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# Comparing integers

Number Foundations · Integers

<!-- Speaker note: Students place integers on a number line, find that further left is less (so −7 < −2), and compare and order integers with < and > (AC9M7N07, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** on a number line, the number further left is less. So −7 is less than −2, even though 7 is more than 2.

- I can put a set of integers, including negatives, in order from least to greatest.
- I can write a comparison between two integers with &lt; or &gt;.
- I can explain why −10 is less than −1, even though 10 is more than 1.

<!-- Speaker note: Before class: Do the Run board and the Make task yourself first. Students should be able to find negative numbers on a number line (Year 6) and know what a temperature below zero means. The one idea of the lesson is "further left is less". The common mistake is to compare negatives by their size without the sign (−8 > −2 because 8 > 2), and it is planted only after the method, in Ari's working. The lesson says "further from zero" for size without the sign, not "absolute value". The primer has the word if you want it. -->

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## Why it matters

> Thredbo at −9 °C, Canberra at −3 °C: which is colder? Below zero, the number that looks bigger is the smaller one.

<!-- Speaker note: Numbers below zero are **negative**: $-1, -2, -3$ and so on. Together with zero and the positive whole numbers they are the **integers**. On a number line, negatives sit to the left of zero, with $-1$ next to zero. In this lesson you find out how to tell which of two integers is less, and write it with $<$ ("is less than") or $>$ ("is greater than"). -->

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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

**Which is greater, $-7$ or $-2$?**

- $-2$
- You cannot compare two negative numbers
- $-7$, because 7 is more than 2
- Neither: they are both negative, so they are equal

<!-- Speaker note: What happens: −2 is greater. On the number line −2 is to the right of −7: further right is greater. −7 is further below zero, so it is less. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose −7 usually say 7 is bigger. Do not correct it yet: the board will. -->

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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%2Fcomparing-integers%2Fstep-2&code=widget%3A+numberline%0Arange%3A+-10%2C+10%0Aslider+a%3A+-10%2C+10%2C+0%0Aslider+b%3A+-10%2C+10%2C+0%0Apoint+A%3A+a%0Apoint+B%3A+b%0Acompare%3A+A%2C+B%0A&fresh=1}}

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

<!-- Speaker note: Move $A$ with slider $a$ and $B$ with slider $b$ (or the − and + buttons). The board writes the comparison under the line. Try a few pairs first: two positives, a positive and a negative, then $-7$ and $-2$. -->

<!-- Speaker note: Teacher: ask students to try two positives first, then a positive and a negative, then −7 and −2. Ask: what decides which is less every time? -->

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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?

* −2 is greater. On the number line −2 is to the right of −7: further right is greater. −7 is further below zero, so it is less.

<!-- 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>

## Ordering on the line

Here is how to put integers in order, using where each one sits on the number line.

$$-2, \; -8, \; 3, \; 0$$

1) $\text{place each on the line}$ — Negatives go left of zero: $-2$ is 2 left, $-8$ is 8 left.
2) $-8 \text{ is furthest left}$ — Furthest left is least, so $-8$ comes first.
3) $-8, \; -2, \; 0, \; 3$ — Read the line from left to right.
4) $-8 < -2$ — Between two negatives, the one further from zero is less.

<!-- 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, -8, 3, 0 and say "Put these in order from least to greatest." Line 1: write place each on the line and say "Negatives go left of zero: -2 is 2 left, -8 is 8 left." Line 2: write -8 is furthest left and say "Furthest left is least, so -8 comes first." Line 3: write -8, -2, 0, 3 and say "Read the line from left to right." Line 4: write -8 < -2 and say "Between two negatives, the one further from zero is less." -->

<!-- Speaker note: Teacher: model the worked example live. Draw a line on the board, place −2, −8, 3 and 0 one at a time, saying 'left of zero, 8 places' and so on, then read left to right. 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>

## Ordering on the line

**Which is true?**

- $-5 = -1$
- $-5 &lt; -1$
- $-5 &gt; -1$

* Answer: $-5 &lt; -1$

<!-- Speaker note: −5 is further left than −1, so −5 < −1. -->

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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

**Ari puts four numbers in order. Click the line where the reasoning goes wrong.**

```
 1  Put these in order from least to greatest: −2, −8, 3, 0
 2  The numbers without signs are 2, 8, 3 and 0.
 3  8 is the biggest of those, so −8 is the greatest negative.
 4  Least to greatest: −2, −8, 0, 3
```

