138. Expanding Brackets and Simplifying Expressions

Learning Intentions

• To understand that the distributive law can be used to Expand brackets
• expand brackets Use the distributive law
• Use expansion together with combining like terms to Simplify expressions

Pre-requisite Summary

• Know that multiplication can be written beside brackets, for example
• Understand that a term is a part of an expression separated by or signs
• Be able to Identify like terms
• Be able to combine like terms such as
• Understand that simplifying means writing an expression in a shorter equivalent form
• Recall that the distributive law means multiplying the factor outside the bracket by each term inside the bracket

Worked Examples

Worked Example 1

Use the distributive law to expand:

a)

b)

c)

Worked Example 2

Use the distributive law to expand:

a)

b)

c)

Worked Example 3

Expand, then simplify by combining like terms:

a)

b)

c)

Worked Example 4

Expand, then simplify by combining like terms:

a)

b)

c)

Worked Example 5

Expand and simplify:

a)

b)

c)

Worked Example 6

Write an expression from the words, then expand and simplify:

a) Three times the sum of and , then add

b) Four times the difference between and , then subtract

c) Twice the sum of and , then add three times

Problems

Problem 1

Use the distributive law to expand:

a)

b)

c)

Problem 2

Use the distributive law to expand:

a)

b)

c)

Problem 3

Expand, then simplify by combining like terms:

a)

b)

c)

Problem 4

Expand, then simplify by combining like terms:

a)

b)

c)

Problem 5

Expand and simplify:

a)

b)

c)

Problem 6

Write an expression from the words, then expand and simplify:

a) Two times the sum of and , then add

b) Five times the difference between and , then subtract

c) Three times the sum of and , then add two times

Exercises

Understanding and Fluency

Exercise 1.

Expand each expression:

a)

b)

c)

Exercise 2.

Expand each expression:

a)

b)

c)

Exercise 3.

Expand and simplify:

a)

b)

c)

Exercise 4.

Expand and simplify:

a)

b)

c)

Exercise 5.

Expand and simplify:

a)

b)

c)

Exercise 6.

Expand and simplify:

a)

b)

c)

Exercise 7.

Write an expression from the words, then simplify:

a) Four times the sum of and

b) Three times the difference between and

c) Two times the sum of and , then add

Exercise 8.

Write an expression from the words, then simplify:

a) Five times the sum of and , then subtract

b) Three times the difference between and , then add

c) Two times the sum of and , then add three times the difference between and

Reasoning

Exercise 9.

Explain why .

Exercise 10.

A student says that . Explain the mistake.

Exercise 11.

Noah says that because only the first term is multiplied by . Is he correct? Explain.

Exercise 12.

Explain why expanding brackets often creates like terms that can then be combined.

Exercise 13.

A student expands as . Describe the error.

Problem-solving

Exercise 14.

A rectangle has width cm and length cm. Write and simplify an expression for the area.

Exercise 15.

A student buys packs of pencils. Each pack contains pencils. Write and simplify an expression for the total number of pencils.

Exercise 16.

A pattern has squares in one part and more squares in another part. Write and simplify an expression for the total number of squares.

Exercise 17.

A taxi fare is calculated as dollars, where is the number of kilometres, and then a discount of dollars is subtracted. Write and simplify the expression.

Exercise 18.

A garden has two sections. One section has area square metres and the other has area square metres. Write and simplify the total area.

Exercise 19.

A builder uses bricks in one row and bricks in another row. Write and simplify the total number of bricks used.

Potential Misunderstandings

• Students may multiply the number outside the bracket by only the first term inside the bracket
• Students may forget to multiply both terms inside the bracket
• Students may change the sign incorrectly when expanding brackets with subtraction
• Students may think means instead of
• Students may combine unlike terms after expanding
• Students may not Recognise like terms created by expansion
• Students may confuse expanding with factorising
• Students may make arithmetic errors when multiplying the coefficient by the constant term

Next: 139. Factorising Expressions