1. Making Predictions
Assume our class has 25 people, and the entire Sophomore class has 100 people –therefore it is 4 times larger.
Predict that next Monday, how many people in our class will:
Have red hair
Have Black or Brown hair
Have Blonde hair
Now predict next Monday how many people in the Sophomore class will:

2. Unit 4
This unit Introduces inductive and deductive reasoning, along with logic statements, converse/inverse/contrapositive, values of true/false and the Laws of Syllogism and Detachment.
It also addresses proofs, and sequences such as the Fibonacci sequence and the Golden Ratio.

3. Standards SPI’s taught in Unit 4:
SPI Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems.
SPI Analyze, apply, or interpret the relationships between basic number concepts and geometry (e.g. rounding and pattern identification in measurement, the relationship of pi to other rational and irrational numbers)
SPI Analyze different types and formats of proofs.
SPI Solve problems involving congruence, similarity, proportional reasoning and/or scale factor of two similar figures or solids.
CLE (Course Level Expectations) found in Unit 4:
CLE Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely in mathematical reasoning.
CLE Apply and adapt a variety of appropriate strategies to problem solving, including testing cases, estimation, and then checking induced errors and the reasonableness of the solution.
CLE Develop inductive and deductive reasoning to independently make and evaluate mathematical arguments and construct appropriate proofs; include various types of reasoning, logic, and intuition.
CLE Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial, graphical, tabular, and symbolic), to solve problems, to model mathematical ideas, and to communicate solution strategies.
CLE Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies.
CLE Use technologies appropriately to develop understanding of abstract mathematical ideas, to facilitate problem solving, and to produce accurate and reliable models.
CLE Establish the relationships between the real numbers and geometry; explore the importance of irrational numbers to geometry.
Recognize and apply real number properties to vector operations and geometric proofs (e.g. reflexive, symmetric, transitive, addition, subtraction, multiplication, division, distributive, and substitution properties).
CFU (Checks for Understanding) applied to Unit 4:
Check solutions after making reasonable estimates in appropriate units of quantities encountered in contextual situations.
Use inductive reasoning to write conjectures and/or conditional statements.
Use proofs to further develop and deepen the understanding of the study of geometry (e.g. two-column, paragraph, flow, indirect, coordinate).
Identify and explain the necessity of postulates, theorems, and corollaries in a mathematical system.
Analyze properties and aspects of pi (e.g. classical methods of approximating pi, irrational numbers, Buffon’s needle, use of dynamic geometry software).
Approximate pi from a table of values for the circumference and diameter of circles using various methods (e.g. line of best fit).
Compare and contrast inductive reasoning and deductive reasoning for making predictions and valid conclusions based on contextual situations.
Identify, write, and interpret conditional and bi-conditional statements along with the converse, inverse, and contra-positive of a conditional statement.
Analyze and create truth tables to evaluate conjunctions, disjunctions, conditionals, inverses, contra-positives, and bi-conditionals.
Use the Law of Detachment, Law of Syllogism, conditional statements, and bi-conditional statements to draw conclusions.
Use counterexamples, when appropriate, to disprove a statement.
Identify similar figures and use ratios and proportions to solve mathematical and real-world problems (e.g., Golden Ratio).

4. Inductive Reasoning
Inductive reasoning is based upon patterns you observe.
Inductive Reasoning is used to draw a General Conclusion based upon Specific Examples. Your conclusion may or may not be correct however.
For example: if you had the pattern 3,6,12,24, what would you predict would be the next number?
It would be 48. Each number in the pattern is multiplied times 2 to get the next number in the pattern.
Or would it. What other patterns could you create?

5. Conjecture
When you make a prediction, or a conclusion, based upon inductive reasoning, you are making a Conjecture
You can test a conjecture, but you can never 100 percent prove it is true.
For example, what is the next number in this pattern?
1,2,3,4,?
You use inductive reasoning (looking at these specific examples) and make a conjecture that the next number is 5
But what if the pattern really went like this 1,2,3,4,3,2,1?
You can predict the next number based upon the pattern, but you cannot prove it, because there may be parts of the pattern you have not seen yet.

6. Examples What are the next two terms in this pattern?
1,2,4,7,11,16,22…?,?
Monday, Tuesday, Wednesday…?,?
5,10,20,40…?,?
O,T,T,F,F,S,S,E…?,?
J,F,M,A,M…?,?

