Thinking Mathematically Making Sense, Making Connections Arithmetic Laws: Shanghai Style Nicola Spencer Sue Smith Jenny Stratton (Primary Teaching for Mastery Specialists)

Arithmetic Laws Shanghai Style It all makes sense when you… ‘ move forward from a solid starting point consistently’ (the philosophy of Shanghai mathematics teaching – Professor Gu)

Do you remember this example? 0.62 x 37.5 + 3.75 x 3.8 How do we get British children to this destination?

0.62 x 37.5 + 3.75 x 3.8

Critical Prior Knowledge A consistently solid starting point that begins in EYFS/ KS1

Do your pupils do this? 45 + 23 = 40 + 20 = 60 + 5 = 65 + 3 = 68

Let’s consider some key steps…

Vocabulary Addend + Addend = Sum Minuend – subtrahend = difference Multiplicand x multiplier = product Multiplier x multiplicand = product Dividend ÷ divisor = quotient

Dong Naojin 5＋2= 7 +30 +30 37 35＋2=

Dong Naojin B Which one do you select? （ ） 13 + 5 3 + = 15 A. 18 B. 15 C. 5

= 13 + 5 3 + 15 Dong Naojin -10 +10 The sum is the same. One addend takes away 10 Another addend adds 1 0 The sum is the same.

Which numbers are friends? Addition Bonds of 10 (multiples of 10) Bonds of 100 (multiples of 100) Bonds of 1000

make hundreds 9 10 9 10 9 10 9 10 9 10 3 9 10 1 8 9 + （ ） 4 0 0 5 4 +（ ） 1 0 0 2 1 1 4 6 7 9 10 8 9 10 3 4 5 +（ ） 8 0 0 （ ） + 1 8 4 9 0 0 7 1 6 4 5 5

Which numbers are friends? Multiplication What applications will this knowledge have? 2 x 5 = 10 2 x 50 = 100 25 x 4 = 100 125 x 8 = 1000

Can you define the 5 laws? When were you were taught them?

5 Laws of Arithmetic Commutative addition Commutative multiplication Associative addition Associative multiplication Distributive law

Define the 5 laws Commutative addition a+ b = b + a Commutative multiplication a x b = b x a Associative addition (a + b) + c = a + (b + c) Associative multiplication (a x b) x c = a x (b x c) Distributive (a+b) x c = a x c + a x b (a-b) x c = a x c – b x c

Commutative addition a + b = b + a a b b a

Using Commutative Law 214 y △ 1 1 5 5 …… Fill in the blanks by using commutative law of addition 256＋214=           ＋256 X+Y=          ＋X △+ =          ＋ 十 367=367 ＋       214 y △ 1 1 5 5 ……

True or False：(√or×)： 56+38=83+56 is using commutative law of addition.（ ） (2) A×B=B+A。(A ≠B) （ ） (3) □+△+○= □+○+△。（ ） × × √

Associative addition (a + b) + c = a + (b + c) a b c a b c

This has caused some debate... 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

Consider... 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 1 + 9 + 2 + 8 + 3 + 7 + 4 + 6 + 5 =

Then consider... 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 1 + 9 + 2 + 8 + 3 + 7 + 4 + 6 + 5 = (1 + 9 ) + (2 + 8) + (3 + 7) + (4 + 6) + 5 =

Clarity “Once you use three or more numbers in the number sentence you always use the commutative law and associative law together; they can’t be used individually.” Chinese Exchange Partners

a x b = b x a Commutative law X We help children to make sense of this with arrays (numicon and Cuisenaire helpful too)

Using the commutative law of multiplication 34×71= × 45× =55× ×▲= ×■ × =C×D 71 34 55 45 ■ ▲ D C Shanghai teachers get children to use this relationship and play with it.

