Understanding Number and

Calculating

Understanding number is the underpinning of all Maths. It develops:

an understanding of place value a sense of the order of numbers and their relative size

a basis for further calculation

By the end of Year 3 the children should:

• count on or back in tens or hundreds, starting from any two- or three- digit number

• read and write whole numbers to at least 1000• know what each digit represents and partition three-digit

numbers into multiples of 100, multiples of 10 and ones• order whole numbers to at least 1000

Always refer to the actual values of the digits e.g. 43, the 4 needs to be referred to as 40 not 4

Visual representation is vital…..

• bundles of straws in tens• cubes in bags of ten • lollipops in bags of ten• bead strings• partitioning cards• place value charts

Multibase equipment doesn’t provide a true visual image

1 2 3 4 5 6 7 8 9

10 20 30 40 50 60 70 80 90

100 200 300 400 500 600 700 800 900

Language

It is imperative that children are encouraged to talk to each other about their maths. It helps clarify their thinking.• Buzz partners• Think, pair and share are two popular methods

The NNS Vocabulary book needs to be referred to at the start of each topic and reinforced throughout.All new vocabulary should be explored thoroughly.

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Understanding Number

Counting is the starting point for all children in their learning and understanding of Maths.

Progression through counting:• the stable order principle• the one-to-one principle• the cardinal principle• the abstraction principle• the order irrelevance principle

By year 3 children should be counting objects efficiently in tens, fives, twos - not one at a time.

Practice these skills during O & M Starters e.g. counting stick, pendulum, click, clap, slap

Negative NumbersIn Year 4 the children start counting back through zero. They need to:

• order positive and negative integers• plot negative numbers onto number lines• Complete a sequence of numbers e.g. ? -2 ? 2 4• Read and solve problems to do with temperatures

It is a good idea to show them numbers below zero from Year 2, so that they know numbers go back that way. A class number line that goes from – 20 to 100 is a good idea.

Place value

In Year 3, children work with numbers to at least 1000, in Year 4 10 000 and then up to millions.All children need to know • what each digit represents. • that numbers are infinitely big and equally infinitely small.• that positive integers are mirrored equally by negative numbers.

Ways to promote the understanding of these:• digit card work• white board writing• calculators• partitioning strips• number line work

Rounding

Rounding links very closely with estimating.It is often hard for the children to understand why it is necessary.

Useful to start with a context e.g. I go into a shop with £40. I see a jumper for £23.99 and a pair of jogging pants for £19.99. Do I have enough money? Round to £24 and £20, add and quite clearly I don’t.It is about 238 miles to Newcastle Sam travels there and back. He thinks it is a 500 mile round trip. Is he right? 238 is nearly 240, double that and you get 480, so no Sam is not right.

Rounding links well with the mental calculation strategy of +/- 8, 9, 11, 12 etc by adding nearest multiple of ten and adjusting.

Mental Calculation

• Counting of objects and mental counting• Early stages of mental calculation and learning of number facts, with recording• Work with larger numbers and informal jottings• Expanded written methods • Compact written methods• Use of calculators for more difficult calculations

Reflect on these questions……

• When asked to calculate what is your initial response? Do you begin to calculate mentally, reach for pencil and paper or a calculator?

• Were you taught mental methods at school or were you taught standard written algorithms?

• As a result of your learning of mathematics at school, do you think this has equipped you for adult life?

Place value and partitioning Knowledge of number facts, such as number bonds to 10 and 100 The size of numbers and where they fit into the number system The relationship between operations

Mental calculation requires familiarity with:

NC states that by the end of KS2 children should be able to:

• Add or subtract mentally, combinations of one-digit and two digit numbers and develop written methods to record and explain these calculations • Add or subtract mentally, pairs of two-digit whole numbers • Counting on or back from the largest number • Reordering numbers in addition • Partitioning • Bridging through 10 and multiples of 10 • Adding, subtracting 9, 11 etc. by compensating • Doubling by partitioning and near doubles • Find differences by counting up

Mental calculation strategies

• Partition and recombine• Near doubles• Adding near multiples of ten and adjusting• Using patterns of similar calculations• Using known number facts• Bridging though ten, hundred, tenth• Inverse operations• Counting on • x4 by doubling and doubling again• x5 by x10 and halving• X20 by x10 and doubling

Do you provide plenty of opportunities for children toexplore methods of calculating?

24

Use all four operations to make:

Kamrin: Can 8 be divided into 2? 5yrs 7months

Barney Subtracting beans5 yrs 7 months

Frances: The train6yrs 1month

Alison: Multiplying by 997 yrs 3 months

Faced with any calculation children should: Decide whether or not they can use a mental method to get an answer including with jottings Decide, if they can’t, is a written method or calculator more appropriate? Use mental methods for estimating roughly what the answer should be even if pencil and paper or calculators are needed for the exact answer.

Teachers should encourage these approaches to any calculation:

• mental strategies• informal jottings to support mental calculations• standard or efficient written form• calculator

Record steps in addition using partitioning:47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123Then:47 + 76 = 47 + 70 + 6 = 117 + 6 = 123

8 + 7 = 15

48 + 36 = 84

or:

4776

110 13

123

4776

13 110

123

Write the numbers in columns.Adding the tens first:

Adding the ones first:

Progression through addition

1 1

366458824

Column addition

Progression through subtraction

74 - 27

+ 40 +7

27 67 74

Leading to:5 463 – 2 657 +343 + 2 463

2 657 3000 5 463

2 806

xx xx xx xxGroups of:4 x 2

Repeated addition:4 x 2 = 2 + 2 + 2 + 2 = 8

Partitioning:43 x 6

5

38 7266

Progression through multiplication

Sharing6 ÷ 2

Grouping6 ÷ 2

Repeated subtraction:6 ÷ 2 = 6 – 2 – 2 - 2 = 0

Partitioning. 84 ÷ 7 might be:

Progression through division

6 196

60 6 10136 60 6 10

76 60 6 10

16 12 6 2

4 32Answer: 32R 4

6 196

180 6 3016

12 6 24 32

Answer: 32 R 4

Leading to larger groups of the divisor: