File - Gobowen Primary School

Progression in calculation Year N to Year 2: MULTIPLICATION AND DIVISION



National Curriculum statements 2014 (statutory) in blue and bold

Nursery Reception

EYFS

Mathematics development involves providing children with opportunities to practise and improve their skills in counting numbers, calculating simple addition and subtraction problems, and to describe shapes, spaces, and measures.

Early Learning Goal 11: Numbers

[Expected outcomes] Children count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer. They solve problems, including doubling, halving and sharing.

EYFS Profile exemplification for the level of learning and development expected at the end of the

EYFS Mathematics

Examples of doubling, halving and repeated addition for ‘expected’ achievement related to ELG:11 taken from exemplification

No examples of these aspects in the ‘expected’ exemplification for ELG 11

Taken from: Non-statutory guidance: Development Matters 30 to 50 months A unique child

• Uses some number names and number language spontaneously.

• Uses some number names accurately in play.

• Recites numbers in order to 10.

• Knows that numbers identify how many objects are in a set.

• Beginning to represent numbers using fingers, marks on paper or pictures.

• Sometimes matches numeral and quantity correctly.

• Shows curiosity about numbers by offering comments or asking questions.

• Compares two groups of objects, saying when they have the same number.

• Shows an interest in number problems.

• Separates a group of three or four objects in different ways, beginning to recognise that the total is still the same.

• Shows an interest in numerals in the environment.

• Shows an interest in representing numbers.

• Realises not only objects, but anything can be counted, including steps, claps or jumps.

Multiplication and division structures : synonymous at this stage combining two or more groups of the same size and counting to establish total ( repeated addition or repeated aggregation );

Also division as uequal sharing and also as grouping

Recording: teacher demonstration of calculation of pictorial recording. Children begin using pictorial recording.

See children’s-mathematics link below.

VOCABULARY: more, and, make, sum, total, altogether, take away, leave,

Equipment : every day objects, counters, fingers http://www.childrens-mathematics.net/ see ‘galleries’ http://www.foundationyears.org.uk/wp-content/uploads/2011/10/Numbers_and_Patterns.pdf

Taken from: Non-statutory guidance: Development Matters 40 to 60+ months A unique child

• Recognise some numerals of personal significance.

• Recognises numerals 1 to 5.

• Counts up to three or four objects by saying one number name for each item.

• Counts actions or objects which cannot be moved.

• Counts objects to 10, and beginning to count beyond 10.

• Counts out up to six objects from a larger group.

Selects the correct numeral to represent 1 to 5, then 1 to 10 objects.

• Counts an irregular arrangement of up to ten objects.

• Estimates how many objects they can see and checks by counting them.

• Uses the language of ‘more’ and ‘fewer’ to compare two sets of objects.

• Finds the total number of items in two groups by counting all of them.

• Says the number that is one more than a given number.

• Finds one more or one less from a group of up to five objects, then ten objects.

• In practical activities & discussion, beginning to use the vocabulary involved in adding & subtracting.

• Records, using marks that they can interpret and explain.

• Begins to identify own mathematical problems based on own interests and fascinations.

Multiplication and division structures : synonymous at this stage combining two or more groups of the same size and counting to establish total ( repeated addition or repeated aggregation );

Also division as equal sharing and also as grouping

Recording: teacher demonstration of calculation to match pictorial recording using numberlines

Children begin using pictorial recording and numberlines. See children’s-mathematics link below.

VOCABULARY : double; half, halve; pair; group; divide into groups of; count out, share out; left, left over

Equipment and resources: every day resources but also ‘maths’ resources such as number tracks, number lines, counters, fingers, straws organised into groups of 10 http://www.childrens-mathematics.net/ see ‘galleries’ http://www.foundationyears.org.uk/wp-content/uploads/2011/10/Numbers_and_Patterns.pdf

Exemplification and activities taken from NCETM National Curriculum toolkit

Year 1

Programme of Study statement: Pupils should be taught to:

 solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of a teacher

Children should be able to:

Use practical apparatus, arrays and images to help solve multiplication and division problems such as:

 Ben had 5 football stickers. His friend Tom gave him 5 more, how many does he have altogether?



Share 12 sweets between two children. How many do they each have?



Find half of and double a number or quantity:



16 children went to the park at the weekend. Half that number went swimming. How many children went swimming?



I think of a number and halve it. I end up with 9, what was my original number?

NCETM Actvity A -

Noah’s Ark

Give the children the opportunity to count in twos, finding the total number of animals on board the Ark. As the children gain fluency counting in twos, start at different numbers and perhaps changing from using concrete objects, to jumping in twos along a number line. Further uses could be to find the number of groups of two on the ark, again initially using tangible objects, then moving on to using a number line and demonstrating repeated subtraction.

Activity B - Arrays powerpoint

This resource is based in ‘PowerPoint’. The teacher can set simple multiplication word problems for the children to solve. It is also useful for modelling arrays and how to write a multiplication sentence.

Activity C - Whole class counting sessions

For this activity the children themselves are the objects to count. You can count in twos to find the number of shoes in a group, count fingers on hands in fives and number of toes in tens. To extend the children you can ask them to model how to write down this calculation or alter it to practise their division facts from the 2, 5 and 10 times tables.

Activity D - NRICH Share Bears:

A lovely investigation involving the children in division by sharing, and early introduction to the concept of remainders.

Non-statutory guidance

Through grouping and sharing small quantities, pupils begin to understand: multiplication and division; doubling numbers and quantities; and finding simple fractions of objects, numbers and quantities.

Year 2

Programme of Study statement: Pupils should be taught to:

 calculate mathematical statements for multiplication and division within the multiplication tables and write them using the multiplication (x), division (÷) and equals signs

Children should be able to: Find missing numbers or symbols in a calculation:

4 x __ = 20, __ ÷ 10 = 3

Anna has 3 boxes of cakes. Each box contains 5 cakes. How many cakes does she have altogether? Show how you worked this out.

NCETM Activity B



BBC KS1 Starship Number Jumbler game , a simple animation where children choose the missing operation

 sign in the calculation.

Write the calculation: give the children different pictures of groups of items or arrays. They then have to write the multiplication sentence to match the picture. 2p, 5p and 10p coins could be used for this activity. The children can write down how many of each coin they have and the total amount. Using coins, the children could also write division calculations to match the images.

 show that multiplication of two numbers can be done in any order (commutative) and division of one number by another cannot

Children should be able to: Use their knowledge of triangles of numbers to show related number facts. e.g. If 6 x 2

= 12 then 2 x 6 = 12 and 12 ÷ 6 = 2.

NCETM Activity C



Triangle of numbers : a good activity to use as a starter or plenary, demonstrating the commutativity of

 multiplication. It can be used to demonstrate one number fact and the children can suggest the others.

