Multiplying and dividing integers

Classroom resources

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Students will be able to

• I can multiply and divide integers, including two negatives.
• I can explain why a negative times a negative is positive, from a pattern or the number line.
• I can tell the rule for multiplying apart from the rule for adding negatives.

Timing

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

Before class

Do the Run board yourself first. Students need Adding and subtracting integers (subtracting goes the opposite way), which this lesson echoes: a negative number of jumps goes the opposite way.

Two justifications are offered: the pattern (3 × (−2), 2 × (−2), … goes up by 2 each time, so it keeps going past zero) and the line. Use whichever lands; the debt model (taking away 3 debts of $2 leaves you $6 better off) is a good third in class.

The common mistake is to use the addition rule for multiplication ("two negatives make a bigger negative"); it is planted after the method, in Kai's working, and the review question puts −4 + (−3) next to −4 × (−3).

This lesson is Year 8 (AC9M8N04). It sits in Number Foundations so that Expanding brackets and the Year 8 Linear relations lessons can rely on it.

Running the lesson

1. Predict: take a show of hands. Students who chose −6 usually say 'two negatives make a negative'. Do not correct it yet.
2. Run: have the class say each answer aloud before the click: −6, −4, −2, 0, … and then 2. Ask: why would it suddenly stop going up?
3. Investigate: model the worked example live. Write the pattern down the board at writing pace, then draw 3 jumps of −2 and the reverse for −3 × (−2). Reveal the prepared lines afterwards to compare.
4. Modify: each task changes one sign. Ask for the sign before the number.
5. Practise: (−1)³ is a stretch for some; let them multiply two at a time.
6. Make: collect pairs. Ask: could one of them be positive? (Then the product would be negative.)

When students get stuck

Answers to the checks

1. −3 × (−2): What is $-3 \times (-2)$?
   6. 3 jumps of −2 would go left to −6; a negative number of jumps goes the opposite way, to 6. The pattern 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0 goes up by 2 each time, so −1 × (−2) = 2 and −3 × (−2) = 6.
2. follow the pattern: The board shows $3 \times (-2)$. Make $a$ smaller one step at a time until it is $-3$. Was your prediction right?
   
3. The pattern: Fill in the pattern for $a \times (-2)$.
   0, 2, 4, 6
   Each step down in a adds 2 to the answer, so past zero the answers are positive: −3 × (−2) = 6.
4. Signs from the pattern: Which is $-12 \div 3$?
   $-4$
   −12 ÷ 3 = −4, because 3 × (−4) = −12. Different signs give a negative.
5. Find the mistake: Kai works out $-4 \times (-3)$. Click the line where his reasoning goes wrong.
   line 3
   Line 3: that is the rule for adding (−4 + −3 = −7). For multiplying, two negatives give a positive: −4 × (−3) = 12.
6. Positive times negative: Work out $4 \times (-3)$. Set the board to check.
   -12
   4 jumps of −3: left 12, so −12.
7. Negative times positive: Work out $-4 \times 3$.
   -12
   Different signs: −12. The same as 3 × (−4).
8. Negative times negative: Work out $-4 \times (-3)$.
   12
   Same signs: 12.
9. Times zero: Edge case: work out $-5 \times 0$.
   0
   −5 jumps of 0 stay at 0: anything times 0 is 0, whatever its sign.
10. Dividing: Work out $-20 \div (-4)$.
    5
    Same signs: 5. Check: −4 × 5 = −20.
11. Multiply: Work out $-6 \times (-5)$.
    30
    Same signs: 30.
12. Divide: Work out $18 \div (-3)$.
    -6
    Different signs: −6. Check: −3 × (−6) = 18.
13. Powers of −1: What is $(-1)^3$, which is $(-1) \times (-1) \times (-1)$?
    $-1$
    (−1) × (−1) = 1, then × (−1) again: −1. An odd number of negatives gives a negative.
14. Review: adding is different: Review: work out $-4 + (-3)$.
    -7
    Adding a negative moves left: −7. Compare: −4 × (−3) = 12.
15. a positive from negatives: Set $a$ and $b$ so that both are negative and $a \times b = 12$.
    
    −3 × (−4) or −4 × (−3) (and −2 × (−6), off this board): same signs give a positive.

Solutions

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

one change at a time › Positive times negative

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

one change at a time › Negative times positive

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

one change at a time › Negative times negative

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

one change at a time › Times zero

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

one change at a time › Dividing

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

a positive from negatives

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

Stretch and support

Stretch: Work out (−2)⁴ and (−2)⁵. Then: what is the sign of the product of 7 negative numbers? Why?

Support: Use the board for every product: set a and b and read R. A printed sign table (+ × + = +, and so on) can come out after the pattern, not before.

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 8 student in Australia. Help me understand ONE idea only, from the lesson "Multiplying and dividing integers".

The idea: Multiplication and division of integers: I can Students establish and apply the sign rules for multiplication and division of integers, including negative × negative = positive.

What understanding looks like:
- Correctly applies sign rules for all combinations of integer multiplication and division.
- Explains the sign rule for negative × negative.
- Does not confuse multiplication sign rules with addition sign rules.

Mistakes students often make here:
- A student says −4 × (−3) = −12 (Kai's mistake).
- A student gets the size right but always the wrong sign for division.
- A student adds instead of multiplying (−6 × −5 = −11).

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 "Multiplying and dividing integers".

Exit ticket

Work out −7 × (−4), 24 ÷ (−6) and (−1)⁴.

28, −4 and 1.

What each step is for

• −3 × (−2): Committing to a sign for −3 × (−2) before the board shows it tests whether they carry over the addition rule (two negatives make a bigger negative).
• follow the pattern: Making a smaller one step at a time shows the answer going up by 2 each time, through zero and past it: the pattern forces a negative times a negative to be positive.
• the sign rules: The pattern and the line give one rule: the same signs give a positive, different signs a negative, for multiplying and for dividing. Kai's mistake uses the addition rule for multiplication.
• one change at a time: One change at a time: positive times negative, negative times positive, negative times negative, the edge case of zero, then division with each pair of signs.
• all four signs: Multiplying, dividing and powers of −1, mixed, with the addition rule as the planted mix-up; the review keeps adding negatives apart from multiplying them.
• a positive from negatives: Choosing two negatives whose product is a given positive reverses the lesson: from the answer back to the signs.

Where it leads

Next in Number: squares, powers and primes (AC9M7N01, AC9M7N02), then fractions. Negative fractions and decimals (node 2) close the fractions unit.

Where it leads: expanding brackets with negatives (−2(x − 5) = −2x + 10), gradients and rules with negative values in Linear relations, and powers of negative numbers.