Finding an unknown

When a formula's answer is known but one letter is not, substitute what you know. That leaves an equation; solve it by undoing, then check.

You know how much fence you have and how long the garden bed must be. How wide can it be? A formula answers that too, worked backwards.

So far you knew the letters and worked out the answer. Sometimes you know the answer, and one letter is the unknown. Substitute what you know: what is left is an equation, and you can solve it by undoing, as in the Equations unit.

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.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

How wide?

P = 2l + 2w

Which letters do you know? Which one is unknown?

A rectangle has perimeter

P = 20

cm and length

l = 6

cm. Using

P = 2l + 2w

, what is its width

w

?47814

TryTry it on the board and compare the result with your prediction.

Find the width

Change $w$ one step at a time. How much does $P$ go up for each 1 cm of width? Why?

Try it yourselfYour edits are saved in this browser.

Keep $l = 6$ . Change $w$ until the perimeter is 20 cm. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Substitute, then solve

Here is how to find an unknown in a formula.

Worked exampleBefore each step, say what you think comes next.

$P = 2l + 2w$ Write the formula.
$P = 20, \; l = 6$ Write what you know. The unknown is $w$ .
$20 = 2 \times 6 + 2w$ Substitute. Now it is an equation with one unknown.
$20 = 12 + 2w$ Work out what you can.
$8 = 2w$ Undo the + 12: take 12 from both sides.
$w = 4$ Undo the × 2: divide both sides by 2.
$2 \times 6 + 2 \times 4 = 20 \;\checkmark$ Check by substituting back into the formula.

The rectangle with P = 20 and l = 6: the width is 4. — The rectangle with $P = 20$ and $l = 6$ : the width is 4.

Read Mia's working one line at a time. Does her answer check?

A rectangle has area $A = 24$ cm² and length $l = 6$ cm. Using $A = lw$ , find its width $w$ .

A plumber charges $C = 5h + 20$ dollars. A job cost 45 dollars. How many hours, $h$ ?

Mia finds the width of a rectangle. Click the line where she goes wrong.

ModifyChange the problem on purpose and see what happens.

Other unknowns

Each task loads a new formula. Write the equation first, solve it, then set the board to check.

Need a hint?

150 = 60t. Undo × 60.

Need a hint?

30 = l × 5. Undo × 5.

Need a hint?

60 = 100 − 4m. How much charge has been used?

Another hint

40 has been used, at 4 a minute.

Try it yourselfYour edits are saved in this browser.

A car travels at 60 km/h: $d = 60t$ . How long does it take to go 150 km? Set $t$ to find out.

Use the board. This ticks when you get there.

A rectangle is 5 cm wide with an area of 30 cm². Keep $w = 5$ and find its length.

Use the board. This ticks when you get there.

A phone battery has $B = 100 - 4m$ percent left after $m$ minutes of gaming. When is it at 60%?

Use the board. This ticks when you get there.

A box has volume $V = lwh$ . $V = 60$ , $l = 5$ and $w = 3$ . Find $h$ .

PractiseA few on your own, to check you have it.

Find the unknown

Concert tickets cost $T = 12n + 6$ dollars. An order cost 66 dollars. How many tickets?

Using $T = 12n + 6$ , what do 3 tickets cost?

Average speed is $s = d / t$ . A cyclist rides at 20 km/h for $t = 3$ hours. How far, $d$ , in km?

Review: solve $4x - 3 = 17$ .

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A garden bed

The bed must use exactly 24 m of edging, and be twice as long as it is wide.

Stretch: with the same 24 m, which rectangle gives the biggest area?

Try it yourselfYour edits are saved in this browser.

Need a hint?

The length is twice the width: P = 2(2w) + 2w = 6w.

Another hint

6w = 24. What is w?

You have 24 m of edging for a garden bed. Make the length twice the width, using all 24 m.

Use the board. This ticks when you get there.

For teachers

Students substitute the known values into a formula, solve the equation that is left for the unknown, and check by substituting back (AC9M7A01, node 3). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can substitute the values I know into a formula to make an equation.
I can solve the equation by undoing each operation, last first.
I can check my answer by substituting it back into the formula.

Before class

Do the Run and Modify boards yourself first. The board lets students hunt for the answer by moving a slider: that is fine at Run, but in Investigate and Modify they should write the equation and solve it first, then use the board to check.

The new move is putting together substitution (last lesson) and backtracking (Equations unit). The common error is repeating the operation instead of undoing it (Mia's mistake: 2w = 8, so w = 16). The check catches it every time, so make checking non-negotiable.

One practise question needs only substitution, on purpose: students must decide which job each question asks for.

Running the lesson

Predict: ask students who chose 14 or 8 to check their width in P = 2l + 2w.
Run: ask why P goes up 2 for each 1 cm of width.
Investigate: model the worked example live. Ask after each line: what did we undo?
Modify: for the battery, ask what the minus means before anyone calculates.
Make: ask how they found it: guessing with the slider, or writing 6w = 24?

When students get stuck

A student repeats the operation instead of undoing it (Mia's mistake). Ask: Does your answer check? Put it back into the formula. Fix: Undo × with ÷, and + with −. Check every answer by substituting back.
A student puts the known answer in for the wrong letter (45 as h). Ask: Which letter is the cost? Which is the hours? Fix: Write each known value next to its letter before substituting.

Stretch and support

Stretch: A rectangle's length is 3 cm more than its width, and its perimeter is 30 cm. Find both sides.

Support: Give a template: formula; what I know; substitute; undo; check.

Where it leads

Next: the Expressions unit continues with writing expressions from words. Solving for an unknown is the Equations unit, and in Linear relations it becomes finding x for a given y on a graph.

What each step is for

how wide?: Committing to a width exposes subtracting the length once instead of twice, and forgetting to halve.
find the width: Moving w until P reaches 20 shows the answer; the working shows why only one width fits.
substitute, then solve: Substituting the known values turns a formula into an equation with one unknown; solving undoes the operations in reverse order; the check substitutes back.
other unknowns: One change at a time: a decimal answer, a different letter unknown, a formula that subtracts, and three letters with one unknown.
find the unknown: Substituting and solving in new formulas, mixed with one plain substitution so students must decide which job each question needs; the review is from the Equations unit.
a garden bed: Meeting two conditions at once (a fixed perimeter and a shape rule) reverses the lesson: the student chooses the unknowns.

Now you can

Substitute the values I know into a formula to make an equation.
Solve the equation by undoing each operation, last first.
Check my answer by substituting it back into the formula.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson4. Words into expressionsWords such as "more than", "less than", "twice" and "divided by" each mean an operation. Write the operations in the order the words say them, then check with a number.

Source: content/maths/algebra/finding-unknowns/lesson.md