GANITA PRAKASH · GRADE 7 · PART II · CHAPTER 2

PATTERN — Discover sign rules and multiplication by ±1

Think first. Use AI purposefully. Verify always.

DISCOVER

Begin with your own idea.

As the first factor falls by one, what changes in the products with −3?

Write or sketch a prediction before you read on. Use the model below to test it.

Understand

The token model and consistent product patterns explain the sign rules. For two nonzero factors, equal signs give a positive product and different signs give a negative product. A zero factor gives zero.

Key Ideas

1 \times a = a; (- 1) \times a = - a

Five factors of −1 give −1, not +1. Pair four of the factors and keep the unpaired factor.

Visualise

Continue the product pattern

Move the first factor down one step. Predict the next product before moving again.

First factor: 4

Fixed second factor: (−3)

$5 \times (- 3) = - 15$
$4 \times (- 3) = - 12$
$3 \times (- 3) = - 9$

For each decrease of 1 in the first factor, the product changes by 3.

The −1 mirror

Integer to reflect: (−5)

(−5)↔ 0 ↔5

$1 \times (- 5) = - 5; (- 1) \times (- 5) = 5$

Multiplying by −1 gives the additive inverse. Zero stays zero.

Method

Continue a product pattern through zero. Check the change at each step. Use multiplication by −1 to connect a number with its additive inverse.

Worked Example

Given

The sequence 2 × (−3), 1 × (−3), 0 × (−3), (−1) × (−3).

To Find

The value of (−1) × (−3) in the product pattern.

Reasoning

Decreasing the first factor by 1 increases this product by 3.

Solution

The products begin −6, −3, 0.
The next product must be 3.
This agrees with removing three negatives once from zero pairs.

Result

(−1) × (−3) = 3.

AIMS Method

A — Attempt independently
Predict before calculating.
I — Investigate the idea
Identify signs, order and operation.
M — Make connections
Connect movements, tokens and expressions.
S — Scrutinise your reasoning
Check the sign, value and original conditions.

Think first. Use AI purposefully. Verify always.

Focused Practice

Find the product of five factors of −1 and explain.

Your reasoning

Continue with Multiply →

Check & Feedback

Explain the key idea to someone who made the mistake described above. Use a model or a checked example.

Your reasoning

Textbook connection: Ganita Prakash · Grade 7 · Part II · Operations with Integers · printed page 32