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Foundations of Math 11
Chapter 1 – Logical Reasoning
Notes Booklet
Homework Assignments
(Nelson Foundations of Mathematics 11 Workbook)
Chapter 1 Logical Reasoning
Section Topic
1.1
1.2
1.3
1.4
Review
1.5
1.6
1.7
Review
Assignment
Making Conjectures
Exploring Validity of Conjectures
Finding Counterexamples
Proving Conjectures
P5: 1-7, 10
P9: 1-4 + worksheet (included)
P11: 1-10
P15: 1-5, 7-8, 10-13
Invalid Proofs
Reasoning to Solve Problems
Analyzing Puzzles and Games
P19: 1-6
P24: 1-3, 5-6, 8-9, 12-13
P30: 1a, 2a, 3-10, 13
P34-35 + review handout
FOM11
1.1 Notes:
Making Conjectures: Inductive Reasoning
Definitions
conjecture:
inductive reasoning:
“Non-math” examples
Tables of Data
Examining an object & speculating about meaning or function
Math examples
Number Patterns
Geometry
Writing out your conjecture and your argument
Terms to Know
FOM11
1.2 Notes:
Exploring the Validity of Conjectures
Definitions:
counterexample
Disproving vs. Proving
Ex1 Every quadrilateral (4-sided shape) has diagonals that intersect each
other
The importance of definitions
Ex2
Every integer can be expressed as the sum of two prime numbers
FOM 11
1.1-1.2 Worksheet
1. The first three numbers in a pattern are 1, 2, 4. At this point, Jamie makes
a conjecture about the pattern of numbers and predicts that the next
number will be 8.
Explain what Jamie thought the pattern was.
2. In fact, Jamie guessed wrong and the next number is actually 7.
Make a new conjecture to explain the pattern of numbers: 1, 2, 4, 7…
What will the next three numbers be if your conjecture is correct?
3. Yasmin conjectures that when you multiply a two-digit number by 11, the
result is equal to just adding the two digits together, and inserting this
number in between the two digits:
11 12  132
11 17  187
For example:
11 23  253
11 36  396
Try more examples. Does your evidence support this conjecture or not?
If it does, explain why it appears to work. If not, explain at what point the
pattern breaks down.
4. In a triangle, an altitude is the line drawn straight down from one of the 3
corners (vertices) so that it is perpendicular to the opposite side, as shown
below:
Sasha conjectures incorrectly that, for any triangle, the three altitudes will
meet at a single point inside the triangle.
Use a ruler and a protractor to test some examples. In the space below,
show two examples that support Sasha’s conjecture and two
counterexamples.
Can you refine Sasha’s conjecture so that it appears valid?
FOM11
1.3 Notes:
Ex1
Finding Counterexamples
Michelle notices that 2 x 3 = 6; 5 x 6 = 30; and 8 x 9 = 72.
She conjectures that multiplying consecutive integers always results in a
multiple of 3.
Do you agree or disagree? Justify your decision.
FOM11
1.4 Notes:
Proving Conjectures: Deductive Reasoning
Definitions
proof:
generalization:
deductive reasoning:
Inductive vs. Deductive
17
18
19
Using unknowns
Ex1
representing even #s, odd #s
Ex2
“Number tricks”
Ex3
divisibility questions / expanded form
Classical Logic
Ex4
All Math teachers are short-sighted.
Mr. Nelson is a Math teacher.
Therefore, Mr. Nelson is short-sighted.
FOM11
1.5 Notes:
Invalid Proofs
Errors in Logical Arguments
Ex1
one of the premises isn’t true
Ex2
the conclusion doesn’t follow logically from the premises
All birds lay eggs. A turtle is not a bird. Therefore, turtles do not lay eggs.
Mathematical errors
Ex3
An Oxford professor (seriously!) on CBC radio’s show commemorating the
10th-year anniversary of the Sept 11 attacks actually said (although I am
paraphrasing): “23% of Americans believe that Barack Obama is secretly
a Muslim, and 46% of Americans believe Barack Obama is not a good
Christian, so right there you have almost 70% of Americans who question
the faith of the president.”
What basic logical mistake did he make?
Ex4
dividing by zero
Ex5
geometry riddles
FOM11
1.6 Notes:
Reasoning to Solve Problems
FOM11
1.7 Notes:
Analyzing Puzzles and Games