7. Summing Odd Numbers
What if you wanted to add the first four odd numbers?
In other words, you want to add
Well, you could easily add those numbers and get 16
But what if you wanted to add the first 83 odd numbers? ( …etc…)
Would you sit there and type … etc. on your calculator?
Hopefully, you would try to find a pattern

8. What’s special about each of these numbers?
Summing Odd Numbers
Numbers Total How many counted
Try to find a pattern
Here we see that we can rewrite the sums of odd numbers as “squares”
Remember, we wanted to add the first 4 odd numbers
Here, our answer is 42
So what would be the sum of the first 83 odd numbers? 1 = 12
1+3 4 22
1+3+5 9 32 16 42
What’s special about each of these numbers?
It would be 83 squared, or 6889

9. Summing Even Numbers That would be 75 x 76, or 5700
Numbers Total How
many counted
We can also sum even numbers in a similar pattern:
If we look at this, we see the sum of the first five even numbers is 5 x 6, or n (the number we counted) x (n+1) –the number we counted plus one
What if we wanted to sum the first 75 even numbers?
That would be 75 x 76, or 5700

10. Karl Gauss Karl Gauss is a famous German Mathematician (1777-1855)
When he was in 3rd grade, he figured out how to add all the numbers from 1 to 100 in ten seconds
How did he do it?
Hint, He figured out a pattern…
What was the pattern?

11. Summing All Numbers Therefore, Karl made this equation: N/2 x (N+1)
Of course you know I wrote a program for this. It is called SUMINT2 in programs on the calculator…
Let N = the number of numbers we add
Karl added 100 numbers, so N = 100
The real question is:
HOW MANY TIMES DID HE ADD 101?
He added “101” 50 times, until he got to the middle 
What looking at N = 100, what is “50” in relation to that?
It is ½ of it, or N/2
Therefore, Karl made this equation:
N/2 x (N+1)
Or  100/2 x (101) = 50 x 101 or 5050

12. Goldbach’s Conjecture
In the early 1700’s, a Prussian Mathematician named Goldbach noticed that even numbers greater than 2 can be written as the sum of two prime numbers.
Again, this is an example of Inductive Reasoning
Can we ever prove this Conjecture to be true?

13. Assignment
Page (Skip #2-5)

14. Unit 4 Quiz 1 Define Inductive Reasoning
Calculate the sum of the first 29 odd numbers
Calculate the sum of the first 37 even numbers
Calculate the sum of the first 46 numbers (both odd and even)
What is the next number in this sequence? Don’t start over, predict the next higher number) 1,1,2,3,5,8,13,21,34
Is Inductive Reasoning always accurate?
What is a conjecture?
What is the equation to calculate the sum of odd numbers?
What is the equation to calculate the sum of even numbers?
What is the equation to calculate the sum of all numbers?

15. Unit 4 Quiz 2
True/False: Inductive Reasoning is drawing a specific conclusion based on general reasoning.
Calculate the sum of the first 120 odd numbers
Calculate the sum of the first 150 even numbers
Calculate the sum of the first 200 numbers (both odd and even)
What is the next number in this sequence? 1,1,2,3,5,8,13,21,34
True/False: Inductive reasoning will always give you the correct answer if you do it right.
(fill in blank): When you make a prediction based on inductive reasoning , you are making a ___________
This equation calculates the sum of ______ numbers: n(n+1)
This equation calculates the sum of ______ numbers: n/2(n+1)
This equation calculates the sum of ______ numbers: n2

16. Fibonacci Sequence
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio".) Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics.
0,1,1,2,3,5,8,13,21,34,55,89

17. Another Application of Fibonacci
Fibonacci proposed a problem of rabbits: assuming that: a newly-born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on
Therefore –after one month you only have your first set of rabbits, but after two months you now have two sets of rabbits
The puzzle that Fibonacci posed was: how many pairs will there be in one year?

18. The Rabbit Problem
NOTE: For this to work, you have to start at F2 –where you begin increasing numbers the very next month. Therefore, the “4th” month is really F5
At the end of the first month, they mate, but there is still one only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
At the end of the nth month, the number of new pairs of rabbits is equal to the number of pairs in month n-2 plus the number of rabbits alive last month. This is the nth Fibonacci number. F0 F1 F2 F3 F4 F5 F6 F7 F8 F9
F10
F11
F12
F13
F14
F15
F16
F17
F18
F19
F20 1 2 3 5 8 13 21 34 55 89
144
233
377
610
987
1597
2584
4181
6765

19. Fibonacci and the Golden Ratio
One of the unique ideas found in math is called the golden ratio. This ratio is 1.618/1
One of the most common places we find this ratio is rectangles. A rectangle with the sides in ratio of – 1 is found to be pleasing to the eye
We will do a more in depth lesson on the golden ratio later, but it is interesting to note that the Fibonacci sequence can create the golden ratio

20. Fibonacci and The Golden Ratio1
Undefined 2 3
1.5 5 8
1.6 13
1.625 21 34 55 89
144
233
377
610
987
1597
2584
Here is a short series of Fibonacci numbers:
To calculate the golden ratio, you divide the number you want by the number before it in the sequence –for example 8 is divided by 5 ( = 1.6)
The higher you go, the closer you get to the exact golden ratio
Try this on the calculator

21. Fibonacci In Nature Romanesque Broccoli, Conch Shell, Pine Cone

22. Deductive Reasoning
Deductive Reasoning is also called logical reasoning
Deductive Reasoning is the process of taking a generally known fact, theorem or postulate (something we hold to be true) and applying it to a specific example.
For example, we know that gravity makes things fall. If I throw a ball into the air, I will logically reason that this one, specific ball will fall to the ground. I have applied a general theorem to a specific example.

