Transcript Lattice multiplications ct`d

Andrea Romano
Jean-Yves Théron
LET’S MULTIPLY
How to teach multiplications
and divisions using different
strategies
 Standard: Numbers: Working mathematically
 Focus: Multiplication and Division
 Outcomes:
Students use the mathematical structure of problems to
choose strategies for solutions. They explain their
reasoning and procedures and interpret solutions.
They create new problems based on familiar problem
structures.
They explain and use mental and written algorithms for
the addition, subtraction, multiplication and division
of natural numbers (positive whole numbers).
Introduction Activity
17 x 4 =
Can you work out the answer in your head?
How did your work it out?
Use a worksheet showing the operation as a last resort
Introduction Activity continued
116 ÷ 4 =
Can you work out the answer in your head?
How did your work it out?
Use a worksheet showing the operation as a last resort
Introduction Activity continued
8x9=
What is the result of this calculation ?
Introduction Activity continued
9x7=
What is the result of this calculation ?
Introduction Activity continued
56 ÷ 7 =
What is the result of this calculation ?
Introduction Activity continued
32 ÷ 4 =
What is the result of this calculation ?
Introduction Activity continued
800 - 300=
What is the result of this calculation ?
How did you do it ?
Introduction Activity continued
4 x 50=
What is the result of this calculation ?
How did you do it ?
Introduction Activity continued
6 x 300=
What is the result of this calculation ?
How did you do it ?
Introduction Activity continued
What is cost of
four items
at $ 12 each?
Introduction Activity continued
You have $15 to spend.
Which of the following
shopping lists could you
buy?
Introduction Activity continued
a) a book for $8.95, a drink for $2.10,
a magazine for $4.50
b) a CD for $9.50, a pen for $1.15, a
card for $3.50
c) a hot dog for $2.50, chocolate
biscuits for $2.80, a scarf for
$10.95
Pen and Paper Methods
 Encourage students to develop and share efficient pen
and paper methods of multiplication and division.
 In groups of four, student are asked to share the
methods they use. This information can be displayed
in the classroom and referred to during the term.
 They are encouraged to practice, compare and explain
the strategies they have used.
 How can these strategies be made more efficient as the
term progresses?
Strategies:
1. Simpler related problems
2. Double digit numbers
multiplications
3. Lattice multiplication
4. The magic of elevens
Simpler related problems
Focus:
Students use the mathematical structure of
problems to choose strategies for solutions.
They explain their reasoning and procedures
and interpret solutions.
They create new problems based on familiar
problem structures.
Simpler related problems ct’d
Method for multiplying a two digit
number by a single digit number
43
X 7
.
Simpler related problems ct’d
Explore the extended notation of the
problem:
(3X7) + (40 x7)
Simpler related problems ct’d
Extended notation can lead us to
other simpler related problems
eg. (50 x 7) — (7 X 7)
Simpler related problems ct’d
6x18 = (6x12)+(6X6)
= (72) + (36)
= 108
8x23=(8X20)+(8x3)
= (160)+ (24)
= 184
Simpler related problems ct’d
7 X 28 =
(7x30)- (7x2) =
(7x20) + (7x8) =
( 7x 12) + (7 X 12) + (7 X 2) =
(7 x 25) + (7 x 3) =
Simpler related problems ct’d
7 X 28 =
(7x30)- (7x2) =
(7x20) + (7x8) =
( 7x 12) + (7 X 12) + (7 X 2) =
(7 x 25) + (7 x 3) =
Simpler related problems ct’d
Assessment
Have the children write which of the
simpler related problems they
would use to solve the problem and
why.
