Calculations Guidance on pencil and paper methods
Dale Community Primary School
Steps to Success Information for Parents The long-term aim is for children to be able to select an efficient method of their choice (whether this be mental, written or in upper Key Stage 2, using a calculator) that is appropriate for a given task. They should always be asking themselves: What is a rough answer (estimate) to this question? Shall I do this in my head? Shall I do this in my head using drawings or jottings? Should I use a pencil and paper procedure? Shall I use a calculator?
Those Very First Steps in Calculations Children, at an early age, should be encouraged through, practical experiences, to: Show an interest in number problems. Separate a group of three or four objects in different ways, beginning to recognise that the total is still the same. Compare two groups of objects, saying when they have the same number. Count repeated groups of the same size. Share objects into equal groups and count how many in each group. Find one more or one less than a number from one to ten. Use the vocabulary involved in adding and subtracting. Say the number that is one more than a given number. Select two groups of objects to make a given total of objects. Use language such as ‘more’ or ‘less’ to compare two numbers. Begin to relate addition to combining two groups of objects and subtraction to ‘taking away’. Use own methods to work through a problem. Find the total number of items in two groups by counting all of them. NUMICON SHOULD BE USED AT ALL AVAILABLE OPPORTUNITIES TO SUPPORT CHILDREN’S UNDERSTANDING OF CALCULATIONS. IT SHOULD BE AN INTEGRAL PART OF EARLY CALCULATION AT DALE.
Steps to Success Higher Steps of Calculation When children reach the higher steps of calculation (step 5 and above) they should be estimating the answer before calculating. They record these estimates in a bubble:
20 4=1 x 0 3
“28 children are given 4 sweets each, how many are there altogether?�
T
U
x
20
8
4
80
32
+ 1
8
0
3
2
1
2
Progression through Calculations
Addition
Language Add together; plus; add; sum Mental recall of number bonds 6 + 4 = 10 + 3 = 10 25 + 75 = 100 19 + = 20 Use near doubles 6 + 7 = double 6 add 1 = 13 Addition using partitioning and recombining 34 + 45 = (30 + 40) + (4 + 5) = 79 Counting on or back in repeated steps 1, 10, 100, 1000 86 + 57 = 143
(by counting on in tens and then in ones)
460 + 300 = 760
(by counting on in hundreds)
Add the nearest multiple of 10, 100, and 1000 and adjust 24 + 19 = 24 + 20 – 1 = 43 458 + 71 = 458 + 70 + 1 = 529 Use the relationship between addition and subtraction 36 + 19 = 55 19 + 36 = 55 55 – 19 = 36 55 – 36 = 19
Progression through Calculations
Subtraction
Language Take away; subtract; minus; find the difference Mental recall of number bonds 10 – 6 = 4 17 - = 11 20 – 17 = 3 10 - = 2 Find a small difference by counting up 82 – 79 = 3 Counting on or back in repeated steps 1, 10, 100, 1000 86 - 52 = 34
(by counting back in tens and then in ones)
460 – 300 = 160
(by counting back in hundreds)
Subtract the nearest multiple of 10, 100, and 1000 and adjust 24 - 19 = 24 – 20 + 1 = 5 458 – 71 = 458 – 70 – 1 = 387 Use the relationship between addition and subtraction 36 + 19 = 55 19 + 36 = 55 55 – 19 = 36 55 – 36 = 19
Addition Step 1: Children will start by counting each and every object to arrive at the total. “Peter has 3 balloons and Jasdeep has 2, how many balloons are there altogether ?”
Children will need to know the correct sequence of numbers. Step 2: Children are encouraged to develop a mental picture in their heads of the number system. Children develop ways of recording calculations using a pictorial format. “If I add 2 ice lollies to 3 ice lollies how many have I got altogether ?”
2
+
3
4
5
Addition Step 3: Children develop addition skills by using number lines to count on in ones. “What is the total of 4 sweets added to 8 sweets?”
