GM Lesson 048 Substitution into Simple Non-linear Expressions

Learning Intentions

By the end of this lesson, students will be able to:

• Substitute numerical values into simple non-linear expressions.
• Evaluate expressions involving powers.
• Apply order of operations when evaluating algebraic expressions.

Prerequisites

Students should already be able to:

• Substitute values into linear expressions such as .
• Use positive and negative integers in arithmetic calculations.
• Apply the order of operations:
  • brackets
  • powers
  • multiplication and division
  • addition and subtraction
• Understand that means , not .

Key Idea Summary

A non-linear expression contains a pronumeral raised to a power, such as or .

When substituting into a non-linear expression:

1. Replace the pronumeral with the given value.
2. Use brackets around substituted values, especially negative values.
3. Evaluate powers before multiplication, division, addition or subtraction.
4. Simplify carefully.

For example, if , then:

If , then:

The brackets are important because .

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Introduction: Linear compared with non-linear expressions

A linear expression such as contains only to the first power.

A non-linear expression such as contains a power of the pronumeral.

Non-linear expressions often arise in measurement contexts. For example, the area of a square with side length is:

This is non-linear because the side length is squared.

Worked Example 1: Substituting into a squared expression

Evaluate when .

Substitute :

Evaluate the power first:

Therefore, when , the value of is .

Worked Example 2: Substituting into an expression with multiplication and powers

Evaluate when .

Substitute :

Evaluate the power first:

Multiply:

Add and subtract from left to right:

Therefore, when , the value of is .

Worked Example 3: Substituting a negative value

Evaluate when .

Substitute using brackets:

Evaluate the power and multiplication:

Simplify:

Therefore, when , the value of is .

Worked Example 4: Substituting into a cubic expression

Evaluate when .

Substitute :

Evaluate powers first:

Multiply:

Therefore, when , the value of is .

Worked Example 5: Applying a non-linear expression in context

The area of a square is given by:

where is the side length.

Find the area of a square with side length .

Substitute :

Therefore, the area of the square is .

Understanding Checks

Understanding Check 1

Evaluate when .

Understanding Check 2

Evaluate when .

Understanding Check 3

Evaluate when .

Understanding Check 4

Which step should be done first in the expression after substituting a value for ?

A. Add
B. Square the substituted value
C. Subtract
D. Multiply by only after all addition is complete

Understanding Check 5

Explain why brackets are important when evaluating for .

Exercises

Simple Familiar Exercises

Exercise 1

Evaluate when .

Exercise 2

Evaluate when .

Exercise 3

Evaluate when .

Exercise 4

Evaluate when .

Exercise 5

Evaluate when .

Exercise 6

Evaluate when .

Exercise 7

Evaluate when .

Exercise 8

Evaluate when .

Complex Familiar Exercises

Exercise 9

Evaluate when .

Exercise 10

Evaluate when .

Exercise 11

Evaluate when .

Exercise 12

Evaluate when .

Exercise 13

Evaluate when .

Exercise 14

Evaluate when .

Exercise 15

Evaluate when .

Exercise 16

Evaluate when .

Homework Problems

Problem 1

Evaluate when .

Problem 2

Evaluate when .

Problem 3

Evaluate when .

Problem 4

Evaluate when .

Problem 5

Evaluate when .

Problem 6

The area of a square is given by . Find the area of a square with side length .

Problem 7

The volume of a cube is given by . Find the volume of a cube with side length .

Problem 8

A simple model for the braking distance of a bicycle is:

where is the braking distance in metres and is the speed in kilometres per hour.

Find the braking distance when .

Problem 9

Evaluate when .

Problem 10

Explain in one or two sentences why and do not have the same value.

Next: GM Lesson 049 Evaluating Formulae