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 8 x 9 = What is the result of this calculation ?

Introduction Activity continued 9 x 7 = 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 = 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)8x23=(8X20)+(8x3) = (72) + (36)= (160)+ (24) = 108 = 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 4 3 x ( 43 x 7 ) (43 X 20 )

Double digits numbers ct’d 4 3 x ( 43 x 7 ) 8 6 0(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 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

Record produce for each pair using the first section of box for tens digit, second section for ones digit

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 = = 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 = 11

Magic of 11s ct’d If the answer is 0 or 11, then the original number is divisible by /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