understanding place value are the most important concepts children need to learn in order to calculate quickly and efficiently.

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We teach children to multiply by 10 by moving digits one place to the left on a place value board. They need to use zero as a placeholder. Once they have understood this, they can visualise what to do in their heads, and can work the answer out mentally.
So 70 x 4 would look like this

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Partitioning is splitting a number into other numbers
Eg 5 can be partitioned into
4 + 1, 3 + 2
52 can be partitioned into
50 + 2
246 can be partitioned into
200 + 40 + 2
Children need to be able to partition to carry out the grid method.

The grid method is a method of multiplying using partitioning.
2-digits (or more) are split up and each number multiplied separately, before being added together again.
It is a step on the way to short and long multiplication.

84 x 7 =

Use chunking on an empty number line to work out the division
72 ÷ 3

Multiplication of a 2-digit (or more) number by a single-digit number.
Children need to understand when they need to carry tens and hundreds, and to add them on once carried.
H T U
Numbers need to be lined up in columns according to place value.
Place value headings should be used at first.

Chunking – becoming more formal
85÷ 5
85
-50 10 x 5
35
-25 5 x 5
10
-10 2 x 5
0 17
85 ÷ 5 = 17
Chunking involves repeatedly taking away
‘chunks’ of easily calculated numbers from the number being divided
(dividend – here the dividend is 85). They need to be chunks of the divisor
(here the divisor is 5).
Once zero is reached, the number of chunks added together gives the answer.
Your turn!
Try chunking like this with the calculation on the board.

Try these calculations using short division
2 1 18 r 2
3)6 3 5)9 2

Write the answer.
24 × 4 =

Solve problems, including missing number problems, involving multiplication and division, including integer scaling problems and correspondence problems in which n objects are connected to m objects.
Children need to know that if you are scaling up you will be using multiplication for example:
To make 7 kg of concrete you need 1kg of cement, 2kg of sand and 4kg of gravel. So how much cement, sand and gravel would you need to make 21 kg of concrete?
Can you work it out? What do you need to do?
There are 3 lots of 7 in 21, so you need to multiply all the amounts by 3. So 3kg cement, 6kg sand,
12kg gravel. All those added together make 21kg.

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Use the inverse operation – division. So
12 ÷ 3 = 5 5 x3 = 12
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Divide 45 by 9 = 5
60 ÷ 5 = 9

Solve multiplication and division problems in context including measuring and scaling contexts, for example: 4 times as high, 8 times as long, 12 sweets shared between 4 children.
Apple pudding – serves 4
• 400g cooking apples
• 50g soft brown sugar
• 1 egg
• Grated rind of 1 lemon
• 80g self-raising flour
• 90g caster sugar
• 75g butter
• 100ml milk
If this recipe makes enough for 4, how much flour would you need to make enough for 12 people?
How much sugar?

Make use of opportunities to give your child practical experience of mathematics in the home and everyday life, such as:
• following recipes and changing them for different numbers of people;
• working out quantities of different parts to make diluted drinks, colours of paint or cement;
• comparing prices for single and multibuy packs to decide which is better value.

Understand how to solve a correspondence problem, for example: you have 3 hats and 4 coats. How many different outfits can you make?
Each coat could have 3 different hats. So 4 coats multiplied by 3 hats equals 12 different outfits.

Talk to someone else – what do you think?

Why do children find fractions difficult?
Difficulties with fractions often stem from the fact that they are different from natural numbers in that they are relative rather than a fixed amount - the same fraction might refer to different quantities and different fractions may be equivalent (Nunes, 2006).
Would you rather have one quarter of £20 or half of £5?
The fact that a half is the bigger fraction does not necessarily mean that the amount you end up with will be bigger. The question should always be, 'fraction of what?';
'what is the whole?'. Fractions can refer to objects, quantities or shapes, thus extending their complexity.

• A fraction is made up of 2 numbers. The top number is called the NUMERATOR and the bottom number is called the
DENOMINATOR. In the fraction ¾, 3 is the numerator and 4 is the denominator.
• DENOMINATOR
This number shows how many equal ‘pieces’ something has been divided into. In the fraction ¾, 4 is the denominator showing that there are 4 equal pieces making up the whole.
• NUMERATOR
This number shows how many of those pieces there are. In the fraction ¾ there are 3 pieces out of the total of 4.

Numerators and Denominators
For example, if a pizza is cut into 4 equal slices there will be 4 pieces on the plate. This makes a fraction of 4/4 (1 whole).
If I eat one of those pieces, ( ¼) then there are 3 pieces left. ( ¾ ).
The denominator stays the same, there are still 4 parts that made up the whole pizza, but the numerator has changed, as there are only 3 parts of the pizza left.

Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10
Children need to practise counting in tenths and relate tenths to place value.
1/10
How many tenths do these pictures show?
4/10

Recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators
What is a unit fraction?
A fraction with 1 as the numerator eg ½ ¼ 1/3 1/10
What is a non-unit fraction?
A fraction with any other number as the numerator eg 2/3
7/10 3/4
Find fractions of a discrete set of objects, for example: you have a basket containing 24 cubes. If you take out 1/4 of the cubes how many cubes have you taken? If you take out 1/3 of the cubes how many cubes have you taken? If you take out 3/4 of the cubes how many cubes have you taken? If you take out 2/3 of the cubes how many cubes have you taken?

Recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators
Children need to be able to order fractions on a number line. Eg put in order 1/3 3/4 ¼ ½
¼ 1/3
½
3/4

Make some different fraction strips.
Use your strips of paper to:
What fractions can you find that are equivalent to 1/4?
Which is larger, 5/8 or ¾?

Recognise and show, using diagrams, equivalent fractions with small denominators

Pictures showing equivalent fractions
=

What fraction is each part of the whole?
What other fractions can you make?
What equivalences can you find?

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Some fractions with the same denominator will add to a total of 1, eg 1 /
3
+ 2 /
3
= 1
If fractions have the same denominator then you just need to add the numerators together.
Some answers will be less than 1 eg
3 /
10
+ 4 /
10
= 7 /
10
Remember not to add the denominators.
In the same way, to subtract fractions with the same denominator, just subtract the smaller denominator from the larger.
9
10
2
10
7
10

Compare and order unit fractions, and fractions with the same denominator
Children need to compare two fractions with the same denominator and recognise which is larger/smaller
Try this! Put these fractions in order:
3/8 1/8 6/8 4/8 1/8 3/8 4/8 6/8
They need to realise, that the larger the denominator, the smaller the unit fraction.
Put these in order smallest first: 1/9 1/3 1/5 1/4
1/9 1/5 ¼ 1/3

Solve problems that involve all of the elements of the above.

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 http://www.bbc.co.uk/bitesize/ks2/maths/number/multiplicatio n_division/read/1/ http://resources.woodlandsjunior.kent.sch.uk/maths/timestable/interactive.htm
http://www.theschoolrun.com/times-tables http://www.topmarks.co.uk/maths-games/7-11years/multiplication-and-division http://www.topmarks.co.uk/maths-games/7-11years/fractions-and-decimals http://www.bbc.co.uk/bitesize/ks2/maths/number/fractions_ba sic/play/ http://www.bgfl.org/custom/resources_ftp/client_ftp/ks2/math s/fractions/index.htm
http://www.st-maryschurchdown.gloucs.sch.uk/files/8314/2350/5383/Fractions_Gui dance.pdf