multiplication - stage 1 · multiplication of 2 digit numbers with partitioning (regrouping) using...

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Multiplication - Stage 1

Solve one –step problems involving multiplication by calculating the answer using

concrete objects, pictorial representations and arrays with the support of the

teacher.

Count in multiples of two, fives and tens from zero (to the 10th multiple)

Skip counting in multiples of 2,5 and 10 from zero

The representation for the amount of groups supports pupils’ understanding of the written

equation. So two groups of 2 are 2, 4. Or five groups or 2 are 2, 4, 6, 8, 10.

Work through the PVA idea – real object, picture of the object, cube or bead string or number line

representation of the object.

Count the groups as the pupils are skip counting.

Number lines could be used in the same way as the bead string.

Pupils can use their fingers as they are skip-counting.

“Six pairs of socks. How many socks altogether? 2, 4, 6, 8, 10, 12”

“Three pots of ten crayons. How many crayons altogether? 10, 20, 30”

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Making Equal Groups and counting the total

How this would be represented as an equation will vary. This could be 2 x 4 or 4 x 2. The importance

should be placed on the vocabulary used alongside the equations. So this picture represents 2

groups of 4, 2 lots of 4 or 4 twice, double 4.

Solve multiplication using repeated addition

Key Vocabulary – groups or, lots of, times, array, altogether, multiply, count, double, twice

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Recall and use multiplication facts for the 2, 5 and 10 multiplication tables

Calculate mathematical statements for multiplication within the multiplication tables and

write them using the multiplication (x) and equals (=) signs

Solve problems involving multiplications using materials, arrays, repeated addition, mental

methods and multiplication facts, including problems in contexts.

Show that multiplication of two numbers can be done in any order (commutative)

Children will use a range of vocabulary to describe multiplication and use practical resources,

pictures, diagrams and the X sign to record equations.



Remember to use a range of vocabulary!

Skip Counting

Children can use their fingers as they are skip counting, to develop an understanding of groups of.

Dotted paper could be used to create a visual representation of the different multiplication facts.

Also bead strings and number lines are useful representations.

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Combining groups

Numicon, arrays, unilink towers or Cuisenaire rods can be used to represent multiplications.

3 groups of 10 crayons. How many groups altogether?

3 groups of 10 is equal to?

3 times 10 is?

3 x 10 = 30

5 groups of 3

5 lots of 3

3 multiplied by 5

3 x 5 =

3, 5 times

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Multiplication as repeated addition

Pupils will apply skip-counting to help find the totals of these repeated additions

Arrays to represent multiplication equations

Concrete manipulatives and images of familiar objects begin to be organised into arrays, and later,

are shown alongside dot arrays. It is important to discuss the benefit of array organisations.

Pupils begin to understand multiplication in a more abstract fashion, applying their skip-counting

skill to identify the multiples of 2, 5 and 10x tables.

When using the array, it is important that the children are shown both orientations to further

reinforce the fact that multiplication follows the commutative law.

It also helps to teach children to understand the missing box questions, such as 4 x = 8.

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Multiplication is commutative

Pupils should understand that an array can represent different equations and that, as multiplication

is commutative, the order of the multiplication does not affect the answer.

Key Vocabulary – groups or, lots of, times, array, altogether, multiply, count, double, twice,

multiplied by, repeated addition, column, row, commutative, sets of, equal groups,

--- times as big as, once, twice, three times ….

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Recall and use multiplication facts for the 3, 4 and 8 multiplication tables (continue to

practise the 2, 5 and 10 multiplications tables)

Write and calculate mathematical statements for multiplication using the multiplication

tables that they know, including for two digit numbers times one digit numbers, using

mental methods.

Solve problems, including missing number problems, involving multiplication, including

positive integer scaling problems and correspondence problems in which n objects are

connected to m objects

Children continue to use arrays, number lines and a variety of imagery to support multiplication, as

appropriate.



