096r. Reviewing Multiplication and Division Strategies

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 Solve the product and quotient of whole numbers

Pre-requisite Summary

• Know that multiplication represents equal groups or repeated addition. See 005. Multiplication Laws and Algorithms
• Know that division represents sharing equally or grouping. See 007. Division with Remainders and Short Division
• Recall multiplication facts and related division facts
• Understand place value of whole numbers. See 002. Place Value and Comparing 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. See 003. Mental Strategies and Laws for Addition and Subtraction
• Be able to line up digits by place value in written algorithms. See 005. Multiplication Laws and Algorithms

Worked Examples

Worked Example 1

Use the commutative law to rewrite:

a)

b)

Worked Example 2

Use the associative law to make the multiplication easier:

a)

b)

Worked Example 3

State the product, quotient or remainder:

a) In , Identify the product

b) In , identify the quotient

c) In remainder , identify the quotient and remainder

Worked Example 4

Use a mental strategy to calculate:

a)

b)

Worked Example 5

Use a mental strategy to calculate:

a)

b)

Worked Example 6

Use the multiplication algorithm:

a)

b)

Worked Example 7

Use the division algorithm:

a)

b)

Problems

Problem 1

Use the commutative law to rewrite:

a)

b)

Problem 2

Use the associative law to make the multiplication easier:

a)

b)

Problem 3

State the product, quotient or remainder:

a) In , identify the product

b) In , identify the quotient

c) In remainder , identify the quotient and remainder

Problem 4

Use a mental strategy to calculate:

a)

b)

Problem 5

Use a mental strategy to calculate:

a)

b)

Problem 6

Use the multiplication algorithm:

a)

b)

Problem 7

Use the division algorithm:

a)

b)

Exercises

Understanding and Fluency

Exercise 1.

Rewrite each multiplication Use the commutative law:

a)

b)

c)

Exercise 2.

Use the associative law to multiply more easily:

a)

b)

c)

Exercise 3.

Identify the product, quotient or remainder:

a) In , state the product

b) In , state the quotient

c) In remainder , state the quotient

d) In remainder , state the remainder

Exercise 4.

Use a mental strategy to calculate:

a)

b)

c)

Exercise 5.

Use a mental strategy to calculate:

a)

b)

c)

Exercise 6.

Use the multiplication algorithm:

a)

b)

c)

Exercise 7.

Use the division algorithm:

a)

b)

c)

Exercise 8.

Choose a sensible mental strategy and calculate:

a)

b)

c)

d)

Reasoning

Exercise 9.

Explain why and have the same product.

Exercise 10.

Noah says the associative law means . Is he correct? Explain your answer.

Exercise 11.

Which mental strategy would be most efficient for ? Explain why.

Exercise 12.

A student says that in remainder , the quotient is . Describe the mistake.

Exercise 13.

A student calculates and writes remainder . Explain why this cannot be correct.

Problem-solving

Exercise 14.

A farmer packs boxes with oranges in each box. How many oranges are packed altogether?

Exercise 15.

A teacher has pencils and shares them equally among tables. How many pencils does each table get?

Exercise 16.

A shop receives cartons with cans in each carton. How many cans arrive altogether?

Exercise 17.

A bus carries students in groups of seats. How many full groups of can be made, and how many students are left over?

Exercise 18.

A gardener plants rows of flowers. How many flowers are planted?

Exercise 19.

A library has books to place equally on 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

Next: 097r. Reviewing Factors, Multiples and Powers