Download Reasoning Student Notes

CHAPTER 1: INDUCTIVE AND DEDUCTIVE REASONING
1. Making Conjectures – pg. 6-15 (1 class)
Assignment: pg. 12-15 #1-5, 7-13,16, 19, 20
2. Using Reasoning to Find a Counterexample – pg. 18-26 (1 class)
Assignment: pg. 22-26 #1-5, 8, 10, 12, 14, 17, 18
3. Deductive Reasoning – pg. 27-33 (1 class)
Assignment: pg. 31-33 #2-5, 7-10, 16, 17
4. Mid-Unit Review – pg. 34-35 (1 class)
Assignment: pg. 35 #1-9, 11
5. Proofs that are not Valid – pg. 36-44 (1 class)
Assignment: pg. 42-44 #1-3, 5-8, 10
6. Problem Solving – pg. 45-51 (1 class)
Assignment: pg. 48-51 #1-3, 6ab, 7-9, 11, 12, 14
7. Analyzing Puzzles and Games – pg. 52-57 (1 class)
Assignment: pg. 55-57 #3-5, 6ab, 7-9, 11, 12, 14
8. Chapter Quiz and Learning Outcomes
9. Chapter Review – pg. 61-62
Assignment: pg. 61-62 #1-11, 13-16
10. Chapter Exam
1
LESSON 1: INDUCTIVE REASONING
Learning Outcome: Learn to use reasoning to make predictions.
Conjecture: A testable expression that is based on available evidence but is not yet
proved.
Inductive reasoning: Draw a general conclusion by observing patterns and identifying
properties in specific examples.
Complete the following investigation with a partner:
Frank has been patterning with equilateral triangles.
Figure 1
Figure 2
Figure 3
Frank thinks that Figure 10 in this pattern will have 100 triangles, and all these triangle will be
congruent to the triangle in figure 1.
How did Frank arrive at this conjecture? (Discuss with a partner what conjecture means)
a. Organize the information about the pattern the table below:
Figure
Number of Triangles
1
1
2
4
3
4
b. What do you notice about the data in the table?
c. Extend the pattern for two more figures.
d. What numeric pattern do you see in the table?
2
5
6
7
8
e. Is Frank’s conjecture reasonable? Explain.
f. How did Frank use inductive reasoning to develop his conjecture?
g. Is there a different conjecture you could make based upon the pattern you see? Explain.
The difference between the number of triangles increases each time by consecutive odd
numbers.
What conjecture could you make for the following situations:
a. Eli has lunch at the same deli every day for a week. Each day, the “Special of
the Day” has included fries. Based on this, what can Eli conclude about his next visit to
the Deli?
b. Justin arrives at his first block class at 8:10 am. Every day for the past three
days, his teacher has arrived between 8:12 and 8:14 am and the class starts at 8:15.
Therefore we can conclude that tomorrow his block one teacher…….
With a partner, use inductive reasoning to make a conjecture about the following
situations:
a. Suppose you add two different odd numbers; note the result. Repeat the
experiment several times. What conclusion can you make?
3
b. Multiply an even number by an odd number and note the result. Repeat the
experiment several times. What conjecture can you make?
.
c. Construct several scalene triangles. Measure each of the angles and each of
the sides. Note the relationship between the smallest angle and the measure of the
smallest side. What conjecture can you make?
d. Construct several pairs of intersecting lines. Measure the pairs of vertically
opposite angles. What conjecture can you make?
Does a conjecture prove a theory?
Can you prove it for every case?
Assignment: pg. 12-13 #1-5, 7-13,16, 19, 20
4
LESSON 2: COUNTEREXAMPLE TO CONJECTURES
Learning Outcome: Learn to invalidate a conjecture by finding a contradiction.
What is a counterexample?
With a partner, find a counterexample for each of the following situations:
a. If you live in a country bordering the United States, then you live in Canada.
b. If a quadrilateral has four right angles, then it is a square.
c. Odd whole numbers less than 10 are prime.
d. All birds can fly
e. All cats are born with a tail.
Ex. Consider the conjecture:
The sum of two prime numbers is an even number.
a. Give two examples for which the conjecture is true.
b. Give one counterexample that shows the conjecture is false.
5
Ex. Matt found a interesting numeric pattern:
1x8+1=9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
Matt thinks that this pattern will continue.
Search for a counterexample to Matt’s conjecture.
1
2
3
4
5
6
7
8
9
10
A
1x8+1
12 x 8 + 2
123 x 8 + 3
1234 x 8 + 4
B
9
98
987
9876
Ex. Give two examples that support each conjecture, and then one counterexample
that shows the conjecture is false.
a. The sum of two perfect squares is an even number.
b. The square root of a number is smaller than the number.
Assignment: pg. 22-25 #1-5, 8, 10, 12, 14, 17, 18
6
LESSON 3: DEDUCTIVE REASONING
Learning Outcome: Learn to prove mathematical statements using a logical argument.