+1+1+1+1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A child would use the number line to say “8” and then add on the four “1,2,3,4” giving total of 12. Children should be encouraged to put the greater number first. Step 4: Children develop skills by taking the first whole number and adding the individual tens of the second and then adding on individual units. “Hamzah wants to find the sum of 27 pence and 34 pence, how much has he altogether?”
27 +10 +10
0
+1+1+1+1+1+1+1
34 44 54 61
Addition Step 5: Children now need to develop partitioning the addition into larger groups of tens and units. “Jessica spends 36 pence on sweets and Gaganpreet spends 28 pence. How much have they spent altogether ?”
28 +20
0
36
+8
56 64
Step 6: Children develop partitioning of the problem by the tens and then the units. “Ben adds 42 seeds to a jar already containing 35 sunflower seeds, he wants to know how many he has altogether ?”
+40
0
+30
40
+5 +2
70 75 77
42 = 40 + 2 and 35 = 30 + 5
Addition Step 7: Children now move onto the more formal way of recording, but need to be very secure on the previous stages before trying to record in this manner. “Shamsa adds 67 buttons to Justyna’s 45. How many altogether ?”
60 + 7 40 + 5 100 + 12 = 112 Step 8: Children now progress to addition using columns. Adding the units/ ones column first then tens column. Ingrid adds 56 stars to 83 squares. “How many shapes altogether ?”
TU 56 83 99 130 139
units (6+3) tens (50+80)
Addition Step 9: Children have now reached a formal method showing numbers carried underneath the correct column. “Katie scores 165 points in her first game which she adds to her second score of 58. How many points did she score altogether ?�
H T U 1 6 5 5 8 2 2 3 1
1
Addition Step 10: Children need a good knowledge of place value to make sure numbers are in the correct column. They now extend to numbers with any number of digits and various decimal places. “Qasim measures 3 distances, 53.42m; 362.8m and 2.984m. What is the total distance measured?�
5 3 3 6 2 2 + 4 1 9 1
2
4 2 8 9 8 4 2 0 4 1
Subtraction Step 1: As with addition, children learn to use objects, and remove “ones” from a group. “James has 7 marbles, Asad takes away 3, how many marbles has James now got?”
-1 -1 -1 7 Marbles take away
-1 -1 -1
leaves 4
Step 2: Children slowly move from the concrete apparatus to using a number line as well. “Menaz has 14 pence she buys a bag of sweets for 6 pence, how much has she got left?”
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-6 -5 -4 -3 -2 -1
Using number lines to count back in ones.
Subtraction Step 3: Counting on, using a number line, to find the difference between numbers (known as the shopkeeper’s method). Mandla asks, “What is the difference between 15 and 12?”
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
+1 +2 +3
Step 4: Children find a small difference by counting up. “James has a money box containing 53 pence he takes away 47 pence, how much has he got left?”
+3
47
+3
50
53
Subtraction Step 5: Children start to develop skills in counting on in tens and units/ones. Gurpreet works out the question, “56 subtract 23 equals what?”using both a number line and by jottings.
+10
23
+10
33
+10
+3
43
53 56
+10 + 10 + 10 + 3 = 33
Step 6: Children start to develop greater skills by partitioning groups of tens and groups of units/ones. “Eleisha wants to find the difference between 78 sweets and 43 sweets. She counts on from 43 up to 78.”
+30
43
+5
73 30 + 5 = 35 so 78 - 43 = 35
78
Subtraction Step 7: Children use complementary addition - counting up from the smaller number to the larger one including bridging through tens/hundreds. Luqman asks, “What is the answer to 164 minus 68?”Frantisek works it out using a number line adding together the jumps ~ 2+30+60+4 gives an answer of 96.
