A. What is Proof?
Math 20: Foundations
FM20.2
Demonstrate understanding of inductive and deductive
reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
Time to Get Started!
 The Mystery of the Mary Celeste
p.4
 Work through with a partner. You have 15 minutes.
 What DO YOU Think?
p.5
1. What is Enough Proof?
 FM20.2
 Demonstrate understanding of inductive and deductive




reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
1. What is Enough Proof?
 Conjecture – A testable expression that is based on available
evidence but is not yet proved.
 Inductive Reasoning - Drawing a general conclusion by
observing patterns and identifying properties in specific
examples.
 Investigate the Math
p.6
Example 1
Example 2
 Is this conjecture convincing? Why or why not?
Example 3
Example 4
 Summary p. 12
Practice
 Ex. 1.1 (p.12) #1-14
#3-19
2. How to Prove Conjectures
 FM20.2
 Demonstrate understanding of inductive and deductive




reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
2. How to Prove Conjectures
 Turn to page 16.
 Choose 2 illusions and develop a conjecture by looking at the
illusions
 Next prove the conjecture.
 Your brain can be deceived that is why Inductive Reasoning
can lead you to a conjecture but can not prove it for all cases.
Practice
 Ex. 1.2 (p.17) #1-3
3. What is a Counterexample?
 FM20.2
 Demonstrate understanding of inductive and deductive




reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
3. What is a Counterexample?
 When points on the circumference of circle are joined by
chords the circle is divided into sections
 Conjecture: As the number of connected points on the
circumference of a circle increases by 1, the number of
regions created within the circle increases by a factor of 2.
 How can we prove this?
 When we try 6 points on the circumference of a circle it does
not increase by a factor of 2
 This is called a counterexample because it disproves our
conjecture.
 Why is only one counterexample enough to disprove a
conjecture?
Example 1
Previously we made 2 conjectures about the difference between
consecutive perfect squares.
 The difference between two consecutive perfect squares is a
prime number.
 The difference between two consecutive perfect squares is an
odd number.
Find a counterexample for each conjecture.
 Find another counterexample for the first conjecture.
 Can you find a counterexample for the second conjecture?
Example 2
 Summary p.22
Practice
 Ex. 1.3 (p.22) #1-16
#3-19
4. Proving for All Cases
 FM20.2
 Demonstrate understanding of inductive and deductive




reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
4. Proving for All Cases
 Proving something to be true for all cases by drawing a
specific conclusions through logical reasoning by starting
with general assumptions that are known to be valid is called
Deductive Reasoning
 So what does this mean? Lets look at the following
conjecture.
𝑆 = 𝑥 − 2 + 𝑥 − 1 + 𝑥 + 𝑥 + 1 + (𝑥 + 2)
This is referred to as a generalization because it is something we
know to be true, in general.
 What type of reasoning did Jon use?
 What type of reasoning did we use? How does this differ
from what Jon used?
Example 1
 We have been looking at the difference between 2
consecutive perfect squares. We last time came up with the
conjecture that the difference between 2 consecutive perfect
squares is an odd number.
 Let’s prove this for all cases.
Example 2
a)
b)
 Transitive Property: If two quantities are equal to the same
quantity, then they are equal to each other. If a = b and
b = c, then a = c.
Example 3
 The process we just used to complete the previous example
is referred to as a Two Column Proof
Example 4
 Summary
p. 31
Practice
 Ex. 1.4 (p.31) #1-14
#4-18
5. Finding Holes in you Proofs!
 FM20.2
 Demonstrate understanding of inductive and deductive




reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
5. Finding Holes in you Proofs!
 Invalid Proof – A proof that contains an error in reasoning or
that contains invalid assumptions.
 Premise – A statement assumed to be true.
 Circular Reasoning – An argument that is incorrect because
it makes use of the conclusion to be proved.
 Investigate the Math
p.36
Example 1
Example 2
Example 3
Example 4
 Is there a number that will not work in Hossai’s number
trick? Explain.
Example 5
 Summary
p.41
Practice
 Ex. 1.5 (p.42) #1-7
#3-10
6. Solving Problems with Reasoning
 FM20.2
 Demonstrate understanding of inductive and deductive




reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
6. Solving Problems with Reasoning
 Try the Trick for some different numbers and see if it is true.
 Use deductive reasoning to try and prove the number trick.
 Investigate the Math
p.45
 Investigate the Math
p.45
Example 1
 Did we use inductive reasoning or deductive reasoning?
Example 2
 Did we use inductive reasoning or deductive reasoning?
 Summary
p.48
Practice
 Ex. 1.6 (p.48) #1-14
#3-16
7. Some Puzzles and Games
FM20.2
Demonstrate understanding of inductive and deductive
reasoning including:
analyzing conjectures
analyzing spatial puzzles and games
providing conjectures
solving problems.
7. Some Puzzles and Games
 Start with one marker each, then 2, then 3 and continue until 5
markers. Find the minimum number of moves for each situation and
record the results in a table
 Investigate the Math
p. 52
Example 1
 Summary p. 55
Practice
 Ex. 1.7 (p. 55) #1-13
#4-15