101. Order of Operations and Substitution

Learning Intentions

• To understand the rules for order of operations
• Evaluate numerical expressions Use the order of operations
• Substitute integers for pronumerals in order to evaluate expressions

Pre-requisite Summary

• Know the four basic operations: addition, subtraction, multiplication and division
• Be able to perform basic calculations with integers accurately
• Understand that brackets group parts of an expression together
• Know that a pronumeral stands for a number
• Be able to replace a pronumeral with a given value
• Understand the difference between an expression and an equation
• Be familiar with using integers, including negative integers, in calculations

Worked Examples

Worked Example 1

Evaluate using the order of operations:

a)

b)

Worked Example 2

Evaluate using the order of operations:

a)

b)

Worked Example 3

Evaluate using the order of operations:

a)

b)

Worked Example 4

Substitute the given integer and evaluate:

a) Solve when

b) Find when

Worked Example 5

Substitute the given integer and evaluate:

a) Find when

b) Find when

Worked Example 6

Substitute the given integer and evaluate using the order of operations:

a) Find when

b) Find when

c) Find when

Problems

Problem 1

Evaluate using the order of operations:

a)

b)

Problem 2

Evaluate using the order of operations:

a)

b)

Problem 3

Evaluate using the order of operations:

a)

b)

Problem 4

Substitute the given integer and evaluate:

a) Find when

b) Find when

Problem 5

Substitute the given integer and evaluate:

a) Find when

b) Find when

Problem 6

Substitute the given integer and evaluate using the order of operations:

a) Find when

b) Find when

c) Find when

Exercises

Understanding and Fluency

Exercise 1.

Evaluate each expression using the order of operations:

a)

b)

c)

Exercise 2.

Evaluate each expression with brackets:

a)

b)

c)

Exercise 3.

Evaluate each numerical expression:

a)

b)

c)

Exercise 4.

Evaluate each numerical expression:

a)

b)

c)

Exercise 5.

Substitute and evaluate:

a) Find when

b) Find when

c) Find when

Exercise 6.

Substitute and evaluate:

a) Find when

b) Find when

c) Find when

Exercise 7.

Substitute and evaluate using the order of operations:

a) Find when

b) Find when

c) Find when

Exercise 8.

Substitute and evaluate using the order of operations:

a) Find when

b) Find when

c) Find when

Reasoning

Exercise 9.

Explain why is not equal to .

Exercise 10.

A student says that in , you subtract first because subtraction comes before division when reading from left to right. Explain the mistake.

Exercise 11.

Noah substitutes into and writes . Explain why this is incorrect.

Exercise 12.

Explain why brackets are important when substituting a negative integer into an expression such as or .

Exercise 13.

A student evaluates as . Describe the error.

Problem-solving

Exercise 14.

A game score is calculated using the expression , where is the number of bonus stars. Find the score when .

Exercise 15.

The cost of hiring bikes is given by , where is the number of bikes hired. Find the cost when .

Exercise 16.

A machine uses the rule . Find the output when the input is .

Exercise 17.

A student evaluates the expression to find the total number of stickers in identical packs. How many stickers are there altogether?

Exercise 18.

The temperature at noon is modelled by . Find the value when .

Exercise 19.

A pattern is described by . Find the value of the pattern when and when .

Potential Misunderstandings

• Students may think operations should always be completed strictly from left to right
• Students may forget that multiplication and division are completed before addition and subtraction
• Students may ignore brackets or complete operations outside the brackets first
• Students may think multiplication must always be done before division, instead of working left to right for operations of equal priority
• Students may think addition must always be done before subtraction, instead of working left to right for operations of equal priority
• Students may confuse an expression with an equation and try to “Solve” instead of evaluate
• Students may substitute the wrong value for a pronumeral
• Students may forget that means
• Students may substitute a negative integer without brackets and lose the negative sign
• Students may perform substitution correctly but then Apply the order of operations incorrectly
• Students may think brackets are only used for grouping operations, rather than also helping show substitution clearly