Chapter Overview
Integers include positive numbers, zero and negative numbers. Multiplication and division rules can be understood through repeated change, patterns and inverse operations.
This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.
Learning Objectives
After studying this chapter, you should be able to:
- explain and apply sign of a product;
- explain and apply multiple factors;
- explain and apply integer division;
- explain and apply properties;
- communicate the reasoning behind a solution clearly;
- check answers using estimation, substitution, or a second method.
Key Concepts
| Concept | Meaning | |---|---| | Sign of a product | Factors with the same sign give a positive product; factors with different signs give a negative product. | | Multiple factors | A product with an even number of negative factors is positive; an odd number is negative. | | Integer division | Division follows the same sign rules as multiplication and is the inverse of multiplication. | | Properties | Integer multiplication is closed, commutative and associative, and distributes over addition; division is not commutative. |
Detailed Explanation
Sign of a product
Factors with the same sign give a positive product; factors with different signs give a negative product.
Multiple factors
A product with an even number of negative factors is positive; an odd number is negative.
Integer division
Division follows the same sign rules as multiplication and is the inverse of multiplication.
Properties
Integer multiplication is closed, commutative and associative, and distributes over addition; division is not commutative.
Do not memorise a rule without testing it on examples. Ask what each number, operation, line, or symbol represents and whether the result fits the original situation.
Important Rules and Formulae
- Zero multiplied by any integer is zero.
- Division by zero is undefined.
- For non-zero integers,
a ÷ bneed not be an integer, so integers are not closed under division.
Worked Examples
Example 1
Problem: Evaluate (-7) × 6 + (-4) × (-3).
Solution: -42 + 12 = -30.
Example 2
Problem: A temperature falls 3°C each hour for 5 hours. Represent the change.
Solution: 5 × (-3°C) = -15°C.
Show the mathematical statement, substitution or construction step before writing the final answer. Include units wherever the quantity is measured.
Common Mistakes
- Applying addition sign rules to multiplication.
- Saying a negative times a negative is negative.
- Treating division by zero as zero.
Quick Revision
- Sign of a product: Factors with the same sign give a positive product; factors with different signs give a negative product.
- Multiple factors: A product with an even number of negative factors is positive; an odd number is negative.
- Integer division: Division follows the same sign rules as multiplication and is the inverse of multiplication.
- Properties: Integer multiplication is closed, commutative and associative, and distributes over addition; division is not commutative.
Practice Questions
- Find
(-12) × (-8). - Find
84 ÷ (-7). - What is
(-1)^7? - Is integer division commutative? Give a reason.
Answers and Explanations
-
- −12.
- −1.
- No; for example,
8 ÷ 2 = 4but2 ÷ 8is not 4.
Self-Check
Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.