+2 +30
68 70
+60
+4
160 164
100
Children can progress to taking jumps even larger from 70 to 160 giving you: +2 + 90 + 4. Step 8: Children move onto the more formal way of recording, but need to be very secure on the previous stages before trying to record in this manner. “Paul wants to find 74 subtract 27.” 60
14
70 +
4
70 +
4
- 20 +
7
- 20 +
7
40 +
7
Subtraction Step 9: Children now use the formal method of “decomposition”, but need to be very secure on the previous stage before trying to record in this manner. Kieran asks Noman, “Can you work out the subtraction of 256 from 725?”
-
6
11
15
7
2
5
2
5
6
4
6
9
Children should be in the habit of checking their answers by adding the answer to the number being taken away. 469 + 256 = 725
Subtraction Step 10: Children can progress to using decomposition with decimals. “Henna has £34.60 and spends £15.36, how much has she got left?”
-
2
14
.
5
10
3
4
.
6
0
1
5
.
3
6
1
9
.
2
4
Multiplication Step 1: Children will use pictures and symbols of every object to arrive at the total. “There are 2 sweets in a jar. How many sweets, are there altogether, in 4 jars?”
Children will first count individual sweets: 1, 2, 3, 4, 5, 6, 7, 8.
Step 2: Children will use pictures and symbols of every object to arrive at the total. “There are 3 sweets in a jar. How many sweets, are there altogether, in 5 jars?”
Children can use a number line to jump in threes. +3
+3
+3
+3
+3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Multiplication Step 3: From pictures to notation using repeated patterns and repeated addition. “Eva has 4 lots of 3 pebbles. How many has she altogether ?�
+3
+3
+3
+3
0 1 2 3 4 5 6 7 8 9 10 11 12
Multiplication Step 4: Children recognise that repeated patterns can be represented in different ways. “Zaryab has 6 lots of 3 pebbles. How many has she altogether ?�
6 lots of 3 (3+3+3+3+3+3) +6
+3
+6
+3
+3
+6
+3
+3
+3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
or 3 lots of 6 (6+6+6)
Multiplication Step 5: Children use number lines to work out multiplication questions. “8 children each have 4 playing cards. How many do they have altogether ?”
0 4 8 12 16 20 22 28 32 8 lots of 4, can be written as, 8 x 4 Step 6: Children use partitioning to work out multiplication questions involving two digit numbers. “28 children are given 4 sweets each, how many sweets are there altogether?”
28 = 20 + 8 (Partition the tens and units)
T
U
x
20
8
4
80
32
=
8
0
+
3
2
1
2
1
Multiplication Step 7: Children use the grid method for multiplication of three digit figures by a single digit figure. “What is the product of 296 and 4?”
This is the same as “What is 296 multiplied by 4?” “What is 296 times by 4?” “What is 296 x 4?”
H
T
U
x
200
90
6
4
800
360
24
=
8
0
0
3
6
0
2
4
8
4
+ 1
1
Children should be confident in the use and knowledge of times tables.
Multiplication Step 8: Children use the grid method for multiplication of three digit numbers by two digit numbers. “A school has 182 children, each child saves 48 pence. How much does the whole school collect?”
H
T
U
x
100
80
2
40
4000
3200
80
8
800
640
16
4
0
0
0
3
2
0
0
8
0
0
6
4
0
8
0
1
6
3
6
+ 8
7
1
1
Change 8,736 pence to pounds and pence. Equals £87.36
Multiplication Step 9: Children progress to using the grid method for decimals. What is the product of 3.6 x 2.7?”
This is the same as “What is the product of 2.7 x 3.6?”
4x
3=1
2
Estimate your answer 4 x 3 = 12
U
t
x
3
0.6
2
6
1.2
0.7
2.1
0.42
6
+
2
.
1
1
.
2
0
.
4
2
9
.
7
2
Division Step 1: They will start by sharing items out equally, to work out how many each child gets. “Share 6 sweets equally between 3 children. How many does each child get?�
Division Step 2: Children start to draw pictures or make marks. “Mr Kooner divides 12 children into teams of 3. How many teams are there?�
Division Step 3: Children start to recognise the symbol for division ÷ (8 ÷ 2). “8 Sweets are shared between 2 children. How many do they both get?”
