Manley Park Primary School Calculation Policy Handbook

Multiplication

MultiplicationThe strategies that children will use to support them

with multiplication all rely on a secure understanding of multiplication facts. This knowledge can be developed through children having an understanding that multiplication is:

• Repeated Addition. 5 x 3 = 5 + 5 + 5• Commutative. Multiplication can be done in any

order. 5 x 3 = 3 x 5• Associative. Multiplication can be separated into

easier parts. 15 x 3 = (10 x 3) + (5 x 3)

Examples of efficient mental strategies:

• 24 x 2• 21 x 4• 204 x 10• 32 x 100• 4.3 x 1000• 7 x 19• 74 x 5• 42 x 4

• Doubling• Double, double again• Moving Digits• Moving Digits• Moving Digits• Compensation• X 10 and halve• Partition

Multiplication Facts• Multiplication Facts need to be

taught through concrete experiences, images, and associative hooks.

• Arrays and Counting On can support children’s understanding of Multiplication Facts.

• Children need to be taught to derive new facts from known ones.

• Teachers need to dedicate time to teaching children multiplication facts. This will allow children to know that 2 x 3 means 2, 3 times, or 2 + 2 + 2.

• Children need regular opportunities to rehearse multiplication facts using the actions mentioned above.

• Children need to have ‘quick recall’ of multiplication facts.

Multiplication Facts• 3 x 0 = • 3 x 1 = • 3 x 2 =• 3 x 3 = • 3 x 4 =• 3 x 5 =• 3 x 6 =• 3 x 7 =• 3 x 8 =• 3 x 9 = • 3 x 10 =

Multiplication Facts – Properties of multiples to support learning facts

• 2 – numbers are always even• 10 – numbers always end in a 0• 5 – numbers always end in a 5 or 0• 3 – digits in numbers always total a multiple of 3• 4 – double the two times table• 6 – double the three times table• 8 – double the four times table• 9 – digits in numbers always total a multiple of 9

Multiplication Facts – Progression and Examples

• 2 Times Table (2 x 3, 2 x 4, 2 x 5 etc.)• 10 Times Table (10 x 3, 10 x 4, 10 x 5 etc.)• 5 Times Table (5 x 3, 5 x 4, 5 x 5 etc.)• 3 Times Table (3 x 3, 3 x 4, 3 x 5 etc.)• 4 Times Table (4 x 3, 4 x 4, 4 x 5 etc.)• 6 Times Table (6 x 3, 6 x 4, 6 x 5 etc.)• 8 Times Table (8 x 3, 8 x 4, 8 x 5 etc.)• 9 Times Table (9 x 3, 9 x 4, 9 x 5 etc.)• 7 Times Table (7 x 3, 7 x 4, 7 x 5 etc.)

Array• Arrays are an important tool

to enhance children’s understanding of multiplication.

• Arrays develop children’s understanding of multiplication as ‘repeated addition’.

• Arrays provide children with a concrete representation or an image of the process of multiplication.

• Arrays are useful for highlighting the ‘commutativity’ of facts.

• Arrays provide a link between multiplication and division.

Array•Coins, Counters and Dienes Cubes can all be used to support Arrays.

•For example, 4 x 6 = ?

•First, identify what quantity will go into each row of the array. 4 x 6 means repeatedly adding ‘4’, 6 times.

•Place rows of 4, (dots, Coins, Counters etc.) each time indicating the total quantity of the array at the end of each row.

•When you have added ‘4’, 6 times, the array is complete. The total number of dots, Coins, Counters etc in the array gives the answer.

•So, 4 x 6 = 24

Partitioning•Partitioning is a mental strategy that can be used to support multiplication. It allows a child to use known multiplication facts to find unknown ones.

•Dienes cubes, counters, money and Beadstrings can all be used to support partitioning.

Partitioning• For example, if a child is

unsure of their 7 times table, they can use their knowledge of the 5 times table and 2 times table to find the 7 times table fact.

• For 7 x 8, the calculation can be completed as 5 x 8 = 40 and 2 x 8 = 16, with the totals added together.

• So 40 + 16 = 56, meaning 7 x 8 = 56

• Similarly, a 2-digit number can be partitioned to be calculated mentally. The Tens digit and the Units digit can be partitioned and multiplied separately.

