096r. Multiplication and Division Strategies

Learning Intentions

To understand the commutative and associative laws for multiplication
To know the meaning of the terms product, quotient and remainder
use mental strategies to calculate simple products and quotients mentally
use the multiplication and division algorithms to find the product and quotient of whole numbers

Pre-requisite Summary

Know that multiplication represents equal groups or repeated addition
Know that division represents sharing equally or grouping
Recall multiplication facts and related division facts
Understand place value of whole numbers
Be able to partition numbers into tens, hundreds and ones
Know that addition can be done in any order, and connect this idea to multiplication
Be able to line up digits by place value in written algorithms

Worked Examples

Worked Example 1

Use the commutative law to rewrite:

a) $4 \times 7$
b) $13 \times 5$

Worked Example 2

Use the associative law to make the multiplication easier:

a) $(2 \times 5) \times 6$
b) $4 \times (25 \times 2)$

Worked Example 3

State the product, quotient or remainder:

a) In $8 \times 6 = 48$ , identify the product
b) In $35 \div 5 = 7$ , identify the quotient
c) In $17 \div 4 = 4$ remainder $1$ , identify the quotient and remainder

Worked Example 4

Use a mental strategy to calculate:

a) $6 \times 14$
b) $84 \div 7$

Worked Example 5

Use a mental strategy to calculate:

a) $25 \times 4$
b) $96 \div 3$

Worked Example 6

Use the multiplication algorithm:

a) $34 \times 6$
b) $128 \times 4$

Worked Example 7

Use the division algorithm:

a) $96 \div 4$
b) $145 \div 6$

Problems

Problem 1

Use the commutative law to rewrite:

a) $3 \times 9$
b) $11 \times 8$

Problem 2

Use the associative law to make the multiplication easier:

a) $(5 \times 2) \times 7$
b) $8 \times (5 \times 2)$

Problem 3

State the product, quotient or remainder:

a) In $9 \times 4 = 36$ , identify the product
b) In $42 \div 6 = 7$ , identify the quotient
c) In $22 \div 5 = 4$ remainder $2$ , identify the quotient and remainder

Problem 4

Use a mental strategy to calculate:

a) $7 \times 13$
b) $63 \div 9$

Problem 5

Use a mental strategy to calculate:

a) $50 \times 2$
b) $84 \div 4$

Problem 6

Use the multiplication algorithm:

a) $47 \times 5$
b) $213 \times 3$

Problem 7

Use the division algorithm:

a) $84 \div 3$
b) $167 \div 8$

Exercises

Understanding and Fluency

Rewrite each multiplication using the commutative law:
a) $8 \times 3$
b) $12 \times 4$
c) $7 \times 15$
Use the associative law to multiply more easily:
a) $(2 \times 5) \times 9$
b) $3 \times (4 \times 25)$
c) $(10 \times 2) \times 6$
Identify the product, quotient or remainder:
a) In $6 \times 8 = 48$ , state the product
b) In $54 \div 9 = 6$ , state the quotient
c) In $19 \div 4 = 4$ remainder $3$ , state the quotient
d) In $19 \div 4 = 4$ remainder $3$ , state the remainder
Use a mental strategy to calculate:
a) $4 \times 16$
b) $5 \times 18$
c) $72 \div 8$
Use a mental strategy to calculate:
a) $25 \times 8$
b) $99 \div 9$
c) $48 \div 6$
Use the multiplication algorithm:
a) $56 \times 7$
b) $142 \times 5$
c) $309 \times 4$
Use the division algorithm:
a) $93 \div 4$
b) $128 \div 5$
c) $246 \div 6$
Choose a sensible mental strategy and calculate:
a) $16 \times 5$
b) $18 \times 5$
c) $64 \div 8$
d) $120 \div 3$

Reasoning

Explain why $6 \times 9$ and $9 \times 6$ have the same product.
Noah says the associative law means $8 \times 3 = 3 \times 8$ . Is he correct? Explain your answer.
Which mental strategy would be most efficient for $25 \times 4$ ? Explain why.
A student says that in $23 \div 5 = 4$ remainder $3$ , the quotient is $3$ . Describe the mistake.
A student calculates $132 \div 4$ and writes remainder $4$ . Explain why this cannot be correct.

Problem-solving

A farmer packs $6$ boxes with $24$ oranges in each box. How many oranges are packed altogether?
A teacher has $96$ pencils and shares them equally among $8$ tables. How many pencils does each table get?
A shop receives $7$ cartons with $35$ cans in each carton. How many cans arrive altogether?
A bus carries $143$ students in groups of $5$ seats. How many full groups of $5$ can be made, and how many students are left over?
A gardener plants $4$ rows of $125$ flowers. How many flowers are planted?
A library has $218$ books to place equally on $6$ shelves. How many books go on each shelf, and how many are left over?

Potential Misunderstandings

Students may think the commutative law applies to division because it applies to multiplication
Students may confuse the commutative law with the associative law
Students may think the associative law changes the order of factors, when it actually changes the grouping
Students may confuse the product with one of the factors instead of the answer to a multiplication
Students may confuse the quotient with the remainder in a division question
Students may think a remainder can be greater than or equal to the divisor
In mental multiplication, students may partition a number incorrectly by place value
Students may not use known facts to simplify products and quotients mentally
In written multiplication, students may misalign digits by place value
In written division, students may record the quotient in the wrong place
Students may forget to interpret a remainder in the context of a problem
Students may not see the inverse relationship between multiplication and division