Download Inductive Reasoning shorter

Math 20-2
Reasoning: Lesson #1 (alt – shorter)
Making Conjectures: Inductive Reasoning
Objective:
-
By the end of this lesson, you should be able to:
make conjectures by observing patterns in data.
solve a contextual problem involving inductive reasoning.
explain why inductive reasoning may lead to a false conjecture.
provide and explain a counterexample to a given conjecture.
Vocabulary:
 Inductive Reasoning –

Conjecture –
e.g. 1) The following table shows the total precipitation by month for Edmonton from 2006 to
2010.
Total Precipitation in Edmonton, AB (mm)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
2.1 17.8 34.5 9.6 63.2 52.2 61.1 55.9 80.4 38.9 46.0 11.8
2006
12.0 20.2 6.7 37.3 81.9 87.0 50.6 39.4 31.1 2.2 13.0 15.2
2007
16.0 10.5 10.2 52.8 50.4 46.2 35.8 27.4 21.8 10.6 2.6 24.6
2008
18.8 15.3 26.8 18.7 16.4 15.2 65.6 30.2 5.8 29.5 12.0 38.9
2009
8.6 1.2 4.2 41.0 105.9 79.4 146.7 38.8 40.5 7.8 15.8 21.2
2010
Average 11.5 13.0 16.5 31.9 63.6 56.0 72.1 38.3 35.9 17.8 17.9 22.3
Source: Environment Canada, National Climate Data and Information Archive
a) Make a conjecture based on the data.
b) Describe a situation in which making a conjecture based on this data would be useful.
e.g. 2) Make a conjecture about the product of two odd integers.
Math 20-2
Reasoning: Lesson #1 (alt – shorter)
e.g. 3) Consider the pattern of multiplication:
1 1 
11 11 
111 111 
1111 1111 
a) Use your answers above to predict the answers to 11111 11111 
111111 111111 
b) Use this information to make a conjecture.
Conclusions reached by inductive reasoning may or may not be true. Inductive reasoning can be
misleading or even destructive.
For example:
- optical illusions
- stereotypes/prejudices
Vocabulary:
 Counterexample –
Only _______ counterexample is required to prove that a conjecture is false.
e.g. 4) Find a counterexample for the following conjectures:
a) Every month has at least 30 days.
b) Math teachers are always women.
c) Every quadrilateral with four equal sides is a square.
d) When you multiply any number by itself, the result is always bigger than or equal to
the original number.
Math 20-2
Reasoning: Lesson #1 (alt – shorter)
* VERY IMPORTANT: Inductive reasoning does NOT prove a conjecture, unless you check
every possible example (which is usually not practical). There may be another example
that you didn’t check that could be false. Finding more examples that support a
conjecture strengthens a conjecture, but does not prove it.
In other words, a conjecture is not proven simply because you can’t find a counterexample.
e.g. 5) Goldbach’s Conjecture:
In 1742, a German mathematician named Christian Goldbach conjectured that every even
number greater than 2 can be written as the sum of two prime numbers. Show Goldbach’s
conjecture to be true for the following integers:
a) 14
b) 66
No counterexample has been found, but no one has been able to prove that it is true for
every even number greater than 2. It is a famous unsolved problem in mathematics!
So… it is still called a conjecture, not a true mathematical fact (theorem).
e.g. 6) Consider the following fourth powers:
24 
44 
64 
84 
a) What conjecture could you make based on these examples?
b) Is this conjecture true? Why or why not?
c) How could you modify the conjecture to make it true?
Assignment:
p. 12-14 #2-3, 5, 7-9, 11
p. 22-25 #1-5, 7, 12, 16-18