At a glance. One lesson of about 60 minutes. Equations with x on both sides, with a bracket, a subtraction and a negative solution, by balancing and checking.
Calculator: no calculator. Nothing here needs one. The numbers are small, and the structure is the point.
What students will learn
Know
- Take the same x term from both sides.
- Collect the x terms on one side and the numbers on the other.
- Check by substituting into both sides.
Understand
- The solution is where two quantities are equal.
- An equation can have no solution.
Do
- Solve equations with x on both sides.
- Include a bracket, subtraction and a negative solution.
- Form and solve an equation from two rates.
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 6 and 8).
- Criterion B. Investigating patterns: the first figure. Ask students to predict where two plans meet from their fees and rates, and test it.
- Criterion C. Communicating: writing each balancing step, and the exit ticket's explanation of the error.
- Criterion D. Applying mathematics in real-life contexts: comparing two rental or data plans (question 7).
The one thing to do with the figure
Before moving x, ask: which plan is cheaper for very little data? And for lots? Let students say where they think the plans meet, then move x to test.
Then connect: the plans meet where 3x + 12 = 7x + 4. Ask which step in the second figure finds that x without trying values.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 2. |
|---|---|
| 10 min | Hook, prediction and the first figure. |
| 10 min | Key idea; the second figure, step by step. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 5x + 3; 4; 5. |
|---|---|
| Worked examples | x = 4; x = 7; x = −3. |
| Faded example | x = 6. |
| 1. Removing x from both sides | A, 4x + 5 = 21. B changes the left only; C adds; D changes the right only. |
| 2. Collect x | 5. 20 ÷ 4. |
| 3. Both sides, two steps | 6. (26 − 8) ÷ 3. |
| 4. Subtraction on both sides | 7. (28 + 7) ÷ 5. |
| 5. Bracket and unknown on both sides | 4. (20 − 12) ÷ 2. |
| 6. Which equation | C. Substituting x = 3 into each side is the quickest check; A, B and D have solutions 2, 4 and 3.5. |
| 7. Two shops | 6 hours. 60 ÷ 10. |
| 8. Brackets on both sides | 8. 6x − 2 = 4x + 14. |
| 9. Find the error | A. B misses the failed check; C subtracts where it should add; D divides one term only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Changed one side, or added instead of subtracted. | Draw both bars and remove the same piece from each. |
| 2 or 17 in question 2 | Added the x terms, or subtracted the plain number. | Ask which side the 3x is on and where it must go. |
| 18 or −6 in question 3 | Stopped at 3x, or subtracted the numbers the wrong way. | Substitute their answer into both sides. |
| 35 or 4.2 in question 4 | Stopped at 5x, or subtracted the 7. | Say it aloud: the −7 is undone by adding 7. |
| 8 or 16 in question 5 | Expanded wrongly, or added the 12. | Expand on its own line first and check by substituting. |
| A, B or D in question 6 | Did not substitute. | Put x = 3 into each side of each option. |
| 60, 10 or 1.5 in question 7 | Stopped at 10h, or added fees or rates. | Ask what the plans cost at their answer; they should be equal. |
| 7.5 or 14 in question 8 | Mishandled a sign in the expansion. | Expand each bracket on its own line, one at a time. |
Questions worth asking
- What does the solution of 3x + 12 = 7x + 4 mean for the two plans? The amount of data at which they cost the same.
- Why subtract the smaller x term? So the x term that remains is positive, and dividing by a positive number keeps the working free of signs.
- Could we have taken 7x from both sides? Yes: −4x + 12 = 4 gives the same x, with a negative coefficient.
Differentiation
A way in for anyone stuck
- Use two strips of paper for the two sides and remove the same pieces from both.
- Give a table: the equation, what was taken from both sides, the new equation, a check.
- Start with equations with a single x on each side and no negatives.
Stretch
- Solve 5(x − 2) = 3(x + 4). (x = 11.)
- Make up an equation with x on both sides whose solution is −2, and check it. (For example 3x + 8 = x + 4.)
Students learning in English as an additional language
Both sides, collect, equal and plan are the key words. The phrase take the same from both sides needs repeating and a gesture.
| Word | Plain meaning | Example |
|---|---|---|
| both sides | the left and the right of the equals sign | take 3x from both sides |
| collect | bring terms of the same kind together | collect the x terms |
| equal | the same in value | 18 = 18 |
| solution | the value that makes both sides equal | x = 2 |
| fee | a fixed charge that does not change | 36 taka a month |
Sentence starters for explaining aloud or in writing:
- I took … from both sides, so …
- The two sides are equal when x is …
- I checked by putting x = … into the left side, …, and the right side, …
Two things not to say
“Move the x's to the left.” It hides why. Say: take the same number of x's from both sides.
“Always take the x from the left.” Take the smaller x term, so the remaining coefficient is positive.
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.
- SolveSolve 6x + 1 = 2x + 17.
x = 4. - SolveSolve 9x = 5x + 28.
x = 7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTaxi A in Abuja charges 500 naira plus 100 naira per km. Taxi B charges 200 naira plus 160 naira per km. After how many km do they cost the same?
5 km. 500 − 200 = 300 and 160 − 100 = 60. - Earlier work, bracketsSolve 3(x − 2) = 18.
x = 8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve 4x + 6 = 4x + 9, and say what it means.
6 = 9 is never true, so there is no solution. - Earlier work, integersSolve −2x = 14.
x = −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 and by touch. Under each, “Show me the steps” gives the same states in words.
No device? Two strips of squared paper, with the x boxes and number boxes cut from both sides.