Solving by backtracking

Undo what was done to x, last step first, to find the number you started with.

Retracing your steps to find a lost key is backtracking: you undo your route, most recent first. Equations work the same way.

An equation like $2x + 3 = 11$ is a machine: $x$ goes in, gets multiplied by 2, then 3 is added, and 11 comes out. Backtracking runs the machine backwards to find $x$ .

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.

What number went in?

How did you work it out? Which step did you undo first?

I think of a number. I multiply it by 2, then add 3, and I get 11. What was my number?

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

Run the machine backwards

Start from the output, 11. Choose an operation and a number that undoes the last step, then keep going back towards $x$ .

Try it yourselfYour edits are saved in this browser.

Undo every step to find $x$ .

Use the board. This ticks when you get there.

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

Undo in reverse

To undo a step, do the opposite.

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

$3x + 7 = 19$ The machine multiplies $x$ by 3, then adds 7, and gives 19.
$19 - 7 = 12$ Undo the last step first: $+\,7$ is undone by $-\,7$ .
$12 \div 3 = 4$ Then undo $\times\,3$ with $\div\,3$ .
$x = 4$ That is the number that went in.
$3(4) + 7 = 19 \checkmark$ Check by running the machine forwards.

Undo the machine one step at a time. Which step do you undo first? Type ÷ as / if it is easier.

Mia's answer fails the check. Which step did she undo first?

Match each step with the step that undoes it.

+ 7

× 3

− 4

÷ 2

Backtrack $5x - 4 = 21$. The machine multiplies by 5, then takes away 4. Write each undo step (such as + 2 or ÷ 3) and the value it gives.

step	undo	value
start	–	21
1
2

Mia backtracks $2x + 3 = 11$ , but her check fails. Click the first line that goes wrong.

ModifyChange the problem on purpose and see what happens.

New machines

Each task loads a new machine. Undo every step to find $x$ .

The machine takes away last this time.

Need a hint?

The machine took 5 away last. What undoes taking away?

This machine divides first.

Need a hint?

The machine divided by 3 first, so that is the last thing to undo.

This machine adds 4 before it multiplies by 3. As an equation that is $3(x + 4) = 27$ : the brackets show the adding happens first.

Need a hint?

This machine adds first and multiplies last.

Another hint

Undo the × 3 first.

Try it yourselfYour edits are saved in this browser.

Solve $x + 9 = 15$ .

Use the board. This ticks when you get there.

Solve $2x - 5 = 9$ .

Use the board. This ticks when you get there.

Solve $\frac{x}{3} + 2 = 6$ .

Use the board. This ticks when you get there.

Solve $3(x + 4) = 27$ .

Use the board. This ticks when you get there.

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

Backtrack these

Solve $4x + 5 = 33$ .

Solve $\frac{x}{2} - 3 = 4$ .

Solve $6x - 8 = 34$ .

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

Your own machine

Make a machine a friend could backtrack. Run 6 through your two steps, then write the equation.

Stretch: make one where the machine adds first, so your equation needs brackets.

Need a hint?

Start with 6 and choose two steps, such as × 3 then + 2.

Another hint

Run 6 through them: 6 × 3 = 18, 18 + 2 = 20. So 3x + 2 = 20.

Design a two-step machine that turns 6 into a new number. Write it as an equation, such as $3x + 2 = 20$ .

For teachers

Students solve one- and two-step equations by backtracking, undoing each step of a function machine in reverse order and checking forwards (AC9M7A03, node 2). About 45 minutes.

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

Students will be able to

I can solve a two-step equation by backtracking.
I can name the step that undoes another (the inverse operation).
I can undo steps in the right order, last step first, and check my answer forwards.

Before class

This is a hinge lesson: the next lesson's balance method needs the same idea of order. Students who undo steps in the wrong order here will set up every later equation wrongly, so watch for it.

Try the last Modify machine yourself ( $3(x + 4) = 27$ ). It adds first and multiplies last, so the first undo is $\div\,3$ . That surprises students who always "take away the number first".

Running the lesson

Predict: take a few answers and ask how each student got there. Someone will describe undoing; name it 'backtracking'.
Run: the machine only accepts the step that undoes the last arrow. Wrong steps get a nudge, not the answer.
Investigate: before the worked example, ask the class for the opposite of each step (the matching task). Then reveal the example one line at a time.
Mia's mistake is the most common error. Ask: what did the machine do last? That is the first thing to undo.
Modify: the brackets task is the point. Ask students to say the machine's order aloud before undoing anything.
Practise: insist on the forwards check every time. It catches the order mistake on its own.

When students get stuck

A student divides first in 2x + 3 = 11 and gets 2.5. Ask: What did the machine do last? What undoes that? Fix: Undo the + 3 first (11 − 3 = 8), then the × 2 (8 ÷ 2 = 4). Check forwards.
A student undoes − 5 by taking away 5 again. Ask: If I take 5 away and want it back, what do I do? Fix: The opposite of taking away is adding: 9 + 5 = 14.
A student stops after one step on a two-step equation. Ask: Is that the number that went in, or a number partway through the machine? Fix: Draw the machine with a box between the steps; keep undoing until you reach x.

Stretch and support

Stretch: Make a three-step machine, such as × 2, + 5, × 3, and write its equation with brackets. Swap with a partner and backtrack each other's.

Support: Write the machine as arrows on paper (x → × 2 → + 3 → 11), then draw the backwards arrows underneath with the opposite steps before calculating.

Where it leads

Next: the balance method, which does the same undoing to both sides of an equation, and still works when $x$ is on both sides (where backtracking cannot).

Where it leads in class: "think of a number" puzzles, and working back from a total, such as the price before a fee was added.

What each step is for

what number went in?: Students can already reverse a two-step calculation in their heads; naming that as undoing is the bridge to backtracking.
run the machine backwards: Undoing has an order: the last thing the machine did is the first thing to undo.
undo in reverse: Each operation has an opposite (inverse), and the steps are undone in reverse order.
new machines: The order of the machine decides the order of undoing, including when the machine adds before it multiplies (brackets).
backtrack these: Fluent backtracking needs the order and the inverse right every time, checked forwards.
your own machine: Building a machine from its input shows that backtracking and running forwards are the same steps in reverse.

Now you can

Solve a two-step equation by backtracking.
Name the step that undoes another (the inverse operation).
Undo steps in the right order, last step first, and check my answer forwards.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson9. The balance methodDo the same to both sides to get x on its own, then check your answer in the first equation.

Source: content/maths/algebra/backtracking/lesson.md