At a glance. One lesson of about 60 minutes. Students solve one-step and two-step equations by doing the same to both sides, use bar models, and check every answer by substitution.
Calculator: no calculator. Nothing here needs one. The numbers are small, and the method is the point.
What students will learn
Know
- An equation says two expressions are equal.
- Do the same to both sides, using opposite operations.
- A solution can always be checked by substituting.
Understand
- Each step keeps the two sides balanced, so the solution of the equation does not change.
- The steps are undone in the reverse order to how they were built.
- A solution is a value that makes both sides equal, not a procedure result.
Do
- Solve one-step equations of the form x + b = c, ax = c and x ÷ a = c.
- Solve two-step equations of the form ax + b = c.
- Check a solution by substitution and form an equation from a context.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving and checking equations (questions 2 to 6).
- Criterion B. Investigating patterns: the second figure. Ask students to find, for several a, b and c, when there is a whole-number solution, and to predict it from c − b and a.
- Criterion C. Communicating: writing each step with the operation done to both sides, and explaining why a wrong answer fails its check (question 9).
- Criterion D. Applying mathematics in real-life contexts: fares and thinking-of-a-number puzzles written as equations (questions 7 and 8).
The one thing to do with the figure
Pause on each step. Before pressing Next step, ask: “What are you going to do to both sides, and why does the balance stay?” Let students say it first, then see the bar change.
On the second figure, ask for a prediction of the solution before the guess is made, then use too small and too big to narrow it down. Finish by asking how the balancing method would have found it at once.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure on testing guesses. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose A, go back to what the opposite of adding is. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 6; 8x + 2; 30. |
|---|---|
| Worked example | x = 8. |
| Faded example | x = 5. |
| 1. The first step | B, subtract 9. A adds where the opposite is needed; C and D use the wrong operation. |
| 2. One-step, add | 3. 10 − 7. |
| 3. One-step, multiply | 4. 20 ÷ 5. |
| 4. One-step, divide | 40. 4 × 10. |
| 5. Two-step | 3. (28 − 13) ÷ 5. |
| 6. Two-step with subtraction | C, 6. A subtracts the 2; B stops one step early; D multiplies where it should divide. |
| 7. Equation from a context | 11. (205 − 40) ÷ 15. |
| 8. Thinking of a number | 17. (49 − 15) ÷ 2. |
| 9. Checking an answer | B. A misreads the check; C adds where the opposite is subtract; D divides one side only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the same operation, or the wrong operation, instead of the opposite. | Ask what was done to x. Then ask what would undo that. |
| 17 or 70 in question 2 | Added or multiplied the numbers on the right instead of subtracting. | Write m + 7 = 10 as a bar of m and 7, and see what is left. |
| 100 or 15 in question 3 | Multiplied, or subtracted, where the equation multiplies. | Say 5k as five lots of k, and ask how to find one lot. |
| 0.4 or 14 in question 4 | Divided where the equation divides, or added. | Write p ÷ 10 = 4 as 'a number shared into 10 parts gives 4'; what was the number? |
| 8.2 or 15 in question 5 | Added 13 to both sides, or stopped after one step. | Write each step on its own line with the operation done to both sides. |
| A, B or D in question 6 | Subtracted the 2, stopped early, or multiplied. | Substitute each option into 3x − 2 and compare with 16. |
| 165 or 245 in question 7 | Gave the cost of the distance, or added the fixed charge. | Ask what 165 stands for in the story, and what is being asked. |
| 34 or 32 in question 8 | Stopped at the doubled number, or added 15 instead of subtracting it, then halved: (49 + 15) ÷ 2. | Write the story as a machine and run it backwards. |
Questions worth asking
- Why do we do the same thing to both sides? To keep them equal. Listen for “balanced” or “equal”, and for the bar picture.
- Why do the steps come off in reverse order? They were put on in an order, and the last put on is the first taken off. Link to function machines.
- How can you be sure you are right? Substitute. If both sides are equal, it is a solution, whatever method you used.
- Can an equation have no whole-number solution? Yes: 3x + 4 = 20. Then x is 5.33…, so no whole number works. Students can see it in the second figure.
Differentiation
A way in for anyone stuck
- Use real balances, or draw two bars with a unit square grid, and write the equation under the diagram.
- Give a four-column table: the equation, what was done, the new equation, a check.
- Always start with one-step equations written as a story: a number plus 7 is 10.
Stretch
- Solve 4x + 3 = 5 and check your answer. (x = 0.5: 5 − 3 = 2, then 2 ÷ 4. Check: 4 × 0.5 + 3 = 5.)
- Solve 3(x + 2) = 27 by two different routes: dividing first, or expanding first. Which is shorter? (x = 7; divide first, 27 ÷ 3 = 9, then 9 − 2.)
Students learning in English as an additional language
Words such as equation, solve, solution, balance and check are new in this sense. Pair each with a gesture or a diagram. The phrase both sides is the key idea and needs repeating.
| Word | Plain meaning | Example |
|---|---|---|
| equation | a statement that two expressions are equal | 3x + 4 = 19 |
| solve | find the value of the letter that makes the equation true | solve 7x = 21 |
| solution | the value that makes both sides equal | x = 3 |
| inverse | the step that undoes another | divide undoes multiply |
| balance | keep both sides equal | take 4 from both sides |
| check | put the answer back in the equation | 3 × 5 + 4 = 19 |
Sentence starters for explaining aloud or in writing:
- To solve this I … both sides, because …
- The solution is x = …. I checked: … × … + … = …
- I undo … first because it was done last.
Two things not to say
“Move it across and change the sign.” It works, but it hides the reason. Say what is done to both sides.
“Just try numbers until it fits.” Guessing is a way to start; the balancing method always finds the answer.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- One-stepSolve x + 12 = 30.
x = 18. - One-stepSolve 9y = 54.
y = 6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA phone plan in Nairobi costs 250 shillings a month plus 8 shillings for each minute. Anna paid 450 shillings. How many minutes did she use?
25 minutes. 450 − 250 = 200, and 200 ÷ 8 = 25. - Earlier work, position rulesFind the 12th term of the sequence with position rule 5n + 2.
62. 5 × 12 = 60, then 60 + 2.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWrite an equation of the form ax + b = c using the steps × 4 and then − 2 whose solution is 9.
4x − 2 = 34, since 4 × 9 − 2 = 34. - Earlier work, function machinesA machine multiplies by 7 and then adds 3. The output is 66. What was the input?
9. 66 − 3 = 63, then 63 ÷ 7.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use two strips of squared paper for the bars and cut the box of b off both strips.