Solving by backtracking

Classroom resources

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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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise 8 min · Make 5 min

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

1. Predict: take a few answers and ask how each student got there. Someone will describe undoing; name it 'backtracking'.
2. Run: the machine only accepts the step that undoes the last arrow. Wrong steps get a nudge, not the answer.
3. 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.
4. Mia's mistake is the most common error. Ask: what did the machine do last? That is the first thing to undo.
5. Modify: the brackets task is the point. Ask students to say the machine's order aloud before undoing anything.
6. Practise: insist on the forwards check every time. It catches the order mistake on its own.

When students get stuck

Answers to the checks

1. what number went in?: I think of a number. I multiply it by 2, then add 3, and I get 11. What was my number?
   4. Take away the 3 first (11 − 3 = 8), then halve (8 ÷ 2 = 4).
2. run the machine backwards: Undo every step to find $x$.
   
3. Opposites: Match each step with the step that undoes it.
   + 7 → C. − 7; × 3 → D. ÷ 3; − 4 → A. + 4; ÷ 2 → B. × 2
   Adding and subtracting undo each other. Multiplying and dividing undo each other.
4. Backtrack it: 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.
   + 4, 25, ÷ 5, 5
   21 + 4 = 25 undoes the − 4. Then 25 ÷ 5 = 5 undoes the × 5. Check: 5 × 5 − 4 = 21.
5. Find the mistake: Mia backtracks $2x + 3 = 11$, but her check fails. Click the first line that goes wrong.
   line 2
   Line 2: Mia undid the × 2 first. The machine added 3 last, so that is undone first: 11 − 3 = 8, then 8 ÷ 2 = 4. Check: 2 × 4 + 3 = 11.
6. One step: Solve $x + 9 = 15$.
   
7. Taking away: Solve $2x - 5 = 9$.
   
8. Dividing: Solve $\frac{x}{3} + 2 = 6$.
   
9. Adding first: Solve $3(x + 4) = 27$.
   
10. Two steps: Solve $4x + 5 = 33$.
    7
    33 − 5 = 28, then 28 ÷ 4 = 7. Check: 4 × 7 + 5 = 33.
11. With a fraction: Solve $\frac{x}{2} - 3 = 4$.
    14
    4 + 3 = 7, then 7 × 2 = 14. Check: 14 ÷ 2 − 3 = 4.
12. Taking away: Solve $6x - 8 = 34$.
    7
    34 + 8 = 42, then 42 ÷ 6 = 7. Check: 6 × 7 − 8 = 34.
13. your own machine: Design a two-step machine that turns 6 into a new number. Write it as an equation, such as $3x + 2 = 20$.
    any equation that is true when $x = 6$
    Your equation is true when x = 6, so backtracking it leads back to 6.

Solutions

One possible solution for each task. Other correct programs are fine.

new machines › One step

Open task: any program that does what the step asks.

new machines › Taking away

Open task: any program that does what the step asks.

new machines › Dividing

Open task: any program that does what the step asks.

new machines › Adding first

Open task: any program that does what the step asks.

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.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Solving by backtracking".

The idea: Solving by backtracking: I can Students solve one-step and two-step linear equations by backtracking (undoing operations in reverse order).

What understanding looks like:
- Correctly solves a two-step linear equation using backtracking.
- Identifies the inverse operations used.
- Applies backtracking to multi-operation equations in the correct order.

Mistakes students often make here:
- A student divides first in 2x + 3 = 11 and gets 2.5.
- A student undoes − 5 by taking away 5 again.
- A student stops after one step on a two-step equation.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Equation as a balance statement), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Solving by backtracking".

Exit ticket

Solve 5x + 6 = 41 by backtracking. Show each step and check your answer.

41 − 6 = 35, then 35 ÷ 5 = 7. Check: 5 × 7 + 6 = 41.

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.

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.