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Early Years and Primary School Calculation Policy Parents’ Guide October 2011
Contents 1.
Introduction 2.
Mathematics Essentials 3.
Addition 4.
Subtraction 5.
Multiplication 6.
Division 7.
Glossary 8.
Resources Introduction We have recently produced a policy for teaching calculation strategies (addition, subtraction, multiplication and division) throughout the Early Years and Primary School. The aim of this policy is to ensure that our students are taught using a progressive and consistent approach in both English and French classes. The policy has been based upon research and best practice from around the world and has involved close consultation amongst all teachers. The policy emphasises the importance of learning mental methods and the need for students to secure their understanding of these before moving on to more formal written methods. We, as teachers at ISM, greatly value the role of parents in supporting their children’s learning and this parents’ guide has been produced to enable teachers and parents to work collaboratively to ensure that ISM students receive the best possible support in their learning of Mathematics. Chris Benson Leader of Learning Curriculum and Assessment Early Years and Primary Please note: Any child who enters the Primary School having already been taught alternative formal methods/strategies will not be dissuaded from using these. Where it is deemed to further their understanding, they will, however, be taught further methods from the ISM calculation policy. Mathematics Essentials At ISM our students will be taught mathematics in order to be able to:
• Choose and use appropriate number operations and ways of calculating to solve problems • Secure understanding of a range of informal methods (including the use of number lines), leading on to the use of formal methods once the necessary prerequisites (see below) are secure. • Solve mathematical puzzles and problems, recognise and explain patterns and relationships. • Explain methods and reasoning orally and in writing. • Use estimation to ensure that answers are reasonable. • Use the four operations to solve problems involving numbers in “real life” (single and multi-­‐step). • Make connections between what they learn and their everyday existence • Applying what they learn to practical situations Addition and Subtraction Children are ready for written methods of addition and subtraction if they: • know addition and subtraction facts to 20 • understand place value and can partition numbers into hundreds, tens and units • use and apply the commutative and associative laws of addition and they can: • add at least 3 single digit numbers mentally • add and subtract any pair of two-­‐digit numbers mentally • explain their mental strategies orally and record them using informal jottings
Multiplication and Division Children are ready for written methods of multiplication and division if they: • know the 2, 3, 4, 5 and 10 times-­‐tables and corresponding division facts • know the result of multiplying by 0 or 1 • understand place value • understand 0 as a place holder and they can: • multiply two and three-­‐digit numbers mentally by 10 and 100 • use their knowledge of all the multiplication tables to approximate products and quotients using powers of 10 • use and apply the commutative and associative laws for multiplication, and the distributive law of multiplication over addition and subtraction • double and halve two-­‐digit numbers mentally • use multiplication facts they know to derive mentally other multiplication facts they don’t know • explain their mental strategies orally and record them using informal jottings Addition
Kindergarten and Transition
Children are encouraged to develop a mental picture of the number system in their
heads to use for calculation.
They develop ways of recording calculations using pictures
Making 6
3 and 3
2 and 4
0 and 6
4 and 2
1 and 5
5 and 1
Bead strings or bead bars can be used to illustrate addition
8
2
8+2=10 Year 1 Bead strings or bead bars can be used to illustrate addition including bridging
through ten by counting on 2 then counting on 3.
8 + 5 =13
8
2
3
Number lines can be used to illustrate addition including bridging through ten by
counting on 2 then counting on 3.
bridging through ten
1. Count on to the next ten
2. Add the rest
They use number lines and practical resources to support calculation and teachers
demonstrate the use of the number line.
Counting on in ones
Year 2
Children will begin to use ‘empty number lines’ themselves starting with the larger
number and counting on.
•
First counting on in tens and ones.
•
Then helping children to become more efficient by adding the units in one
jump
•
Followed by adding the tens in one jump and the units in one jump.
•
Bridging through ten can help children become more efficient.
Year 3
Children will continue to use empty number lines with increasingly large numbers,
including compensation where appropriate.
•
Count on from the largest number irrespective of the order of the
calculation.
Compensation
Children will use horizontal partitioning (with brackets) for addition of 2 digit + 2
digit addition problems.
