1. Using formulas

2. Using units
Formulas deal mainly with real-life quantities such as length, mass, temperature or time, and the variables often have units.
Units are usually defined when writing formulas, but they are not included in the calculations. d
For example:
s = t
This does not mean much unless we say:
Mathematical practices
6) Attend to precision.
Students should understand that a formula is a specific type of equation where the variables are defined and may have units. They should use the mathematical terms “formula” and “equation” appropriately so that they can be easily understood by others. They should ensure that when they write formulas, they give appropriate units for the quantities.
s is the average speed in m/s
d is the distance travelled in meters
t is the time taken in seconds. 2

3. Speed, distance and time
If a car travels 2 km in 1 min 40 secs, at what speed is it traveling?
s = d t
Use the formula:
Write the distance and the time using the correct units before substituting them into the formula:
2 kilometers = 2000 meters 1 min and 40 seconds = 100 seconds
Teacher notes
When students find the average speed of the car in km/h rather than m/s this should help them understand how the units are related to the structure of the formula. They may find it helpful to write in the units throughout their calculations.
There are two ways students could calculate the speed in kilometers per hour. As they have already found the speed in meters per second, they could multiply this by 60 × 60 to find meters per hour, and divide by 1000 to find kilometers per hour.
However, advise students that this method will preserve any errors they made in the original calculation.
The more reliable method is to perform the calculations from scratch.
We already have the distance in kilometers. Convert the time to hours: 40 seconds is 40/60 = 2/3 of a minute, or … so 1 minute and 40 seconds is 5/3 or … minutes. Divide this by 60 to find the fraction of an hour: (5/3)/60 = …
Then distance / time = 2km / … hours = 72 km per hour. Advise students that to ensure accuracy they could perform all the calculations in a single step on their calculators.
Mathematical practices
6) Attend to precision.
Students should be careful to convert the values they are given for speed and time into the units appropriate for the formula, in this case meters and seconds.
Now substitute these numerical values into the formula:
s =
2000
100
Write the units in at the final step in the calculation:
= 20 m/s
Now find the speed in kilometers per hour. 3

4. Rearranging formulas
Formulas are usually (but not always) arranged so that a single variable is written on the left-hand side of the equal sign. For example, the formula below has v as the subject, and is said to be solved for v:
v = u + at
What happens if we are given values of v, u and a, and asked to find t?
Rearrange the formula to make t the subject, then substitute values in to solve the equation.
v – u
t = a

5. Inverse operations
To rearrange a formula, find the inverse operations that undo the operations in the formula.
Here is a formula for the area of a rectangle, A, using its length, L, and width, W:
A = LW
The formula is written with A as the subject. To rearrange it to make L the subject:
A = LW
rewrite the formula with L on the left:
Teacher notes
To check students’ understanding, have them rearrange the formula for W. They should get the result W = A ÷ L. Test the rearranged formula by checking that the same numbers work in both formulas. For example, put L = 5 cm, W = 2 cm into the first formula to find that A = 5 × 2 = 10 cm2. Then put A = 10 cm2 and W = 2 cm into the bottom formula to check that L = 10 ÷ 2 = 5 cm.
Mathematical practices
2) Reason abstractly and quantitatively.
In order to rearrange the formula for the area of a rectangle, students should be able to manipulate the symbols without attending to their referents, while also realizing that this is done so that the length of a rectangle can be found from its area and width.
solving for A
LW = A
L = A ÷ W
remove W from the left-hand side by dividing both sides by W:
solving for L A W
L =

6. Problems leading to equations
The formula used to find the area, A, of a trapezoid with parallel sides a and b and height h is: a h
A = (a + b)h 1 2 b
By substituting values into the formula and rearranging the equation, it is possible to calculate an unknown.
Teacher notes
In this example, we are told the area and the lengths of the parallel sides and we need to find the height. Explain that we can substitute the values we are given into the formula to make an equation with one unknown, h. The solution and calculations using this method are shown on the next slide. An alternative would be to rewrite the formula with h as the subject, and then substitute values in after the formula has been rearranged. These two methods are very similar but some students may prefer one or the other.
What is the height of a trapezoid with an area of 40 cm2 and parallel sides of length 7 cm and 9 cm? 6

