GM Lesson 049 Evaluating Formulae
Learning Intentions
By the end of the lesson, students will be able to:
- Identify pronumerals and constants in a formula.
- Substitute given values into formulae.
- Evaluate formulae using correct arithmetic order.
Prerequisites
Students should already be able to:
- Substitute numbers into algebraic expressions.
- Evaluate powers such as
, and using a calculator. - Apply order of operations.
- Round decimal answers appropriately.
- Recognise that a pronumeral represents a quantity whose value may change.
Key Idea Summary
A formula shows a relationship between quantities.
In a formula:
- A pronumeral is a letter that represents a variable quantity.
- A constant is a fixed value that does not change.
- To evaluate a formula, substitute each given value carefully, then follow order of operations.
The general process is:
- Write the formula.
- Identify the value of each pronumeral.
- Substitute values using brackets.
- Evaluate powers and roots first.
- Multiply and divide.
- Add and subtract.
- State the answer with suitable units.
For this lesson, formulae will come from physics contexts. The main mathematical skill is not physics theory; the main skill is accurate substitution and evaluation.
Direct Instruction and Worked Examples
Time Allocation
Time Allocation
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- 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.
Worked Example 1: Kinetic Energy
The kinetic energy of a moving object is given by
where:
is kinetic energy in joules is mass in kilograms is speed in metres per second
Find
Substitute:
Evaluate the power first:
Multiply:
Therefore, the kinetic energy is
Worked Example 2: Gravitational Potential Energy
The gravitational potential energy of an object is given by
where:
is gravitational potential energy in joules is mass in kilograms is gravitational acceleration is height in metres
Find
Substitute:
Multiply:
Therefore, the gravitational potential energy is
Worked Example 3: Wave Speed
The speed of a wave is given by
where:
is wave speed in metres per second is frequency in hertz is wavelength in metres
Find
Substitute:
Evaluate:
Therefore, the wave speed is
Worked Example 4: Period of a Pendulum
The period of a simple pendulum is approximated by
where:
is the period in seconds is the length of the pendulum in metres is gravitational acceleration
Find
Substitute:
Evaluate inside the square root:
Take the square root:
Multiply:
Therefore, the period is approximately
Worked Example 5: Newton’s Law of Universal Gravitation
The gravitational force between two objects is given by
where:
is gravitational force in newtons is the gravitational constant and are masses in kilograms is the distance between the objects in metres
Find
Substitute:
Evaluate the numerator:
Evaluate the denominator:
Substitute these values:
Evaluate:
Therefore, the gravitational force is approximately
Worked Example 6: Stopping Distance Formula
A simple physics formula for stopping distance under constant acceleration is
When evaluating the final speed
where:
is final speed is initial speed is acceleration is distance
Find
Substitute:
Evaluate powers and multiplication:
Add:
Evaluate:
Therefore,
Understanding Checks
Check 1
In the formula
identify:
- the pronumerals
- the constant
Answer:
The pronumerals are
The constant is
Check 2
A student substitutes
and writes
What should be evaluated first?
Answer:
The power
Check 3
Evaluate:
when
Answer:
Check 4
Evaluate:
when
Answer:
Check 5
Why are brackets useful when substituting into formulae?
Answer:
Brackets make it clear which values are being substituted and help prevent errors with negative numbers, powers and multiplication.
Exercises
Simple Familiar Exercises
Exercise 1
Evaluate the kinetic energy formula
when
Exercise 2
Evaluate the gravitational potential energy formula
when
Exercise 3
Evaluate the wave speed formula
when
Exercise 4
Evaluate the electrical power formula
when
Exercise 5
Evaluate the density formula
when
Exercise 6
Evaluate the pressure formula
when
Complex Familiar Exercises
Exercise 7
Evaluate
when
Exercise 8
Evaluate
when
Round to
Exercise 9
Evaluate
when
Round to
Exercise 10
Evaluate
when
Exercise 11
Evaluate
when
Round to
Exercise 12
Evaluate
when
Homework Problems
Complete the following problems. This homework should take no more than
Problem 1
Evaluate
when
Problem 2
Evaluate
when
Problem 3
Evaluate
when
Problem 4
Evaluate
when
Problem 5
Evaluate
when
Round to
Problem 6
Evaluate
when
Problem 7
Evaluate
when
Round to the nearest whole number.
Problem 8
A student evaluates
using
They write:
Evaluate the formula correctly and round to