What are the limitations of inductive reasoning?
How do you think deductive reasoning will differ?
Deductive Reasoning: Drawing a specific conclusion through logical reasoning by
starting with general assumptions that are known to be valid.
Examples:
a. Sam transferred to a new high school. A student must attend school for at
least six months before being eligible to play on a school team. Therefore, before he
can play on a school team,… (he must attend the school for six months).
b. Between the hours of noon and 2:00 pm., a bank offers to give you $5 if no
one serves you within 5 minutes. Sarah arrived at 1:30 pm, and no one helped her until
1:42 pm. Therefore, …
c. Paulette lives in Medicine Hat. Medicine Hat is in Alberta. Therefore, …
With a partner, create an example of deductive reasoning.
7
Opposite Angles Theorem:
When two lines intersect the opposite angles are equal.
1
2
3
4
∠1 = ∠4
∠2 = ∠3
We can deductively prove that this is the case:
∠1 + ∠3 = 180˚ (supplementary angles)
∠1 + ∠2 = 180˚ (supplementary angles)
∠1 = ∠ 1
∴ ∠2 = ∠3
Similarly, we can use deductive reasoning to show that ∠1 = ∠ 4:
∠1 + ∠2 = 180˚ (supplementary angles)
∠2 + ∠4 = 180˚ (supplementary angles)
∠2 = ∠ 2
∴ ∠1 = ∠4
8
Ex. Mike approached his friend Jack saying that he had a mathematical magic trick.
He handed Jack a piece of paper and a pencil and had him complete the following
steps:






Select a positive number
Triple the number
Subtract two
Add eight
Divide by three
Subtract the original number
Mike said the answer will always be 2.
a. Use inductive reasoning to test the trick to see if Mike’s magic trick will work:
b. Use deductive reasoning to show that the answer will always be two regardless of
the original number.






Select a positive number
Triple the number
Subtract two
Add eight
Divide by three
Subtract the original number
x
By using a variable, x, to represent the chosen number, we have proven deductively
that this algorithm will produce an answer of two regardless of the number chosen.
9
Write an expression for an even number:
Write an expression for an odd number:
Class discussion:
What was the learning outcome for today’s class?
State the difference between inductive and deductive reasoning.
Assignment: pg. 31-33 #2-5, 7-10, 16, 17
Mid-Unit Review: pg. 35 #1-9, 11
10
LESSON 4: PROOFS THAT ARE NOT VALID
Learning Outcome: Learn to identify errors in proofs.
Given the following argument, can you identify any errors or faulty logic?
Professional athletes do not compete in more than one sport. Bo Jackson played
football for the Oakland Raiders. Therefore, Bo Jackson has not played in any other
professional sports.
Invalid Proof: A proof that contains an error in reasoning or that contains invalid
assumptions.
Ex. Jane claims she has proved that -5 = 5.
Jane’s proof:
I assumed that -5 = 5
Then I squared both sides: (-5)² = 5²
I got a true statement: 25 = 25
This means that my assumption, -5 = 5, must be correct.
Where is the error in Jane’s proof?
If an assumption is not true, then any argument that was built on the assumption is not
valid.
Circular Reasoning: An argument that is incorrect because it makes use of the
conclusion to be proved.
11
Ex. Wayne is trying to prove the following number trick:
Choose any number. Add 3. Double it. Add 4. Divide by 2. Take away the number
you started with.
Each time Wayne tries the trick, he ends up with 5. His proof, however, does not give
the same result.
Wayne’s proof:
Choose any number:
n
Add 3:
n+3
Double it:
2n + 6
Add 4:
2n + 10
Divide by 2:
2n + 5
Take away the number you started with:
n+5
Try the number trick yourself. Does it work?
Where is the error in Wayne’s proof?
Ex. Critique the following “questionable proof” pointing out errors and then use the
direct proof (just get started and keep going) method to come up with a correct proof.
a) Conjecture: the square of an odd number is an odd number
Proof 1
12 = 1
Therefore, no matter what odd number we take the squares and the
32 = 9
answer will end in a 1, 5 or 9. Therefore an odd number
52 = 25
72 = 49
92 = 81
…2 = 1
Critique:
12
Proof 3: Correct Proof
Let x be any integer
Then we can write an odd integer as 2x + 1
(odd)2
=(2x+1) 2
Since 2 multiplied by any number is even,
=4x2+4x+1
add 1, result is an odd number
=2(2x2+2x)+1
When evaluating proofs, there are several common errors. They are,
1.
2.
3.
4.
5.
Dividing by zero
Taking the square root of both sides and not considering the + and – possibilities.
Faulty assumption
Using only Inductive reasoning
Problems in basic math (add, sub, mult, divide)
Assignment: pg. 42 -44 #1-3, 5-8, 10
13
LESSON 5: REASONING TO SOLVE PROBLEMS
Learning Outcome: Learn to solve problems using inductive or deductive reasoning.
