At a glance. One lesson of about 60 minutes. Multiplication of integers from repeated addition and a pattern; division as its inverse; several negative factors.
Calculator: no calculator. Times-table facts up to 12 × 12 carry the arithmetic; the reasoning about signs is the learning.
What students will learn
Know
- Same signs: positive. Different signs: negative.
- An even number of negative factors gives a positive product; odd gives negative.
- Any product with a factor of 0 is 0.
Understand
- The sign rules are forced by the pattern of a times table, not chosen.
- Division follows the same rules because it undoes multiplication.
Do
- Multiply and divide integers.
- Find the sign of a product of several integers.
- Interpret a product of two negatives in a context of rate and time.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: questions 2 to 7 across the four operations with signs.
- Criterion B. Investigating patterns: the staircase. Ask students to state the change per row for any a and to predict the row for b = −5 that is off the figure.
- Criterion C. Communicating: the exit ticket asks students to judge an argument; ask for the multiply-back check in writing.
- Criterion D. Applying mathematics in real-life contexts: the descending walker and diver, where the positive product means “higher in the past”.
The one thing to do with the figure
Start with a = 3, a positive, and cover everything below the 0 row. Ask how the products change each row (down by 3), then reveal: 3 × (−1) = −3 continues the pattern, which students already accept.
Now set a = −3 and repeat. The products go up by 3 each row, so after 0 they must be 3, 6, 9. Ask: “If −3 × (−1) were −3, what would happen to the pattern?” It would break. That is the argument.
A possible order
| 5 min | Warm-up: two from Lesson 3, one times-table fact with its division. |
|---|---|
| 15 min | The staircase, a = 3 then a = −3; students try a = −2 and a = 4 on their own devices. |
| 5 min | Division as the inverse; multiply back. |
| 10 min | Worked examples; the walker going back in time; the faded division in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn and exit ticket (question 9). |
The answers
| Warm-up | 5; −7; 48 and 8. |
|---|---|
| Worked examples | −28; 30; −9; −30; the walker was 12 m higher. |
| Faded example | 8, checked by 8 × (−6) = −48. |
| 1. Sign of a product | B. Anyone choosing D may think division follows different rules. |
| 2. Different signs | −24. |
| 3. Same signs | 21. |
| 4. Dividing | −4. |
| 5. Dividing by a negative | −9. |
| 6. Going back in time | 10 m: (−2) × (−5) = +10. |
| 7. Three factors | −30. |
| 8. Product and sum | −4, the pair being −4 and 6. |
| 9. Dividing two negatives | B. A shows the rule half-remembered; ask for the multiply-back check. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 24 in question 2, −21 in question 3 | The digits are right; the sign rule was applied the wrong way round. | Back to the staircase for that a. |
| −2, −10 or 40 anywhere | Added or subtracted instead of multiplying or dividing. | Ask them to read the operation aloud before calculating. |
| −10 in question 6 | Sees the negative rate and keeps the answer negative. | Ask: going back in time on the way down, were you higher or lower? |
| 30 in question 7 | Two of the negatives cancelled, the third forgotten. | Count the negative signs and write the count down. |
| A in question 9 | “Two negatives make a negative”, a mixed-up slogan. | Multiply back: −4 × (−2) is 8, not −8. |
Questions worth asking
- Why can't we just say “minus three groups of minus two”? Because it does not mean anything yet. The pattern gives it a meaning.
- What is (−1) × (−1) × … with 10 factors? With 11? 1 and −1: even and odd counts of negatives.
- Is there an integer that gives a negative answer when it is squared? No: a number times itself has two equal signs. This leads straight into Lesson 5.
Differentiation
A way in for anyone stuck
- Repeated addition first: 3 × (−2) as three jumps of −2 on a number line.
- Write the sign decision before the arithmetic: a two-column table, “sign” then “size”.
- Leave question 8 until questions 2 and 3 are secure.
Stretch
- Question 8, then: find all pairs of integers with product −12 (six pairs).
- Extend the staircase idea to explain why (−1) × a is the opposite of a for any a.
Students learning in English as an additional language
“Product” and “groups of” need explicit attention: a product is not a thing that is sold here.
| Word | Plain meaning | Example |
|---|---|---|
| product | the answer to a multiplication | the product of −3 and 4 is −12 |
| quotient | the answer to a division | −20 ÷ 4 has quotient −5 |
| factor | one of the numbers being multiplied | −2 is a factor in (−2) × 7 |
| same signs | both positive or both negative | (−6) and (−5) |
| pattern | a rule that repeats, so you can predict the next step | −9, −6, −3, 0, 3 |
Sentence starters for explaining aloud or in writing:
- Each row down, the product changes by …, so the next row is …
- The answer is positive / negative because the signs are …
- I checked by multiplying back: … × … = …
Two things not to say
“Two negatives make a positive.” It is true for multiplying and dividing only, and the shortened slogan causes −3 + (−5) = 8. Say “same signs give a positive product”.
“It is just the rule.” The staircase gives a reason students can rebuild for themselves in a test.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- MultiplyingWork out (−9) × 6 and (−9) × (−6).
−54 and 54. - DividingWork out (−72) ÷ 8 and (−72) ÷ (−8).
−9 and 9.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- TemperatureThe temperature falls by 2 °C every hour for 6 hours. What is the total change?
(−2) × 6 = −12 °C. - Shared lossA team of 4 shares a score of −28 points equally. What does each person get?
−7 points.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningThe product of five integers is negative. How many of them could be negative? List every possibility.
1, 3 or 5 (an odd number). - From Lesson 3Work out −4 − (−10), then multiply your answer by −3.
6, then −18.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard; “Show me the steps” lists every row for a = 3 and a = −3.
On paper: write the table for a = −3 as a column and ask students to extend it past zero.