It is also important that the commutative law is continued to be developed in Stage 3 (see Stage 2)

It is also vital that missing number fact equations are done in Stage 3 (arrays is a good tool to

support children’s understanding of this (see Stage 2)

Remember to use a range of vocabulary!

Skip Counting in 2, 5 10, 3, 4 and 8’s

Rehearsal of previously learnt tables as well as new content for stage 3 should be practised regularly.

They could be incorporated into transition activities, e.g. lining up, coming into and going out of the

classroom etc.

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Doubling to derive new multiplication facts

Pupils learn that known facts from easier tables can be used to derive facts from related times

tables.

Ten times Bigger

Pupils’ work on this must be firmly based on concrete manipulatives - the language of 10 times

bigger must be well modelled and understood to prevent the numerical misconception of “adding

0”.

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Using known facts for multiplying by 10

Pupils growing understanding of place value allows them to make use of known facts to derive

multiplication facts which are ten times bigger.

It is important to use tables with which they are already familiar ( i.e not 6, 7 and 9 by year 3)

2 x 3 = 6 2 x 30 = 60

6 = 2 x 3 60 = 2 x 30

Partitioning of 2 digit numbers with no regrouping

Children should always consider whether partitioning is the best strategy – if it is possible to use

strategies such as doubling (some may use twice 2 for x4). They need to use the most efficient

strategy.

Children may wish to make jottings, including using a full grid as exemplification. The grid, however,

is not a formal method and its only purpose is to record mental calculations. This supports the

development of the necessary mental calculating skills, but does not hinder the introduction of

formal written methods in Stage 4. Concrete manipulatives are essential to develop understanding.

10 x 3 2 x 3

12 x 3 = 36

12 x 3 = or 3 x 12 =

12 = 10 + 2

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Multiplication of 2 digit numbers with partitioning (regrouping)

Using concrete manipulatives and later moving to using images that represent them, supports

pupils’ early understanding, leading towards formal written methods in Stage 4. Once again, this is a

mental strategy, which they may choose to support with informal jottings, including a full grid.



--- times as big as, once, twice, three times …. partition, grid method, multiple, produce, tens, unit,

value

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Count in steps of 6, 7, 9, 25 and 1000

Recall multiplication facts to 12 x 12

Write and calculate mathematical statements for multiplication using the multiplication

tables that they know, including two-digit numbers times one –digit numbers, using mental

and progressing to formal written methods.

Use place value, known and derived facts to multiply mentally, including: multiplying by 0

and 1; multiplying together 3 numbers.

Recognise and use factors and commutativity in mental calculations

Multiply two digit and three digit numbers by a one digit number using formal written

layout.

Use known facts and place value for mental multiplication involving multiples of 10 and 100

Pupils use their growing knowledge of multiplication facts, place value and derived facts to multiply

mentally. Emphasis is placed on understanding the relationship (10 times and 100 times greater)

between a known number fact and one to be derived, allowing for larger “fact families” to be

derived from a single known fact. Knowledge of commutativity is further extended and applied to

find a range of related facts.

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Short multiplication with 2 digit numbers

Initially, they work with calculations that require no regrouping and then only regrouping from the

ones to the tens. Their conceptual understanding is supported by the use of Dienes or place value

counters, both during teacher demonstrations and during children’s independent practise. With

practice, pupils will be able to regroup from any column, including from the hundreds into the

thousands, including being able to multiply numbers containing zero and regrouping through

multiples columns in a single calculation.

Grid method

(See Stage 3)

Once children are competent with Dienes, move onto place value counters.

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To watch grid method being taught to a year 3 class, using place value counters click on the link

below or type ‘YouTube grid multiplication as an interim step’ into google

http://www.youtube.com/watch?v=qyTRtoqYi7Q&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiix&i

ndex=4

Leading onto abstract recording

Expanded short multiplication

Move onto the expanded short multiplication if and when the children are confident and accurate in

multiplying numbers using the grid method.

Again the conceptual understanding will be supported through the use of Dienes or place value

counters.

Use the language of place value to ensure understanding.