67 + 24 = 91
80
11
(60 + 20) + (7 + 4) = 91
Year 4
Children will continue to use empty number lines with increasingly large numbers,
including compensation where appropriate.
•
Count on from the largest number irrespective of the order of the
calculation.
Compensation
Children will continue to use horizontal partitioning (with brackets) for addition of
2 digit + 2 digit addition problems.
67 + 24 = 91
80
11
(60 + 20) + (7 + 4) = 91
Formal Addition
• Carry above the line.
1
6 2 5
+
4 8
6 7 3
Adding with decimals
ex: money
6
+
.
0
.
6
.
1
2
5
€
4
8
€
7
3
€
All decimals points lined-up
before starting the calculation
Year 5
Children will continue to use empty number lines with increasingly large numbers,
including compensation where appropriate.
•
Count on from the largest number irrespective of the order of the
calculation.
Compensation
Children will continue to use horizontal partitioning (with brackets) for addition of
2 digit + 2 digit addition problems.
67 + 24 = 91
80
11
(60 + 20) + (7 + 4) = 91
Children should extend the carrying method to numbers with at least four digits
and including decimals.
1
5
+
4
1 0
1
8 7
7 5
6 2
1 1 1
3 5 8 7
+
6 7 5
4 2 6 2
1
2 . 6 3
+3 . 4 2
6 . 0 5
Partitioning with decimals
Each number has 2 decimal places, therefore, remove the decimal point, calculate
the answer and replace the decimal point, counting back 2 decimal places.
2.63 + 3.42
263 + 342 =
500
100
5
(200 + 300) + (60 + 40) + (3+2) = 605
= 6.05
Year 6
Children will continue to use empty number lines with increasingly large numbers,
including compensation where appropriate.
Count on from the largest number irrespective of the order of the
calculation.
•
Compensation
Children should extend the carrying method to number with any number of digits
and adding with decimals.
1
6
+
1
4
1
2
4
7
1
4
3
8
6
9
8
4
2
2
6
3
1
4
+
1
3
2
3
9
.
.
.
.
1
4
1
4
0
3
7
5
5
Subtraction
Kindergarten and Transition
Children are encouraged to develop a mental picture of the number system in
their heads to use for calculation.
They develop ways of recording calculations using pictures etc.
Bead strings or bead bars can be used to illustrate subtraction including
bridging through ten by counting back 3 then counting back 2.
6-2=4
Year 1
Bead strings or bead bars can be used to illustrate subtraction including
bridging through ten by counting back 3 then counting back 2.
13-5=8
Children then begin to use numbered lines to support their own calculations using a numbered line to count back in ones.
Year 2
Children will begin to use empty number lines to support calculations.
Counting back: First counting back in tens and ones.
Then helping children to become more efficient by subtracting the units in one
jump (by using the known fact 7 – 3 = 4).
Subtracting the tens in one jump and the units in one jump.
Bridging through ten can help children become more efficient.
Counting on: The number line should still show 0 so children can cross out the
section from 0 to the smallest number. They then associate this method with
‘taking away’.
30 – 16 = 14
+4
xxxxxxxxxxxxxxxxxx 16
+10
20
30
Year 3
Children will continue to use empty number lines with increasingly large
numbers.
Decomposition - base 10 materials`
TENS
e.g. 25 - 13
ONES
25
TENS
ONES
13
Year 4
Decomposition
Formal Subtraction
76
+
6
5 14
8
11
4
6
6
8
Carry above the line.
Continue using number lines – counting up from the smaller number
to the larger one through multiples of 10 and 100
Year 5
Decomposition
Formal Subtraction
76
+
6
5 14
8
11
4
6
6
8
Carry above the line.
Continue using number lines – counting up from the smaller number
to the larger one through multiples of 10, 100 and 1 000
Year 6
Decomposition
Formal Subtraction
76
+
6
5 14
8
11
4
6
6
8
Carry above the line.
Continue using number lines – counting up from the smaller number
to the larger one through multiples of 10, 100 and 1 000
Multiplication
Kindergarten and Transition
Children will experience equal groups of objects.
They will work on practical problem solving activities involving
There are 6
pairs of socks.
