037. Multiplication and Division in Algebraic Expressions

Learning Intentions

• To know the different ways multiplication and division can be written
• Simplify expressions with multiplication
• simplify expressions with division

Pre-requisite Summary

• Understand that letters can represent numbers in algebra
• Recognise terms, coefficients and variables in simple expressions
• Recall multiplication facts and division facts
• Know that multiplication in algebra is often written without a sign, for example
• Understand that division can be written Use a division sign, a fraction bar or slash notation
• Be able to combine numbers using order of operations
• Understand that a product means multiply and a quotient means divide

Worked Examples

Worked Example 1

Identify different ways multiplication and division can be written:

a)

b)

c)

d)

Worked Example 2

Simplify expressions with multiplication:

a)

b)

c)

Worked Example 3

Simplify expressions with multiplication:

a)

b)

c)

Worked Example 4

Simplify expressions with division:

a)

b)

c)

Worked Example 5

Simplify expressions with division:

a)

b)

c)

Worked Example 6

For each expression:

a) rewrite it in simplest algebraic form

b) State whether multiplication or division has been simplified

For , , and

Problems

Problem 1

Identify different ways multiplication and division can be written:

a)

b)

c)

d)

Problem 2

Simplify expressions with multiplication:

a)

b)

c)

Problem 3

Simplify expressions with multiplication:

a)

b)

c)

Problem 4

Simplify expressions with division:

a)

b)

c)

Problem 5

Simplify expressions with division:

a)

b)

c)

Problem 6

For each expression:

a) rewrite it in simplest algebraic form

b) state whether multiplication or division has been simplified

For , , and

Exercises

Understanding and Fluency

Exercise 1.

Rewrite each multiplication expression in algebraic form:

a)

b)

c)

Exercise 2.

Rewrite each division expression in another form:

a)

b)

c)

Exercise 3.

Simplify expressions with multiplication:

a)

b)

c)

Exercise 4.

Simplify expressions with multiplication:

a)

b)

c)

Exercise 5.

Simplify expressions with division:

a)

b)

c)

Exercise 6.

Simplify expressions with division:

a)

b)

c)

Exercise 7.

Simplify each expression:

a)

b)

c)

Exercise 8.

Simplify each expression:

a)

b)

c)

Reasoning

Exercise 9.

Explain why and mean the same thing.

Exercise 10.

A student says should be written as . Explain the mistake.

Exercise 11.

Explain why simplifies to .

Exercise 12.

A student simplifies to . Explain why this is incorrect.

Problem-solving

Exercise 13.

A shop sells pens for $3 each. Write and simplify an expression for the cost of buying pens.

Exercise 14.

A factory packs boxes of items each. Write and simplify an expression for the total number of items.

Exercise 15.

A ribbon of length cm is cut into equal pieces. Write and simplify an expression for the length of one piece.

Exercise 16.

A farmer has kg of feed shared equally into bags. Write and simplify an expression for the mass of each bag.

Exercise 17.

A machine makes groups of parts. Write and simplify an expression for the total number of parts.

Exercise 18.

A quantity of litres is poured equally into containers. Write and simplify an expression for the amount in each container.

Potential Misunderstandings

• Students may think multiplication in algebra must always use the sign
• Students may confuse with instead of
• Students may think division can only be written with the sign and not as a fraction
• Students may add coefficients when they should multiply them
• Students may divide the variable by the number incorrectly when simplifying expressions such as
• Students may think instead of dividing only the numerical coefficient
• Students may not keep the variable part attached when simplifying multiplication or division
• Students may change the meaning of an expression by rewriting it with the wrong operation

Next: 038e. Expanding Brackets