GANITA PRAKASH · GRADE 7 · PART II · CHAPTER 2
Chapter Summary
Think first. Use AI purposefully. Verify always.
Quick Revision · Return to a full lesson whenever a model or explanation needs another look.
Concept Map
Key Ideas
- A sum combines the two values. A difference compares the first value with the second; order matters.
- A signed movement records magnitude and direction: positive is right, negative is left. Final position is the sum of the movements.
- A +1 token and a −1 token form a zero pair. Adding a zero pair changes the representation, not the value. Subtracting an integer has the same effect as adding its additive inverse.
- In this model the first factor tells how many groups to place or remove. A positive first factor places groups; a negative first factor removes groups. The second factor describes each signed group. Start every operation from zero.
- 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.
- Division asks for a missing factor. Check a quotient by multiplying it by the divisor. Division by zero is undefined, and dividing two integers need not give an integer.
- Multiplication of integers is commutative and associative, and distributes over addition. These properties help reorganise expressions without changing their value.
- Use signed quantities to model a situation. State what positive and negative mean, form an expression, then interpret the result. Testing several cases can support a conjecture but does not prove every case.
Sign Rule Bank
For two nonzero factors: equal signs give a positive product, different signs give a negative product. A zero factor gives zero. These rules follow from the token and pattern models.
For division with a nonzero divisor and nonzero dividend, the same sign comparison applies to the quotient. A zero dividend gives zero; a zero divisor is never allowed.
Visual Recall
Zero-pair subtraction
Start with the given positive tokens.
Additive Inverse
Subtract an integer by adding its additive inverse. The inverse of −17 is 17; the inverse of 0 is 0.
Multiplication/Division Relationship
A quotient is a missing factor, provided the divisor is not zero. It need not be an integer: 1 ÷ 2 = 1/2.
Properties
These multiplication properties hold for integers. Division is not commutative.
Common Mistakes
- Swapping numbers preserves their sum but reverses their difference.
- A negative position is not a negative distance. The distance from zero is its magnitude.
- The additive inverse of a negative integer is positive. Zero is its own additive inverse.
- Removing negative tokens is different from adding negative tokens. Zero pairs make removal from an empty bag possible.
- Five factors of −1 give −1, not +1. Pair four of the factors and keep the unpaired factor.
- Do not swap dividend and divisor. For example, 8 ÷ 2 = 4 but 2 ÷ 8 = 1/4.
- Multiplication properties do not automatically transfer to division. Do not distribute to only one term.
- Distinguish signed displacement from distance travelled. Evidence from a few patterns is not a universal proof.
Mini Examples
GUESS · Two numbers have sum 25 and first minus second equal to 11.
Given
Two numbers have sum 25 and first minus second equal to 11.
To Find
The first and second integers whose sum is 25 and ordered difference is 11.
Reasoning
Test both the sum and the ordered difference.
Solution
- 18 + 7 = 25.
- 18 − 7 = 11.
- Reversing the pair gives 7 − 18 = −11.
Result
First 18, second 7.
MOVE · First movement −4; final position 5.
Given
First movement −4; final position 5.
To Find
The signed second movement from −4 to the final position 5.
Reasoning
The two signed movements add to the final position.
Solution
- From −4 to 0 is 4 units right.
- From 0 to 5 is another 5 units right.
- (−4) + 9 = 5.
Result
Second movement: +9 units.
CANCEL · 7 positive tokens; subtract 18 positive tokens.
Given
7 positive tokens; subtract 18 positive tokens.
To Find
The value of 7 − 18.
Reasoning
A zero pair has value zero.
Solution
- Add 11 zero pairs: there are 18 positive and 11 negative tokens.
- Remove all 18 positive tokens.
- 11 negative tokens remain.
Result
7 − 18 = −11.
MODEL · An empty bag; remove two negative tokens four times.
Given
An empty bag; remove two negative tokens four times.
To Find
The value of (−4) × (−2).
Reasoning
Adding zero pairs preserves the starting value of zero.
Solution
- Add 8 zero pairs: 8 positives and 8 negatives.
- Remove 2 negatives in each of 4 groups.
- All 8 positives remain.
Result
(−4) × (−2) = 8.
PATTERN · The sequence 2 × (−3), 1 × (−3), 0 × (−3), (−1) × (−3).
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.
REVERSE · −100 divided by 25.
Given
−100 divided by 25.
To Find
The quotient (−100) ÷ 25.
Reasoning
25 × q = −100.
Solution
- 25 × 4 = 100.
- 25 × (−4) = −100.
- Check: divisor × quotient = dividend.
Result
(−100) ÷ 25 = −4.
CONNECT · 4 groups, each containing 2 positive and 3 negative tokens.
Given
4 groups, each containing 2 positive and 3 negative tokens.
To Find
The value of 4 × (2 + (−3)), checked by grouping and distribution.
Reasoning
Distribute multiplication over the addition inside each group.
Solution
- Each group totals −1; 4 × (−1) = −4.
- Separately there are 8 positives and 12 negatives.
- 8 + (−12) = −4.
Result
4 × (2 + (−3)) = 4 × 2 + 4 × (−3) = −4.
MASTER · Mala answers 30 questions correctly and 20 incorrectly; +5 and −2 marks respectively.
Given
Mala answers 30 questions correctly and 20 incorrectly; +5 and −2 marks respectively.
To Find
Mala’s total marks from 30 correct and 20 incorrect answers.
Reasoning
Add the positive and negative contributions.
Solution
- 30 × 5 = 150.
- 20 × (−2) = −40.
- 150 + (−40) = 110.
Result
Mala earns 110 marks.
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