maths methods for blogs

Minchinhampton Primary School

Calculation Methods

+ - x ÷ Do your children start talking a strange language whilst doing their maths homework? Do

words such as ‘chunking’, ‘grid method’, or ‘partitioning’ baffle you? The aim of this PowerPoint is to enable parents and carers to understand more about the

methods used in our school and to assist them in supporting children with maths. The traditional, formal methods of teaching children mathematics are now being used less during the stages whilst children develop and acquire number skills and understanding. This PowerPoint demonstrates the methods in the progressive order that pupils will experience them at Minchinhampton. Methods are taught in a way that will aid the pupils’ conceptual understanding and support their mental calculations. Although pupils may be able to master more advanced methods, they are taught methods appropriate to their understanding of number: it is essential that they fully comprehend every stage of the process.

+ - x ÷

Some Year 5/6 pupils are in the process of creating short videos to demonstrate these methods being used. To see these videos, please visit your child’s class blog.

Contents

Click on the operation that you would like to find out about, or use the forward arrow key to view the PowerPoint from start to finish.

• Addition• Subtraction• Multiplication• Division

Addition Section Contents

• Pictures/objects/symbols• Number lines/tracks – jumps of 1• Number lines – efficient jumps• Vertical (column) addition – expanded• Vertical (column) addition – compact

Click here to return to main contents.

Pictures/objects/symbols

Pupils use concrete resources and drawings/symbols to solve the problem. Objects may include real items (such as small plastic toys, money), counting cubes (Unifix, Multilink) or Numicon. As they progress, symbols, and eventually numbers will be used to show working and answers.

I eat 3 cherries and my friend eats 2 cherries. How many cherries do we have altogether?

Numicon will be used to support calculations.

2 cherries and 3 cherries make 5 cherries altogether.

II III2 + 3 = 5

9 + 1 = 10

20 + 3 = 23

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Pupils show and calculate addition problems on a number line or number track. They begin by counting in ones and progress to larger, more efficient, jumps. When counting on, pupils always count from the larger number.

18 + 5 = 23 (Counting in ones)

18 19 20 21 22 23

+1 +1 +1 +1 +1

Number Lines/Tracks


Before moving on to more traditional methods, pupils use the number line method with more complex numbers (including decimals). This reinforces mental calculations and understanding of number.

Number Lines – efficient jumps

80 7747

+30

+3

47 + 35 = 82

+2

82

18 23 20

+2+318 + 5 = 23

18 38

+2018 + 19 = 37

-137Click here to return to contents for addition section. Click here to return to main contents.

This method links the number line with the more traditional column method. It encourages children to consider the value of the digits and aids conceptual understanding. This method will be revisited as numbers become more complex, for example as pupils solve written addition problems with decimals.

H T O 3 3 6 + 1 8 7 1 3 Add the ones (6 + 7) 1 1 0 Add the tens (30 + 80)

4 0 0 Add the hundreds (300 + 100) 5 2 3 The new columns are added, starting with the digit of least value.

Vertical (Column) Addition - Expanded


When confident with using the expanded method, pupils learn the compact method. To calculate in this way, digits are added in order of value (least to greatest). For this sum, the order of value is: ones, tens, hundreds. Tens and hundreds are carried by writing a digit below the line.

H T O 3 3 6 + 1 8 7

5 2 3 1 1

Vertical (Column) Addition - Compact


Subtraction Section Contents

• Pictures/objects/symbols• Number lines/tracks – counting back• Number lines – counting on • Vertical (column) subtraction – no exchanging• Vertical (column) subtraction – with

exchanging



Pupils use concrete resources and drawings/symbols to solve the problem. Objects may include real items (such as small plastic toys, money), counting cubes (Unifix, Multilink) or Numicon. As they progress, symbols, and eventually numbers will be used to show working and answers.

I have 5 cherries and eat 2 of them. How many do I have left?

Numicon can be used for subtraction by arranging the shapes on a number line or laying pieces on top of each other.

5 cherries take away 2 cherries leaves 3 cherries.

III II5 – 2 = 3

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Pupils begin to show and calculate subtraction problems on a pre-drawn number line or number track by counting back. They begin by counting back in ones and progress to larger, more efficient, steps. For subtraction, the jumps are drawn below the line (this is a very recent change to our previous policy and older pupils will still record subtraction jumps above the line). The larger number is always written on the right. As they progress, pupils draw their own number lines, including only the relevant numbers. They also begin to take more efficient jumps as in the bottom example. Once competent with efficient jumps, children apply this method to find the difference between any numbers of any size, including decimals.

