At a glance. One lesson of about 60 minutes. Linear equations with brackets and negative numbers, solved by doing the same to both sides, and checked by substitution.
Calculator: no calculator. Nothing here needs one. The numbers are small, and the structure of each step is the point.
What students will learn
Know
- Do the same to both sides.
- Undo operations in reverse order.
- A solution is checked by substituting it.
Understand
- A bracket equation has two routes that give the same solution.
- Negative coefficients are divided by, not ignored.
Do
- Solve one-step and two-step equations with negatives.
- Solve equations with a bracket by dividing or expanding first.
- Check any solution by substitution.
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 (questions 2 to 8).
- Criterion B. Investigating patterns: with the figure, and then on paper with 2(x + 7) = 25, ask students to find when dividing first is shorter than expanding first, and to state a rule.
- Criterion C. Communicating: writing each step with what was done to both sides, and the exit ticket's explanation of the error.
- Criterion D. Applying mathematics in real-life contexts: shared costs written as a bracket equation (question 7).
The one thing to do with the figure
Run both routes on the first equation, then ask which was shorter. They will say dividing first. Move to equation 3, 2(x + 7) = 26, and ask the same.
Then ask: “What if the right side were 25?” Dividing first gives 12.5, a decimal; expanding first gives 2x = 11. Both are correct, so the choice is about convenience.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 3. |
|---|---|
| 10 min | Hook and the prediction, then the first figure on both routes. |
| 10 min | Key idea and the two slips; worked examples. |
| 10 min | The faded example in pairs; checking by substitution. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 5; 6x + 12; −6. |
|---|---|
| Worked examples | x = 9; x = 4; x = −5. |
| Faded example | x = −3. |
| 1. First step | A, x + 2 = 8. B expands wrongly; C divides one side only; D subtracts a number that is inside the bracket. |
| 2. Negative number, one step | 4. −3 + 7. |
| 3. Negative coefficient | −7. 28 ÷ (−4). |
| 4. Two steps | 7. (13 + 8) ÷ 3. |
| 5. Bracket | 8. 52 ÷ 4 − 5. |
| 6. Which value works | C, 3. A stops at 3x; B and D mishandle the −4. |
| 7. Equation from a context | 50 rupees. 600 − 2750 ÷ 5. |
| 8. Negative solution | −6. (17 − 5) ÷ (−2). |
| 9. Find the error | A. B misses that the answer fails its check; C and D mishandle the 4 inside the bracket. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Expanded wrongly, divided one side only, or subtracted inside the bracket. | Ask: what did you do to the right-hand side? |
| −10 or 10 in question 2 | Subtracted instead of added, or lost the sign. | Draw −3 on a number line and move 7 to the right. |
| 7, 24 or −112 in question 3 | Ignored the sign of the coefficient, subtracted, or multiplied. | Substitute their answer into −4x. |
| 21 or 5 in question 4 | Stopped at 3x, or subtracted 8. | Put the answer in the equation and see the left side. |
| 13 or 11.75 in question 5 | Did not take away the 5, or expanded wrongly. | Check: 4 × (13 + 5) is 72, not 52. |
| A, B or D in question 6 | Stopped early or mishandled the −4. | Substitute each option. |
| 550 or 2150 in question 7 | Gave the price each friend pays, or subtracted 600 once. | Ask what the 550 stands for in the story. |
| 6 or 11 in question 8 | Lost the sign, or added the 5. | Check: 5 − 12 is −7. |
Questions worth asking
- Why does the 3 multiply the 4 in 3(x + 4)? Three equal groups each have a 4.
- When is dividing first better? When the number outside divides the right side exactly.
- How could you have known x = 7.5 was wrong? Substitute it: 4 × 13.5 is 54, not 36.
Differentiation
A way in for anyone stuck
- Give a table: the equation, what was done, the new equation, a check.
- Use the bar picture for every bracket equation before writing steps.
- Start with one bracket equation per student using positive numbers, then add a negative.
Stretch
- Solve 4(3x − 2) = 6x + 10. (x = 3: 12x − 8 = 6x + 10, 6x = 18.)
- Solve −3(x − 4) = 21. (x = −3: divide by −3 to get x − 4 = −7.)
Students learning in English as an additional language
Equation, solve, bracket, inverse and check are the key words. The phrase both sides is the key idea and needs repeating.
| Word | Plain meaning | Example |
|---|---|---|
| equation | a statement that two expressions are equal | 3(x + 4) = 27 |
| solve | find the value that makes the equation true | x = 5 |
| both sides | the left and the right of the equals sign | divide both sides by 3 |
| inverse | the step that undoes another | subtract undoes add |
| check | put the answer back in the equation | 3 × (5 + 4) = 27 |
Sentence starters for explaining aloud or in writing:
- First I … both sides, because …
- I checked: … × (… + …) = …
- I chose to divide / expand first because …
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.
“Always expand first.” Both routes are valid; dividing first is shorter when the numbers allow it.
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 y + 9 = 4.
y = −5. - Two stepsSolve 6x − 5 = 19.
x = 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA taxi in Dubai charges 12 dirhams to start plus 3 dirhams for each km. A trip cost 45 dirhams. How many km?
11 km. 45 − 12 = 33, and 33 ÷ 3 = 11. - Earlier work, expandingExpand 4(2x − 5).
8x − 20.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve 2(3x − 1) = −14.
x = −2. Divide by 2: 3x − 1 = −7. - Earlier work, integersWork out −6 − (−9) + 2.
5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider and buttons) and by touch. “Show me the steps” gives the states in words.
No device? Two strips of squared paper: one cut into three equal groups with a gap for the notebook.