GM Lesson 053 Finding a Pronumeral in Simple Non-linear Equations

Learning Intentions

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

• Find the value of a pronumeral in simple non-linear equations.
• Use square roots or other inverse operations where appropriate.
• Check whether solutions are reasonable in context.

Prerequisites

Students should already be able to:

• Substitute values into formulae.
• Evaluate expressions involving powers.
• Solve simple linear equations using inverse operations.
• Use square roots as the inverse operation of squaring.
• Recognise that a length, radius, height, area or volume must usually be positive in context.

Key Idea Summary

A non-linear equation contains a pronumeral with a power, such as , or another non-linear operation.

To solve simple non-linear equations, use inverse operations in the reverse order.

For example:

Since squaring and square rooting are inverse operations:

In pure algebra, has two solutions:

In measurement contexts, negative lengths are not reasonable, so only the positive solution is used.

Important syllabus formulae used in this lesson include:

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

Part A: Solving Equations of the Form

For an equation such as:

Use the square root:

If there is no practical context, also consider the negative solution:

because:

and

Worked Example 1: Finding a Pronumeral in a Simple Non-linear Equation

Solve:

Use the inverse operation of squaring:

So:

Check:

Both values satisfy the equation.

Part B: Solving Equations of the Form

If a squared term has been multiplied by a number, divide first.

For example:

Divide both sides by :

Then square root:

Worked Example 2: Solving After Dividing First

Solve:

Divide both sides by :

Take the square root:

So:

Check:

Both values satisfy the equation.

Part C: Using Pythagoras’ Theorem

For a right-angled triangle:

where is the hypotenuse and and are the two shorter perpendicular sides.

If the unknown is the hypotenuse, find first, then square root.

Worked Example 3: Finding the Hypotenuse

A right-angled triangle has shorter sides cm and cm. Find the hypotenuse .

Use:

Substitute:

Square root:

So the hypotenuse is cm.

Reasonableness check: the hypotenuse should be longer than both shorter sides. Since and , the answer is reasonable.

Worked Example 4: Finding a Shorter Side

A right-angled triangle has hypotenuse m and one shorter side m. Find the other shorter side .

Use:

Substitute:

Subtract from both sides:

Square root:

So the missing side is m.

Reasonableness check: the shorter side must be less than the hypotenuse. Since , the answer is reasonable.

Part D: Finding a Radius from Area

The area of a circle is:

If is known, divide by first, then square root.

Worked Example 5: Finding the Radius of a Circle

A circle has area cm. Find its radius, correct to decimal place.

Use:

Substitute:

Divide by :

Square root:

So the radius is approximately cm.

Reasonableness check: the radius must be positive, so the negative square root is rejected.

Part E: Finding a Radius from Cylinder Volume

The volume of a cylinder is:

If and are known, divide by first, then square root.

Worked Example 6: Finding the Radius of a Cylinder

A cylinder has volume cm and height cm. Find the radius, correct to decimal place.

Use:

Substitute:

Divide by :

Square root:

So the radius is approximately cm.

Reasonableness check: substituting gives a volume close to cm, so the answer is reasonable.

Understanding Checks

Understanding Check 1

Solve:

Questions for students:

• What operation undoes squaring?
• Are there one or two algebraic solutions?
• If represented a length, which solution would be reasonable?

Expected answer:

If is a length:

Understanding Check 2

Solve:

Expected process:

Understanding Check 3

A right-angled triangle has shorter sides cm and cm.

Find the hypotenuse.

Expected process:

Understanding Check 4

A circle has area m.

Estimate the radius.

Expected process:

Understanding Check 5

Explain why is not a reasonable answer for the radius of a circle.

Expected response:

A radius is a distance from the centre of a circle to the edge. Distances cannot be negative in this context, so is rejected.

Exercises

Simple Familiar Exercises

Exercise 1

Solve:

Exercise 2

Solve:

Exercise 3

Solve:

Exercise 4

Solve:

Exercise 5

Solve:

Exercise 6

Solve:

Exercise 7

A square has area cm.

Find its side length.

Exercise 8

A circle has area cm.

Use:

Find the radius, correct to decimal place.

Complex Familiar Exercises

Exercise 9

A right-angled triangle has shorter sides cm and cm.

Use:

Find the hypotenuse.

Exercise 10

A right-angled triangle has hypotenuse m and one shorter side m.

Find the other shorter side.

Exercise 11

A circle has area m.

Use:

Find the radius, correct to decimal place.

Exercise 12

A cylinder has volume cm and height cm.

Use:

Find the radius, correct to decimal place.

Exercise 13

A cylinder has volume m and radius m.

Use:

Find the height.

Exercise 14

A right-angled triangle has hypotenuse cm and one shorter side cm.

Find the other shorter side.

Homework Problems

Homework 1

Solve:

Homework 2

Solve:

Homework 3

A square has area cm.

Find its side length.

Homework 4

A right-angled triangle has shorter sides cm and cm.

Find the hypotenuse.

Homework 5

A right-angled triangle has hypotenuse m and one shorter side m.

Find the other shorter side.

Homework 6

A circle has area cm.

Use:

Find the radius, correct to decimal place.

Homework 7

A cylinder has volume cm and height cm.

Use:

Find the radius, correct to decimal place.

Homework 8

A square paddock has area m.

Find the side length of the paddock.

Then find its perimeter.

Homework 9

A right-angled triangular sail has hypotenuse m and one shorter side m.

Find the other shorter side.

Homework 10

A circular table top has area m.

Find its radius, correct to decimal places.

Then find its diameter.

Next: GM Lesson 054 Transposing Simple Non-linear Formulae