Mental and Written Calculation: Multiplication and Division

1. SCITT Day 4 2016-17 Mental and Written Calculation Multiplication and Division

2. Calculator digits

3. Today we will… • Study the NC2014 approach to multiplication and division using manipulatives and/or recording mental methods, informal jottings and formal written methods • Consider how the use and complexity of arrays progress through the primary years • Explore the importance of developing fluency of times tables and deriving division facts Associated Issues • The role of calculators in the classroom • Making links – measurement x/ ÷ by 10, 100, 1000…

4. Words/ phrases x and ÷ What different words/ phrases can you list… ?

5. Day 4 - Multiplication and Division Multiplication as repeated addition Arrays  grid method Standard written methods Arrays Multiplication as scaling Build it Language Models and images White Rose Calculation guidance Understanding x/÷ Fluency Learning x tables Wave 3 charts Division as sharing Links between x/÷ Division as grouping

6. Multiplication Julia Anghileri (2009): “Children’s first experiences of multiplication arise when they make groups with equal numbers of objects and recognise the possibility of counting groups rather than counting individual items.”

7. Understanding Arrays • What different facts are represented by this picture? 3 x 5 = 15

8. Multiplication - Repeated Addition ‘3 x 5’ 3 + 3 + 3 + 3 + 3 We read 3 x 5 as ‘3 multiplied by 5’ (since the 5 is operating on the 3) 0 3 6 9 12 15

9. Understanding Arrays - Language & relationships • What different vocabulary is represented by this picture? “3 times 5” “3 groups of 5” “3 lots of 5” “3 multiplied by 5” “the product of 3 and 5”

10. ‘3x5’ The power of mathematics lies in this compactness of representation … • A tray of muffins, five rows by three • Three bags, each containing five apples • If I have three t-shirts and five pairs of shorts, the number of days that I can go out and not wear the same combination twice • The number of sprats a mother shark eats if she eats five for every one her baby eats and the baby eats three • The length of a rope that is needed to make five skipping ropes each of which are 3m long • The weight of a bag of cement that is five times as heavy as a 3kg bag • …

11. Multiplication - Scaling Increasing a quantity by a scale factor. Multiplying by 5 makes it 5 times bigger. ‘Scaling 3 by a factor of 5’ ‘Scaling 3 by a factor of ½ ’ Y3 NC2014 ‘Positive integer scaling’ Y5 NC2014 ‘Scaling by simple fractions’

12. X grid

13. Six two’s… 2, 4, 6, 8, 10, 12 12 dots can be represented as… Make cards showing calculation on one side, the answer on the other

14. Secrets for Cheating at x tables Secret 1: If you know one fact, then you know a second. Secret 2 X tables always give you a buy one, get three free offer. Secret 3: If you can double, then you can multiply by 2, 4 and 8. Secret 4: If you can multiply by 1, then you can multiply by 10. Secret 5: If you can halve, then you can multiply by 5. Secret 6: If you use fingers, then you can multiply by 9 Nines in your head: Multiply by 10 and then adjust Secret 7: If you know the 3 times table, then you know the 6 times table.

15. White Rose Maths Hub Calculation Guidance http://tinyurl.com/z26lags

16. White Rose Maths Hub - Calculation Guidance

17. NCETM Teaching for Mastery books • Look at the examples in your year group • How could you use any of the examples? YEAR 1

18. Making ‘fact family’ cards… 3 x 4 =12 4 x 3=12 12÷3=4 12 shared between 3 equals 4 One third of 12 = 4 ¼ of 12 = 3

19. You Tube • http://www.youtube.com/watch?v=yXdHGBfoqfw • Search: ‘Times tables in 10 mins’

20. What skills are required for written methods of multiplication? Recognise the size and position of numbers Count on in different steps 2s, 5s, 10s Double numbers up to 10 Recognise multiplication as repeated addition Quick recall of multiplication facts Use known facts to derive associated facts Multiplying by 10, 100, 1000 and understanding the effect Multiplying by other multiples of 10 (e.g. 20, 30…)

21. H T U100 10 1

22. Multiplication Written Methods How would you work this out in your head? 13 x 6 =

23. White Rose Maths Hub - Calculation Guidance

24. Your turn… Use grid multiplication 64 x 8 = 234 x 6 = 34 x 78 =

25. Napier’s Bones The Egyptian Method 17 x 32 = 345 x 12 = Start with one lot (1 x …) , then just keep doubling. 1x32=32 2x32=64 4x32=128 8x32=256 16x32=512 So… 17 x 32 = (1 x 32) + (16 x 32) 32+512=544 So now try with 18 x 56 = ? Can you explain why/ how do they work?

26. Division – Equal Sharing This corresponds to finding a fraction… Each group is 1/3 of 12 ⅓ of 12 = 12 ÷ 3 ‘Divide 12 into 3 equal parts or groups’

27. Division - Grouping • How many groups of a given size… ‘How many groups of 3 are there in 12?’ This grouping model may involve repeated addition or repeated subtraction ‘How many 3’s equal 12?’

28. White Rose Maths Hub - Calculation Guidance

29. The language of division • Should always use the language ‘divided’ 18 ÷ 3 = “18 divided by 3” Sharing – “18 divided between 3” Fraction – “18 divided into 3 equal parts” (or groups) Grouping – “18 divided into 3’s” What are the limitations?

30. 94 ÷ 4 = • Chunking v Short Division

32. Task 3 - Mental Calculation [for +/- OR x/÷] • Work with a group of children on task involving MENTAL calculation (examples from relevant year group expectations). What different mental calculation strategies/ methods are they familiar with? Discuss with pupils different models & images they utilise to explain HOW they did the calculations? • (With KS2 children you might be able to explore alternative mental and written methods and discuss ‘efficiency’ and ‘fluency’). • Hand in day 5.

33. GROUP READING for Day 5: • D Haylock ‘Mathematics Explained for Primary Teachers’ Chapters 14 and 16 • SEARCH Nrich • ‘Understanding Fractions’ • ‘Teaching Fractions with Understanding: Part-whole Concept’ • ‘Issues in Teaching Numeracy in Primary Schools’ [Second Edition] Chapter 11 • Measurement background reading • Mental Fluency (Primary Magazine 70, December 2014)