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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 35+2= 37 Dong Naojin Which one do you select? ( B ) 13 + 5 = A. 18 B. 15 3 +15 C. 5 Dong Naojin -10 +10 13 + 5 15 3 + = 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 5 4 +(4 6) 1 0 0 7 99 10 3 4 5 +(4 5 5) 8 0 0 3 9 10 1 8 9 + (2 1 1) 4 0 0 8 9 10 (7 1 6) + 1 8 4 9 0 0 Which numbers are friends? Multiplication 2 x 5 = 10 2 x 50 = 100 25 x 4 = 100 125 x 8 = 1000 What applications will this knowledge have? 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 Commutative multiplication a+ b = b + a axb=bxa 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 Fill in the blanks by using commutative law of addition 256+214= 214 +256 X+Y= △+ y +X = + △ 十 367=367 + 1 5 …… 1 5 True or False:(√or×): (1) 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 Commutative law X axb=bxa Using the commutative law of multiplication 34×71= 71 × 34 45×55 =55× 45 ■ ×▲= ▲ ×■ D × C =C×D 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 = 75×4 = 300 3×25×4 =3×(25×4) =3×100 =300 Which method is easier? 3×25×4= 3×(25×4) (a × b)×c = a×(b × c) 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: Fill in the blanks by using associative law of multiplication 20 × _____ 50 ) (36×20)×50 = 36×( ____ (57×125)×8 = 57×( ____ 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) √ Solve in simpler way – think about which laws you are using 25×19×4 25×43×40 =25×4×19 =100×19 =19,000 =25×40×43 =1,000×43 =43,000 Solve in simpler way: 8×23×125 125×13×4 =8×125×23 =1000×23 =23000 =125×4×13 =500×13 =6500 Solve in simpler way: 125×5×2×8 25×125×4×8 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 Distributive Law (a + b) x c = a x c + b x c (a - b) x c = a x c - b x c 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 + 35) 3×25 + 3×35 =3X60 =75 + 105 =£180 =£180 solve in easier way (1)4×12 + 6×12 = (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 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 Choose the right answer: 24×12+24=( C) A. 24×(12+24) B. 24×12+24×24 C. 24×(12+1) solve in easier way (1)201× 25 (2)101× 125 solve in easier way (3)99×12 (4)39× 25 Oscar want to buy something in the supermarket. Product chocolate sweets Unit price £12 £8 Quantity 11 bags 11 bags (1)How much is that altogether? (2)How much more did he spend on chocolate than on sweets? Dong Naojin: solve in easier way 25×28 (2) 25×28 (1) 25×28 =(20+5)×28 =25×(20+8) =20 × 28+5 × 28 =25 × 20+25 × 8 =560+140 =500+200 =700 =700 Which Year (3) 25×28 Group is this =25×(4×7) from in China? =(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?