• 13 x 7. 13 can be partitioned into 10 and 3.

(10 x 7 = 70) (3 x 7 = 21)70 + 21 = 91, so 13 x 7 = 91

Partitioning – Progression and Examples

• 6 x 3 = (5 x 3) + (1 x 3)• 7 x 5 = (5 x 5) + (2 x 5)• 14 x 5 = (10 x 5) + (4 x 5)• 142 x 4 = (100 x 4) + (40 x 4) + (2 x 4)• 41 x 6 = (41 x 5) + (41 x 1)• 32 x 12 = (32 x 10) + (32 x 2)• 32 x 26 = (32 x 20) + (32 x 6)• 3.2 x 4 = (3 x 4) + (0.2 x 4)

Doubling•Doubling is used for multiplying any number by 2.

•Doubling builds on children’s knowledge of early doubles.

•Doubling builds on children’s knowledge of number facts and number bonds.

•Doubling relies on children’s understanding of the relative position of numbers, place value and the ability to partition and recombine.

•Children need to be able to double all numbers up to 20 to be able to access further doubling.

Doubling• For example, 16 x 2 = ?• First partition the 16 into

10 and 6. Then multiply 10 by 2 (10 x 2 = 20).

• Next multiply 6 by 2 (6 x 2 = 12).

• Finally add the totals together. (20 + 12 = 32) So 16 x 2 = 32

• Dienes cubes, Arrays, Coins, Counters and 100 squares can all be used to support Doubling.

Doubling – Progression and Examples

• 23 x 2• 227 x 2• 1044 x 2• 73 x 20• 176 x 20• 300 x 200• 4.6 x 2• 4.36 x 2• 4.6 x 20

Moving Digits• When multiplying a number by 10,

100 and 1000, the calculation can be completed efficiently using knowledge of place value.

• Moving digits builds on children’s understanding of counting in steps of 10, 100 and 1000.

• When multiplying a positive whole number by 10, all of the digits move one place to the left.

• The new number will be 10 times the size of the old number.

• To multiply a positive whole number by 100, each digit is moved two places to the left.

• To multiply a positive whole number by 1000, each digit is moved three places to the left.

Moving Digits• For example, 42 x 100 = ?• First draw a place value chart, placing

the digit 4 in the Tens column and the digit 2 in the Units column.

• Next identify how many places to the left each digit has to move. (As 42 is being multiplied by 100 each digit will need to move two places to the left.

• Then move each digit two places to the left. The 4 will move to the Thousands column and the 2 will move to the Hundreds column.

• Finally move two 0s from after the decimal point each two places to the left, into the Tens and Units column. Each 0 acts as a place holder.

• 42 x 100 = 4200 • Place value charts, Dienes cubes,

Coins (£1, 10p and 1p) 100 square with cubes and Place value cards can be used to support Moving Digits.

Moving Digits – Progression and Examples

• 4 x 10/100• 42 x 10/100• 142 x 10/100• 1042 x 10/100• 40 x 20 (Before 4 x 1000?)• 400 x 200• 4 x 1000• 42 x 1000• 4.2 x 10/100/1000• 4.02 x 10/100/1000• 0.4 x 7• 42 x 0.1• 42 x 0.01

Compensation• Compensation is used

for multiplying numbers by near multiples of 1, 10, 100 and 1000.

• Compensation builds on children’s understanding of moving digits to multiply by 1, 10, 100 and 1000.

• Compensation builds on children’s understanding of multiplication as repeated addition.

• Compensation builds on children’s understanding of counting on and back from any number.

Compensation• 23 x 9 = ?• First multiply 23 by 10.

23 x 10 = 230. (Instead of doing ‘23 nine times’ do ‘23 ten times’ because it is easier.)

• Next subtract 23 from 230 because there is 23 too many. 230 – 23 = 207

• So 23 x 9 = 207• Counters, Beadstrings,

Coins, Dienes cubes, Arrays and 100 squares with cubes can be used to support compensation.