Function box : reinforcing the relationship of division being the inverse of multiplication. To build up to this online activity, a function box could be made and used as a visual resource in class. For example, model how a number goes in and doubles, but put the number back through the machine it will halve. There are

 lots of different ways to use the function box in class to deepen the understanding of the relationship between multiplication and division.

Class number sentence: using digit cards and x and

÷ and = cards. Get the children to show how we can take one known fact and find others using those numbers. As the children move the digits around, they will demonstrate their understanding of using and applying their tables knowledge. What numbers can we move around in a division sentence? Can they spot the relationship between multiplication and division? Ensure they really understand the concept behind this activity, by encouraging them to show you with practical apparatus

 solve problems involving multiplication and division, using materials, arrays, repeated addition, mental methods, and multiplication and division facts, including problems in contexts

Children should be able to: Use various methods and apparatus to help them solve word problems such as:

There are 10 lollies in a bag. Charlie needs 30 lollies for his party. How many bags does he need to buy? Show how you worked this out.

NCETM Activity D



TES word problems with differentiated questions : resources submitted by a Year 2 teacher, focusing on problem solving using multiplication.



BBC Class Clips: problem solving - how many chairs?

organise their tables ready f or their guests’ arrival.

Use multiplication facts to help the Chuckle Brothers



Give the children a multiplication or division fact. Can they write a word problem to match it? Now swap calculations with a partner and talk about how you would solve the problem.

 recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers

Children should be able to: Recognise a multiple of 2, 5 or 10 and use their knowledge of multiplication facts to find corresponding division facts. They can say which numbers are odd and which are even. e.g. 2 x 5 = 10, show me three more number facts using these numbers. Is 34 an odd number? How do you know?

NCETM activity A:



Multiplication tables - pelmanism style matching cards. Find the pairs of cards showing the product and multiplication fact in a timed activity.



Counting stick: using a metre stick with ten divisions use sticky notes to mark multiples of 2, 5 or 10. Practise counting on and back from different numbers. As the children become more fluent, remove more sticky notes and see how they can recall multiples starting at different points counting on and back.

BBC KS1 Bitesize: Camel Times Tables and Division Mine . These two games are a super resource for the children to practise their recall of multiplication and division facts independently

Non-statutory guidance

Pupils use a variety of language to describe multiplication and division.

They make connections between arrays, number patterns, and counting in twos, fives and tens.

Pupils are introduced to the multiplication tables. They practise to become fluent in the 2, 5 and 10 multiplication tables and connect them to each other. They connect the 10 multiplication table to place value, and the 5 multiplication table to the divisions on the clock face. They begin to use other multiplication facts, including using related division facts to perform written and mental calculations.

Pupils work with a range of materials and contexts in which multiplication and division to relate to grouping and sharing discrete and continuous quantities, and relating these to fractions and measures (e.g. 40 ÷ 2 = 20, 20 is a half of 40). They use commutativity and inverse relations to develop multiplicative reasoning (e.g. 4 x 5 = 20 and 20

÷ 5= 4).

NB: In Y1 most strategies are based on counting but mental strategies are developing

Multiplication and division structures : repeated aggregation (addition) of groups of the same size and recorded as an array; begin to recognise sharing equally; also repeated addition of groups of the same size i.e. divided into groups of ….

Recording: Following teacher demonstration, children continue to develop pictorial recording including an array; beginning to record on their own empty number line.

VOCABULARY : double, groups of, sets of, lots of; groups, share, left over, half, each

Equipment and resources : number lines, counters, fingers, straws, Base 10 equipment, every day objects, counters, fingers,

ITPs : Multiplication Array ; Grouping ;

NCETM video examples:

‘Multiple Representations’

and

‘

Reinforcing tables facts

The commutative law of multiplication at KS1 Sharing and grouping

NB: In Y2 mental and written strategies are still largely interchangeable

Multiplication structures : to recognise repetitive addition of groups of the same size; as an array; counting in steps of 2, 5 & 10

Division structures: recognise division as sharing equally; and, also repeated addition or subtraction of groups of the same size i.e. grouping

Recording: Following teacher demonstration of calculation to match pictorial recording (number line & array) , c hildren to continue to develop recording especially using pictures and their own number lines. They begin to use standard notation (including the symbols x ÷ and =)

VOCABULARY: lots of, groups of; times, multiply, multiplied by; multiple of; once, twice, three times… ten times…; times as (big, long, wide… and so on); repeated addition; array; row, column; double, halve; share, share equally; one each, two each, three each…; group in pairs, threes… tens; equal groups of; ¸, divide, divided by, divided into; left, left over

Equipment and resources : number lines, hundred squares, counters, array, pegboards, straws, squared paper, cubes etc. & everyday objects e.g. bars of chocolate, egg boxes,

ITPs : Multiplication Array ; Multiplication facts ; Number dials ; Grouping

NCETM video examples:

‘Multiple Representations’

and Reinforcing tables facts and The commutative law of multiplication at KS1 Sharing and grouping Also, BBC Learning Zone Class Clips

©

Shropshire Council Cathryn.Hardy@shropshire.gov.uk

C. Hardy Calculation policy version 12/04/2020 1

Progression in calculation Year N to Year 2: MULTIPLICATION AND DIVISION



National Curriculum statements 2014 (statutory) in blue and bold

Nursery

School examples of multiplication

Example from exemplification


How many feet have these three teddy bears got altogether?

Strategy: begin to recognise repetitive addition of groups of the same size

Recording: teacher demonstration of appropriate pictorial recording where appropriate.

Vocabulary: groups, sets

Equipment: every day objects, counters, fingers


Example from exemplification

Reception


How many wheels do we need for these three lego cars?

4 + 4 + 4 = 12

Strategy: begin to recognise repetitive addition of groups of the same size; counting in steps of 10 or 2

Recording: teacher demonstration of calculation to match pictorial recording using standard notation of + and =. Demonstrate on number line.

Vocabulary: double, groups of, sets of, lots of

Equipment: every day objects: e.g. cars, chairs, bears, children, fingers, gloves, toy cars, pairs of socks. Also ‘maths’ objects e.g. counters,

Exemplification and activities taken from NCETM National Curriculum toolkit

Year 1

Multiplication: example from exemplification (NCETM)



Ben had 5 football stickers on each page. He has two pages of stickers. How many does he have altogether? ARRAY and

0 1 2 3 4 5 6 7 8 9 10

Show two groups of 5 football stickers. Model this on a bead bar/string.

Demonstrate recording of two groups of 5 stickers on a number line.

Also show as a rectangular ‘array’ i.e. two groups of 5 or 5 + 5 = 10

Year 2

Multiplication: example

Example from exemplification: Anna has 3 boxes of cakes. Each box contains 5 cakes. How many cakes does she have altogether? Show how you worked this out.

For example, ‘ practically

’ and counting in 5s.

