How children learn (and we teach) the Four Operations in KS2 BPS November 2015

 To understand expectations of the new curriculum in mathematics for KS2.  To be familiar with methods used for addition and subtraction  To become familiar with methods used for multiplication and division  To clarify progression Aims

The New Curriculum for mathematics aims to ensure that all pupils:  become fluent in the fundamentals of mathematics  reason mathematically  can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication The New Curriculum for mathematics

4 Your feelings of maths

  312 x 54  1850 ÷ 25  2786 – 1899  x 13  4782 ÷ 13  Consider for a moment…  15 – 5 =  29+11=  18 – 16 =  2007 – 1992 =  359 – 211 =  =  =  24.6 – 12.4 = How would you approach these questions?

We want our children to consider Which method to use and why:  mental calculation skills  written methods of calculation  calculator skills Ways of doing

 We may have been taught different ways at school ADDITION

 Number order  Number bonds – to and within  Place value understanding Place value understanding  Mental maths skills are key  Vocabulary can be a barrier The Must Haves …

Counting on Number Lines (Progression) Numbered lines Bead strings Partly numbered lines

Counting On Not crossing tens boundary e.g Jumps of 10s and 1s Encourage children to become more efficient Add 3 in 1 jump Add 20 in 1 jump

Crossing tens boundary e.g Jumps of 10s and 1s Help children become more efficient:  Add 5 in jumps of 3 and 2 (bridging over 10)  Add 20 in 1 jump Counting on

= ( ) 140 ( ) 9 (6 + 3) Column addition = We only use these if children are able to manipulate numbers and have a good understand of place value

Compact (Formal) method New Curriculum Accurate use of language is key (These are carried)

SUBTRACTION

The must haves… Number order Number bonds Place value Lots of Practical (concrete) Purpose Mental maths Vocabulary

Using a numberline for larger numbers 74 – 27 = – 27 = = 47

No ‘borrowing’ Formal written method

Compact Formal method New Curriculum

New Curriculum expectations + - Lower key stage 2 (Years 3 and 4) Ensure pupils become increasingly fluent with the 4 operations Develop efficient written and mental methods Perform calculations accurately with increasingly large whole numbers. (fractions and decimals) Year 3 Pupils should be taught to: add and subtract numbers mentally add and subtract numbers with up to 3 digits, using formal written methods estimate the answer to a calculation and use inverse operations to check answers solve problems Year 4 Pupils should be taught to: add and subtract numbers with up to 4 digits using the formal written methods estimate and use inverse operations to check answers to a calculation solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

Year 4 examples = = 7624 − 931 − 87 = 1492 − 605 = Dev and Joe each buy a book. Dev pays with a £5 note and gets £1.05 change. Joe’s book costs £7 How much more does Joe’s book cost than Dev’s book?

New Curriculum expectations + - Upper key stage 2 (years 5 and 6) Develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. Pupils are introduced to algebra as a means for solving a variety of problems. Year 5 add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction) add and subtract numbers mentally with increasingly large numbers use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why Year 6 solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Year 6 examples Nicola has £50. She buys 3 flowerpots and a spade. How much money does she have left? A shop makes 100 sandwiches. All the sandwiches are either cheese or tuna. Some of the sandwiches also have salad with the cheese or tuna. 30 sandwiches have cheese with salad. 15 sandwiches have tuna without salad. 75 sandwiches have salad. How many sandwiches have cheese without salad?

Multiplication Slide 2.23

 Counting sticks Counting sticks  Repeated addition  Rapid recall of multiplication tables facts  Games  Good as oral / mental starters  Using known facts (e.g. doubling)  Arrays Arrays Early Multiplication

 Grid method Grid method 26 x 7 Moving on with Multiplication Progression 2 digit x 1 digit 3 digit x 1 digit 3 / 4 digit x 2 digit x

3 4 2 x Formal Methods – New curriculum x

Apply times tables facts This represents the multiplication of a 4-figure number by 3. The whole calculation uses each of the digits 0−9 once and once only.

Year 3  recall and use multiplication and division facts for 3, 4 and 8 x  write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods  solve problems, including missing number problems, involving multiplication and division Year 4  recall multiplication and division facts up to 12 × 12  use place value and known facts to multiply and divide mentally  multiply two-digit and three-digit numbers by a one-digit number using formal written layout  solve problems involving multiplying and dividing New curriculum Expectations - x

5 × 4 × 7 = Year 4 examples 12 × 12 = 729 × 4= Two children take part in a sponsored silence. Laura said ‘I kept quiet from 10:30 to 10:35 in the morning. I got £3.42 a minute.’ Carl said ‘I kept quiet for half an hour. I got 35p a minute.’ How much money did each child get?

Year 5  identify multiples and factors, prime numbers, square numbers, cube numbers  multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers  multiply and divide numbers mentally, drawing upon known facts  multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000  solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates New curriculum expectations x

Year 6  multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication  perform mental calculations, including with mixed operations and large numbers  solve problems involving addition, subtraction, multiplication and division and use estimation to check answers to calculations

Year 6 examples 678 × 54 Write the two missing digits to make this long multiplication correct. The area of a rugby pitch is 6,108 square metres. A football pitch measures 112 metres long and 82 metres wide. How much larger is the area of the football pitch than the area of the rugby pitch?

33 Division

 Children appear to find sharing (partitive) tasks easier but this becomes inefficient with larger numbers, e.g. 196 Smarties divided by 6 children – on a one for you, one for me basis this takes a while!!! We therefore need to help children to move from sharing individual items to groups of items, e.g. We could do 20 for you, 20 for me etc, we are taking away groups of the divisor from the dividend (CHUNKING). Dividing - Sharing and grouping

Children’s working A tent holds 6 children. How many tents are needed to hold 70 children?

48 ÷ 6 = Number line division 1 x

To divide 81 by 3

Remainders 17 ÷ 5 = r2

Formal method – short and long division New curriculum r

4 ) 848 no exchange, no remainder 3 ) 635 no exchange, with remainder 7 ) 994 with exchange, no remainder 3 ) 470 with exchange, with remainder 7 ) 287 empty place at start of quotient 4 ) 816 noughts in the quotient 5 ) 61.5 decimal dividend Progression of difficulty

Year 3  recall and use multiplication and division facts for 3, 4 and 8 x  write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods  solve problems, including missing number problems, involving multiplication and division Year 4  recall multiplication and division facts up to 12 × 12  use place value and known facts to multiply and divide mentally  multiply two-digit and three-digit numbers by a one-digit number using formal written layout  solve problems involving multiplying and dividing New curriculum expectations

Year 4 examples Calculate 48 ÷ 3 Alan has 45 beans. He plants 3 beans in each of his pots. How many pots does he need? Leila puts 4 seeds in each of her pots. She uses 6 pots and has 1 seed left over. How many seeds did she start with? Write in the missing numbers. 37 × = 111 ? ÷ 4 = 21

Year 5  identify multiples and factors, prime numbers, square numbers, cube numbers  multiply and divide numbers mentally, drawing upon known facts  divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context  multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000 New curriculum expectations 43

Year 6  divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context  divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context  perform mental calculations, including with mixed operations

Year 6 examples Chen bought three jars of honey and a dipper. The total cost was £5.40 The dipper cost 75p. How much did each jar of honey cost? Calculate 544 ÷ 32 Eggs are put in trays of 12. The trays are packed in boxes. Each box contains 180 eggs. How many trays are in each box?

 To understand expectations of the new curriculum in mathematics for KS2.  To be familiar with methods used for addition and subtraction  To become familiar with methods used for multiplication and division  To clarify development Revisiting our aims