Associative Law of Multiplication (a x b) x c = a x (b x c)

Danny's father bought 3 boxes of juice, 25 cans per box, each can cost £4，how much did his father pay in total? 3×25 ×4

Danny's father bought 3 boxes of juice, 25 cans per box, each can cost £4 ，how much did his father pay in total? 3×25×4 =3×（25×4） =3×100 =300 3×25×4 = 75×4 = 300 Which method is easier? On this slide we can refer back to the making connections, making sense title and for each method draw out what each number represents – modelling how Lily and Lin kept asking what each number represents So lhs the 75 represents the total number of cans and the 4 is the cost per can On the rhs the 100 represents the cost of one box and the 3 the number of boxes 3×25×4= 3×（25×4）

(a × b)×c = a×(b × c) associative law of multiplication Multiply three numbers. Multiply the first two numbers and then multiply the third number. Or multiply the last two numbers and then multiply the first number. Their product remains the same. associative law of multiplication

Follow-up exercises: (36×20)×50 = 36×（ ____ × _____ ） Fill in the blanks by using associative law of multiplication (36×20)×50 = 36×（ ____ × _____ ） (57×125)×8 = 57×（ ____ × ____ ） (●×▲)×★ =___ ×（▲× __ ） 20 50 125 8 ● ★

True or False Which ones conform to the associative law? (1) a×(b×c)=(a×b)×c  (2) 15＋(7＋3)=(15＋2)＋3 (3) (23+41)+72+28=(23+41)+(72+28) √ × √ ? Not sure if we need these examples

Solve in simpler way – think about which laws you are using        25×19×4 25×43×40 =25×40×43 =1,000×43 =43,000 =25×4×19 =100×19 =19,000

Solve in simpler way: 8×23×125 125×13×4 =125×4×13 =8×125×23 =500×13        8×23×125 125×13×4 =125×4×13 =500×13 =6500 =8×125×23 =1000×23 =23000 Dont think we need this one

Solve in simpler way: 125×5×2×8 25×125×4×8        125×5×2×8 25×125×4×8 Dont think we need this either

Factorising for a purpose: Learning how to use and apply knowledge of factors to make calculations easier. 25 x 24 = 25 x 4 x 6 = 100 x 6 Do we need to put in more examples for them to try here? YES PLEASE

Distributive Law (a + b) x c = a x c + b x c

3×25 + 3×35 3× (25 + 35) =75 + 105 =3X60 =£180 =£180 On sale: The discount price of the jacket is £25. The discount price of a pair of trousers is £35. How much in all for 3 sets of jackets and trousers? 3×25 + 3×35 =75 + 105 =£180 3× (25 + 35) =3X60 =£180 Introduced the concept through a real context to help children make sense of the distributive law Again – lots of asking what each number represents so that we can all make sense of the numbers. Does it change the answer?

（1）4×12 + 6×12 solve in easier way = (4 + 6) x 12 = 10× 12 = 120 Steps： 1、find the same factor 2、put the same factor out of the bracket 3、calculate the sum of different factors. a×c + b×c =(a+b)×c

Can you find the same factor? Next steps – using to solve in an easier way Can you find the same factor? (1) 35×23 + 65×23 =(35 + 65)×23 (2) 52×16 + 48×16 =(52+ 48)×16 (3) 55×12 － 45×12 =(55 － 45)×12 (4) 19×64 － 9×64 =(19 － 9)×64

C 24×12+24=（ ） A. 24×(12+24) B. 24×12+24×24 C. 24×(12+1) Choose the right answer： 24×12+24=（ ） A. 24×(12+24) B. 24×12+24×24 C. 24×(12+1) C

（1）201× 25 （2）101× 125 solve in easier way If we have a board we can show these (200+1) x 25 = 200 x 25 + 1 x 25 = 5000 + 25 What is different about this step? Or we can just put in this slide to animate. They could try e.g. 2

（3）99×12 （4）39× 25 solve in easier way (100-1) x 12 = 100 x 12 – 1 x 12

（1）How much is that altogether? Oscar want to buy something in the supermarket. Product Unit price Quantity chocolate £12 11 bags sweets £8 （1）How much is that altogether? （2）How much more did he spend on chocolate than on sweets?

Which Year Group is this from in China? 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 Which Year Group is this from in China? (3) 25×28 =25×（4×7） =（25×4）×7 =100×7 =700

Key learning points Vocabulary: addend + addend = sum etc Importance of equals sign Explicit recording is key Explicit teaching of laws through careful examples in meaningful contexts The answer is only the beginning – reasoning is key Early introduction of algebraic thinking Application to both real life contextual and more complex problem solving Students must observe numbers and operations then choose the best way.

Implications for our pedagogy and practice… DISCUSS

Thank you Any questions?