Why everyone should learn the number laws Starter questions on board What are the number laws? What do you know? Name that law Jo Boaler film – go through how these link to the number laws Why teach them? Importance of number fluency – example of Y13 resit student Use of generalisation early on – introduces algebra Using proper mathematical language from early age How did we teach them? Family pairs How we structured the lessons: one key concept; variation; concept / non-concept; Dong Nao Jing Worked examples on board Worksheet to try Go through answers Benefits and links to other topics 5 min think they appear EVERYWHERE What staff and kids think Recap question from start

What Shanghai has taught us Mastery Maths Please have a go at these questions… 9999 + 999 + 99 + 9 + 5 = Worksheet to do on entry 0.62 x 37.5 + 3.75 x 3.8 = What Shanghai has taught us

Why everyone should learn the number laws What are the number laws? Why teach them? How did we teach them? Benefits and links to other topics

Name that law! a + b = b + a a × b = b × a (a + b) + c = a + (b + c) Starter on board (a × b) × c = a × (b × c) a - b - c = a - (b + c) a - b - c = a – c – b a ÷ b ÷ c = a ÷ (b x c) (a + b) × c = a × c + b × c

Distributive law a x b + a x c = a x (b+c) Commutative law of addition and multiplication a + b = b + a; a x b = b x a Associative law of addition and multiplication (a+b) + c = a + (b + c); (a x b) x c = a x (b x c) Distributive law a x b + a x c = a x (b+c) Laws of subtraction a – b – c = a – (b + c); a – b – c = a – c – b Law of division a ÷ b ÷ c = a ÷ (b x c) In the new national curriculum they start to be explicitly mentioned in Year 4 – though even year 1 children will be using them. Important to think about how much teachers at secondary know these laws.

Why explicitly teach the Number Laws? https://www.youcubed.org/what-is-number-sense/ Play and stop after 5th example before extra question comes up Model better working on the board Feedback Play the question – 5 mins to work out different ways of doing this using laws Resit Y13 Important to bring in generalisations even really early on, makes algebra less scary – no moans about algebra this year

From GCSE paper

How did we teach the Number Laws? Part of the Secondary Mastery Workgroup project – decided to teach to Year 7s, in a few years we won’t need to, actually ended up teaching to all years pretty much Needed to go back to basics – aims to teach the laws but also develop good mathematical habits and written working etc

5 big ideas 5 big ideas from Primary Mastery Specialist Programme

Family pairs 2 x 5 = 10 4 x 25 = 100 8 x 125 = 1000 One key concept Proper vocabulary True / false Dong Nao Jin 2 x 5 = 10 4 x 25 = 100 8 x 125 = 1000 Used family pairs – all students familiar with them and know to try and spot them Used techniques that we had seen modelled in Shanghai whilst planning and teaching the lessons We are going to model a few examples and then give you some to work on yourselves Need to learn these!!!!

Can you find the same factor? Warming-up: Can you find the same factor? (1) 35×23 + 65×23 =(35 + 65)×23 (deb) modelling correct use of language and the answer is not important (2) 52×16 + 48×16 =(52+ 48)×16 (3) 55×12 － 45×12 =(55 － 45)×12 (4) 19×64 － 9×64 =(19 － 9)×64

Which law of subtraction would make these easier? (Don’t calculate!) 100-(35+45) (1) 100-35-45= (2) 447-190-47= (3) 189-28-72= (4) 321-45-121= Calculating is not the point! 447-47-190 189-(28+72) 321-121-45

Which one is easier to calculate? (1) 3200÷25÷4 3200÷（25×4） (2) 7000÷125÷8 7000÷（125×8） (3) 800÷8÷20 800÷（8×20） (4) 140÷7÷4 140÷（7×4） a÷b ÷ c = a ÷ ( b × c )

Have a go yourselves Pay attention to: The questions themselves The nature of the task