Double digits numbers
multiplications
Focus:
Students develop and share efficient methods
of computation - long multiplication and
extended notation
Double digits numbers ct’d
43
x 27
______
Double digits numbers ct’d
Solve using simpler related
problem (43 x 7) + (43 x 20 )
Use investigation to highlight the
recording ‘0’ when multiplying
by 20
Double digits numbers ct’d
43
x 27
. . . . . . . . . . . ( 43 x 7 )
. . . . . . . . . . . (43 X 20 )
Double digits numbers ct’d
43
x 27
301
860
1161
( 43 x 7 )
(43 X 20 )
Double digits numbers ct’d
Explore the extended notation of
this problem to ensure that the
children understand the concept
and aren’t just learning the
process.
Double digits numbers ct’d
Extension work through
open ended investigation:
I have multiplied four digits
- - x - - and this answer is 2080.
The four digits were 2 , 3 , 5 & 6
How were the digits arranged?
Double digits numbers ct’d
How many other answers can I
get by multiplying the same
digits 2, 3,5 and 6?
 A x B x C x D = 6x5x3x2
 AB x CD = 65x 32, 65 x 23, 56 x 23, 56 x 32, 62 x 35,
62 x 53, 25 x 63, 25 x 36, 36 x 52.
 ABC x D = 532 x 6, 523 x 6, 235 x 6, 253 x 6, 325 x6,
etc
Double digits numbers ct’d
Assessment
 Keep work samples
 Anecdotal Records noting estimation / trial and
error strategies during open ended problem solving
 Have the children demonstrate long multiplication
using extended notation and the ‘short cuts’ as in
the stages shown in this exercise.
Lattice multiplications
Focus:
Students use the mathematical structure of
problems to choose strategies for solutions.
They explain their reasoning and procedures
and interpret solutions.
They create new problems based on familiar
problem structures.
Lattice multiplications ct’d
Focus :
Students develop and share efficient pen and
paper methods of computation
Lattice multiplications ct’d
Multiplying 34 x 26:
Lattice multiplications ct’d
Lattice multiplications ct’d
Record produce
for each pair
using the first
section of box
for tens digit,
second section
for ones digit
Lattice multiplications ct’d
Lattice multiplications ct’d
Add the
numbers in
each diagonal,
starting ‘from
the right and
regroup into the
next diagonal if
necessary.
Lattice multiplications ct’d
Ask the children to use lattice
multiplication to find the
answer but to cross check their
work using long multiplication
(or vice-versa)
Lattice multiplications ct’d
Assessment:
Have the children write which
method they think is best for
solving multiplication problems
(lattice or long).
They need to provide reasons with
their answer.
Lattice multiplications ct’d
Extension work interdisciplinary activity:
Design and produce Napier’s bones
interdisciplinary activity involving the domain of
Design, Creativity and Technology.
Lattice multiplications ct’d
The children can design and produce their own Napier’s
bones to practice their maths skills and engage in an
interdisciplinary activity involving the domain of
Design, Creativity and Technology. In doing so they
would link Maths to the Producing dimension
The Producing dimension involves students in the management of the production phase
and includes the appropriate selection and safe manipulation and use of tools,
equipment, materials/ingredients and components to carry out processes appropriate to
the materials/ingredients or assembly of systems components to produce a quality
product or technological system.
Magic of 11s
To multiply a number by 11
First multiply the number by 10
and then add the original number to it.
865 X 11 = ?
856 X 10 = 8650
8650 + 865 = 9515
Magic of 11s
ct’d
To find out if any number
is divisible by 11:
start with the digit on the left, subtract
the next digit from it, add the next
digit, and subtract the next and so on.
Magic of 11s
ct’d
53746
5-3+7-4+6 = 11
Magic of 11s
ct’d
If the answer is 0 or 11, then
the original number is
divisible by 11.
53745/11 = 4886
References
 Victorian Essential Learning Standards,2007 Victorian Curriculum and
Assessment Authority, State Government of Victoria,
 Success in Numeracy Education (SINE) , 2001 Catholic Education
Commission of Victoria
 EXTENDED SINE Interview testing kit 5 – 6, , 2001 Catholic Education
Commission of Victoria
 Heinemann Maths Zone 7 CSF II, Reed international Books Australia
Pty Ltd
 www.curriculum.edu.au/math300