With a partner, complete the following investigation:
Emma was given this math trick:





Choose a number
Multiply by 6
Add 4
Divide by 2
Subtract 2
Emma was asked to use inductive reasoning to make a conjecture about the
relationship between the starting and ending numbers, and then use deductive
reasoning to prove that her conjecture is always true. Here is her response to the
problem:
Inductive Reasoning:
#
5
-3
x6
+4
÷2
30
34
17
-18
-14
-7
0
4
2
24 144
148
74
What conjecture do you think Emma made?
-2
15
-9
0
72
Deductive Reasoning:
I chose d.
Then I multiplied, added, divided, and subtracted to get an expression.
6𝑑 + 4
(
)−2
2
It simplified to 3d.
Explain why Emma might have chosen the values she did?
14
What details are missing from the deductive reasoning Emma used to arrive at the
expression 3d?
How does it help to understand the mathematics when both symbols and words are
used in an explanation?
Ex. The members of a recently selected basketball team met each other at their first
team meeting. Each person shook the hand of every other person. The team had 10
players and 2 coaches. How many handshakes were exchanged?
What do we need to know about handshakes?
Is it easier to use deductive or inductive reasoning?
Inductive, observing patterns.
15
Ex. Sue signed up for games at her school’s fun night. Seven other people were
assigned to her group, making up four pairs of partners. The other members of her
group were Dave, Angie, Josh, Tanya, Joy, Stu, and Linus. When the games started,
Dave and his partner were to the left of Stu. Across from Dave was Sue, who was to
the right of Josh. Dave’s brother’s partner, Tanya, was across from Stu. Joy was not
on Stu’s right. Name the four pairs of partners.
Make a rectangle for the table and a list of the student’s names.
Dave
Angie
Josh
Tanya
Joy
Stu
Linus
Sue
Was inductive or deductive reasoning used?
Deductive, based on facts.
Assignment: pg. 48-51 #1-3, 5-9, 15, 17
16
LESSON 6: ANALYZING PUZZLES AND GAMES
Learning Outcome: Learn to determine, explain, and verify a reasoning strategy to
solve a puzzle or to win a game.
Discuss the following scenario with a partner:
Three students are playing a game. Two of the students flip a coin, and the third
student records their scores. Student 1 gets a point if the result is two heads, student 2
gets a point if the result is two tails, and student 3 gets a point if the result is a head and
a tail. The first student to get 10 points wins. Explain whether you would prefer to be
student 1, student 2 or student 3.
Try playing the game 2-3 times before making your final conclusion.
Did your reasoning involve inductive or deductive reasoning?
With a partner, try the game below then analyze the game to see if there can be any
strategy to use.
Toothpick Game: (2 players)
Place a pile of 20 toothpicks on a desk. On each turn a player can take one or two
toothpicks from the pile. The player to remove the last toothpick is the winner. Flip a
coin to determine the starting player. Is there a strategy you can use to ensure you win
the game?
Did your reasoning for strategy involve inductive or deductive reasoning?
17
Magic Squares
Around 2250 B.C. Emperor Yu of China discovered a turtle in the yellow river. On its
shell were some curious markings. On closer inspection they formed a 3 x 3 square
with the numbers 1 to 9 inclusive. At least, so goes the story of the first magic square
known at the Lo Shu or yellow river writing.
The “magic” in a magic square is that each row and each column add up the same
number…15 as do the two diagonals.
What are the numbers of Lo Shu’s 3 x 3 magic square? (only use each number once)
4
7
8
Sudoku
The game Sudoku dreamt up in 1979 by Howard Garns in New York is a type of latin
square.
Underlying a puzzle is a 9 x 9 latin square, into which the digits 1 to 9 must be written.
The extra rule comes from the grid being subdividing into nine 3x3 blocks each of these
too must contain the numbers 1 to 9.
The sudoku begins with a few numbers already in place, these are the clues. The
challenge is to complete the entire grid.
An interesting fact one needs a minimum of 17 clues to get a unique (only 1) solution.
That means <17 clues – multiple solutions
1
4 2
1
8 6
3
6 7
2 5
3
7 2 5
5 6 9
8
1 2 7 3
6 5
9
5 2 4
3 6
3
5
4 1 2 6
7
8
3
4 2
1 8 7
18
6
4
9
8
7
The Strimko Object
The object of the puzzle is to fully fill in the given grid with missing numbers observing
three simple rules. You have numbers 1 through 3 for a 3 x 3 grid; 1 through 4 for a 4 x
4 grid; 1 through 5 for a 5 x 5 grid; 1 through 6 for a 6 x 6 grid; and so on.
Rule #1
Each row must contain different numbers.
Rule #2
Each column must contain different numbers.
Rule #3
Each stream must contain different numbers.
Assignment: pg. 55-57 #3-5, 6ab, 7-9, 11, 12, 14
Chapter Quiz
Chapter Review pg. 61-62 #1-11, 13-16
Chapter Exam
19