Each child gets 4 sweets.
Division Step 4: Children start to use a number line in grouping. “There are 10 sweets. How many children can have two sweets each?”
1 2 3 4 5
0 2 4 6 8 10 How many 2’s make 10? (Inverse operation to multiplication & using knowledge of tables).
Step 5: Children use the multiple subtraction method to work out division with remainders. “Liam has 17 sweets to share between 4 friends and himself. How many sweets do they each get and how many are left over?”
1 lot of 5
2 lots of 5
5
3 lots of 5
10
17 ÷ 5 = 3 remainder 2
Remainder 2
15
17
Division Step 6: Children use ‘chunking up’ on a number line to work out division. Satinder works out the question, “135 divided by 4 equals what?” using both a number line and by jottings.
x10
0
x10
40
x10
80
x2
120
x1
128
Don’t forget to count any remainder =3
132 135
Add up how many ‘lots of 4’ you jumped: 10 + 10 + 10 + 2 + 1 = 33 r3
Step 7: Children continue to use the ‘chunking up’ method on a number line to work out division. They now use greater chunks:
Use easy times tables facts to help you: x4 facts: 1x4=4 2x4=8 5 x 4 = 20 10 x 4 = 40
Emily works out the question: “135 divided by 4 equals what?”
x30
0
x3
120
remainder = 3
132 135
Add up how many ‘lots of 4’ you jumped: 30 + 3 = 33 r3
Division Step 8: Children progress to a more formal way of recording (called chunking): “If 100 pence is shared equally between 7 children, how much does each child get?”
1 -
-
0
0
7
0
3
0
2
8
(10 x 7)
(4 x 7)
2 So 100 ÷ 7 = 14 r2 Leading to: 518 ÷ 7 =
-
-
-
5
1
8
3
5
0
1
6
8
1
4
0
2
8
2
8
(50 x 7)
(20 x 7)
(4 x 7)
So 518 ÷ 7 = 74
Times Tables 1x1= 1
1x2=2 1x3=3 1x4=4 1x5=5
2x1= 2
2x2=4 2x3=6 2x4=8
2 x 5 = 10
3x1= 3
3x2=6 3x3=9
3 x 4= 12
3 x 5 = 15
4x1= 4
4x2=8
4 x 3 = 12
4 x4 = 16
4 x 5 = 20
5x1= 5
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20
5 x 5 = 25
6x1= 6
6 x 2 = 12
6 x 3 = 18
6 x 4 = 24
6 x 5 = 30
7x1= 7
7 x 2 = 14
7 x 3 = 21
7 x 4 = 28
7 x 5 = 35
8x1= 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
9x1= 9
9 x 2 = 18
9 x 3 = 27
9 x 4 = 36
9 x 5 = 45
10 x 1 = 10 10 x 2 = 20 10 x 3 = 30 10 x 4 =40 10 x 5 = 50 1x6= 6
1x7=7
1x8=8
1x9=9
1 x 10 = 10
2 x 6 = 12
2 x 7 = 14
2 x 8 = 16
2 x 9 = 18
2 x 10 = 20
3 x 6 = 18
3 x 7 = 21
3 x 8 = 24
3 x 9 = 27
3 x 10 = 30
4 x 6 = 24
4 x 7 = 28
4 x 8 = 32
4 x 9 = 36
4 x 10 = 40
5 x 6 = 30
5 x 7 = 35
5 x 8 = 40
5 x 9 = 45
5 x 10 = 50
6 x 6 = 36
6 x 7 = 42
6 x 8 = 48
6 x 9 = 54
6 x 10 = 60
7 x 6 = 42
7 x 7 = 49
7 x 8 = 56
7 x 9 = 63
7 x 10 = 70
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80
9 x 6 = 54
9 x 7 = 63
9 x 8 = 72
9 x 9 = 81
9 x 10 = 90
10 x 6 =60 10 x 7 = 70 10 x 8 = 80 10 x 9 = 90 10x10=100
Dale Community Primary School