Adding and subtracting integers

Classroom resources

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Students will be able to

• I can add and subtract integers by moving along a number line.
• I can explain why adding a negative number moves left.
• I can explain why 5 − (−3) is 8, not 2.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board and the Make task yourself first. Students need the last lesson: further left is less.

The one rule of the lesson: start at the first number; adding moves the way the second number points (right if positive, left if negative); subtracting moves the opposite way. The board draws the "adding" direction faintly above a subtraction, so students can see the turn-round.

The lesson does not teach "two minuses make a plus". If students bring it, ask them to explain it with the line: subtracting −3 turns round a move to the left.

Running the lesson

1. Predict: take a show of hands. Most will say 2 (5 − 3). Do not correct it yet.
2. Run: ask students to make b smaller one step at a time from 3 and say the answer before they look: 2, 3, 4, 5, then past zero, 6, 7, 8. Every step of b smaller adds one to the answer. Point at the dashed arrow when b goes negative.
3. Investigate: model the worked example live. Draw a number line, then for each line say 'start at…, which way, how far' before drawing the jump. For 5 − (−3), draw the dashed 'adding −3' arrow first, then turn round. Reveal the prepared lines afterwards to compare.
4. Modify: each task changes one thing. Ask students to say the direction before they set the board.
5. Practise: the subtracted negative question plants Mia's mistake (−8). A student who gets −8 can draw the dashed arrow first.
6. Make: collect a few answers. Ask: what do all the b values have in common, and why?

When students get stuck

Answers to the checks

1. subtracting a negative: What is $5 - (-3)$?
   5 − (−3) = 8. Adding −3 would move left 3; subtracting goes the opposite way, so it moves right 3, from 5 to 8.
2. subtract on the line: The board shows $5 - 3$. Change $b$ to $-3$. Was your prediction right?
   
3. Moves on the line: Work out $4 + (-6)$.
   -2
   Start at 4. Adding −6 moves left 6: 3, 2, 1, 0, −1, −2.
4. Find the mistake: Mia works out $5 - (-3)$. Click the line where her working goes wrong.
   line 3
   Line 3: adding −3 would move left 3, but this is subtracting, so the move is the opposite way: right 3. 5 − (−3) = 8.
5. Right or left?: From the first number, does each sum move right or left?
   2 + 3 → Moves right; 2 + (−3) → Moves left; 2 − 3 → Moves left; 2 − (−3) → Moves right; −5 + 4 → Moves right; −2 − 4 → Moves left
6. Adding across zero: Work out $-6 + 4$. Set the board to check.
   -2
   From −6, right 4: −5, −4, −3, −2.
7. Adding a negative: Work out $3 + (-7)$. Set the board to check.
   -4
   From 3, left 7 lands on −4.
8. Subtracting past zero: Work out $2 - 5$. Set the board to check.
   -3
   From 2, left 5: 1, 0, −1, −2, −3.
9. Subtracting a negative: Work out $-4 - (-6)$. Set the board to check.
   2
   Subtracting −6 moves right 6: from −4 to 2.
10. A number minus itself: Edge case: work out $-5 - (-5)$.
    0
    From −5, right 5 lands on 0: any number minus itself is 0.
11. Adding: Work out $-7 + 3$.
    -4
    From −7, right 3 lands on −4.
12. Subtracting: Work out $3 - 8$.
    -5
    From 3, left 8 lands on −5.
13. A negative subtracted: Work out $-2 - (-6)$.
    4
    Subtracting −6 moves right 6: from −2 to 4.
14. Temperature: At 6 am it is $-4$ °C. By noon it is 9 degrees warmer. What is the temperature at noon?
    5
    −4 + 9 = 5 °C.
15. Review: Review: which is greater, $-3$ or $-8$?
    $-3$
    −3 > −8: −3 is to the right of −8.
16. a subtraction that grows: Make a subtraction that gives 10, where the number you subtract is negative.
    
    Subtracting a negative moves right, so a − b = 10 with b negative, such as 4 − (−6) = 10 or 2 − (−8) = 10.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Adding across zero

Open task: any program that does what the step asks.

one change at a time › Adding a negative

Open task: any program that does what the step asks.

one change at a time › Subtracting past zero

Open task: any program that does what the step asks.

one change at a time › Subtracting a negative

Open task: any program that does what the step asks.

one change at a time › A number minus itself

Open task: any program that does what the step asks.

a subtraction that grows

Open task: any program that does what the step asks.

Stretch and support

Stretch: Which is bigger, 3 − (−5) or 3 + 5? Explain with the line. Then work out −2 − (−2) − (−2).

Support: Use the board for every question: set a and b, choose + or −, and say the move before reading R. A paper number line from −10 to 10 on every desk helps.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Adding and subtracting integers".

The idea: Addition and subtraction of integers: I can Students add and subtract integers using a number line and understand the effect of adding or subtracting a negative number.

What understanding looks like:
- Correctly adds and subtracts integers including cases with negative operands.
- Interprets adding a negative number as movement in the negative direction.
- Does not treat subtraction of a negative as subtraction (e.g. 5 − (−3) = 8, not 2).

Mistakes students often make here:
- A student says 5 − (−3) = 2.
- A student starts at zero instead of the first number.
- A student moves the right distance but lands on the wrong side of zero (−6 + 4 = 2).

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Comparing and ordering integers), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Adding and subtracting integers".

Exit ticket

Work out −3 + 7, 2 − 9 and 4 − (−5).

4, −7 and 9.

What each step is for

• subtracting a negative: Committing to an answer for 5 − (−3) before the board shows it makes the move of subtracting a negative a test of their idea.
• subtract on the line: Changing b from 3 to −3 while subtracting shows the jump turn round: subtracting moves the opposite way to adding b, which is drawn faintly above.
• which way to move: One rule covers all four cases: adding moves the way the number points (right for positive, left for negative), subtracting moves the opposite way. Mia's mistake treats the minus sign of the number as the operation.
• one change at a time: One change at a time: crossing zero when adding, adding a negative, subtracting past zero, subtracting a negative, then the edge case of subtracting a number from itself.
• mixed sums: Adding and subtracting with negatives in mixed forms and a context; the review question keeps ordering from the last lesson fresh.
• a subtraction that grows: Choosing numbers to make a given answer reverses the lesson: from the result back to the move, and it needs subtracting a negative.

Where it leads

Next in Number: negative fractions and decimals on the same line, then multiplying and dividing integers.

Where it leads: tables of values with negative inputs (Tables from rules in Linear relations has a quick check of this), gradients and intercepts below zero, and changes in temperature, height and money.