GM Lesson 050 Formulae with Several Pronumerals

GM Lesson 050: Formulae with Several Pronumerals

Learning Intentions

By the end of this 45 minute lesson, students will be able to:

• Substitute values into formulae containing several pronumerals.
• Identify which values are known and which value is required.
• Evaluate formulae in practical contexts, including physics and measurement contexts.

Prerequisites

Students should already be able to:

• Substitute values into simple algebraic expressions.
• Apply order of operations correctly.
• Evaluate expressions involving powers, fractions and square roots.
• Identify pronumerals and constants in a formula.
• Use a calculator accurately for formula evaluation.

Key Idea Summary

A formula shows a relationship between quantities.

When using a formula with several pronumerals:

1. Identify the required value.
2. Identify the known values.
3. Substitute each known value into the correct position.
4. Evaluate using correct order of operations.
5. Include appropriate units.

Common formulae used in this lesson include:

• Kinetic energy:
• Gravitational potential energy:
• Force:
• Pressure:
• Wave speed:
• Electrical power:
• Heat energy:
• Pythagoras’ theorem:
• Volume of a cylinder:
• Simple interest:

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

Direct Instruction

A formula with several pronumerals can look difficult, but the method is always the same.

For example, in the formula :

• is kinetic energy.
• is mass.
• is speed.
• The number is a constant.

If and , then we substitute into the formula:

The power is evaluated before multiplication:

Therefore, the kinetic energy is joules.

Worked Example 1: Kinetic Energy

A cyclist and bicycle have a combined mass of kg and travel at m/s. Calculate the kinetic energy using:

Known values:

Required value:

Substitute:

Evaluate powers first:

The kinetic energy is J.

Worked Example 2: Gravitational Potential Energy

A kg object is lifted to a height of m. Use m/s. Calculate the gravitational potential energy using:

Known values:

Required value:

Substitute:

The gravitational potential energy is J.

Worked Example 3: Pressure

A force of N acts on an area of m. Calculate the pressure using:

Known values:

Required value:

Substitute:

The pressure is Pa.

Worked Example 4: Heat Energy

A kg metal block has a specific heat capacity of J/kgC. Its temperature increases by C. Calculate the heat energy using:

Known values:

Required value:

Substitute:

The heat energy is J.

Worked Example 5: Cylinder Volume

A cylindrical tank has radius m and height m. Calculate its volume using:

Known values:

Required value:

Substitute:

Evaluate powers first:

The volume is approximately m.

Worked Example 6: Simple Interest

A loan of $ earns simple interest for years at an annual interest rate of . Calculate the interest using:

Known values:

Required value:

Substitute:

The simple interest is $ .

Understanding Checks

Check 1

In the formula , identify the required value if the question asks:

“A kg object accelerates at m/s. Find the force.”

Expected response:

The required value is .

Check 2

In the formula , identify the known values if:

, and .

Expected response:

The known values are , and .

Check 3

Evaluate when and .

Expected response:

Check 4

Evaluate when and .

Expected response:

Check 5

A student substitutes into as follows:

What operation must be completed first?

Expected response:

Evaluate first.

Exercises

Simple Familiar Exercises

Exercise 1

Use to calculate when and .

Exercise 2

Use to calculate when and .

Exercise 3

Use to calculate when and .

Exercise 4

Use to calculate when , and .

Exercise 5

Use to calculate when and .

Exercise 6

Use to calculate when , and .

Complex Familiar Exercises

Exercise 7

A car has mass kg and is travelling at m/s. Use to calculate its kinetic energy.

Exercise 8

A wave has frequency Hz and wavelength m. Use to calculate the wave speed. Round to one decimal place.

Exercise 9

A concrete slab experiences a force of N over an area of m. Use to calculate the pressure.

Exercise 10

A cylindrical water tank has radius m and height m. Use to calculate its volume. Round to two decimal places.

Exercise 11

A kg object is lifted to a height of m. Use with to calculate the gravitational potential energy.

Exercise 12

A heater has voltage V and current A. Use to calculate the electrical power.

Homework Problems

Problem 1

Use to calculate when and .

Problem 2

Use to calculate the kinetic energy of a kg runner moving at m/s.

Problem 3

Use to calculate the gravitational potential energy of a kg object lifted m above the ground. Use .

Problem 4

Use to calculate the speed of a wave with frequency Hz and wavelength m.

Problem 5

Use to calculate the heat energy needed to warm kg of water by C. Use .

Problem 6

Use to calculate the volume of a cylinder with radius cm and height cm. Round to two decimal places.

Problem 7

Use to calculate the simple interest on $ invested for years at per year.

Problem 8

A speaker produces a sound wave with frequency Hz and wavelength m.

a. Use to calculate the speed of the sound wave.

b. State whether your answer is reasonable if sound travels at about m/s in air.

Problem 9

A person pushes on a surface with force N. The area of contact is m.

a. Use to calculate the pressure.

b. State the units for pressure.

Problem 10

A skateboarder has mass kg and speed m/s.

a. Use to calculate the kinetic energy.

b. Identify the known values and the required value before substituting.

Next: GM Lesson 051 Finding a Pronumeral in Linear Equations