How many socks
are there
altogether?
Year 1
Children will experience equal groups of objects.
They will count in 2s and 10s
They will work on practical problem solving activities involving equal sets or
groups.
There are 4
horses. How
many legs do
they have
altogether?
Year 2
They will count in 2s, 5s and 10s
Children will develop their understanding of multiplication and use jottings to
support calculation:
Repeated addition
+
5
3 times 5
is
5 + 5 + 5 = 15
5
+
5
or 3 lots of 5 or 5 x3
Repeated addition can be shown easily on a number line:
and on a bead bar:
Commutativity
Children should know that 3 x 5 has the same answer as 5 x 3. This can also be
shown on the number line.
Arrays
Children should be able to model a multiplication calculation using an array.
This knowledge will support with the development of the grid method.
Year 3
Children will continue to use:
Repeated addition
4 times 6 is 6 + 6 + 6 + 6 = 24 or 4 lots of 6 or 6 x 4
Children should use number lines or bead bars to support their understanding.
Arrays
Children should be able to model a multiplication calculation using an array.
This knowledge will support with the development of the grid method.
Scaling
e.g. Find a ribbon that is 4 times as long as the blue ribbon
Using symbols to stand for unknown numbers to complete equations using
inverse operations
£ x 5 = 20

Partitioning
3 x r = 18
£ x š = 32
38 x 5 = (30 x 5) + (8 x 5)
= 150 + 40
= 190
Year 4
Children will continue to use arrays where appropriate leading into the grid
method of multiplication.
Partitioning
38 x 5 = (30 x 5) + (8 x 5)
= 150 + 40
= 190
Grid method
TU x U
(Short multiplication – multiplication by a single digit)
23 x 8
Children will approximate first
23 x 8 is approximately 25 x 8 = 200
Year 5
Grid method
HTU x U
(Short multiplication – multiplication by a single digit)
346 x 9
Children will approximate first
346 x 9 is approximately 350 x 10 = 3500
TU x TU
(Long multiplication – multiplication by more than a single digit)
72 x 38
Children will approximate first
72 x 38 is approximately 70 x 40 = 2800
1
Using similar methods, they will be able to multiply decimals with one decimal
place by a single digit number, approximating first. They should know that the
decimal points line up under each other.
e.g. 4.9 x 3
Children will approximate first
4.9 x 3 is approximately 5 x 3 = 15
Formal Multiplication (Carrying Above)
x
3
4
11
1
5
7
2
21
2
3
0
5
5
5
4
0
0
0
By end of class 5 - formal multiplication 3 digit x 2 digit
Year 6
ThHTU x U
(Short multiplication – multiplication by a single digit)
4346 x 8
Children will approximate first
4346 x 8 is approximately 4346 x 10 = 43460
HTU x TU
(Long multiplication – multiplication by more than a single digit)
372 x 24
Children will approximate first
372 x 24 is approximately 400 x 25 = 10000
1
Using similar methods, they will be able to multiply decimals with up to two
decimal places by a single digit number and then two digit numbers,
approximating first. They should know that the decimal points line up under
each other.
For example:
4.92 x 3
Children will approximate first
4.92 x 3 is approximately 5 x 3 = 15
Formal Multiplication (Carrying Above) – including decimal numbers.
x
3
4
11
1
5
7
2
21
2
3
0
5
5
5
4
0
0
0
Division
Kindergarten and Transition
Children will understand equal groups and share items out in play and problem
solving.
Year 1
Children will understand equal groups and share items out in play and problem
solving. They will count in 2s and 10s
Year 2
Children will understand equal groups and share items out in play and problem
solving activities.
They will count in 2s, 5s and 10s
Year 3
The emphasis in Y3 is on grouping rather than sharing.
Children will continue to use:
Repeated subtraction using a number line, leading on to the use of repeated
addition
20 ÷ 4 = 5
Children should also move onto calculations involving remainders.
22 ÷ 4 = 5r2
0
4
8
r2
12
16
20 22
Year 4
Children will develop their use of repeated subtraction (and then addition) to
be able to subtract (then add) multiples of the divisor. Initially, these should
be multiples of 10s, 5s, 2s and 1s – numbers with which the children are more
familiar.
