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New Mills Primary School Maths Calculation Policy January 2014

Counting

New Mills Primary School Maths Calculation Policy January 2014 Counting supports all understanding and should be included in every lesson. Counting skills need to be kept sharp as they support all areas of mathematical

learning.

Counting Stage Details Strategies/Equipment to support

Stable-order

Children can count and up and down in order.

Children know the counting words and can recite them in the correct order.

One-one

Children can count objects one for one.

One and only one number word has to be matched to each and every object.

NB: Lack of coordination can be a problem.

Helps if children move objects, use large rhythmic movements or clap as they count.

Cardinality

Children understand that the number name allocated to the last object tells you how many objects have been counted.

Counting out a number of objects from a larger collection shows that children understand cardinality because they know the number they stop counting at will give the total number.

NB: Children who cannot distinguish cardinal from ordinal numbers may, when asked to give you three, give you the third objects.

Subitising

Children can recognise a small number of objects or dots etc. without counting them.

Can be achieved for quantities up to approximately 5.

Dot patterns on dice, dominoes and playing cards. Small groups of randomly arranged shapes stuck on cards.

Abstraction

Children can count anything – visible objects, hidden objects, imaginary objects, sounds etc.

Children find it harder to count things they cannot move, touch, see or that move around. Children also find it difficult to count a mix of different objects, or similar objects of very different sizes.

Conservation

Children understand that when objects are rearranged the number of them stays the same.


Mental Calculations


Expectations for Each Year

Note: These expectations are cumulative – i.e. in Year 3, children should also be secure in Year 1 and 2 objectives etc.

Year Group Pupils should be taught to:

1

Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number.

Count, read and write numbers to 100 in numerals; count in multiples of twos, fives and tens.

Given a number, identify one more and one less.

Represent and use number bonds and related subtraction facts within 20.

Add and subtract one-digit and two-digit numbers to 20, including zero.

2

Count in steps of 2, 3, and 5 from 0, and in tens from any number, forward and backward.

Recognise the place value of each digit in a two-digit number (tens, ones/units).

Identify, represent and estimate numbers using different representations, including the number line.

Compare and order numbers from 0 up to 100; use <, > and = signs.

Recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100.

Add and subtract numbers using concrete objects, pictorial representations, and mentally, including: - A two-digit number and ones/units. - A two-digit number and tens. - Two two-digit numbers. - Adding three one-digit numbers.

Recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems.

Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers.



3

Count from 0 in multiples of 4, 8, 50 and 100. Find 10 or 100 more or less that a given number.

Recognise the place value of each digit in a three-digit number (hundreds, tens, ones/units).

Compare and order numbers up to 100.

Identify, represent and estimate numbers using different representations.

Read and write numbers up to 1000 in numerals and in words.

Add and subtract mentally, including: - A three-digit number and ones/units. - A three-digit number and tens. - A three digit number and hundreds.

Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables.

Count and up and down in tenths. Recognise that tenths arise from dividing an object into 10 equal parts and dividing one-digit numbers or quantities by 10.



4

Count in multiples of 6, 7, 9, 25 and 1000.

Find 1000 more or less than a given number.

Count backwards through zero to include negative numbers.

Recognise the place value of each digit in four-digit numbers.

Order and compare numbers beyond 1000.

Identify, represent and estimate numbers using different representations.

Round any number to the nearest 10, 100 or 1000.

Recall multiplication and division facts multiplication tables up to 12 x 12.

Use pace value, known and derived facts to multiply and divide mentally, including: multiplying by 0 and 1, dividing by 1, multiplying together three numbers.

Recognise and use factor pairs and commutativity in mental calculations.

Count up and down in hundredths. Recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten.

Find the effect of dividing a one or two digit number by 10 and 1000, identify the value of the digits in the answers as ones, tenths and hundredths.

Round decimals with one decimal place to the nearest whole number.

Compare numbers with the same number of decimal places up to two decimal places.



5

Read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit.

Count forwards and backwards in steps of powers of 10 for any given number up to 1 000 000.

Interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers, including through zero.

Round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000.

Add and subtract numbers mentally with increasingly large numbers.

Multiply and divide numbers mentally, drawing upon known facts.

Read, write, order and compare numbers with up to three decimal places.

6

Read, write, order and compare numbers to at least 10 000 000 and determine the value of each digit.

Round any whole number to a required degree of accuracy.

Use negative numbers in context, and calculate intervals across zero.

Perform mental calculations, including with mixed operations and large numbers.


Models for Addition


Adding two sets of objects (less than 20)

Step Details Model Apparatus

1 Count each group then both

together.

Count 5, then 7, then 12 Note: Encourage children to visualise numbers in 10s, e.g. boxes of 10 to put counters in to give order to numbers

Counters Counting bears Beans Coins Dienes apparatus Number cards Bead bars Number lines

2 Counting on – hide one set of

objects and replace with a number card.

3 Adding two numbers by

counting out on a bead bar.

4 Adding two numbers on a

number line.

NB: Always show transition of apparatus when moving between steps, e.g. show bead bar and number line side by side.



Adding two sets of objects (two digits greater than 20)

1 Show numbers as all units, and

then separate into tens and units.

e.g. 25 + 37 becomes

Counters, unit cubes, coins. Dienes apparatus. Place value counters.

2 Children group into tens and

units, and exchange any groups of ten units to reach final answer.

3

Compacted: Children sort tens and units apparatus into columns.