Ensure the digit carried over is written under the line in the correct column.

Help the children to see that the equation can be done in any order as this helps children see the link

between expanded methods and compact methods.

Encourage the children to do rounding to the nearest 10 to approximate the answer.

Ensure the multiplication sign is written to the left of the equation.

Get the children to compare the grid method with the expand method with the expanded method.

Ask what is similar? What is different?

23 x 4 = 266 20 and 3

20 and 3 x 4 12 (3 x4) 80 ( 20 x 4) 92

80 +12 = 92

http://www.youtube.com/watch?v=qyTRtoqYi7Q&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiix&index=4

http://www.youtube.com/watch?v=qyTRtoqYi7Q&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiix&index=4

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x 4 80 ( 20 x 4) 12 (3 x4) 92 Leading to 38 x 4 120 (30 x 4) 12 (3x4) 122

Efficient short multiplication

Ensure the multiplication sign is written to the left of the equation

The children should also be presented with missing number boxes.

23 x 4 = 23 X 4 9 2 1

2 2 X 4 x 4 8 8 9 2

Short multiplication of a 3 digit numbers

Make sure the children do rounding to the nearest 100 and the nearest 10 to get an approximation.



38 x 4 12 (3x4) 120 (30 x 4) 122

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value, inverse

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Multiply numbers with up to 4 digits by a one or two digit number using a formal written

method, including long multiplication and two digit numbers.

Build on the work covered in Year 4 with the grid method, the expanded short method, leading to

the compact carry methods.

Short multiplication of a 3 digit numbers

(See Stage 4)

Ensure the children do rounding to the nearest 100 to get an approximation.



The language of place value must be use eg 236 x 4 = 30, 3 tens not 3.

Long multiplication of a 2 digit number by a 2 digit number

Encourage use of column addition to add up the parts in the grid.

Begin using teens numbers and then progress to using a two digit number multiplied by a teen

number. Finally, move on to a two digit number by a 2 digit number.

Use the language of place value to ensure understanding.

Ensure the digit carried over is written under the line in the correct column.

Help the children to see that the equation can be done in any order as this helps children see the link

between expanded methods and compact methods.

Encourage the children to do rounding to the nearest 10 to approximate the answer.

.

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Grid method

18 × 13

x 10 8

10 100 80

3 30 24

130 104 234

Apply to bigger numbers

56 × 27

x 20 7

50 1000 350

6 120 42

1120 392 1512

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Expanded method

Once the children are confident with the grid method, introduce them onto the expanded method.

Start with a 2 digit number multiplied by a teens. Get the children to compare the different

methods. What is the same? What is different?

This leads to using larger numbers (two–digit numbers multiplied by a two-digit number)

Compact long multiplication

Ensure equations with missing number boxes are presented to children.

Use the language of place value to ensure conceptual understanding.

Encourage the children to approximate by rounding to the nearest 10 to get a sense of the size of

the answer.

X 27

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Leading to larger numbers

56 x 27

56

x 27

392 (56 x 7) 1120 (56 x 20) 1512 When the children are confident with long multiplication, extend to three-digit numbers multiplied

by two-digit numbers, returning to the grid method if appropriate.

The prompts in brackets can be omitted it the children no longer need them.




value, inverse, square, factor, integer, decimal, short/long multiplication, “carry”

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Multiply multi-digit numbers (including decimals) uto to 4 digits by a two digit whole number

Continue to practise and develop short multiplication methods and formal long multiplication

methods with larger numbers and decimals throughout Year 6. Return to the expanded forms of

calculations initially, in necessary.

(See Stages 4 and 5)

These expanded methods lead to the formal written method of long multiplication (decimal number

multiplied by a 2 digit number.

It is an option to include a zero as in the example, but not essential. The prompts in brackets can be

omitted it the children no longer need them.

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value, inverse, square, factor, integer, decimal, short/long multiplication, “carry”, tenths,

hundredths, decimals

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multiplication - stage 1 · multiplication of 2 digit numbers with partitioning (regrouping) using...

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