Equations as a balance

Classroom resources

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Students will be able to

• I can check whether a value is a solution by putting it in place of the letter and working out both sides.
• I can tell an expression from an equation.
• I can explain that a solution makes both sides of an equation equal.

Timing

Predict and Run 10 min · Investigate 15 min · Modify 12 min · Make 8 min

Before class

Do the Run and Modify boards yourself first. The surprise is the last Modify task: $2x = 7$ never balances with a whole number, and that is the point. Be ready to ask "what would $x$ have to be?" and accept 3.5 without teaching a method yet: solving step by step is the next two lessons.

Running the lesson

1. Predict: students lock in a value before touching the board. Ask two students with different predictions to explain them.
2. Run: the board shows each side worked out for the current x. Point at the line under each pan: that is substitution, done for them.
3. Investigate: walk through the worked example together, one step at a time. Before each click, ask what comes next.
4. Say 2x aloud as 'two times x', never 'two x', for the whole lesson. It prevents the 24-for-8 slip in the second task.
5. Modify: the last task has no whole-number answer. Let students try every value before discussing.
6. Make: share a few student equations and have the class check them by substitution.

When students get stuck

Answers to the checks

1. which value balances?: Which value of $x$ makes $2x + 3 = 11$ true?
   x = 4. Two lots of 4 is 8, and 8 + 3 = 11, the same as the right side.
2. find the balance: Find the value of $x$ that balances the scale.
   
3. Check by substitution: Is $x = 5$ a solution of $3x - 2 = 13$? Start by working out the left side, $3x - 2$, when $x = 5$.
   13
   3 × 5 − 2 = 15 − 2 = 13. The right side is 13 too, so x = 5 is a solution.
4. Find the mistake: Sam decides that $x = 4$ is not a solution of $2x + 3 = 11$, but it balanced the scale. Click the line where his working goes wrong.
   line 2
   Line 2: Sam wrote 2 and 4 side by side to get 24. 2x means 2 × x, so it is 8, and 8 + 3 = 11. x = 4 is a solution after all.
5. Expression or equation?: Which one is an equation?
   $2x + 3 = 11$
   An equation has an = sign: it says two things are equal. The others are expressions: they have a value, but they do not say that anything is equal.
6. Four bags: Balance $4x = 20$.
   
7. $x$ on both sides: Balance $x + 7 = 3x + 1$.
   
8. A scale that will not balance: Can you balance $2x = 7$ with a whole number?
   No: $2x$ is always even when $x$ is whole, and 7 is odd
   x = 3 gives 6 and x = 4 gives 8, so 7 is skipped. The solution is x = 3.5, which is not a whole number.
9. Work it out: Work out $5x - 4$ when $x = 3$.
   11
   5 × 3 − 4 = 15 − 4 = 11.
10. Which value?: Which value of $x$ solves $3x + 2 = 20$? Try 5, 6 and 7.
    6
    3 × 6 + 2 = 18 + 2 = 20, the same as the right side.
11. Both sides: Is $x = 2$ a solution of $4x + 1 = 3x + 4$?
    No: the left side is 9 and the right side is 10
    4 × 2 + 1 = 9 and 3 × 2 + 4 = 10. The sides are different, so x = 2 is not a solution. (x = 3 is: both sides are 13.)
12. your own equation: Write your own equation whose solution is $x = 6$. Use $x$ at least once.
    any equation that is true when $x = 6$
    Your equation is true when x = 6: both sides have the same value.

Solutions

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

new scales › Four bags

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

new scales › $x$ on both sides

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

new scales › A scale that will not balance

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

Stretch and support

Stretch: Write an equation with x on both sides whose solution is 6, then one whose solution is not a whole number.

Support: Use the board for the Investigate questions too: set up 3x − 2 and 13 by hand on paper as bags and blocks, then count.

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 "Equations as a balance".

The idea: Equation as a balance statement: I can Students understand that an equation is a statement of equality and that the solution is the value that makes it true.

What understanding looks like:
- Correctly identifies whether a given value satisfies an equation by substitution.
- Distinguishes between an expression and an equation.
- Understands that a solution must make both sides equal.

Mistakes students often make here:
- A student says x = 4 gives 2x = 24.
- The student moves the slider at random and stops when it balances, without noticing why.
- A student writes an expression, such as 2x + 3, for the Make task.

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.
- 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 "Equations as a balance".

Exit ticket

Is x = 7 a solution of 3x − 5 = 16? Show how you know.

Yes. 3 × 7 − 5 = 21 − 5 = 16, which equals the right side.

What each step is for

• which value balances?: Committing to a value before trying it makes the balance a test of their thinking, not a guessing game.
• find the balance: The solution is the one value that balances the scale: every other value tips it one way or the other.
• check a solution: Substitution is how you check a solution: replace the letter with the value, work out each side, and see if they are equal.
• new scales: The idea holds for any equation, even with the letter on both sides, and some equations have no whole-number solution at all.
• check a value: Substitution is a habit: every value gets checked by working out both sides.
• your own equation: Building an equation from its solution shows that the solution is the value that makes both sides equal.

Where it leads

Next: solving by backtracking, then the balance method (do the same to both sides). Both lean on the habit built here: check every answer by substitution.

Where it leads in class: missing totals in recipes or budgets ("4 tickets and a $5 booking fee cost $41"), and formulas in science where one value is unknown.