Model on a bead bar/string as ‘ repeated’ addition

i.e. 5 + 5 + 5 = 15

AND ‘5 three times equals 15’ so 5 x 3 = 15 or ‘five multiplied or times by three equals 15’.

Demonstrate recording on a number line. x 5 1 1

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

Model/demonstrate as a rectangular ‘array’ using the cakes or beads (fold bead string) or counters (left). Also show as a rectangular

‘array’ (right) as ‘three rows of 5’ i.e. 5 x 3 =15

Use the ITP Multiplication Facts to demonstrate these processes and also commutativity.

School examples of division

If we share out these cakes so everyone has one each, how many will be left over? If everyone has two cakes, how many children will be able to have cakes today?

School examples of division

Can we share out these cakes fairly? How shall we do it?

If we put two cakes on each plate, how many plates do we need?

Strategy: begin to recognise sharing equally; also repetitive addition or subtraction of groups of the same size

Recording: teacher demonstration of appropriate pictorial recording where appropriate.

Vocabulary: groups, share


Strategy: begin to re cognise sharing equally; also repetitive addition of ‘groups’ of the same size i.e. grouping

Recording: teacher demonstration of calculation to match pictorial recording using standard notation of + and =. Demonstrate on number line.

Vocabulary: groups, share, left over, half


Example from division exemplification (NCETM)



Share 12 sweets between two children. How many do they each have?

0 2 4 6 8 10 12

2 + 2 + 2 + 2 + 2 + 2 = 12

Or ‘sharing’ model to create an array, as above.

EQUAL SHARING Show 12 sweets. Share the sweets equally between two i.e. one for you and one for me

– we’ve used 2 of the sweets. Model this on a bead bar/string and demonstrate recording of one group of 2 sweets on a number line….. Another one for you and one for me – w e’ve used 2 more of the sweets. Model this on a bead bar/string and demonstrate recording of the second group of 2 sweets on a number line….. etc.

ALSO: use as above with GROUPING e.g. We have 12 sweets. If each child has 2 sweets, how many children will have 2 sweets? i.e. two for you

– that’s one group of two – that’s one person - model on bead bar and record on a number line as above.

Opportunities for links with other mathematics domains:

Cross-curricular and real life connections

Learners will encounter multiplication and division in:

Money - when shopping and recognising prices of items, ordering items by price, finding quantities in multiple purchases, sales prices, sharing costs.

Measurement - calculating area and perimeter, finding journey distances, reading and calculating scales, adjusting recipe quantities.

Data - interpreting and evaluating data, calculating amounts from pie charts and pictograms.

Grouping ITP

Example from division exemplification (NCETM)

Other example: How many sticks of 4 cubes can you make from 20 cubes? OR If 20 cubes are shared equally between 4 people, how many cubes do they each get?

GROUPING Show 20 cubes. Divide the cubes into groups of 4 i.e. one group of 4, two groups of 4 and so on…..

Model this on a bead bar/string and demonstrate recording of one group of 4 cubes on a number line….. And another group of 4 cubes etc. So, 4 + 4 + 4 + 4 + 4 = 4 x 5 = 20 so 20 ÷ 4 = 5 i.e. 20 divided into groups of 4 equals 5.

÷4

1

0 4

1

8

1

12

1

16

1

20

EQUAL SHARING use same model as above. e.g. We have 20 cubes. Share the cubes equally between each person

– we’ve used 4 of the cubes. Model this on a bead bar/string and demonstrate recording of one group of 4 cubes on a number line….. Another one for each person etc. So, 4 + 4 + 4 + 4 + 4 = 4 x 5 = 20 so 20 ÷ 4 = 5 i.e. 20 shared equally between 4 equals 5.

Find missing numbers or symbols in a calculation:

 ÷ 10 = 3 Represent e.g. using the Singapore bar:

The bar above represents the unknown quantity.

If we divide the bar into 10 equal sections and we know that each section has a value of 3.

3

3 3 3 3 3 3 3 3 3

So, 3 +3 +3 +3 +3 +3 +3 +3 +3 +3 = 30 OR 3 x 10 = 30

So the value of the empty box here

 ÷ 10 = 3 must be 30 because 30

÷ 10 = 3 OR as above but divided into GROUPS of 10.




Money – shopping: finding quantities in multiple purchases, sales prices, sharing costs.

Measurement - calculating area and perimeter, finding journey distances, reading and calculating scales, adjusting recipe quantities.

Data

– interpreting and evaluating data, calculating amounts from pie charts and pictograms

©



Progression in calculation Year 3 to Year 6: MULTIPLICATION AND DIVISION National Curriculum statements 2014 (statutory) in blue and bold Exemplification and activities taken from NCETM National Curriculum toolkit

Year 3

Programme of Study Pupils should be taught to:

 recall and use multiplication and division facts for the 3,4 and 8 times tables.

 multiply seven by three; what is four multiplied by nine? Etc.

 Circle three numbers that add to make a multiple of 4: 11 12 13 14 15 16 17 18 19



Leila puts 4 seeds in each of her pots. She uses 6 pots and has 1 seed left over. How many seeds did she start with?

 At Christmas, there are 49 chocolates in a tin and Tim shares them between himself and 7 other members of the family. How many chocolates will each person get?

 write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods.



One orange costs nineteen pence. How much will three oranges cost?

 Mark drives 19 miles to work every day and 19 miles back. He does this on Mondays, Tuesdays,

Wednesdays, Thursdays and Fridays. How many miles does he travel to work and back in one week?

 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.



Miss West needs 28 paper cups. She has to buy them in packs of 6 How many packs does she have to buy?

NCETM ACTIVITES

Non-statutory guidance

Pupils continue to practise their mental recall of multiplication tables when they are calculating mathematical statements in order to improve fluency. Through doubling, they connect the 2, 4 and 8 multiplication tables.

Pupils develop efficient mental methods, for example, using commutativity (e.g. 4 x 12 x 5 = 4 x 5 x

12= 20 x 12 = 240) and multiplication and divi sion facts (e.g. using 3 x 2 = 6, 6 ÷ 3 = 2 and 2 = 6 ÷3) to derive related facts (30 x 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).

Pupils develop reliable written methods for multiplication and division, starting with calculations of two-digit numbers by one-digit numbers and progressing to the formal written methods of short multiplication and division.

Pupils solve simple problems in contexts, deciding which of the four operations to use and why, including measuring and scaling contexts, and correspondence problems in which m objects are connected to n objects (e.g. 3 hats and 4 coats, how many different outfits?; 12 sweets shared equally between 4 children; 4 cakes shared equally between 8 children).

Multiplication structures : to recognise repeated aggregation (addition of groups of the same size); and scaling (increase/decrease a number of times/by a scale factor of)

– both structures should be represented as an array.