Repeated subtraction
Repeated addition
75 ÷ 6 = 12 r3
10 x 6
2x6
60
r3
72 75
Any remainders should be shown as integers, i.e. 14 remainder 2 or 14 r 2.
Children need to be able to decide what to do after division and round up or
down accordingly.
They should make sensible decisions about rounding up or down after division.
Year 5
Continued use of number lines – See Year 4
Children will continue to use written methods to solve short division TU ÷ U.
Children can start to subtract larger multiples of the divisor, e.g. 30x
Short division HTU ÷ U
154 ÷ 8 = 19 r2
8
0
1
1
5
9
74
r2
a) 8 into 1 doesn’t go.
b) 8 into 15 goes once remainder 7
c) 8 into 74 goes 9 times
d) remainder 2
e) = 19 remainder 2
Long Division – Chunking Method
Any remainders should be shown as integers, i.e. 14 remainder 2 or 14 r 2.
Children need to be able to decide what to do after division and round up or
down accordingly.
They should make sensible decisions about rounding up or down after division.
Year 6
Continued use of number lines – See Year 4
Children will continue to use written methods to solve short division TU ÷ U
and HTU ÷ U.
Short division HTU ÷ U
154 ÷ 8 = 19 r2
8
0
1
1
5
9
74
r2
a) 8 into 1 doesn’t go.
b) 8 into 15 goes once remainder 7
c) 8 into 74 goes 9 times
d) remainder 2
e) = 19 remainder 2
Long division HTU ÷ TU – Chunking Method
Any remainders should be shown as fractions, i.e. if the children were dividing
32 by 10, the answer should be shown as 3 2/10 which could then be written as
3 1/5 in its lowest terms.
Extend to decimals with up to two decimal places. Children should know that
decimal points line up under each other.
Glossary Addition and subtraction facts : An Addition Fact is any 2 whole numbers added together, up to and including 10+10. A Subtraction Fact is any 2 numbers subtracted one from the other, from 20 down. Facts should be committed to memory for quick and easy recall. Array : Items (such as objects, numbers, etc.) arranged in rows and columns. Associative Law : In addition and multiplication, no matter how the numbers are grouped, the answer will always be the same. Addition and multiplication are associative. Subtraction and division are not. Carrying : To transfer (a number) from one column of figures to the next, as from units to tens 1
6 2 5
+
4 8
6 7 3
carrying Commutativity : The commutative law is the law that says you can swap numbers around and still get the same answer when you add or multiply. Examples: 3 + 6 = 6 + 3 2 × 4 = 4 × 2 Compensation : In order to perform a calculation on two numbers, we sometimes round one of the numbers to make the calculation simpler. Then we perform an addition or subtraction, to compensate. Decimal places : The number of digits a number contains after the decimal point. e.g. 4.7 has one decimal place 4.75 has two decimal places Decomposition : A vertical method of subtraction. The number in the top line is broken down to aid calculation. Example: For 754 – 86 the calculation is written as 76
6
5 14
14
8
6
6
8
Digit : One of the symbols of a number system most commonly the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Examples: the number 29 is a 2-­‐digit number; there are three digits in 2.95. The position or place of a digit in a number conveys its value. Distributive Law: The Distributive Law means that you get the same answer when you multiply a number by a group of numbers added together as when you do each multiplication separately Example: 3 × (2 + 4) = 3×2 + 3×4 So the "3" can be "distributed" across the "2+4" into 3 times 2 and 3 times 4. Division facts: A division fact is the result of a division of any two whole numbers Divisor : The number by which another is divided. Example: In the calculation 30 ÷ 6 = 5, the divisor is 6. In this example, 30 is the dividend and 5 is the quotient. Estimation: To make an approximate or rough calculation, often based on rounding e.g. David receives 3€ pocket money per week. How much does he receive in one year? Estimate = 3€ x 50 = 150€ Formal methods Written methods (see example below) 1
6 2 5
+
4 8
6 7 3
