Cogito
Integrated Math 2 · Chapter 7 · Lesson 1
Reasoning and Proof
What counts as an argument.
12 problems · about 20 minutes · G-CO.C.9
What this lesson teaches
The student distinguishes inductive from deductive reasoning and identifies counterexamples.
- Inductive reasoning generalises from examples and suggests.
- Deductive reasoning derives from accepted facts and proves.
- One counterexample destroys a general claim.
Warm Up
Straightforward practice. Get the method working first.
5 problemsHow many counterexamples disprove a general claim?
Answer 1
Why 1.
Why is inductive reasoning not a proof?
Answer The next untested case could still fail.
Why Examples never exhaust the possibilities.
Generalising from examples is which kind? 1 inductive, 2 deductive.
Answer 1
Why From cases to a rule.
Deriving from accepted facts is which kind? 1 or 2?
Answer 2
Why A proof.
Do 100 confirming examples prove a general claim? 1 yes, 0 no.
Answer 0
Why The 101st might fail.
Build It Up
The same ideas with more to keep track of.
3 problemsEvery prime is odd. Which number disproves it?
Answer 2
Why The only even prime.
Every square number is even. Which of 4, 9 or 16 disproves it?
Answer 9
Why 3².
Is the converse of a true statement always true? 1 yes, 0 no.
Answer 0
Why Swapping the halves can break it.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each activity by the kind of reasoning it uses.
Answer Inductive: Measuring several triangles and noticing a pattern, Guessing a rule from a sequence · Deductive: Deriving a result from known theorems, Writing a two-column proof
Why One starts from cases and the other from established facts.
Every multiple of 3 is odd. Which of 6, 9 or 15 disproves it?
Answer 6
Why An even multiple.
The Counterexample: Every number ending in 1 is prime. Which of 21, 31 or 41 disproves it?
Answer 21
Why 21.
The Count: How many counterexamples are needed to disprove a general claim?
Answer 1
Why 1.