014i - Multiplying and Dividing Integers

Lesson 14 — Explicit Instruction: Multiplying and Dividing Integers

Strand: Number | Descriptor: AC9M8N04 | Duration: 45 minutes

Learning Intentions

• To multiply and divide positive and negative integers accurately.
• To understand why the sign rules for multiplication and division work, not just memorise them.

Success Criteria

I can:

1. State the sign rules for multiplying and dividing two integers.
2. Multiply and divide integers, including chains of more than two integers.
3. Explain the sign of a product using a pattern-based or repeated-addition argument.
4. Apply the sign rules within order-of-operations expressions.

Warmup

(5 minutes — pattern spotting, mini whiteboards)

Extend each number pattern by two more terms and describe the rule:

Discussion: Pattern 1 should fall out naturally (each term decreases by 2). Pattern 2 is the hook: if the pattern of decreasing by 2 each time continues, then must be , and must be — a negative times a negative gives a positive, purely from continuing a consistent pattern. Do not over-explain yet; return to this logic in Activity 1.

Activities

Activity 1 — Explicit Instruction: the Sign Rules (10 min)

I do / We do / You do.

I do: Formalise the rule the warmup revealed.

First sign	Second sign	Result sign

Same signs give a positive result; different signs give a negative result. This rule holds for both multiplication and division.

Model with repeated addition for the first two rows, and the warmup’s pattern-continuation logic for the negative negative row (repeated addition cannot directly model “negative groups”, which is exactly why the pattern argument matters).

We do: Evaluate together: , , , , , .

You do: Evaluate: , , , , , .

Activity 2 — Guided Practice: Chains of Integers (10 min)

I do: Explain that with more than two factors, count the number of negative factors: an even count gives a positive result, an odd count gives a negative result.

Wait — check by counting negatives directly: four negative factors () is an even count, so the result should be positive. Re-work carefully:

Teacher note: Deliberately model this self-correction live — it demonstrates the value of the “count the negatives” shortcut as a check against step-by-step slips.

We do: Evaluate together, first predicting the sign by counting negatives, then calculating: , , .

You do: Predict the sign, then evaluate: , , .

Activity 3 — Inquiry: Building a Rule for Division Chains (14 min)

Pairs, then whole-class share.

Does the “count the negatives” rule also work for chains involving division, such as ? Investigate using at least three different examples, some all-division and some mixed multiplication/division.

Students should discover the rule extends naturally, since division by a negative behaves exactly like multiplication by a negative for sign purposes.

Socratic scaffolding for pairs who stall:

Prompt	Purpose
Understand: what are you trying to test?	Whether “count the negative signs” predicts the sign of an answer involving division, not just multiplication.
What do you already know?	The sign rule table from Activity 1 works step-by-step, one operation at a time.
Can you try a simple case first?	(positive, one operation, two negatives).
Extend to a chain	.
Count the negative signs in the original expression	Three negatives: — an odd count.
Does the count match the final sign?	Yes — odd count of negatives, negative result.
Looking back — will this always work, even with mixed × and ÷?	Test ; three negatives (odd), negative result — consistent. Conjecture: the rule works regardless of the mix of × and ÷, as long as you count every negative factor or divisor in the whole expression.

Finish with a class challenge: without calculating, predict the sign of , then verify.

Four negative values () — an even count — positive result. Confirmed.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate .
2. Evaluate .
3. Predict the sign, then evaluate: .
4. A student says “dividing by a negative always makes the answer smaller.” Give a counterexample.

Answers: 1. ; 2. ; 3. Four negatives (even) → positive; ; 4. E.g. , which is larger than — dividing a negative by a negative increases the value.

Common Misconceptions

Misconception	How to pre-empt it
”Two negatives always cancel to make things positive, no matter the operation.”	Contrast with — the sign rule is specific to multiplication and division, not addition.
Losing track of sign in chains of 3+ factors by working left to right without counting negatives first.	Teach the “count the negatives” check as a verification step, as modelled in Activity 2’s self-correction.
Believing and are the same.	Though this lesson focuses on × and ÷, flag briefly: (square applies to 4 only) but (square applies to ). Revisit fully when exponents and integers combine.
Assuming dividing by a negative number always shrinks the value.	Use CFU Q4’s counterexample: dividing two negatives gives a larger positive result.
Writing or treating division by zero casually.	State explicitly and briefly: division by zero is undefined, and this is never an allowed step.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). What is the value of ?

Answer

There are 10 negative factors (even count), so the result is positive: .

E2 (Kangaroo style). If and , what can you conclude about ?

Answer

Since the product is negative, and must have different signs. As , it follows that .

E3 (Challenge). Find all integer values of such that is a negative integer.

Answer

For the quotient to be negative, must be positive (different signs). For the quotient to be a whole number, must divide exactly. So , giving quotients respectively.

E4 (Investigation). A sequence starts at and each term is found by multiplying the previous term by . Write the first six terms. What do you notice about the signs?

Answer

. The signs alternate perfectly, since multiplying by a single negative number flips the sign every time.

Homework

1. Evaluate: (a) (b) (c) (d) .
2. Predict the sign, then evaluate: (a) (b) (c) .
3. Evaluate using order of operations: (a) (b) .
4. A submarine descends m per minute (recorded as m/min). If it descends steadily for minutes, what is its total change in depth? Express your working as a multiplication of integers.
5. Reasoning. Explain, using the warmup’s pattern-continuation idea (not a memorised rule), why must equal .
6. Challenge. Two integers multiply to give . Their difference is . Find both integers. (Hint: consider factor pairs of 36 and check signs.)

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) three negatives, negative: (b) five negatives, negative: (c) three negatives, negative: . Q3 — (a) (b) . Q4 — m, i.e. m deeper. Q5 — continuing the pattern , each step up by 5; so continues the pattern by adding another 5, giving . Q6 — factor pairs of 36 with a difference of 13: and differ by (no), and differ by (no), and differ by (no), and differ by (no); re-check: need product and difference — try and : product ✓, difference ✓. The integers are and .