Division structures: recognise division as equal sharing ; the inverse of repeated aggregation

(divided into groups of…) i.e. grouping

Recording: Following teacher demonstration to link pictorial recording (number line & array) , c hildren continue to develop recording especially using pictures and their own number lines. They use standard notation (including the symbols x ÷ and =)

VOCABULARY: lots of, groups of; ´, times, multiply, multiplication, multiplied by; multiple of, product, repeated addition, array; row, column;, halve; share, share equally; one each, two each, three each… group in pairs, threes… tens; equal groups of, divide, division, divided by, divided into, left, left over, remainder

Language of scaling

: once, twice, three times… ten times…; times as (big, long, wide… and so on); double , s caling, scale, factor, doubling, trebling, so many times bigger, than (longer than, heavier than, and so on), so many times as much as (or as many as).


ITPs : Multiplication Array ; Multiplication facts ; Multiplication tables ; Number dials ; Grouping ;

Multiplication Board (tables square); Multiplication grid (grid multiplication)

NCETM video examples: Times Tables in Ten Minutes ; Grid multiplication as an interim step

Year 4


 recall multiplication and division facts for multiplication tables up to 12 × 12


 Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency. e.g.

o One orange costs nineteen pence. How much will three oranges cost? o What is twenty-one multiplied by nine? o How many twos are there in four hundred and forty?

 use place value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1; dividing by 1; multiplying together 3 numbers




 Pupils practise mental methods and extend this to three-digit numbers to derive facts, for example

200 × 3 = 600 into 600 ÷ 3 = 200.

e.g.

o Divide thirty-one point five by ten. o Ten times a number is eighty-six. What is the number?

 recognise and use factor pairs and commutativity in mental calculations


 Pupils write statements about the equality of expressions (e.g. use the distributive law 39 × 7 = 30 ×

7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations e.g. 2 x 6 x 5 = 10 x 6.

 e.g. Understand and use when appropriate the principles (but not the names) of the commutative, associative and distributive laws as they apply to multiplication:  



Example of commutative law

8 × 15 = 15 × 8

 Example of associative law 6 × 15 = 6 × (5 × 3) = (6 × 5) × 3 = 30 × 3 = 90

 Example of distributive law 18 × 5 = (10 + 8) × 5 = (10 × 5) + (8 × 5) = 50 + 40 = 90

 multiply two-digit and three-digit numbers by a one-digit number using formal written layout

 solve problems involving multiplying and adding, including using the distributive law to multiply two-digit numbers by 1 digit, integer scaling problems and harder correspondence problems such as n objects are connected to m objects




Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or three cakes shared equally between 10 children. e.g.

o 185 people go to the school concert. They pay £l.35 each. How much ticket money is collected? o Programmes cost 15p each. Selling programmes raises £12.30. How many programmes are sold?

NCETM ACTIVITES


Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.

Pupils practise mental methods and extend this to 3digit numbers to derive facts, (for example 600 ÷ 3

= 200 can be derived from 2 x 3 = 6).

Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers (see Mathematics Appendix 1 ).

Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 =

30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 =

60.

Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or 3 cakes shared equally between 10 children.

Multiplication structures : to recognise repeated aggregation (addition of groups of the same size); and scaling (increase/decrease a number of times/by a scale factor of)


Division structures: recognise division as equal sharing ; the inverse of repeated aggregation


Recording: Following teacher demonstration to link pictorial recording (number line & array) , c hildren continue to develop recording especially their own number lines, grid multiplication to represent arrays, and vertical recording of expanded vertical layouts of multiplication & division. They use standard notation (including the symbols x ÷ and =)

VOCABULARY: l ots of, groups of’ times, multiply, multiplication, multiplied by, multiple of, product, once, twice, three times… ten times…times as (big, long, wide… and so on), repeated addition, array, row, column, double, halve, share, share equally, one each, two ea ch, three each…, group in pairs, threes… tens, equal groups of, divide, division, divided by, divided into, remainder, factor, quotient, divisible by

Inverse


ITPs : Multiplication Array ; Multiplication facts ; Multiplication tables ; Number dials ; Multiplication

Board (tables square); Multiplication grid (grid multiplication) Moving Digits http://www.wmnet.org.uk/resources/gordon/Chunking.swf

NCETM video examples: Grid multiplication as an interim step Times Tables in Ten Minutes

Year 5


 identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers

 know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers

 establish whether a number up to 100 is prime and recall prime numbers up to 19



Use the vocabulary factor, multiple and product. They identify all the factors of a given number; eg, the factors of 20 are 1, 2, 4, 5, 10 and 20. They answer questions such as: o Find some numbers that have a factor of 4 and a factor of 5. What do you notice? o My age is a multiple of 8. Next year my age will be a multiple of 7. How old am I?



They recognise that numbers with only two factors are prime numbers and can apply their knowledge of multiples and tests of divisibility to identify the prime numbers less than 100. They explain that 73 children can only be organised as 1 group of 73 or 73 groups of 1, whereas 44 children could be organised as 1 group of 44,

2 groups of 22, 4 groups of 11, 11 groups of 4, 22 groups of 2 or 44 groups of 1. They explore the pattern of primes on a 100-sq uare, explaining why there’ll never be a prime number in the 10th column & the 4th column.

 multiply and divide numbers mentally, drawing upon known facts

 Rehearse multiplication facts and use these to derive division facts, to find factors of two-digit numbers and to multiply multiples of 10 and 100, e.g. 40 × 50. They use and discuss mental strategies for special cases of harder types of calculations, for example to work out 274 + 96,< 8006 – 2993, 35 × 11, 72 ÷ 3, 50 × 900. They use factors to work out a calculation such as 16 × 6 by thinking of it as 16 × 2 × 3. They record their methods using diagrams (such as number lines) or jottings and explain their methods to each other. They compare alternative methods for the same calculation & discuss any merits &disadvantages.

 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

 Develop and refine written methods for multiplication. They move from expanded layouts (such as the grid method) towards a compact layout for HTU × U and TU × TU calculations. They suggest what they expect the approximate answer to be before starting a calculation and use this to check that their answer sounds sensible.

For example, 56 × 27 is approximately 60 × 30 = 1800.

 multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000

 Recall quickly multiplication facts up to 10 × 10 and use them to multiply pairs of multiples of 10 and 100. They should be able to answer problems such as: the product is 400. At least one of the numbers is a multiple of 10.

