At a glance. One lesson of about 60 minutes. Why the sign reverses when multiplying or dividing by a negative, solving with a negative coefficient, and testing an answer with a value.
Calculator: no calculator. The numbers are small on purpose: the work is in noticing the negative divisor and reversing.
What students will learn
Know
- Multiplying or dividing by a negative number reverses the sign.
- Adding, subtracting, or using a positive multiplier keeps it.
- A test value from the answer must make the original true.
Understand
- Multiplying by a negative reflects the number line, which reverses the order.
- Testing a value can catch a missing reversal.
Do
- Solve inequalities with a negative coefficient.
- Test a solution with values on each side of the boundary.
- Use the same idea for context problems.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving and testing (questions 2 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to find for which operations and values the order reverses, and to state a rule.
- Criterion C. Communicating: writing the reversal as part of the step, and the exit ticket's justification with a test value.
- Criterion D. Applying mathematics in real-life contexts: a descending drone, written and solved as an inequality (question 8).
The one thing to do with the figure
Start with Add k and k = −4. Ask: did the order change? Then switch to Multiply by k with the same k.
Then ask for a value of k for which multiplying does not reverse the order. Any positive k. Ask about k = 0: both numbers become 0, so there is no order left.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 1. |
|---|---|
| 10 min | Hook, prediction, then the first figure. |
| 10 min | Key idea; the second figure and test values. |
| 10 min | Worked examples, including the route that never divides by a negative. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | No, then yes; −7; x = −3. |
|---|---|
| Worked examples | x > −8; x ≤ −3; x < 3. |
| Faded example | x ≥ −6. |
| 1. Reversing | B. A keeps the sign; C loses the minus signs; D adds instead of multiplying. |
| 2. Divide by a negative | −5. 20 ÷ (−4). |
| 3. Which direction | B. A forgets to reverse; C loses the negative number; D does both. |
| 4. Two steps | −4. (14 − 6) ÷ (−2). |
| 5. Largest whole number | 2. x < 3. |
| 6. Equivalent form | B. A keeps the sign; C loses the minus on the number; D drops the minus. |
| 7. Bracket and negative | −2. (10 − 6) ÷ (−2). |
| 8. Context | 7 seconds. t ≤ 7. |
| 9. Find the error | A. B misses the reversal; C and D lose the minus sign on the number. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Kept the sign, dropped the minus signs, or added. | Use the first figure with p = 8, q = 5. |
| 5 or −80 in question 2 | Lost the sign, or multiplied. | Substitute their answer into the original. |
| A, C or D in question 3 | Forgot to reverse, or lost the sign on the number. | Test x = 0 in the original and in their answer. |
| 4 or −10 in question 4 | Lost the sign, or added the 6. | Check x = 0 in 6 − 2x ≤ 14. |
| 3 or −4 in question 5 | Included the boundary, or divided by 3 not −3. | Substitute x = 3 into the original. |
| A, C or D in question 6 | Kept the sign, or lost the minus. | Test x = 0 and x = 8 in each option. |
| 2 or −8 in question 7 | Lost the sign, or added 6. | Expand the bracket on its own line. |
| 13 or 28 in question 8 | Added the starting height, or stopped at 4t. | At t = 13 the drone would be below the ground. |
Questions worth asking
- Why does the sign not reverse when you add −4? Both numbers move the same way; the gap and order stay.
- How can you check without redoing the whole problem? Substitute a value from the answer, and one outside it.
- What happens if you multiply both sides by 0? Both become 0: the inequality becomes an equality, and the order is lost.
Differentiation
A way in for anyone stuck
- Use a vertical number line so that “reflecting through zero” is a flip you can see.
- Give a rule card: add or subtract keeps; multiply or divide by a positive keeps; by a negative reverses.
- Let students solve with the add-x route first (no negative divisor), then compare.
Stretch
- Solve −3(x − 2) < 12. (x > −2.)
- Show that −x > 3 and x < −3 have the same solutions by dividing by −1, then test three values as a check. (x = −5, −3, 0.)
Students learning in English as an additional language
Reverse, negative, test and boundary are the key words. Pair reverse with a gesture: turn a hand over.
| Word | Plain meaning | Example |
|---|---|---|
| reverse | turn the other way round | < becomes > |
| negative number | a number less than zero | −4 |
| coefficient | the number multiplying a letter | −3 in −3x |
| test value | a number you try to see if it works | x = 0 |
| boundary | the number where the range starts | −7 in x > −7 |
Sentence starters for explaining aloud or in writing:
- I divided by …, which is negative, so I reversed the sign.
- I tested x = … and got …, which is true / false.
- The sign does not reverse here because …
Two things not to say
“Flip the sign when you move something across.” It is not moving, it is dividing by a negative.
“Whenever you see a negative, flip it.” Only a negative multiplier or divisor reverses it: x − 3 > −5 keeps its sign.
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 > 24.
x < −4. - ReverseStarting from 7 > 1, multiply both sides by −2.
−14 < −2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tank holds 90 litres and loses 6 litres a minute, so it holds 90 − 6m after m minutes. For how many whole minutes does it hold at least 30 litres?
10 minutes. −6m ≥ −60 gives m ≤ 10. - Earlier work, equationsSolve 7x + 3 = 3x + 19.
x = 4.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningDoes −2x > 4 have the same solutions as x < −2? Test x = −3, −2 and 0.
Yes: divide both sides by −2 and reverse the sign. As a check, at −3 both are true; at −2 and 0 both are false. - Earlier work, integersWork out (−5)² − (−5).
30: 25 + 5.
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? Write two numbers on cards on a number line on the floor, and flip them through zero.