Worked examples for solving two-step equations can include a short explanation of each line rather than only a finished procedure. In this routine, you model one equation, pause for short explanations, compare solution paths and move directly to a near-match question.
You will be able to plan a complete two-step equations lesson around the routine. You will also have a ready-to-use algebra worked example, self-explanation prompts in maths, comparison tasks and adaptations for pupils who copy, stall or need language support.
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Choose one equation structure and one clear method
Begin with one equation family and use the same representation throughout the first set of examples. For pupils beginning two-step equations, a useful starting structure is ax + b = c, using positive whole numbers that produce whole-number solutions.
For example, use equations such as 3x + 5 = 20 and 4x + 7 = 31. These keep attention on the central idea: undo the addition first, then undo the multiplication.
Write every operation on both sides of the equation. This balance representation makes the preservation of equality visible:
- Start: 3x + 5 = 20
- Subtract 5 from both sides: 3x + 5 − 5 = 20 − 5
- Simplify: 3x = 15
- Divide both sides by 3: 3x ÷ 3 = 15 ÷ 3
- Simplify: x = 5
Avoid switching immediately between balance scales, flow diagrams, inverse-operation arrows and abbreviated symbolic steps. Introduce another representation later, when pupils can connect it to the method they already understand.
Keep the language precise. Say, “Subtract 5 from both sides,” rather than, “Move the 5 and change its sign.” The first wording names an operation that preserves equality; the second can sound like an unexplained symbol-moving rule.
Conceptual understanding and procedural skill both matter in algebra. A Mathematics Teacher article about worked examples and self-explanation describes combining the two approaches to help address algebra misconceptions.
Model one solution a line at a time
Reveal the solution sequentially rather than displaying every line at once. This lets pupils connect the operation you choose with the equation as it currently stands. Research reported in Applied Cognitive Psychology examined worked examples with different levels of detail and static, sequential or dynamic presentation.
Use this ready-to-teach script for 3x + 5 = 20:
- Display only the equation. Say, “Our goal is to leave x alone. What operation is applied to the term 3x last?”
- Reveal 3x + 5 − 5 = 20 − 5. Say, “I subtract 5 from both sides. The two sides were equal, so applying the same operation keeps them equal.”
- Reveal 3x = 15. Ask, “What has been undone, and what still needs to be undone?”
- Reveal 3x ÷ 3 = 15 ÷ 3. Say, “The variable is multiplied by 3, so I divide both sides by 3.”
- Reveal x = 5. Ask pupils to read the result as a sentence: “The value of x is 5.”
- Reveal the check. Substitute 5 into the original equation: 3(5) + 5 = 20, then 15 + 5 = 20.
Keep the unsimplified operation line visible before showing the simplified line. If you jump directly from 3x + 5 = 20 to 3x = 15, pupils can imitate the change without seeing why it is valid.
You can ask pupils to keep pencils down during the first reveal. Their task is to predict and explain each line, not race ahead or copy the completed example.
Pause pupils for short self-explanations
Pause after each line for one short explanation. The response can be spoken to a partner, written beside the equation or selected from two possible explanations. Do not wait until the complete solution to ask for reasoning.
Useful self-explanation prompts in maths include:
- “What are we trying to isolate?”
- “Which operation is attached to the variable term?”
- “Why do we undo the addition before the multiplication?”
- “Why must we subtract the same amount from both sides?”
- “What changed in this line?”
- “What stayed equal?”
- “Which inverse operation did we use?”
- “How could you check the final value?”
Ask for a complete but brief response. For example: “I subtract 5 from both sides because adding 5 is the outer operation, and doing the same thing to equal quantities keeps them equal.”
If pupils answer with “because it goes across,” follow with, “What operation does that movement represent?” If they say only “inverse,” ask them to name the operation and identify what it undoes.
An Instructional Science study of an algebra equation-solving unit incorporated self-explanation prompts with correct, incorrect and incomplete worked examples. This combination supports a lesson design in which pupils explain and inspect examples rather than treat them as notes to copy.
A simple participation routine is:
- You reveal one line.
- Every pupil prepares an explanation.
- Partners take turns to explain the choice and the equality.
- You hear one response and refine the mathematical wording.
- You reveal the next line.
Compare a correct example with an error or gap
Place a correct solution beside an error or incomplete example and ask pupils to identify the first important difference. Ask them to name the line they question, explain the operation used and repair it. This keeps attention on the steps in the solution, not only the final answer.
Use this comparison:
| Correct example | Example containing an error |
|---|---|
| 4x + 7 = 31 | 4x + 7 = 31 |
| 4x = 24 | 4x = 24 |
| x = 6 | x = 20 |
| Check: 4(6) + 7 = 31 | Check: 4(20) + 7 ≠ 31 |
Ask pupils:
- “Which line is the first one you disagree with?”