What two numbers could have been multiplied together? Are there any other possibilities?

 recognise and use square numbers and cube numbers, and the not ation for squared (²) & cubed (³)

 solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes

 use knowledge of multiplication facts to derive quickly s quares of numbers to 12 × 12 & the corresponding squares of multiples of 10. They should be able to answer problems such as: o tell me how to work out the area of the cardboard with dimensions 30cm by 30cm o find two square numbers that total 45

 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

Extend written methods for division to include HTU ÷ U, including calculations with remainders. They suggest what they expect the approximate answer to be before starting a calculation and use this to check that their answer sounds sensible. They increase the efficiency of the methods that they are using. For example: 196 ÷ 6 is approximately 200 ÷ 5 = 40

32 r 4 or

4

6

or

2

3

6 196

Children know that, depending on the context, answers to division questions may need to be rounded up or rounded down. They explain how they decided whether to round up or down to answer problems eg.: Egg boxes hold 6 eggs. A farmer collects 439 eggs. How many boxes can he fill? OR Egg boxes hold 6 eggs. How many boxes must a restaurant buy to have 200 eggs?

 solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign

 solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates

NCETM ACTIVITES


Pupils practise and extend their use of the formal written methods of short multiplication and short division (see

Mathematics Appendix 1 ). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.

They use and understand the terms factor, multiple and prime, square and cube numbers.

Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (eg

, 98 ÷ 4 =

98

= 24 r 2 = 24

½ = 24.5 ≈ 25).

4

Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1000 in converting between units such as kilometres and metres.

Distributivity can be expressed as a(b + c) = ab + ac.

They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (eg, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 92 x 10).

Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example, 13 + 24 = 12 + 25; 33 = 5 x ).

Multiplication structures : to recognise repeated aggregation (addition of groups of the same size); and scaling

(increase/decrease a number of times/by a scale factor of)


Division structures: recognise division as equal sharing ; the inverse of repeated aggregation (divided into groups of…) i.e. grouping ; and, ratio structure for division i.e. comparison of two quantities – how many times less than or more than …

Recording: Following teacher demonstration to link pictorial recording (number line & array) , c hildren continue to develop recording especially grid multiplication to represent arrays, and vertical recording of expanded vertical layouts of multiplication & division. They use begin to use standard compact layouts for multiplication & division.

VOCABULARY: lots of, groups of, times, multiply, multiplication, multiplied by, multiple of, product, once, twice, three times… ten times…, times as (big, long, wide… and so on), repeated addition, array, row, column, double, halve, share, share equally one each, two each, three each…, group in pairs, threes… tens equal groups of, divide, division, divided by, divided into, remainder, factor, quotient, divisible by, inverse

Equipment and resources : a range of mathematics equipment as well as everyday situations especially those that produce arrays e.g. number of seats in an auditorium

ITPs : Multiplication Array ; Multiplication facts ; Multiplication tables ; Number dials ; Multiplication Board (tables square); Multiplication grid (grid multiplication) Moving Digits http://www.wmnet.org.uk/resources/gordon/Chunking.swf

NCETM video examples: Moving from grid to a column method Representing division with place value counters rapid recall of multiplication facts

Year 6


 multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

 Look at long-multiplication calculations containing errors, identify the errors and determine how they should be corrected.

 divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context

 Every day a machine makes 100 000 paper clips, which go into boxes. A full box has

120 paper clips. How many full boxes can be made from 100 000 paper clips?

 Each paper clip is made from 9.2 centimetres of wire. What is the greatest number of paper clips that can be made from 10 metres of wire?

 A DJ has two different sized storage boxes for her CDs. Small boxes hold 15 CDs.

Large boxes hold 28 CDs. The DJ has 411 CDs. How could the DJ pack her CDs?



Solve problems such as: Printing charges for a book are 3p per page and 75p for the cover. I p aid £4.35 to get this book printed. How many pages are there in the book?

Write down the calculations that you did. Seeds are £1.45 for a packet. I have £10 to spend on seeds. What is the greatest number of packets I can buy?

 solve problems involving multiplication and division

 use estimation to check answers to calculations and determine, in the context of a problem, levels of accuracy


 Give the best approximation to work out 4.4 × 18.6 and explain why. Answer questions such as: roughly, what answer do you expect to get? How did you arrive at that estimate? Do you expect your answer to be greater or less than your estimate?

Why?

 perform mental calculations, including with mixed operations and large numbers

 Use mental strategies to calculate in their heads, using jottings and/or diagrams where appropriate. For example, to calculate 24 × 15, they multiply 24 × 10 and then halve this to get 24 × 5, adding these two results together. They record their method as (24

× 10) + (24 × 5). Alternatively, they work out 24 × 5 = 120 (half of 24 × 10), then multiply 120 by 3 to get 360.

 identify common factors, common multiples and prime numbers



Children should be able to answer questions such as: o

How can you use factors to multiply 17 by 12? o Start from a two-digit number with at least six factors, e.g. 72. How many different multiplication and division facts can you make using what you know about 72? What facts involving decimals can you derive? o What if you started with 7.2? What about 0.72? o Which three prime numbers multiply to make 231? o use their knowledge of the order of operations to carry out calculations involving the four operations



Children should be able to find answers to calculations such as 5.6

⬜ = 0.7 or 3 × 0.6, drawing on their knowledge of number facts and understanding of place value. They should be able to approximate, use inverses and apply tests of divisibility to check their results.

 Children should know the square numbers up to 12 × 12 and derive the corresponding squares o f multiples of 10, for example 80 × 80 = 6400.

NCETM ACTIVITES


Pupils use the whole number system, including saying, reading and writing numbers accurately Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division (see Mathematics Appendix 1 ).

They undertake mental calculations with increasingly large numbers and more complex calculations.

Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.

Pupils round answers to a specified degree of accuracy, for example, to the nearest 10,

20, 50, etc, but not to a specified number of significant figures.

Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 +

1) x 3 = 9.

Common factors can be related to finding equivalent fractions.

Multiplication structures : to recognise repeated aggregation (addition of groups of the same size); and scaling (increase/decrease a number of times/by a scale factor of)


Division structures: recognise division as equal sharing ; the inverse of repeated aggregation


; and, ratio structure for division i.e. comparison of two quantities

– how many times less than or more than …

Recording: Following teacher demonstration to link pictorial recording (number line & array) , c hildren continue to develop recording especially grid multiplication to represent arrays, and vertical recording of expanded vertical layouts of multiplication & division.

They use begin to use standard compact layouts for multiplication & division.

VOCABULARY: lots of, groups of, times, multiply, multiplication, multiplied by, multiple of, product, once, twice, three times… ten times…, times as (big, long, wide… and so on), repeated addition, array, row, column, double, halve, share, share equally one each, two each, three each…, group in pairs, threes… tens, equal groups of, divide, division, divided by, divided into, remainder, factor, quotient, divisible by, inverse

Equipment and resources : a range of mathematics equipment as well as everyday situations especially those that produce arrays e.g. number of seats in an auditorium

ITPs : Multiplication Array ; Multiplication facts ; Multiplication tables ; Number dials ;

Multiplication Board (tables square); Multiplication grid (grid multiplication) Moving

Digits http://www.wmnet.org.uk/resources/gordon/Chunking.swf

NCETM video examples: Moving from grid to a column method Representing division with place value counters rapid recall of multiplication facts

©




Year 3 Year 4 Year 5


56 × 27 is approximately 60 × 30 = 1800.


