CLASS 7 MATHEMATICS

Operations with Integers: Addition and Multiplication Using the Token Method

Learn integer addition, zero pairs and multiplication using tokens and the bag method. Understand the rules with easy examples and solved questions from Ganita Prakash.

The Busy Brains | Class 7 Maths Notes

Introduction

Integers include positive numbers, negative numbers and zero. We use them to represent gains and losses, temperatures above and below zero, and movements in opposite directions.

The token method helps you understand operations with integers visually. You can use coloured tokens to represent numbers, cancel opposite tokens and explore multiplication through bags.

These notes cover addition and multiplication, the four sign combinations, the bag method and important questions from the Class 7 Ganita Prakash chapter Operations with Integers.

WATCH THE VIDEO LESSON

Learn the Token Method with Mansi Ma'am

Want to understand the concept before solving the questions? Watch this complete lesson from The Busy Brains. I explain integer addition and multiplication using tokens, demonstrate the bag method and solve the Figure It Out questions from Class 7 Operations with Integers.

What will you learn in this video?

Addition of integers using positive and negative tokens.
How positive and negative tokens form zero pairs.
Multiplication of integers using the bag method.
Why a positive multiplier means placing tokens in a bag.
Why a negative multiplier means removing tokens from a bag.
Solutions to the Figure It Out questions from the chapter.

Study tip: Watch the explanation, pause after each example and try to solve it yourself before continuing.

▶ Watch on YouTube Subscribe to The Busy Brains

1. Understanding Tokens

A token is a small counter or object that represents a number. We use two colours to represent positive and negative integers.

Green Token: Positive One

A green token represents +1.

3 green tokens = +3

Red Token: Negative One

−

A red token represents −1.

3 red tokens = −3

Remember: Every green token represents +1, while every red token represents −1.

2. Zero Pairs

One positive token and one negative token cancel each other. Together, they form a zero pair.

+ + − = 0

(+1) + (−1) = 0

One positive token and one negative token have a total value of zero.

Example: 4 + (−2)

Start with four positive tokens and add two negative tokens. Cancel two positive-negative pairs. Two positive tokens remain.

4 + (−2) = +2

Important: Adding a zero pair does not change the value. Zero pairs are useful when we need to remove tokens from an empty bag.

3. Addition Using Tokens

To add integers, put the tokens for both numbers together. Cancel every positive-negative pair and count the tokens that remain.

Example 1: (+3) + (+2)

Place three positive tokens and add two more positive tokens.

++ ++ ++

(+3) + (+2) = +5

All five tokens are positive, so the sum is +5.

Example 2: (+4) + (−2)

Place four positive tokens and two negative tokens together. Cancel two zero pairs. Two positive tokens remain.

++ ++ −−

(+4) + (−2) = +2

Example 3: (+3) + (−5)

Cancel three zero pairs. Two negative tokens remain.

3 + (−5) = −2

Example 4: (−3) + (−2)

All five tokens are negative.

(−3) + (−2) = −5

Addition rule: Combine the tokens, cancel opposite pairs and count the remaining tokens with their signs.

4. Multiplication Using the Bag Method

The bag method helps us understand multiplication by showing repeated groups of tokens. In an expression a × b, the first factor tells us how many bags or repeated actions to use. The second factor tells us which tokens are involved in each bag.

The Two Most Important Rules

When the multiplier is positive, we place tokens into the bag.

When the multiplier is negative, we remove tokens from the bag.

If a bag is empty and we need to remove tokens, first add zero pairs. Then remove the required tokens and count what remains.

5. Case 1: Positive × Positive

Example: 3 × 2

Step 1: Make three bags because the first factor is 3.

Step 2: The multiplier is positive, so place two positive tokens in each bag.

++ | ++ | ++

Step 3: Count all six positive tokens.

3 × 2 = +6

6. Case 2: Positive × Negative

Example: 3 × (−2)

Step 1: Make three bags.

Step 2: The multiplier is positive, so place two negative tokens in each bag.

−− | −− | −−

Step 3: Count all six negative tokens.

3 × (−2) = −6

7. Case 3: Negative × Positive

Example: (−3) × 2

Step 1: Make three bags. Each bag starts empty.

Step 2: Add two zero pairs to each bag. Each bag now contains two positive and two negative tokens, so its value is still zero.

++ −−

Step 3: The multiplier is negative, so remove two positive tokens from each bag.

−− Each bag has −2 left.

Step 4: Three bags contain six negative tokens altogether.

(−3) × 2 = −6

8. Case 4: Negative × Negative

Example: (−3) × (−2)

Step 1: Make three bags. Each bag starts empty.

Step 2: Add two zero pairs to each bag. Each bag now contains two positive and two negative tokens.

++ −−

Step 3: The multiplier is negative, so remove two negative tokens from each bag.

++ Each bag has +2 left.

Step 4: Three bags contain six positive tokens altogether.

(−3) × (−2) = +6

Why is the answer positive? Removing negative tokens from zero-valued bags leaves positive tokens behind.

9. Sign Rules for Multiplication

Once you understand the bag method, use these sign rules to solve integer multiplication questions quickly.

Signs of the Numbers	Result	Rule
Positive × Positive	Positive	(+a) × (+b) = +
Positive × Negative	Negative	(+a) × (−b) = −
Negative × Positive	Negative	(−a) × (+b) = −
Negative × Negative	Positive	(−a) × (−b) = +

Same Signs

Positive × Positive

Negative × Negative

Same signs = Positive

Different Signs

Positive × Negative

Negative × Positive

Different signs = Negative

Quick rule: Multiply the numerical values. If the signs are the same, the product is positive. If the signs are different, the product is negative.

10. Figure It Out: Class 7 Operations with Integers

Try these questions from the chapter. You can also watch the video lesson above to see the token method explained and the questions solved.

Question 1: Use the token interpretation

Find the values of the following:

3 × (−2)
(−5) × (−2)
(−4) × (−1)
(−7) × 3

Show answers

−6
+10
+4
−21

Question 2: Find the values without calculating

If 123 × 456 = 56088, find the following without performing multiplication again:

(−123) × 456
(−123) × (−456)
123 × (−456)

Show answers with reasoning

(a) One number is negative and the other is positive. The product is negative.

(−123) × 456 = −56088

(b) Both numbers are negative. The product is positive.

(−123) × (−456) = +56088

(c) One number is positive and the other is negative. The product is negative.

123 × (−456) = −56088

Need help solving these questions?

Watch the complete video lesson for the token method, all four multiplication cases and the Figure It Out solutions.

▶ Watch the Explanation Subscribe on YouTube

11. Quick Revision and Practice

Remember These Rules

A green token represents +1.
A red token represents −1.
One positive token and one negative token form a zero pair.
For addition, combine the tokens, cancel opposite pairs and count what remains.
A positive multiplier means placing tokens into the bag.
A negative multiplier means removing tokens from the bag.
Use zero pairs when you need to remove tokens from an empty bag.
Same signs give a positive product.
Different signs give a negative product.

Practice Questions

Find (+5) + (−3).
Find (−4) + (−2).
Find 4 × (−3).
Find (−5) × 2.
Find (−4) × (−3).
Find (−123) × 456 if 123 × 456 = 56088.
Find (−123) × (−456) if 123 × 456 = 56088.
Find 123 × (−456) if 123 × 456 = 56088.

Check the practice answers

+2
−6
−12
−10
+12
−56088
+56088
−56088

Class 7 Mathematics | Ganita Prakash | Operations with Integers | Addition and Multiplication Using the Token Method