At a glance. One lesson of about 60 minutes. Adding and subtracting integers with zero pairs, a number line and a pattern; subtracting is adding the opposite.
Calculator: no calculator. The point is to see why the rules hold; a calculator would hide that.
What students will learn
Know
- A positive and a negative counter make a zero pair, worth 0.
- a − b = a + (opposite of b).
- a + (−b) = a − b.
Understand
- Adding zero pairs changes how a number is shown, not what it is worth.
- Three models (counters, number line, pattern) give the same rule, which is why it can be trusted.
Do
- Add and subtract any integers, with a model if needed.
- Rewrite a subtraction as an addition of the opposite.
- Find a change, such as a change in temperature, as new minus old.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: questions 2 to 8, from simple sums to a two-step context.
- Criterion B. Investigating patterns: the subtraction table. Ask a student to extend it two more rows and state the rule in general.
- Criterion C. Communicating: explaining 5 − (−2) = 7 with counters, in words and a diagram.
- Criterion D. Applying mathematics in real-life contexts: temperature changes and time zones, and judging whether the sign of an answer fits the situation.
The one thing to do with the figure
Set 5 − (−2) and stop on stage 2. Ask: “What are the counters worth now?” The readout says 5, though there are more counters. Ask why. That pause, a bigger picture with the same value, is the point of the lesson.
Then try 2 − 6 and −3 + 5 before any rule is written. Ask students to predict the number of zero pairs needed each time; the rule “subtracting is adding the opposite” should come from them.
A possible order
| 5 min | Warm-up; link question 1 (distance) to subtraction later in the lesson. |
|---|---|
| 15 min | Zero pairs, the figure: 5 − (−2), 2 − 6, −3 + 5, with predictions. |
| 5 min | The pattern table: a second argument for the same rule. |
| 10 min | Worked examples; the temperature change in pairs. |
| 5 min | Question 1, the check. If A is common, return to stage 2 of the figure. |
| 15 min | Your turn and exit ticket (question 9). |
The answers
| Warm-up | 8; 4; 4 °C. |
|---|---|
| Worked examples | 5; −5; 4; 5 °C. |
| Faded example | −12 °C, a fall of 12 degrees. |
| Pattern table | 3, 4, 5, 6, 7. |
| 1. Add the opposite | B. A: the double sign has been dropped, not interpreted. |
| 2. Adding | 3. |
| 3. Subtracting past zero | −7. |
| 4. Subtracting a negative | 3 = −6 + 9. |
| 5. Two negatives | −10. |
| 6. Time zones | 15 hours: 9 − (−6) = 15. |
| 7. Missing number | −13, because −8 − (−13) = −8 + 13 = 5. |
| 8. Two changes | −4 °C: 4 − 11 = −7, then −7 + 3. |
| 9. Subtracting a negative | C, 2. A: “minus means left” for both signs. D: both signs flipped. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in questions 1 or 9; −8 for −3 − (−5) | Reads every minus sign as “move left”. | Counters: ask how many negatives they can take away from 3 negatives when they need 5. |
| −13 or 13 in question 2 | Moved the wrong way, or added sizes. | Number line, finger on −5, then move. |
| −15 in question 4 | Subtracted 9 instead of −9. | Ask them to rewrite the question as an addition first. |
| −4 or 4 in question 5 | −4: took 3 from 7 and kept the sign. 4: moved right for a subtraction. | “Is the answer colder or warmer than −3, and by how much?” |
| 3 in question 6 | Treats +9 and −6 as on the same side. | Link to Lesson 2: zero is between them, so the distances add. |
| −7 in question 8 | Stopped at midnight. | Underline each change in the question before calculating. |
Questions worth asking
- Why can you add a zero pair whenever you like? Because it is worth 0, and adding 0 changes nothing.
- Is 3 − 8 the same as 8 − 3? No: −5 and 5. Same distance, opposite directions.
- When is a − b positive? When a is to the right of b on the number line. Ask for examples with negatives.
Differentiation
A way in for anyone stuck
- Physical two-colour counters, if the class has them, alongside the figure.
- A printed number line for every question, and the rule card: “subtract = add the opposite”.
- Let students write the addition version of each subtraction before answering.
Stretch
- Questions 7 and 8, then: find two integers whose difference is 9 and whose sum is −3 (3 and −6).
- Explain, with counters, why a − b and b − a are always opposites.
Students learning in English as an additional language
“Difference”, “rise” and “fall” carry the sign of the answer, so check them explicitly.
| Word | Plain meaning | Example |
|---|---|---|
| zero pair | a positive and a negative counter together, worth 0 | +1 and −1 |
| add | put together; on a number line, move by the number | −4 + 9 = 5 |
| subtract | take away; the same as adding the opposite | 3 − 8 = −5 |
| rise / fall | go up / go down | a fall of 12 degrees is −12 |
| change | new value minus old value | −9 − 3 = −12 |
Sentence starters for explaining aloud or in writing:
- I added … zero pairs so that I could take away …
- Subtracting … is the same as adding … because …
- The answer is negative because …
Two things not to say
“Two minuses make a plus.” It is a slogan with no reason, and students then apply it to −3 − 7. Say “subtracting a negative is adding its opposite”.
“You can't take 8 from 3.” You can, and this lesson is where students find out how.
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.
- AddingWork out −9 + 4 and −9 + (−4).
−5 and −13. - SubtractingWork out 6 − 10 and 6 − (−10).
−4 and 16.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- DivingA diver at −14 m swims up 9 m, then down 6 m. Where is the diver now?
−11 m. - Game scoreIn a game you have −20 points and lose 15 more. What is your score?
−35.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWithout calculating, which is greater: −17 − (−5) or −17 + (−5)? Explain.
−17 − (−5) = −12 is greater than −22: subtracting a negative moves right, adding one moves left. - From Lesson 2How far apart are −31 and 14? Write it as a subtraction.
45 = 14 − (−31).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure is a step-through, not an animation: every step waits for a button press, and “Show me the steps” gives three examples in words.
Two-colour counters or squares of paper reproduce the figure exactly on a desk.