053e. Order of Operations with Integers

Learning Intentions

• To know the convention for determining order of operations in an expression involving more than one operation
• Evaluate arithmetic expressions involving more than one operation with positive and/or negative numbers

Pre-requisite Summary

• Understand the four operations: addition, subtraction, multiplication and division
• Understand that brackets change the order in which parts of an expression are evaluated
• Know how to add, subtract, multiply and divide positive and negative integers
• Understand that multiplication and division are performed before addition and subtraction
• Be able to work from left to right when operations have the same priority
• Recognise that a negative number is different from subtraction as an operation

Worked Examples

Worked Example 1

Evaluate Use the correct order of operations:

a)

b)

c)

Worked Example 2

Evaluate expressions with negative numbers:

a)

b)

c)

Worked Example 3

Evaluate expressions with brackets:

a)

b)

c)

Worked Example 4

Evaluate expressions with brackets and negative numbers:

a)

b)

c)

Worked Example 5

Evaluate expressions with several operations:

a)

b)

c)

Worked Example 6

Evaluate expressions with integers and brackets:

a)

b)

c)

Problems

Problem 1

Evaluate using the correct order of operations:

a)

b)

c)

Problem 2

Evaluate expressions with negative numbers:

a)

b)

c)

Problem 3

Evaluate expressions with brackets:

a)

b)

c)

Problem 4

Evaluate expressions with brackets and negative numbers:

a)

b)

c)

Problem 5

Evaluate expressions with several operations:

a)

b)

c)

Problem 6

Evaluate expressions with integers and brackets:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Evaluate:

a)

b)

c)

Exercise 2.

Evaluate:

a)

b)

c)

Exercise 3.

Evaluate with brackets:

a)

b)

c)

Exercise 4.

Evaluate with brackets and negative numbers:

a)

b)

c)

Exercise 5.

Evaluate:

a)

b)

c)

Exercise 6.

Evaluate:

a)

b)

c)

Exercise 7.

Evaluate:

a)

b)

c)

Exercise 8.

Evaluate:

a)

b)

c)

Reasoning

Exercise 9.

Explain why .

Exercise 10.

A student evaluates as . Explain the mistake.

Exercise 11.

Explain why gives a different answer from .

Exercise 12.

A student says because they added first. Explain why this is incorrect.

Problem-solving

Exercise 13.

A game score is calculated by the expression . Solve the score.

Exercise 14.

A diver’s position is modelled by . Find the final position.

Exercise 15.

A lift starts at floor and follows the expression . Find the final floor.

Exercise 16.

A bank balance changes according to . Find the final balance.

Exercise 17.

A temperature change is modelled by . Find the final temperature.

Exercise 18.

A pattern is given by . Evaluate the expression.

Potential Misunderstandings

• Students may perform operations strictly from left to right instead of following the order of operations
• Students may not recognise that multiplication and division must be completed before addition and subtraction
• Students may ignore brackets or treat them as decorative symbols
• Students may confuse a negative sign with subtraction as an operation
• Students may not work left to right when multiplication and division appear together
• Students may make sign errors when multiplying or dividing negative integers
• Students may think and have the same meaning in later work, so careful bracket Use should be reinforced
• Students may Simplify part of an expression correctly but then lose track of the remaining operations

Next: 054e. Substituting Integers into Expressions