* Answer: line 3

<!-- Speaker note: Line 3: 8 is bigger than 2, but −8 is further below zero than −2, so −8 is less. The order is −8, −2, 0, 3. -->

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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>

## Either side of −3

**Less than $-3$, or greater than $-3$?**

Groups: Less than −3 · Greater than −3

−6 · −1 · 0 · −10 · 2 · −4

* Answer: −6 → Less than −3; −1 → Greater than −3; 0 → Greater than −3; −10 → Less than −3; 2 → Greater than −3; −4 → Less than −3

<!-- 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



- **Between two negatives:** Each task changes one thing. First: drag $P$ to a number between $-8$ and $-4$.
- **Temperatures:** The same idea in degrees. Warmer means further right.
- **Zero and the negatives:** Zero is greater than every negative number.
- **Further from zero:** When is the number further from zero the smaller one?

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

<!-- Speaker note: What this step is for: One change at a time: a number between two negatives, the same idea in temperatures, then the edge case where the number further from zero is the smaller one. Hints, in order: Which way from −4 is less: left or right? / Go left of −4, but not as far as −8. / Warmer is further right. Is it above or below zero? / Right of Canberra (−3), but still left of 0. / Which way is less: left or right? Is zero left or right of −1? For students who need more support: Use the Run board for every comparison: set a and b to the two numbers and read the sentence. A paper number line from −10 to 10 on every desk helps. For early finishers: Which is greater, −0.5 or −5? Place both on the line. Then: is there an integer between −3 and −2? When students get stuck: A student says −8 is greater than −2. Ask: Which is colder, −8 °C or −2 °C? Fix: Place both on the line. Further left is less, and −8 is further left. Say it as 'eight below zero' and 'two below zero'. / A student puts negatives in the order −1, −2, −3 when asked for least to greatest. Ask: Which of these is furthest left on the line? Fix: Point at the line and read it from left to right: −3, −2, −1. / A student mixes up < and >. Ask: Which number is greater? Which way does the wide end of the sign face? Fix: The wide end faces the greater number: −6 > −9 reads '−6 is greater than −9'. -->

<!-- Speaker note: Teacher: each task changes one thing. Before the temperature task ask: which way is warmer? -->

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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>

## Which sign?

**Which sign goes in the gap? $-6 \;\square\; -9$**

- $&lt;$
- $&gt;$

* Answer: $&gt;$

<!-- Speaker note: −6 > −9: −6 is to the right of −9. -->

<!-- Speaker note: Teacher: the sign question plants the reversed sign. A student who picks < can place both on the line and say which is further left. -->

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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>

## In order

**Which list is in order from least to greatest?**

- $-9, -4, 0, 2, 5$
- $0, -4, -9, 2, 5$
- $-4, -9, 0, 2, 5$

* Answer: $-9, -4, 0, 2, 5$

<!-- Speaker note: Left to right on the line: −9, −4, 0, 2, 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>

## Colder

**At 6 am it was $-5$ °C in Cooma and $-8$ °C in Perisher. Which was colder?**

- Perisher
- Cooma

* Answer: Perisher

<!-- Speaker note: −8 < −5, so Perisher was colder. -->

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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: what number is 4 places to the left of 1 on a number line?**

* Answer: -3

<!-- Speaker note: From 1, four places left: 0, −1, −2, −3. -->

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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 surprising pair

Find a pair where the less number has the bigger size without its sign. The sentence under the line tells you which is less.

Stretch: write three temperatures below zero, then put them in order from warmest to coldest.

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

<!-- Speaker note: What this step is for: Building a pair where the less number is the one further from zero reverses the lesson: from the rule to an example that shows it. Hints, in order: P must be further left than Q. / Try putting P below zero, further from zero than Q is: P at −6 and Q at −2, or Q at 3. -->

<!-- Speaker note: Teacher: collect a few pairs. Ask: could P be positive? (No: a positive further from zero than Q would be greater than Q.) -->

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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

**Put in order from least to greatest: 4, −6, 0, −1, −10.**

* −10, −6, −1, 0, 4.

<!-- 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: adding and subtracting integers, as moves along the same line. Where it leads: plotting points in all four quadrants, reading graphs with negative values, and comparing negative fractions and decimals (AC9M7N04). -->