Transcript Arithmetic Laws Shanghai Style for 30.1.17 FINAL VERSION

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?