In a formula, letters stand for quantities that can change (variables) and numbers stay the same (constants). In an equation, a letter can stand for one unknown number.
Recipes, phone bills and the perimeter of a garden bed all follow a formula. The letters are what changes; the numbers are what stays.
In maths a letter can stand for a number. A formula such as P=2l+2w uses letters for quantities, here the perimeter P, length l and width w of a rectangle. This lesson is about what the letters and numbers in a formula mean.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
What changes?
P=2l+2w
Think of two different rectangles: a long thin one and a square. What is the same about how you find their perimeters?
TryTry it on the board and compare the result with your prediction.
Change the rectangle
Change l and w. Watch the rectangle, the working and P. Which numbers in the formula never change?
Try it yourselfYour edits are saved in this browser.
Make a rectangle 7 cm long and 2 cm wide. What is its perimeter?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Variables, constants and unknowns
Here is how to read the parts of a formula.
Worked exampleBefore each step, say what you think comes next.
P=2l+2wThe perimeter of a rectangle: twice the length plus twice the width.
P,l,wThese letters are variables: quantities that change from one rectangle to the next.
2The 2s are constants: they are the same for every rectangle.
l=5,w=3For one particular rectangle, each variable has a value.
3x+2=14In an equation, a letter can be an unknown: one particular number.
x=4Only one value makes this equation true.
Read Kim's working one line at a time. What does l stand for?
Kim works out the perimeter. Click the line where her thinking goes wrong.
ModifyChange the problem on purpose and see what happens.
Other formulas
Each task loads a new formula. Change the variables and watch what stays the same.
Try it yourselfYour edits are saved in this browser.
In C=5h+20, what is the charge for a job of 0 hours (just the call-out)?
PractiseA few on your own, to check you have it.
Read the formula
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A formula of your own
Name the variables and the constant in your formula.
Stretch: write a formula for the perimeter of a triangle with sides a, b and c.
Need a hint?
A square has four equal sides, each s long.
Another hint
How many lots of s make the perimeter?
A square has sides of length s. Write a formula for its perimeter P. Start with P=.
For teachers
Students tell the variables in a formula from the constants, and see that in an equation a letter can be one unknown number (AC9M7A01, node 1). About 45 minutes.
I can say which letters in a formula are variables and which numbers are constants.
I can say what a constant means in a real formula, such as a call-out fee.
I can tell a variable (can take many values) from an unknown (one value makes the equation true).
Before class
Do the Run board yourself first: change l and w and watch the working.
The new move is reading a letter as a quantity that changes. Students often think each letter has one fixed value, or link a letter to its place in the alphabet (Kim's mistake). Keep saying the quantity aloud: "l, the length", not just "l".
This lesson does not ask students to calculate much; the next lesson is substitution. The board does the arithmetic so attention stays on what changes.
Running the lesson
Predict: ask students who chose 'Nothing changes' to describe two different rectangles. Do they have the same length?
Run: before they reach l = 7, w = 2, ask: which numbers in the working never changed?
Investigate: model the worked example live. Draw the contrast on the board: formula (variables, many values) and equation (unknown, one value).
Modify: for π, point out that a letter is not automatically a variable.
Make: have two students compare 4s with s + s + s + s. Are they the same formula?
When students get stuck
A student gives a letter its alphabet value (Kim's mistake). Ask: What does l stand for in this formula? How would you find its value? Fix: A letter stands for a quantity. Its value is measured or given, and can change from one rectangle to the next.
A student says the 2 in 2l is a variable because it is next to a letter. Ask: Draw a different rectangle. Did the 2 change? Fix: Constants stay the same for every case. 2l means 2 × l: two lengths.
Stretch and support
Stretch: Write a formula for the cost of n pizzas at 15 dollars each with a 5 dollar delivery fee. Name every variable and constant, and say what each constant means.
Support: Give a two-column table, 'changes' and 'stays the same', to sort the parts of each formula into before answering.
Where it leads
Next: substituting into formulas, then finding an unknown in a formula. Later, linear relations use the same letters: in y = mx + c, x and y vary and m and c are constants for one line.
What each step is for
what changes?: Committing to which parts change exposes the idea that each letter stands for one fixed number.
change the rectangle: Moving l and w and watching P change, while the 2s never do, shows what a variable is.
variables, constants and unknowns: A variable is a letter for a quantity that can change; a constant stays the same; in an equation a letter can be an unknown, one particular number.
other formulas: One change at a time: a formula with a constant added, one with a single variable, a product of two variables, and a letter that is a constant.
read the formula: Naming variables, constants and unknowns, and saying what a constant means, in new formulas.
a formula of your own: Writing a formula for a familiar shape reverses the lesson: from the quantities to the letters and constants.
Now you can
Say which letters in a formula are variables and which numbers are constants.
Say what a constant means in a real formula, such as a call-out fee.
Tell a variable (can take many values) from an unknown (one value makes the equation true).