Number Lines/Tracks

13 – 5 = 8(Jumps of one) 1312111098

-1 -1 -1-1-1

13108

-3-2

13 – 5 = 8(Efficient jumps)


When numbers are close together, pupils may choose to find the difference by counting on. As always, the number of least value is recorded at the left of the line. Firstly, they count onto the next multiple of ten, then count on to the multiple of ten before the higher number. Finally they count on to the higher number. Then they add together the jumps that they have taken (40 + 6 + 1 = 47). Some children may split the jump of 40 into a ten and a 30 to ease going over the hundred; others may complete it in two jumps from 389 to 400 and then from 400 to 436. (This method is also taught and used with smaller numbers.)

Number Line – counting on to find the difference between numbers

430 436

+1

+40

+6

390389

486 – 389 = 47


Initially children will be taught column subtraction with numbers that do not involve exchanging (sometimes referred to as borrowing, decomposition or carrying) tens for ones, hundreds for tens and so on. This method will be developed, with the aid of concrete resources, to use exchanging.

Digits are arranged in columns, with the larger number on top. Starting with the digit of least value (in this sum, the ones), children subtract vertically:

Ones: 4 – 3 = 1Tens: 70 – 20 = 50Hundreds: 800 – 500 = 300

The answer to each part of the calculation is recorded in the appropriate column below the line.

H T O

8 7 4

- 5 2 3

3 5 1

Vertical (Column) Subtraction – no exchanging


In the first example below, you cannot do 2 – 7 (for this method), so a ten needs to be exchanged for ten ones, thus making 12 – 7 = 5. Due to the exchanging, there are now 2 tens and not 3 on the top number.20 – 50 cannot be done so (for this method), so a hundred is exchanged for ten tens, thus making 120 – 50 = 70.The second example has been included to show an example with a zero.

Vertical (Column) Subtraction - with exchanging

+ 7 2

8 1 9

-


Multiplication Section Contents

• Pictures/objects/symbols• Arrays• Number lines/tracks• Partitioning and the grid method• Using the grid method to multiply 2 digit num

bers• Vertical (column) multiplication – expanded• Vertical (column) multiplication - compact



Pupils use concrete resources and drawings/symbols to solve the problem.

How many socks in three pairs?

There are five cakes in each bag.How many cakes are there in three bags?

3 pairs, 2 socks in each pair, 6 socks altogether.

II II II

3 bags, 5 cakes in each bag, 15 cakes altogether.

I I I I I

I I I I I

I I I I I

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Multiplication problems can be represented in arrays.

2 x 5 or 5 x 2

6 x 4 or 4 x 6

Arrays


Pupils show and calculate multiplication problems on a number line. This is sometimes referred to as repeated addition.

5 x 3 =15 or 3 x 5 =15

0 1510 5

0 3 6 9 12 15

+3 +3 +3 +3 +3

+5 +5 +5

Number Lines/Tracks


In all areas of maths, pupils are taught to recognise the value of digits and to partition numbers. In the grid method, pupils partition numbers according to their value and multiply each part separately. Even when pupils have progressed to more efficient methods, this method is revisited as numbers become more complex. This method will reinforce mental calculations, including doubling.

36 x 4 = 144

X 30 6

4 120 24

120 + 24 = 144

136 x 4 = 544

X 100 30 6

4 400 120 24

400 + 120 + 24 = 544

X 10 3 0.6

4 40 12 2.4

13.6 x 4 = 54.4

40 + 12 + 2.4 = 54.4

Partitioning and the grid method


The grid method can be used to multiply pairs of numbers that have more than one digit, this also includes decimal numbers.

Using the grid method to multiply 2 digit numbers

X 30 4(Totals)

20 600 80 680 7 210 28 238 918

34 X 27 = 918


As with the grid method, pupils need to be able to partition the digits and understand their value. Even when pupils have progressed to more efficient methods, this method is revisited as numbers become more complex.

H T O 3 6 X 4 2 4 (4 X 6) Digits are multiplied, starting 1 2 0 (4 X 30) with the lowest value digit. 1 4 4 (24 + 120) The new columns are added, starting with the digit of least value.

Vertical (Column) Multiplication - Expanded

Click here for more examples.Click here to return to contents for multiplication section. Click here to return to main contents.