Mark drives 19 miles to work every day. He does this on Mondays, Tuesdays, Wednesdays,

Thursdays and Fridays. How many miles does he travel to work in one week?

Demonstrate using a numberline as ‘ repeated’ addition

using straws or Base 10 as a model i.e.

19 + 19 + 19 + 19 + 19 = partitioned into 10s and 1s as:

10 + 9 + 10 + 9 + 10 + 9 + 10 + 9 + 10 + 9 =

10 9 10 9 10 9 10 9 10 9

0

Then ‘regrouped’ as 10s and 1s:

10 + 10 + 10 + 10 + 10 + 9 + 9 + 9 + 9 + 9 =

10 10 10 10 10 9 9 9 9 9

0

Multiplication: example

The class wants to make 235 spiders for a display. How many legs do they need to make?

235 x 8 is approximately 235 x 10 = 2350

Model using Base 10 equipment or straws. Use the equipment to demonstrate partitioning i.e. 235 x 8 =

200 x 8 + 30 x 8 + 5 x 8 = (200 + 30 + 5) x 8

Record on a number line.

x8 200 30 5

240 40

0 1600 1840 1880

Leading to formal recording of ‘ grid multiplication’ as a representation of the rectangular array. x 200 30

8 1600 240

5

40

1600 + 240 + 40 = 1880 so, 235 x 8 = 1880

Where possible, model using Base 10 equipment. See Y4 example. Use the equipment to show partitioning i.e. 56 x 27 = 50 x 20 + 50 x 7 + 6 x 20 + 6 x 7 = (50 + 6) x 27

Leading to formal recording of ‘ grid multiplication’ as a representation of the rectangular array. Possible to record on a number line

– see Y4 example.

Possible use of Excel to generate array to represent 56 x 27

56

27

This can be rewritten as: 10 x 5 + 9 x 5 = 50 + 45 = 95 x 5

0

10 9

45

50 95

Use informal mental methods of calculation i.e. PARTITIONING

19 x 5 = (10 x 5) + (9 x 5)

ITP: Multiplication array

Leading to formal recording of ‘grid multiplication’ as a representation of the rectangular array. (Link with

‘area’)

50 + 45 = 95 so, 19 x 5 = 95

OR COMPENSATION

i.e. 19 x 5 = 20 x 5

– 5 x

5 10 x 5 = 50

19

9 x 5 = 45

As 235 x 8 = 8 x 235 (commutative), the layout for grid multiplication can also be presented ‘vertically’ as a step towards

‘long multiplication’.

Leading to an expanded ‘vertical’ layout:. x

200

30

5

8

1600

240

40

1880

2 3 5 x 8

4 0 8 multiplied by 5 is 40

2 4 0 8 multiplied by 30 is 240

1 6 0 0 8 multiplied by 200 is 1600

1 8 8 0 To give a total of 1880

As 56 x 27 = 27 x 56

(commutative), the layout for grid multiplication can also be presented

‘vertically’ (largest number on the vertical axis) as a step towards ‘long multiplication’.

x

50

6

20

1000

120

1120

7

350

42

+ 392

So,

1120

+ 392

1512

1

Leading to an expanded ‘vertical’ layout. Either begin with least or most significant figure.

Leading to a compact ‘vertical’ layout: x

2

5

7

6

4 2 6 multiplied by 7 is 42





1 x

2

5

7

6




1

Division: example from exemplification (NCETM)

Miss West needs 28 paper cups. She has to buy them in packs of 6 How many packs does she have to buy?

÷ 6

0

1 1 1 1

6 12 18

r. 4

24 28

GROUPING i.e. 28 ÷ 6 in this context: ‘28 cups divided into groups of 6’.Modelled using cups then a bead string or bar and recorded on a number line as above. So, 28 ÷ 6 = 4 remainder 4

Also use Grouping ITP:

EQUAL SHARING e.g. There are 28 paper cups to share equally between 6 people. So if everyone has one paper cup , that’s one group of 6 paper cups. Demonstrate recording of one group of 6 paper cups on a number line….. Another one for each person etc.

So, 6 + 6 + 6 + 6 + 4 = 6 x 4 + 1 = 28 OR so 28 ÷ 6 = 4 r.4 i.e. 28 shared equally between 6 equals 4 remainder 4.

Division example: There are 87 shopping days to Christmas. How many weeks is that? 87



7

So, ‘87 divided into groups of 7’ is approximately 70 ÷ 7 = 10.

GROUPING: Model using bead bar or string. Demonstrate partitioning into groups of 7 using the interpretation of 87 ÷ 7 as ‘ 87 divided into groups of 7’

Record on a number line alongside bead bar or string.

÷7 1 1 1 1 1 1 1 1 1 1 1 1

r. 3

0 7 14 21 28 35 42 49 56 63 70 77 84

So, ’87 divided into groups of 7 equals 12 remainder 3’ i.e.

87



7 = 12 r.3 or 12

3

7

87

Then begin to divide into

‘groups of the divisor’ using groups of 10, 5 and 2 wherever possible:

÷7 10 2

14 r. 3

0 7 14 21 28 35 42 49 56 63 70 77 84 87

If children’s understanding is secure, introduce vertical recording e.g. during summer term.

Demonstrate vertical recording alongside number line recording. Begin to record information vertically introducing ‘repeated subtraction/chunking’ (or ‘repeated addition/additive chunking’.

8 7 ÷ 7

- 7 0

1 7

10

- 1 4 2

3

so, 87



7 = 12 r.3 or 12

3

7


196 ÷ 6 i.e. ‘196 divided into groups of 6’ is approximately 180 ÷ 6 = 30

Leading on from Year 4, divide into ‘groups of the divisor’ using multiplies of groups of 10, 5 and 2 wherever possible and record on a number line.

÷6

30 2

12 r. 4

0 7 14 21 28 35 42 49 56 63 180 771 192

So, ’196 divided into groups of 6 equals 32 remainder 4’ i.e.

196



6 = 32 r.4 or 32

4

6

or 32

2

3

196

When children’s understanding is secure, use vertical recording. Demonstrate vertical recording alongside number line recording. Begin to record information vertically introducing ‘repeated subtraction/chunking’ (or ‘repeated addition/additive chunking’) lead

Where necessary dividing by ‘groups of the divisor’ using groups of 10, 5 and 2

(see left) and then using ‘multiple groups of 10, 5 and 2 (see right)

1 9 6

- 1 8

1

0

6

÷ 6

30

- 1 2 2

4

so, 196



6 = 32 r.4 or 32

4

6

or 32

2

3

Year 6

Multiplication: example 1328 x 43 1328 x 43 is approximately 1200 × 40 = 48000

Use the equipment e.g. Base 10 materials, to show partitioning i.e. 1328 x 43 =

1000 x 43 + 300 x 43 + 20 x 43 + 8 x 43 = (1000 + 300 + 20 + 8) x 43

Leading to recording using ‘ grid multiplication’ as a representation of a rectangular array. Possible use of Excel to generate array

– see Y5 example.