Make these commutative What goes in the missing box?     Go through answers

Complete the following: A) 17×4 + 17×6 B) 63×34 - 63×24 C) (8+10) X 125 D) 99×12 Students to complete work in books. Go through answers – could get students to write answers up on white board E) 999×99 + 99

True or false? (22 – 17) × 35 = 22 × 35 – 22 × 17 78 × 91 + 91 × 25 = 78 + 25 × 91 True or False. If false then make the correction and write another correct example in the same format. False. Should be 22 x 35 – 17 x 35. False. Should be (78 + 25) x 91 False. Not distributive law. Could be 2x(3x4) = 2 x 3 x 4 (Associative Law). Or could be 2x(3+4) = 2x3+2x4 True. Note 101x99 – 99 is the same as 101 x 99 – 1x99 therefore = (101-1)x99 2 × (3 × 4) = 2 × 3 × 2 × 4

84 x 25 1000 ÷ 40 Now try these: Use different laws to calculate Copy and complete in books. Discuss.

Dong Naojin: choose the right answer The boy was solving the number sentence: （ 25 +50）×4 , he calculated like this, 25×4+50 What’s difference between the wrong answer and the right answer? （25 +50）×4 = 25 ×4+50×4 25×4+50 A. 50 B. 100 C. 150 D.200

(1) 25＋34＋66 = (2) 25×40×78 = (3) 56＋72＋44 = (4) 75×8×2×125 = 1. Rewrite each of these in a way that makes them easier to do 2. Find the answer, writing your workings mathematically (1) 25＋34＋66 = (2) 25×40×78 = (3) 56＋72＋44 = (4) 75×8×2×125 = Could use this an an exit ticket. Remind them to write solutions out as modelled in previous slides – this is as important as the answers (in fact more)

In each pair of calculations, which one would you prefer to work out In each pair of calculations, which one would you prefer to work out? Explain your choices. A 35 × 0.3 + 3.5 × 7 or B 3.5 × 0.3 + 35 × 7 C 6.4 × 1.27 – 64 × 0.1 or D 6.4 × 1.27 – 64 × 0.027 E 52.4 ÷ 0.7 + 524 ÷ 7 or F 52.4 ÷ 0.7 – 524 ÷ 7 G 31.2 ÷ 3 – 2.4 ÷ 6 or H 31.2 ÷ 3 – 1.2 ÷ 0.3

Go through the answers

25×28 (1) 25×28 (2) 25×28 =25×（20+8） =（20+5）×28 =25 × 20+25 × 8 Dong Naojin: solve in easier way 25×28 (1) 25×28 =25×（20+8） =25 × 20+25 × 8 =500+200 =700 (2) 25×28 =（20+5）×28 =20 × 28+5 × 28 =560+140 =700 (3) 25×28 =25×（4×7） =（25×4）×7 =100×7 =700

Benefits and links to other topics Number calculations and fluency Collecting like terms No more BODMAS! Factorising Area of trapezium Angle calculations We have found that they come into everything – as soon as you start thinking about them (especially distributive law) they are everywhere 5min discussion – what are you currently teaching/going to teach – how could you use the number laws

Tick and cross: Are they like terms or not in each group? TWO SAMEs variables power of each variable TWO DOESN'T MATTERs order coefficient

Steps of Collecting like terms ① Combine the coefficient of like terms Link to distributive law ② write down the result of coefficient, and maintain the common factor of both terms

Commutative can be very useful   Family Pairs!   20m 9.5m

What staff / pupils think

Law of Subtration work by students – I have been really really strict on layout, use of visualiser (ipad), class giving instant feedback to each other

What Shanghai has taught us Mastery Maths Would you now do them differently? 9999 + 999 + 99 + 9 + 5 = Worksheet to do on entry 0.62 x 37.5 + 3.75 x 3.8 = What Shanghai has taught us

Sharing of resources Link to all lessons: http://tinyurl.com/shanghaigoogledrive PLEASE look, add, comment – this link gives editing rights http://tinyurl.com/sussexshanghaicpd Contains: all resources from our CPD sessions all lessons that are on Google drive above