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# Adding and subtracting integers

Number Foundations · Integers

<!-- Speaker note: Students add and subtract integers as moves on a number line, find that adding a negative moves left and that subtracting moves the opposite way to adding, so 5 − (−3) = 8 (AC9M7N07, node 2). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** start at the first number and move along the line. Adding a negative moves left; subtracting moves the opposite way to adding, so subtracting a negative moves right.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A temperature that rises 9 degrees from −4 °C, a lift that goes down 5 floors from level 2, a bank balance below zero: all of them are moves along a number line.

<!-- Speaker note: In the last lesson, further left on the number line meant less. Now the line is for working out: **start at the first number, then move**. Adding a positive number moves right. In this lesson you find out what adding a negative does, and what subtracting does, including subtracting a negative. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**What is $5 - (-3)$?**

- $-8$
- $2$
- $8$
- $-2$

<!-- Speaker note: What happens: 5 − (−3) = 8. Adding −3 would move left 3; subtracting goes the opposite way, so it moves right 3, from 5 to 8. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Most will say 2 (5 − 3). Do not correct it yet. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fadding-integers%2Fstep-2&code=widget%3A+numberline%0Arange%3A+-10%2C+10%0Asum%3A+a+-+b%0Aslider+a%3A+-10%2C+10%2C+5%0Aslider+b%3A+-10%2C+10%2C+3%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/adding-integers/?full=step-2#step-2)

<!-- Speaker note: The blue jump starts at $a$ and lands on $R$. When you subtract, the faint dashed arrow shows which way adding $b$ would go. Watch it as you make $b$ smaller, past zero. -->

<!-- Speaker note: Teacher: 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. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 5 − (−3) = 8. Adding −3 would move left 3; subtracting goes the opposite way, so it moves right 3, from 5 to 8.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Moves on the line

Here is how to use the line for each kind of sum.

$$-2 + 5$$

1) $-2 + 5 = 3$ — Adding a positive moves right: 5 to the right lands on 3.
2) $4 + (-6) = -2$ — Adding a negative moves left: from 4, left 6 lands on $-2$.
3) $5 - (-3)$ — Subtracting moves the opposite way to adding.
4) $\text{adding } {-3} \text{ is left 3, so subtracting it is right 3}$ — Find the way adding would go, then turn round.
5) $5 - (-3) = 8$ — From 5, right 3 lands on 8. Subtracting $-3$ is the same as adding 3.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write -2 + 5 and say "Start at the first number, -2." Line 1: write -2 + 5 = 3 and say "Adding a positive moves right: 5 to the right lands on 3." Line 2: write 4 + (-6) = -2 and say "Adding a negative moves left: from 4, left 6 lands on -2." Line 3: write 5 - (-3) and say "Subtracting moves the opposite way to adding." Line 4: write adding -3 is left 3, so subtracting it is right 3 and say "Find the way adding would go, then turn round." Line 5: write 5 - (-3) = 8 and say "From 5, right 3 lands on 8. Subtracting -3 is the same as adding 3." -->

<!-- Speaker note: Teacher: 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. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Moves on the line

**Work out $4 + (-6)$.**

* Answer: -2

<!-- Speaker note: Start at 4. Adding −6 moves left 6: 3, 2, 1, 0, −1, −2. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Mia works out $5 - (-3)$. Click the line where her working goes wrong.**

```
 1  Work out 5 − (−3) on a number line.
 2  Start at 5.
 3  The number is −3, so move 3 to the left.
 4  I land on 2, so 5 − (−3) = 2.
```

* Answer: line 3

<!-- Speaker note: Line 3: adding −3 would move left 3, but this is subtracting, so the move is the opposite way: right 3. 5 − (−3) = 8. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Right or left?

**From the first number, does each sum move right or left?**

Groups: Moves right · Moves left

2 + 3 · 2 + (−3) · 2 − 3 · 2 − (−3) · −5 + 4 · −2 − 4

* Answer: 2 + 3 → Moves right; 2 + (−3) → Moves left; 2 − 3 → Moves left; 2 − (−3) → Moves right; −5 + 4 → Moves right; −2 − 4 → Moves left

<!-- Speaker note:  -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Adding across zero:** Each task changes one thing. First: start below zero and add a positive.
- **Adding a negative:** Now the number you add is negative.
- **Subtracting past zero:** A positive subtracted from a smaller number.
- **Subtracting a negative:** The move from the Predict step.
- **A number minus itself:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/adding-integers/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: Start at −6 and move right 4. / Adding a negative moves left. / Subtracting 5 moves left 5, past zero. / Adding −6 would move left. Subtracting goes the other way. / From −4, right 6. / Where does subtracting a number from itself always land? For students who need more 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. For early finishers: Which is bigger, 3 − (−5) or 3 + 5? Explain with the line. Then work out −2 − (−2) − (−2). When students get stuck: A student says 5 − (−3) = 2. Ask: Which way would 5 + (−3) go? So which way does 5 − (−3) go? Fix: Draw the adding arrow (left 3), then the subtracting arrow the other way (right 3). Then walk the Run board: as b goes 3, 2, 1, 0, −1, −2, −3 the answer goes 2, 3, 4, 5, 6, 7, 8. / A student starts at zero instead of the first number. Ask: Where does the sum start? Fix: Put a finger on the first number before any move. / A student moves the right distance but lands on the wrong side of zero (−6 + 4 = 2). Ask: Count the steps out loud. Where is zero on the way? Fix: Count each step on the line: −5, −4, −3, −2. -->

<!-- Speaker note: Teacher: each task changes one thing. Ask students to say the direction before they set the board. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Adding

**Work out $-7 + 3$.**

* Answer: -4

<!-- Speaker note: From −7, right 3 lands on −4. -->

<!-- Speaker note: Teacher: the subtracted negative question plants Mia's mistake (−8). A student who gets −8 can draw the dashed arrow first. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Subtracting

**Work out $3 - 8$.**

* Answer: -5

<!-- Speaker note: From 3, left 8 lands on −5. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A negative subtracted

**Work out $-2 - (-6)$.**

* Answer: 4

<!-- Speaker note: Subtracting −6 moves right 6: from −2 to 4. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Temperature

**At 6 am it is $-4$ °C. By noon it is 9 degrees warmer. What is the temperature at noon?**

* Answer: 5

<!-- Speaker note: −4 + 9 = 5 °C. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review

**Review: which is greater, $-3$ or $-8$?**

- $-3$
- $-8$

* Answer: $-3$

<!-- Speaker note: −3 > −8: −3 is to the right of −8. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a subtraction that grows

Use the + and − buttons and both sliders. The answer must be 10, and you must subtract a negative number.

Stretch: find two different subtractions that give $-3$, one subtracting a positive and one subtracting a negative.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/adding-integers/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing numbers to make a given answer reverses the lesson: from the result back to the move, and it needs subtracting a negative. Hints, in order: Press − to subtract. / To land above where you start, subtracting has to move right. What kind of number does that? -->

<!-- Speaker note: Teacher: collect a few answers. Ask: what do all the b values have in common, and why? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Work out −3 + 7, 2 − 9 and 4 − (−5).**

* 4, −7 and 9.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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. -->