As 1328 x 43 = 43 x

1328 (commutative), the layout for grid multiplication can also be presented

‘vertically’ as a step towards ‘long multiplication’.

x

1000

300

20

8

40

40 000

12 000

800

320

53 120

3

3000

900

60

24

+ 3984

So,

53 120

+ 3 984

57 104

Leading to an expanded

‘vertical’ layout.

Either begin with least or most significant figure.

1 3 2 8 x 4 3

2

6

4

0

3 multiplied by 8 is 24

3 multiplied by 20 is 60


3 0 0 0 3 multiplied by 1000 is 3000

3 2 0 40 multiplied 8 is 320


1 2 0 0 0 40 multiplied by 300 is 12 000

4 0 0 0 0 40 multiplied by 1000 is 40 000

5 7 1 0 4 To give a total of 57 104

2 1

1 3 2 8 x 4 3

Leading to a compact

‘vertical’ layout. Either begin with least or most significant figure.

5


3 1 2

0 40 multiplied by 1328 is

53 120

5 7 1 0 4 To give a total of 1880

1 1

Also include examples that involve decimals, money and other measures e.g. 18 · 6 x 4 · 4

As

18·6 x 4·4 = 4·4 x

18·6

(commutative), the layout for grid multiplication can also be presented ‘vertically’ as a step towards ‘long multiplication’.

x

10

8

0.6

4

40 · 0

32

·

0

2

·

4

74

·

4

4 · 00

3

·

20

0

+ 7

0

·

·

·

4

24

44

So,

74.40

+ 7.44

81

·

84

Leading to an expanded ‘vertical’ layout. Either begin with least or most significant figure and then a compact ‘vertical’ layout – see above.


Each paper clip is made from 9 · 2 centimetres of wire. What is the greatest number of paper clips that can be made from 10 metres of wire?

10m



9 · 2cm = 1000



9 · 2 i.e. ‘10m or 1000cm of wire divided into lengths (groups) of

9

· 2cm’ is approximately 1000 ÷ 10 = 100

Leading on from

Year 4 and Year 5, divide into ‘groups of the divisor’ using multiplies of groups of 10, 5 and 2 wherever possible and record on a number line.

÷ 9·2

100 8

73·6 r. 6

·4

0 7 14 21 28 35 42 49 920 993·6 1000

So, ’1000 divided into lengths of 9·2 equals 108 paper clips with 6.4cm of wire remaining.

When children’s understanding is secure, use vertical recording.

Demonstrate vertical recording alongside number line recording. Begin to record information vertically introducing ‘repeated subtraction/chunking’ (or ‘repeated addition/additive chunking’).

-

1 0

9

7

0

2

8 1

0

0

0

- 4

3

2

6

3 4 · 1 0

7

6

· 6

· 4

so, 1000



9 · 2 = 108 r.6

· 4



9

·

2

100

5

3

©




Year 3 Year 4 Year 5 Year 6

For Year 5 and Year 6 : When children are secure with ‘chunking’ and tables facts, introduce short division for single digit divisors using Base 10 equipment or place value counters to secure children’s understanding of the process.

For example, 196 ÷ 6 i.e. ‘196 divided into groups of 6’ is approximately 180 ÷ 6 = 30 a. Set out 1 hundred, 9 tens and 6 ones. b.

‘One hundred divided into (groups of) six is difficult using the counters, so exchange 1 hundred for 10 tens. Set out counters as indicated and amend written layout. c.

‘Nineteen tens divided into (groups of) six equals three (groups of six ‘tens counters’) with one ten remaining’. Set out counters as indicated and amend written layout. d.

‘One ten and six ones divided into (groups of) six is difficult using the counters, so exchange 1 ten for 10 ones. Set out counters as indicated and amend written layout. e.

‘Sixteen ones divided into (groups of) six equals two (groups of six ‘ones counters’) with four ones remaining.’ Set out counters as indicated and amend written layout.



Learners will encounter aspects of multiplication and division when working on area, relating to arrays. Problem solving work involving finding all possibilities and combinations also draws on knowledge of multiplication tables facts.

Fractions work within other curriculum areas and in real life links naturally to multiplication and division work.

The notion of equal groups can emerge in many different activities and contexts, e.g. when packing boxes, purchasing quantities of items for several people etc.


Making connections to other topics within this year group

Measurement



Convert between different units of measure [for example, kilometre to metre; hour to minute]

 solve problems involving converting from hours to minutes; minutes to seconds; years to months; weeks to days.

Fractions

 recognise and show, using diagrams, families of common equivalent fractions

 count up and down in hundredths; recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten

 recognise and write decimal equivalents to 1/4 , 1/2 , 3/4

 find the effect of dividing a one- or two-digit number by 10 and 100, identifying the

  value of the digits in the answer as ones, tenths and hundredths



Counting

– Calculating totals by counting small amounts or a proportion and then scaling up e.g. standing against a tree and using your known height to work out ‘How many of me are equal to the height of the tree?’ or counting people on one part of a stadium and multiplying to calculate the total number of spectators.

Money

– shopping: adding multiple products of the same price, adding coins of same value, working out fraction/percentage discounts and special offers, sharing bills.

Measurement

– Scaling quantities (e.g. recipes) to cater for more and less people, reading scales and unlabelled increments on measuring apparatus, calculating area for carpets, decorating etc., scaling shapes to scale geometric artwork e.g. How would you make this triangle three times its size/half its size? Comparing river lengths/building heights e.g. the River Nile is x times longer than the River X.

The height of Snowdon is (fraction) of the height of Everest.

Statistics

– Reading scales and determining appropriate scales for different types of graph relating to weather, temperature, sound etc., Working with proportion, fractions and percentages using pie charts, comparing data using ratio, fractions and scaling such as proportion of children missing breakfast or 1 in 7 children under 10 now has a mobile phone etc.






Fractions (including decimals and percentages)

Requirements include:

 multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams

 solve problems involving number up to 3 decimal places

When working on multiplication and division and/or fractions (including decimals and percentages), there are opportunities to make connections between them, for example:

You could give the children strips of paper and ask them to fold them to show you different proper and mixed fractions, for example,

5 ⁄

8

, 1

3 ⁄

4

. Next ask them to multiply these fractions by single digit numbers. They could use the strips to help them: 1 5 ⁄

8

x 6

1x 6 = 6

5 ⁄

8

x 6 =

30 ⁄

8

or 3

6 ⁄

8

1

5 ⁄

8

x 6 = 6 + 3

6 ⁄

8

= 9

6 ⁄

8

or 9

3 ⁄

4

Numbers with decimals are frequently seen in real life, for example when using money, so give the children opportunities to multiply these in context. For example, you could give them take-away menus and ask them to find out how much it would cost to buy four of a meal deal or a particular course. You could give them the total cost of six of the same dish and ask to work out which dish you chose.