- “What operation appears to have been used?”
- “Why should 4x = 24 be followed by division rather than subtraction?”
- “Repair the line without changing the original equation.”
- “How does substitution expose the error?”
You can also remove a line instead of supplying an error:
- 5x − 3 = 22
- 5x = 25
- [missing line]
- x = 5
Ask pupils to supply 5x ÷ 5 = 25 ÷ 5 and explain why that line belongs in the gap.
Once the standard method is secure, compare two valid paths. For 2x + 6 = 18, one solver can subtract 6 and then divide by 2. Another can divide every term by 2 to obtain x + 3 = 9, then subtract 3. Emphasise that dividing only one term would not preserve equality.
Research on worked-example pairs and language support in Instructional Science treats comparison as an important mechanism for algebra learning. Keep the examples aligned so pupils can compare corresponding lines without searching across unrelated layouts.
Move from example to a near-match question
Follow the model with an equation that has the same structure but different values. This gives pupils a manageable change while requiring them to make each decision themselves.
After modelling 3x + 5 = 20, give the near match 3x + 8 = 26. Both equations require pupils to undo addition and then multiplication, but the new values prevent direct copying.
Use paired roles:
- Solver: writes one line at a time and explains the chosen operation.
- Checker: confirms that the same operation was applied to both sides and asks for clarification when needed.
- Both pupils: substitute the solution into the original equation.
- Switch roles: solve another near match, such as 4x + 3 = 27.
A complete response to the first near match is:
- 3x + 8 = 26
- 3x + 8 − 8 = 26 − 8
- 3x = 18
- 3x ÷ 3 = 18 ÷ 3
- x = 6
- Check: 3(6) + 8 = 26, so 18 + 8 = 26.
Inspect explanations as well as answers. A correct answer with no valid account of the operations may still reflect guessing, mental trial and error or copying.
When most pupils can solve the near match, vary one feature at a time. You might change the coefficient, use subtraction in the original equation or introduce a negative solution. Avoid changing several features at once if your immediate goal is to establish the routine.
Adapt the routine when pupils copy, stall or face language barriers
Adapt the support according to what pupils do, not simply whether their final answer is correct. Preserve the requirement to choose an operation, maintain equality and explain the result.
| What you notice | Likely issue | Adjustment |
|---|---|---|
| Pupils copy every line without looking at the equation. | The full solution is available before they need to think. | Cover later lines. Reveal one line only after pupils predict the operation and give a reason. |
| Pupils stall at the first step. | They may not identify the operations around the variable. | Ask them to mark the multiplication and addition, then identify which operation was applied last. |
| Pupils perform an operation on only one side. | They are treating equations as symbol-moving exercises. | Return to an unsimplified line, such as 3x + 5 − 5 = 20 − 5, and ask what must remain equal. |
| Pupils choose the right operation but make arithmetic errors. | The algebraic decision and calculation load are competing. | Keep the same equation structure but choose simpler values. Discuss the algebraic choice separately from the arithmetic correction. |
| Pupils can solve but cannot explain. | The prompt may be too broad. | Use a sentence frame: “I ___ both sides because ___ is being ___ to the variable term.” Gradually remove the frame. |
| Pupils need more language support. | Reading and producing an explanation may hide their mathematical understanding. | Pair symbols with short phrases such as “subtract from both sides”. Allow pointing, matching and rehearsal with a partner before a spoken or written response. |
| Pupils are overwhelmed by detailed examples. | Too many annotations are competing for attention. | Remove decorative arrows and repeated prose. Retain the equation line, operation on both sides and one focused prompt. |
| Pupils finish quickly and copy the pattern mechanically. | The near matches no longer require enough decision-making. | Provide an incomplete example, an error to repair or two valid methods to compare. |
For language support, keep the mathematical symbols and layout identical across the pair. Pre-teach a small set of terms such as both sides, subtract, divide, coefficient, inverse and substitute. Do not replace all explanation with vocabulary drills.
You can also give pupils a prompt card:
- “My goal is to isolate ___.”
- “First I ___ both sides.”
- “This preserves equality because ___.”
- “Next I ___ both sides.”
- “I checked by substituting ___.”
If the routine still fails, reduce the number of new features. Return to one equation structure, one representation and one type of comparison. Increase complexity only when pupils can explain why the current steps work.
The routine is simple: reveal one line, ask why it is valid, compare it with an error or alternative, and then give pupils a near-match equation. Finish by checking the solution through substitution.
Keep the written method consistent, but adjust how much information you reveal and which prompts you provide. The goal is not silent copying. It is for pupils to explain how each operation preserves equality and use that reasoning independently.
