Exercise Practice
Exercise Set 9.1 · Q11
Source page 6 · MASTER
Distinct prime products and exactly four divisors
Test prime cubes as well as products of different primes.
AIMS reasoning checkpoint
Four divisors do not force two distinct primes.
GANITA MANJARI · GRADE 9 · PART II · CHAPTER 9
MASTER — Reasoning with Propositions and Converses
Think first. Use AI purposefully. Verify always.
01 · DISCOVER
Start by thinking.
A convincing answer needs more than a true/false label. What evidence would let someone check your verdict?
02 · UNDERSTAND
Reasoning with Propositions and Converses
Bring the chapter together: read a claim, identify its direction, test its logic and conclude with a reason or a counterexample. Use the mastery work to identify what needs another look, not to guess a label.
03 · KEY IDEAS
Keep the conditions clear.
- A correct converse preserves every condition.
- One valid counterexample defeats a universal claim; examples alone do not prove one.
- Number, figure and proof questions all need explicit reasons.
04 · VISUALISE
Follow the direction.
- Read the claim
- Test the direction
- Justify or counterexample
Conclude only after checking the condition and the reasoning.
05 · METHOD / REASONING
Make the argument checkable.
Read the claim → identify the direction → test the logic → justify or counterexample → conclude.
06 · ORIGINAL AIMS WORKED EXAMPLE
Follow each step.
A learner claims that a number with exactly four positive factors must be a product of two distinct primes.
- Given: a positive integer with four factors. To decide: whether the proposed conclusion always follows.
- Test 27: its positive factors are 1, 3, 9 and 27.
- But 27 = , not a product of two distinct primes. The claim is false.
- Verify: a product pq of distinct primes does have four factors, 1, p, q, pq; this true direction does not prove the reverse.
07 · AIMS METHOD
Your turn to reason.
- Attempt independently
- Investigate the idea
- Make connections
- Scrutinise your reasoning
What must accompany a false verdict?
Compare after your own attempt
A permitted example that meets the starting condition and fails the conclusion, with those two checks explained.
08 · PRACTICE SPRINT
Use the idea in a new situation.
Ten original AIMS questions: 3 Warm-up, 4 Build, 2 Think and 1 Challenge.
Start Practice Sprint →Check & Feedback: incorrect answers receive a conceptual hint and another attempt; correct answers include reasoning.
TEXTBOOK CONNECTION
Return to the source. Bring your reasoning.
Ganita Manjari · Grade 9 · Part II · Chapter 9. Use your textbook for the complete questions. These are topic references and original AIMS guidance, not reproduced exercises.
Exercise Practice
Exercise Set 9.1 · Q15
Source page 6 · MASTER
A proposed divisibility shortcut for 8
Separate “multiple of 8” from “multiple of both 2 and 4”.
AIMS reasoning checkpoint
A condition required in the forward direction may fail as a reverse test.
Exercise Practice
Exercise Set 9.1 · Q17
Source page 7 · MASTER
Quadrilateral diagonal conditions · Figure 9.2
In the first direction equal sticks are required but do not alone guarantee the category. In the reverse direction equal sticks suffice when a genuine quadrilateral is formed.
AIMS reasoning checkpoint
The second direction alone does not assert that every quadrilateral of the category has equal diagonals.