Once represented using tens and

units apparatus, children move onto place value counters.

tens units tens units

2

5 2

5

3

7 3

7

6

2 6

2



Adding two sets of objects (two digits greater than 20)

4 Column method addition.

2 5

+ 3 7

6 2

1


Models for Subtraction


A model for progression through early skills for subtraction

Counting out e.g. 9 – 3

Hold up 9 fingers, fold down 3, and count the remaining fingers.

Counting back from e.g. 9 – 3

Count back 3 numbers from 9; “8…7…6”

Counting back to

e.g. 11 – 7

Count back from 11 to 7, keeping a tally using fingers;

“10…9…8…7” (4 fingers)

Counting up (complementary addition) Count up from 7 to 11; “8…9…10…11”

Note: not a natural strategy for many children because of the perception of subtraction as ‘taking away’.

Using known facts Rapid response based on facts known ‘by heart’.

Using derived facts e.g. a child who knows 20 – 5 = 15 can adjust for 20 – 6 = 14

Note: more unusual in subtraction than in addition.

Using knowledge of place value e.g. a child who knows 35 – 10 = 25 can use this to calculate 35 – 9.

Subtracting smaller Step Details Model Apparatus


numbers

1

Removing items from a set.

(When this is done on a bead bar, there are links with both counting back and difference

on a number line.)

12 – 5 = 7

Counters Counting bears Beans Coins Number cards Bead bars Number lines

2 Comparing two sets

(comparison or difference).

3

Seeing one set as partitioned.

This is a key image – partitioning needs to be done in a variety of ways, not just

tens and units.

e.g. 12 as made up of 5 and 7, as well as 10 and 2.

helps to see related calculations:

5 + 7 = 12, 7 + 5 = 12

12 – 7 = 5, 12 – 5 = 7

4 Counting on and back on a

number line.

NB: Always show transition of apparatus when moving between steps, e.g. show bead bar and number line side by side.


12 – 7 = 5

7 12 10

3 2


Subtracting larger numbers

1 Show larger number as all units, and then separate into tens and units.

e.g. 35 – 27 becomes

Dienes apparatus Place value counters.

2

Children then subtract the smaller number, by removing units first and exchanging any tens blocks when necessary to find answer.

2

Subtracting with exchanging, using base ten equipment. Children start with units and exchange any tens when needed. Once represented as base ten equipment, children then use place value counters.

tens units tens units

3

5 3

5

2

7 2

7

8

8

4 Column method subtraction.

3 5

- 2 7

8

2 1


Models for Multiplication



Early stages of multiplication

1 Lots of the ‘same thing’.

e.g. 4 lots of 2 = 8

Counters Counting bears Beans Coins Bead bar Number lines Times tables games

2 Counting in groups.

e.g. groups of 3

3 Jumping up number line in equal steps. Note: Steps 1 – 3 all use repeated addition.

4 Recalling multiplication tables.



Advanced stages of multiplication

1 Representing multiplication as an array

Counters Number lines

2 2 digit numbers represented as an array, split into tens and units.

e.g. 16 x 6 = 60 + 36 = 96 e.g. 16 x 12 = 100 + 60 + 20 + 12 = 192

Counters



Advanced stages of multiplication

3 Grid method (note, this is no longer considered a formal written method).

e.g. 16 x 6 = 60 + 36 = 96

e.g. 16 x 12 = 100 + 60 + 20 + 12 = 192

4

Compact method (Children only to move on to this method once they are very secure with the grid method.)


Models for Division


Step

Details Model Apparatus

Early stages of division.

1 Grouping

e.g. 12 into groups of 2 = 6

Hoops Counters Multilink Beans Coins Number lines Table trios

2

Counting backwards on a number line. This can be shown as repeated subtraction to reinforce link with division: e.g. 12 ÷ 2 12 – 2 = 10 10 – 2 = 8 8 – 2 = 6 6 – 2 = 4 4 – 2 = 2 2 – 2 = 0 6 lots of 2 have been subtracted.

e.g. 12 ÷ 2 – count back in jumps of 2.

3 Deriving from knowledge of multiplication facts.

4 Counting backwards on a number line with remainders.

e.g. 19 ÷ 3 = 6 remainder 1



Step

Dividing multiples of 10.

1 Grouping e.g. 120 ÷ 3 = 40 Use ‘tens’ place value counters and divide into 3 groups.

Place value counters (tens only)

2 Grouping into rows (note – this is not the same as an array used for multiplication)

Note – using place value counters, where each counter represents 10

3 Bus shelter method


Step


Dividing larger numbers

1 Grouping

e.g. 138 ÷ 6 = 23 Group hundred, tens and unit separately, exchanging where necessary. 1 hundred cannot be grouped into 6, so exchange for tens. 13 tens can be grouped into 2 groups of 6, with one left over. 18 units can be groups into 3 groups of 6.

Place value counters


2 Grouping into rows (note – this is not the same as an array used for multiplication)

e.g. 138 ÷ 6 = 23 1 hundred cannot be grouped into 6, so exchange for tens.



Step


13 tens can be grouped into 2 groups of 6, with one left over.

18 units can be groups into 3 groups of 6.


3 Bus shelter method e.g. 138 ÷ 6 = 23


Exchange for units.


Step


Progression in short division


No exchange, no remainder

e.g.

No exchange, with remainder

e.g.

With exchange, no remainder

e.g.

With exchange, with remainder

e.g.

Empty place at start of quotient

e.g.

Noughts in the quotient

e.g.

Decimal dividend

e.g.

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