1. 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

2. What Shanghai has taught us
Mastery Maths
Please have a go at these questions… =
Worksheet to do on entry
0.62 x x 3.8 =
What Shanghai has taught us

3. 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

4. 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

5. 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.

7. Why explicitly teach the Number Laws?
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

8. From GCSE paper

9. 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

10. 5 big ideas
5 big ideas from Primary Mastery Specialist Programme

11. 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 = 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!!!!

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

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

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

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

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

17. Complete the following:
A) 17×4 + 17×6
B) 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×

18. True or false?
(22 – 17) × 35 = 22 × 35 – 22 × 17
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 ( ) 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

19. 84 x 25 1000 ÷ 40 Now try these: Use different laws to calculate
Copy and complete in books.
Discuss.

20. Dong Naojin:
choose the right answer
The boy was solving the number sentence: （ ）×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 B C D.200

21. (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)

22. 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 × × 7
or B
3.5 × × 7 C
6.4 × 1.27 – 64 × 0.1
or D
6.4 × 1.27 – 64 × 0.027 E
52.4 ÷ ÷ 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

23. Go through the answers

24. 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 × × 8 =
=700
(2) 25×28
=（20+5）×28
=20 × 28+5 × 28 =
=700
(3) 25×28
=25×（4×7）
=（25×4）×7
=100×7
=700

25. 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

26. 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

27. 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

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

29. What staff / pupils think

30. 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

32. What Shanghai has taught us
Mastery Maths
Would you now do them differently? =
Worksheet to do on entry
0.62 x x 3.8 =
What Shanghai has taught us

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