Comparing integers

Classroom resources

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Students will be able to

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

Timing

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

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.

Running the lesson

1. Predict: take a show of hands. Students who chose −7 usually say 7 is bigger. Do not correct it yet: the board will.
2. Run: 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?
3. Investigate: 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.
4. Modify: each task changes one thing. Before the temperature task ask: which way is warmer?
5. Practise: the sign question plants the reversed sign. A student who picks < can place both on the line and say which is further left.
6. Make: collect a few pairs. Ask: could P be positive? (No: a positive further from zero than Q would be greater than Q.)

When students get stuck

Answers to the checks

1. which is greater?: Which is greater, $-7$ or $-2$?
   −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.
2. two points on the line: Put $A$ at $-7$ and $B$ at $-2$. Was your prediction right?
   
3. Ordering on the line: Which is true?
   $-5 < -1$
   −5 is further left than −1, so −5 < −1.
4. Find the mistake: Ari puts four numbers in order. Click the line where the reasoning goes wrong.
   line 3
   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.
5. Either side of −3: Less than $-3$, or greater than $-3$?
   −6 → Less than −3; −1 → Greater than −3; 0 → Greater than −3; −10 → Less than −3; 2 → Greater than −3; −4 → Less than −3
6. Between two negatives: Drag $P$ to a number that is less than $-4$ and greater than $-8$.
   
   −5, −6 or −7: left of −4 and right of −8.
7. Temperatures: Drag $P$ to a temperature warmer than Canberra but still below zero.
   
   −2 °C or −1 °C: further right than −3, still left of 0.
8. Zero and the negatives: Drag $P$ to a number less than $-1$. Then try zero: is it less or greater than $-1$?
   
   Any number left of −1 is less: −2, −3 and so on. Zero is to the right, so 0 > −1.
9. Further from zero: Edge case: $-10$ is further from zero than $-1$. Which is greater?
   $-1$
   −1 > −10. −10 is further from zero, but it is further left, so it is less.
10. Which sign?: Which sign goes in the gap? $-6 \;\square\; -9$
    $>$
    −6 > −9: −6 is to the right of −9.
11. In order: Which list is in order from least to greatest?
    $-9, -4, 0, 2, 5$
    Left to right on the line: −9, −4, 0, 2, 5.
12. Colder: At 6 am it was $-5$ °C in Cooma and $-8$ °C in Perisher. Which was colder?
    Perisher
    −8 < −5, so Perisher was colder.
13. Review: Review: what number is 4 places to the left of 1 on a number line?
    -3
    From 1, four places left: 0, −1, −2, −3.
14. a surprising pair: Place $P$ and $Q$ so that $P < Q$, but $P$ is further from zero than $Q$.
    
    P is left of Q, so P < Q, and further from zero: it must be a negative, such as P = −6 and Q = −2 (or Q = 3).

Solutions

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

one change at a time › Between two negatives

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

one change at a time › Temperatures

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

one change at a time › Zero and the negatives

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

one change at a time › Further from zero

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

a surprising pair

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

Stretch and support

Stretch: Which is greater, −0.5 or −5? Place both on the line. Then: is there an integer between −3 and −2?

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.

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 "Comparing integers".

The idea: Comparing and ordering integers: I can Students compare and order integers using less-than and greater-than notation, including negative integers.

What understanding looks like:
- Correctly orders a set of integers including negatives.
- Uses < and > notation correctly between integers.
- Does not confuse the magnitude of a negative with its value (e.g. understands −10 < −1).

Mistakes students often make here:
- A student says −8 is greater than −2.
- A student puts negatives in the order −1, −2, −3 when asked for least to greatest.
- A student mixes up < and >.

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.
- 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 "Comparing integers".

Exit ticket

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

−10, −6, −1, 0, 4.

What each step is for

• which is greater?: Committing to an answer for −7 and −2 before using the line makes the board a test of their idea about negatives, not a demonstration.
• two points on the line: Placing −7 and −2 and reading the sentence shows the rule: further left is less, for negatives as well as positives.
• further left is less: Further left is less, so the order of negatives runs the opposite way to their sizes without the sign: −8 is less than −2. Ari's mistake orders by the size alone.
• one change at a time: 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.
• compare and order: Comparing with < and >, ordering, and a context, mixed; the review question keeps locating integers from Year 6 fresh.
• a surprising pair: 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.

Where it 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).