7. Problems leading to equations
What is the height of a trapezoid with an area of 40 cm2 and parallel sides of length 7 cm and 9 cm?
A = (a + b)h 1 2
write 40 instead of A, 7 instead of a, and
9 instead of b
40 = (7 + 9)h 1 2
a = 7 cm
40 = × 16h 1 2
simplify the equation
40 = 8h h
A = 40 cm²
Teacher notes
This slide shows how to solve the problem by substituting the given values to make an equation. Check the solution by substituting h = 5, a = 7 and b = 9 into the original formula to make sure that it gives an area of 40 cm2.
Mathematical practices
2) Reason abstractly and quantitatively.
Students should use their understanding of the meaning of the given formula to identify which value to write for each variable. They should decontextualize, manipulating the equation until it is in form h = a value, without needing to think about the meaning of h (the height of a trapezoid) in order to do this. They should view their final result in context, writing the solution with appropriate units (centimeters).
divide both sides by 8
5 = h
b = 9 cm
The height of the trapezoid is 5 cm. 7

8. Rearrange this formula to make w the subject.
Repeated variables
Sometimes a variable appears more than once in a formula.
For example, three of the variables appear twice in this formula for the surface area of a rectangular prism:
S = 2lw + 2lh + 2hw
S is the surface area of a rectangular prism l
l is its length h
w is its width
Mathematical practices
2) Reason abstractly and quantitatively.
In order to rearrange the formula for the surface area of a rectangular prism, students should be able to manipulate the symbols without attending to their referents, while also realizing that this is done so that the width of a rectangular prism can be found from its length, height and surface area.
h is its height. w
Rearrange this formula to make w the subject.

9. Repeated variables Rearrange this formula to make w the subject:
S = 2lw + 2lh + 2hw
Write the formula so that the terms containing w are on the left:
2lw + 2lh + 2hw = S
Subtract 2lh from both sides so that all terms containing w are together:
2lw + 2hw = S – 2lh
Isolate w by factoring:
w(2l + 2h) = S – 2lh
Teacher notes
There is no need to factor 2lw + 2hw completely. Only factor out the w to isolate it.
Mathematical practices
2) Reason abstractly and quantitatively.
In order to rearrange the formula for the surface area of a rectangular prism, students should be able to manipulate the symbols without attending to their referents, while also realizing that this is done so that the width of a rectangular prism can be found from its length, height and surface area.
w =
S – 2lh
2l + 2h
Divide both sides by 2l + 2h:

10. Stopping distances
This formula can be used to calculate the stopping distance of a car: x2 20
+ x = S
x is the car’s speed in mph
S is the stopping distance in feet.
As an estimate, what will be the stopping distance of a car traveling at 30 mph?
Teacher notes
This formula can be used as an approximation of the stopping distance of a car. Use different values for x to check students’ substituting ability.
x² ÷ 20 + x = S
30² ÷ = S
900 ÷ = S
= S
S = 75 feet
Students should be encouraged to come up with different ways in which they could figure out what speed a car had been traveling at if it took 315 feet to stop. They may choose to use trial and error to figure out that, as an estimate, the car would have been traveling at 70 mph.
Mathematical practices
2) Reason abstractly and quantitatively.
Students should be able to extract the useful information about speed and stopping distances from the questions and write this for the correct terms in the formula. They should be able to manipulate the equations produced to solve it for the stopping distance or speed without needing to think about the meaning of the symbols, and write their answer in the appropriate units for the context (mph or feet).
Photo credit: © Heather Renee, Shutterstock.com 2012
What speed would a car have been doing if it took 315 feet to stop? 10

11. Van hire solution Can you find values for d and m where the cost is
the same for both firms? How did you figure this out?
C = 50d + 0.2m
C = 25d + 0.4m
The cost is the same for both firms, so we can write:
50d + 0.2m = 25d + 0.4m
Teacher notes
The cost is the same for both vans so long as the mean number of miles covered in a day is 125. When d = 1 and m = 125, the cost is the same for both firms. The same is true when d = 2 and m = 250.
Mathematical practices
4) Model with mathematics.
Students should write formulas to model the cost of hiring the vans. They should recognize that these formulas can be manipulated to find values that satisfy both, and be able to perform this mathematical operation. They should interpret the result in the context of the situation, realizing that this means the vans will cost the same amount if they drive an average 125 miles per day.
50d – 25d = 0.4m – 0.2m
25d = 0.2m
125d = m
The cost is the same if the vans drive 125 miles in a day. 11