Compensation – Progression and Examples

• 27 x 9 (19, 29, 39, 49 etc.)• 27 x 99 (199, 299, 399, 499 etc.) • 127 x 9 (19, 29, 39, 49 etc.)• 127 x 99 (199, 299, 399, 499 etc.)• 1027 x 9 (19, 29, 39, 49 etc.)• 1027 x 99 (199, 299, 399, 499 etc.)• 2.7 x 9 (19, 29, 39 etc.)• 2.7 x 99 (199, 299, 399 etc.)• 27 x 0.9 (1.9, 2.9, 3.9 etc.)

X 10 and halve• X 10 and halve is used for

multiplying numbers by 5, 50, 500, 25 and 250.

• X 10 and halve builds on children’s understanding of multiplying numbers by 10, 100 and 1000 by moving digits.

• X 10 and halve builds on children’s understanding of halving both odd and even numbers.

• X 10 and halve builds on children’s understanding of place value and the relative position of numbers.

X 10 and halve• For example, 43 x 5 = ?• First multiply 43 by 10.

(Double the 5 to get 10 to make the calculation easier.) 43 x 10 = 430.

• Next halve the answer (430), as 5 is half of 10. Half of 430 = 215.

• So 43 x 5 = 215• It is vital that children have a

secure strategy for halving numbers before using X 10 and halve.

• Arrays, Counters, Coins and Dienes cubes can all be used to support X 10 and halve.

X 10 and halve – Progression and Examples

• 43 x 5• 143 x 5• 1043 x 5• 43 x 50• 43 x 25• 43 x 500• 5 x 4.3• 5 x 4.32

Number Line• Number lines are a formal

written method used to support children when completing multiplication calculations.

• Number lines build on the concept of repeated addition, counting on and place value.

• Number lines can be supported with Dienes cubes, Beadstrings and Counters.

• Pre-numbered and part-numbered lines can also be used to support children using Number lines.

Number Line• For example, 8 x 7 = ?• Starting at 0, repeatedly add 8

the required number of times (in this case, 7)

(For example 8 x 7: add 8, 7 times.• Each time you add another

‘jump’ of 8 write your new position on the number line. This will support with accuracy and will assist with each step of counting on.

• When you have added 8 the necessary number of times (7) your position on the number line will provide your answer.

• So, 8 x 7 = 56

Grid Method• Grid Method is a formal calculation strategy that is used to support children with multiplication. Grid Method can be used with any multiplication calculation, small or large.

• Grid Method builds on children’s understanding of multiplication facts, moving digits, place value and addition.

• Number lines, arrays, Dienes cubes, counters, Beadstrings and Coins can all be used to support Grid Method.

Grid Method• For example, 24 x 7 = ?• First, partition the calculation into

Tens and Units (24 is partitioned into 20 and 4.)

• Next draw a grid with a multiplication symbol in the corner, placing the partitioned numbers on the outside of the grid. (20 and 4 written at the top of the grid, 7 written at the side).

• Complete each separate calculation writing the answer in the relevant box. E.g. (20 x 7 = 140) and (4 x 7 = 28).

• When all of the multiplications have been completed add up the answers inside the grid to find the final answer. E.g. 140 + 28 = 168.

• So, 24 x 7 = 168

Vertical Method (Expanded)• Vertical Method is a formal

written method used to support children with multiplication calculations.

• Vertical Method builds on children’s understanding of Grid Method.

• Vertical Method builds on children’s understanding of multiplication facts, moving digits, place value and addition.

• There is no requirement for children to be able to access this strategy. It is more important that children have an embedded and secure understanding of how to use Grid Method accurately.

Vertical Method (Expanded)• For example, 32 x 24 = ?• Partition each number in the

calculation into Tens and Units. E.g. Partition 32 into 30 and 2 and 24 into 20 and 4.

• Underneath the calculation, on separate lines write the individual calculations that will need to be completed. E.g. 30 x 20, 30 x 4, 2 x 20, 2 x 4

• Complete each calculation individually, placing the answers in the correct place value columns. E.g. 30 x 20 = 600, 30 x 4 = 120, 2 x 20 = 40, 2 x 4 = 8.

• Add the digits in each column to find the answer. In this example, the Units addition would be 0 + 0 + 0 + 8 = 8 Units. The Tens would be 0 + 2 + 4 + 0 = 6 Tens. The Hundreds would be 6 + 1 + 0 + 0 = 7 Hundreds.

• 7 Hundreds + 6 Tens + 8 Units = 768.

• So, 32 x 24 = 768