Multiply a 2 digit number by a 2 digit number

T H T O 3 6 X 7 4 2 4 (4 X 6) 1 2 0 (4 X 30) 4 2 0 (70 x 6) 2 1 0 0 (70 x 30) 2 6 6 4

Multiply a 2 digit decimal number by a 1 digit number

T O . t 3 . 6 X 4 2 . 4 (4 X 0.6) 1 2 . 0 (4 X 3) 1 4 . 4 (2.4 + 12)

Examples of more complex vertical (column) multiplication - expanded


When children fully understand the maths behind the expanded vertical method, they will be taught a compact vertical method. Initially this will be for multiplying by a 1 digit number and, when ready, for multiplying larger numbers.

T O 3 6 X 4 1 4 4 2

4 x 6 is 24. Pupils write the 4 in the ones column and carry the 2 tens below the line in the tens column. Then they do 4 x 30 which is 120, add on the 2 tens and make 140.

H T O 2 4X 1 6 1 4 4 2

2 4 0 3 8 4

The method for this more advanced calculation is the same as the first one, but first the 24 is multiplied by 6 and then by 10. Then the two totals are added together.

Vertical (Column) Multiplication - Compact


Division Section Contents

• Pictures/objects/symbols• Arrays• Number lines/tracks• Number line - chunking• Bus stop method – short division• Vertical chunking• Bus stop method – long division



Pupils use concrete resources and drawings/symbols to solve the problem, either by sharing or grouping.

6 cakes shared between 2 people. How many cakes each?

6 pencils are put into packs of 2. How many packs are there? (Grouping into twos)

II II II

How many apples in each bowl if I share 12 apples between 3 bowls?

Four eggs fit into a box. How many boxes would you need to pack 20 eggs? (Grouping into fours)

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Arrays can be used to solve division problems, for example:

10 ÷ 5 or 10 ÷ 2

24 ÷ 4 or 24 ÷ 6

Arrays


Links can easily be made to fractions. For example, the first array could be used to calculate ½ of 10 and the second one to calculate ¼ of 24.

Pupils use number lines to count up and see how many lots of a number are in a given number. They count up in one ‘lot’ (jump of the multiple) each time to see how many ‘lots’ there are. Pupils will also be given problems with remainders.

0 1510 5

+5 +5 +5

Number Lines/Tracks

16 ÷ 3 = 5 r1 or 16 ÷ 5 = 3 r1

0 3 6 9 12 15

+3 +3 +3 +3 +3

r1

r1


Number Line - Chunking

As pupils become more confident in the concept of division and their recall speed of times tables increases, they begin to use larger jumps (chunks) or more ‘lots’ of the multiple. This is called chunking.

In this example, the first jump is of 20 lots of 4. The second jump is of 4 lots of 4. Therefore there are 24 (20 + 4) lots of 4 altogether in 96.

98 ÷ 4 = 24 r2

0 80 96

20 x4 x

r2

Chunking on the number line is also used for dividing by 2 digit numbers.

0 240 264

10 x1 x

283 ÷ 24 = 11 r19

r19


Bus Stop – Short Division (Dividing by numbers with only one digit)

When pupils are secure with the concept of division, they are taught the ‘bus stop’ method.

353 ÷ 4 = 88 r1

To work out this sum, divide 353 by 4, one digit at a time, starting from the left. The remainder of each part of the calculation is carried on to the next digit.

4 3 5 3

0 8 83 3

R 1

Before moving on to divide by two digit numbers, pupils will use this method for larger numbers and decimals.

Pupils will also be taught how to calculate the remainder as a fraction: 88 ¼ or 88.25


Vertical Chunking with numbers of more than one digit

For this method, children solve division problems by repeated subtraction. They are taking away chunks (lots of) a number at a time. The size of the chunk (number of lots) is recorded to the left of the sum and the subtraction carried down. At the end the children count up and see how many lots of the number have been taken away and what the remainder is. The subtraction part of the sum is completed as the vertical compact method.


56 13 2 ÷ 1 7

(20 X 17) - 3 4 0

2 9 2

(10 X 17) - 1 7 0

1 112 12

(5 X 17) - 8 5

3 7

(2 X 17) - 3 4

Remainder 3

632 ÷ 17 = 37 r3

Bus Stop – Long Division (dividing by numbers with more than one digit)

2 73 6 9 7 2

- 7 2 0 (20x36) This is the biggest chunk (lot) of 36 that can be taken away. A 2 is recorded on the bus stop (tens column) and the remainder to be divided (252) is brought down.

2 5 2 (7x36) 252 is 7 lots of 36 so this is recorded on the bus stop in the ones column.

Answer: 972 ÷ 36 = 27

During their primary education, some pupils may learn long division. This is a method that is taught.


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