You could ask the children problems that involve multiplying numbers up to 3 decimal places and link to measures, such as:



Jessie had eight lengths of rope. Each was1m 36cm. If he put them side by side what would the total length be?



Paddy had 12 cartons of orange juice. Each carton contained 0.750l. How much juice did he have altogether?



Suzie, the baker, was making 14 loaves of bread for the local supermarket. For each loaf she needed

1.275kg of flour. What is the total amount of flour that she needed?



India took part in a sponsored bike ride at her school. She cycled 25 times around the perimeter of the school playground. The perimeter is 105.34m. How far did she travel?

Measurement

When working on multiplication and division and/or measurement there are opportunities to make connections between them, for example:

You could give the children opportunities to rehearse multiplying by 10, 100 and1000 by converting, for example, millimetres to centimetres, centimetres to metres, metres to kilometres. They could then multiply lengths, masses and capacities of different sizes, for example, 14.75kg by 8. You could then put these into problem format, eg:



Benji, a party organiser, was going to make a fruit punch. For each guest he needed 0.250ml of orange juice and 0.250l of mango juice. If there are 25 guests coming to the party, what is the total amount of juice Benji needs?

You could give the children an approximate equivalence between miles and kilometres, for example1.6km is approximately 1 mile. Then they multiply this amount to find approximate equivalences for other miles, for example 5 miles, 8 miles, 10 miles, 14 miles. The children could make a spider diagram for this and other equivalences.

You could give the children lengths of one side of different regular polygons, for example, pentagon, octagon, decagon, dodecagon and ask them to find their perimeters by multiplying each length by the number of sides the polygon has.

You could also give the children the lengths of different sized rectangles and ask them to find their areas, for example, a rectangle 28cm by 12cm.

Set problems involving time and money for the children to use, for example:



Samir spent 45 minutes completing his homework. It took Pete three times as long. How long did it take Pete to complete his homework?

 It took Carol 1 ½ hours to drive from Oxford to London. It took Lorna a third of that time. How long did it take

Lorna to travel to London?



Harr y is given £3.75 a week as pocket money. He is saving it to buy a computer game. How much will he



Fractions

 identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places

 multiply one-digit numbers with up to two decimal places by whole numbers

 use written division methods in cases where the answer has up to two decimal places

 divide proper fractions by whole numbers [e.g. ⅓ ÷ 2 = ⅙ ]

 multiply simple pairs of proper fractions, writing the answer in its simplest form

When working on multiplication and division and/or fractions there are opportunities to make connections between them, for example:

Multiply numbers such as:



245.25 by 10, 100 and 1000



1.35 by 8

 ¼ x ½

Divide numbers such as:



12 578 by 10, 100 and 1000



237 by 5

 ⅓ ÷ 2

Ratio and proportion

 solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts

When working on multiplication and division and/or ratio and proportion there are opportunities to make connections between them, for example:

Convert the ingredients in this lasagne recipe for 4 people so that it will serve 12:



350g minced beef



1 onion



1 clove garlic



600g tin of tomatoes



2 tablespoons tomato puree



175g lasagne sheets

Algebra

 express missing number problems algebraically

 use simple formulae

When working on multiplication and division and/or algebra there are opportunities to make connections between them, for example:

Solve missing number problems, e.g. 6(a + 12) = 144 multiply out the equation: balance by -72:

6a + 72 = 144

6a + 72

– 72 = 144 – 72

Use known division facts:

6a a

= 72

=

72 ÷ 6 a = 12

Find perimeters and areas of rectangles using the appropriate formulae, e.g. a square field has sides of 24.75m. What is its perimeter? What is its area?

Measurement

 solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places where appropriate

 convert between miles and kilometres

When working on multiplication and division and/or measurement there are opportunities to make connections between them by solving problems such as;



1 pint = 0.57 litres, how many litres in 8 pints? How many pints in 12 litres?



Dan was driving between two cities in France. The sign said the distance was 185km.

He wanted to know what that was in miles. How can he find out? How many miles is it?

Statistics

 calculate and interpret the mean as an average

When working on multiplication and division and/or statistics there are opportunities to make connections between them, for example:

Solve problems, e.g. find the mean monthly temperature for Reykjavik, Iceland

©




Year 3 Year 4 Year 5 have saved over 8 weeks? What about 12 weeks?

 Georgie saved £2.25 of her pocket money each week. How much will she have saved over 9 weeks?

 Penny had saved £75 over a period of 12 weeks. She saved an equal amount every week. How much did she save each week?


Learners will encounter number and place value in:

Within the geography curriculum there are opportunities to connect with multiplication and division, for example in the introduction of the Key Stage 2 Programme of Study it states that pupils should extend their knowledge and understanding beyond the local area to include the United Kingdom and Europe, North and South America. This will include the location and characteristics of a range of the world’s most significant human and physical features. Children could, for example, find out about the currencies used in a selection of countries. They could then make up a currency converter using mental calculation strategies and then check using multiplication, for example:

£1= 1.20 Euros

£2 = 2.40 Euros

£3 = 3.60 Euros

£4 = 4.80 Euros

£5 = 6 Euros

Year 6

Monthly temperatures for Reykjavik

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Art & Design

Within the art and design curriculum there are opportunities to connect with multiplication and division, for example in the introduction of the Key Stage 2 Programme of Study it states that pupils should be taught to develop their techniques, including their control and their use of materials, with creativity, experimentation and an increasing awareness of different kinds of art, craft and design. This could include designing and creating life size models of, for example a Barbara Hepworth sculpture or a Van Gogh painting where the children need to find realistic measurements and then scale them down using division.

Geography

Within the geography curriculum there are opportunities to connect with multiplication and division, for example in the introduction of the Key Stage 2 Programme of Study it states that pupils should extend their knowledge and understanding beyond the local area to include the United Kingdom and Europe, North and South America. This will include the location and characteristics of a range of the world’s most significant human and physical features. Work on multiplication and division could include converting between miles and kilometres and vice versa when looking at distances between countries or famous locations, making currency converters for pounds stirling and the currency in the country they are investigating.

See, for example:



Mathematics and geography

History

Within the history curriculum, there are opportunities to connect with multiplication and division, for example in the introduction of the Key Stage 2 Programme of Study it states that ‘in planning to ensure the progression described above through teaching the British, local and world history outlined below, teachers should combine overview and depth studies to help pupils understand both the long arc of development and the complexity of specific aspects of the content’. The history curriculum requires that pupils should

‘compare aspects of life in different periods’, suggesting comparisons between Tudor and

Victorian periods, for example. Scale models could be one way of learning about life in different periods.

See, for example:



The Tudors







The Victorians

The Ancient Egyptians

The Ancient Greeks

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File - Gobowen Primary School

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