Fractions Part of a whole. A number written with the bottom part (the denominator) telling you how many parts the whole is divided into, and the top part (the numerator) telling how many you have. Grid method : A method of multiplication you break up the number into (for example) hundreds, tens and units. Multiply each separately and then add the answers together. e.g. 4 346 x 8
Horizontal partitioning : To split a number into component parts. Addition 80 11 (60 + 20) + (7 + 4) = 91 67 + 24 = 91 Multiplication 38 x 5 = (30 x 5) + (8 x 5) = 150 + 40 = 190 Informal Methods: Calculation methods used prior to using formal methods. e.g; See Jottings and Number lines Integers : Any of the positive or negative whole numbers and zero. Example: ...-­‐2, -­‐1, 0, +1, +2 Inverse operations : Opposite, inverse operations Addition and Subtraction are inverse operations Multiplication and Division are inverse operations e.g. 9 + 5 = 14 so 14 – 5 = 9 6 x 9 = 54 so 54 ÷ 9 = 6 Jottings : The use of pictures and symbols to aid with calculation. Lowest terms (fractions) : Also known as the lowest common denominator (number at the bottom of a fraction) The smallest number that can be used for all denominators (the bottom number) of 2 or more fractions. Mathematical Puzzles : These are often number-­‐based problem solving activities such as Suduko and the example shown below : e.g. Write the numbers 1-­‐9, so that each row, column and diagonal add up to the same number : 9 2 8 Multiples : The result of multiplying by a whole number. Example: 4 × 5 = 20 20 is a multiple of 4 and also of 5 multiples of 4 = 4, 8, 12, 16, 20 etc… Number lines : A line marked with numbers, used to show operations : e.g. Number system : A writing system for expressing numbers e.g. Roman numerals or 0,1, 2, 3, 4, 5, 6, 7, 8 and 9 Operations: The four operations are addition, subtraction, multiplication and division. Place Value : The value of a digit depending on its place in a number. e.g. In the number 555, the 5 digit can represent 5 hundreds, 5 tens and 5 units Place Holder : In this context, that zero maintains a value within a number e.g. We cannot ignore 0 as a place holder in the number 505 Powers of 10 : Numbers that are formed by multiplying ten by itself a certain number of times e.g.. ten to the third power= 10,000. You add that number of zeros to the end of ten. Problem solving activities : Activities where the student is presented with a problem (sometimes written) which they must interpret, devise a method to solve it and follow mathematical procedures to achieve the result. e.g. To cook rice, you need 5 cups of water for every cup of rice. You cook 3 cups of rice. How many cups of water do you need? Products : The answer when two or more numbers are multiplied together. Quotients: The answer after you divide one number by another Dividend ÷ divisor = quotient e.g. in 12 ÷ 3 = 4, 4 is the quotient Round up or down : To change a number to a more convenient value. e.g. 26 rounded to the nearest 10 equals 30 146 rounded to the nearest 10 0 equals 100 2 575 rounded to the nearest 1 000 equals 3 000 Sets : Collections of items Memebers of a set are called elements Abbreviations : U = UNITS TU = TENS and UNITS HTU = HUNDREDS, TENS and UNITS ThHTU = THOUSANDS, HUNDREDS, TENS and UNITS THOUSANDS 1000s HUNDREDS 100s TENS 10s UNITS (ONES) 1s 4 7 8 2 4 782 = 4 000 + 700 + 80 + 2 Websites : Resources www.mathletics.com Mathletics is the next generation in learning, helping students enjoy maths and improve their results. Mathletics covers Kindergarten to Year 12 curriculum. Includes interactive lessons and a detailed maths glossary. Every child in Primary School has a mathletics account with a designated username and password. If your child doesn’t have their login details, ask their class teacher www.primarygames.co.uk Interactive maths games and supporting maths worksheets. http://nrich.maths.org/public/ The Nrich Maths Project Cambridge,England. Mathematics resources for children ,parents and teachers to enrich learning. www.mathsphere.co.uk For teachers and parents to download practice worksheets and tests to improve children's achievement. http://www.math-­‐aids.com/Number_Lines/ Create your own number lines http://www.helpingwithmath.com/resources/oth_number_lines.htm Create your own number lines