23. Conclusion
Inductive Reasoning: Drawing a general conclusion based upon specific examples. Never 100 percent certain however
Deductive Reasoning: Drawing a specific conclusion based upon general rules or facts that we know to be true. If the facts are true, and the reasoning is sound, the conclusion will always be true too.

24. Examples
The train has been late 3 days in a row. You conclude it will be late today. Is this inductive, or deductive reasoning?
A carpenter calculates what materials he needs to build a shed. What kind of reasoning does he use?
Karl has a Chevrolet Monte Carlo. He races Mr. Bass in his Mustang, and Karl loses. He concludes that Mustangs must be faster than Monte Carlos. What kind of reasoning did Karl use?
Based upon data gathered by NASA over a period of years, Jim Lovell calculates the proper re-entry path for Apollo 13. What kind of reasoning did Mr. Lovell use?

25. Conditional Statement
A Conditional Statement is an If-Then Statement
Every conditional statement has two parts:
The part following the “if” is the hypothesis
The part following the “then” is the conclusion

26. Example -SUMINTGS This is a screen capture from my calculator
You are looking at programming code
Input C asks whether you want to solve an Odd, Even, or All numbers problem
If C = 1, then what type of problem will it solve?
If C does not equal 1, then calculator will not run this problem
This is an example of “If-Then” logic

27. Example Identify the hypothesis and conclusion in these statements:
If today is the first day of fall, then the month is September
If y-3 = 5, then y = 8
If you are not completely satisfied, then you get your money back

28. Writing Conditionals
Write this as a conditional: “A tiger is an animal.”
“If it is a tiger, then it is an animal.”
Write this as a conditional: “A rectangle has four right angles.”
“If it is a rectangle, then it has four right angles.”

29. Truth Values
Conditional statements have truth values of either True or False
To show that a conditional is true, you must show that every time the hypothesis is true, the conclusion is true as well
To show that a conditional is false, all you have to do is prove it is not true once
To do this, we use a “Counterexample”

30. Example
Conditional: “If it is February, then there are only 28 days in the month.”
What is the counterexample?
Leap Year
If you are at Houston County High School, you must live in Houston County
Mr. Bass lives in Clarksville (Montgomery County) and he’s at HCHS

31. Venn Diagrams This is how we read this:
All residents of Chicago are residents of Illinois, but all residents of Illinois are not residents of Chicago
This is how we make it into a conditional statement: If you are a resident of Chicago then you are a resident of Illinois
Residents of Illinois
Residents of Chicago

32. Converse of Conditional
Converse: Switches the hypothesis and the conclusion
Conditional: If two lines intersect to form right angles, then they are perpendicular
Converse: If two lines are perpendicular, then they intersect to form right angles
Write the converse of this conditional:
“If two lines are not parallel and do not intersect, then they are skew”

33. Truth Values of Converses
Consider this conditional statement:
If a figure is a square, then it has four sides.
What is the truth value of this statement?
This is true. There is no square without four sides. There is no counterexample.
Now write the converse:
If a figure has four sides, then it is a square.
This is false. Rectangles have four sides, but they aren’t all square

34. Examples If two lines do not intersect, then they are parallel
What is the truth value of this statement?
If it is false, what is the counterexample?
What is the converse of this statement?
If two lines are parallel, then they do not intersect

35. Examples If x = 2, then |x| = 2 True? What is the converse?
Hint: Absolute value problems always have a counterexample (the negative)

36. Examples If x = 4, then x2 = 16 True? What is the converse?
Hint: “square” problems always have a counterexample (the negative)

37. Assignment
Page
Worksheet 2-1
Worksheet 2-2

38. Unit 4 Quiz 3 -In your Own Words (no copying word for word from notes)
Define these terms:
Inductive Reasoning
Deductive Reasoning
Conjecture
Conditional Statement
Hypothesis
Conclusion
Fibonacci Sequence –list the first 12 terms (start with 1)
Which type of reasoning will have a true conclusion and why
Which type of reasoning may be false reasoning, and why
Write an example of a conditional statement

39. Symbols
Here is the short-hand symbols for conditionals and converses:

40. Assignment
Page odd (Guided Practice)

41. Unit 4 Quiz 2 Define Counter-example Define Converse
Tomas watches “Saving Private Ryan” and is amazed at how good actor Vin Diesel is in this movie. He concludes that Vin Diesel is an awesome actor. What type of reasoning is Tomas using?
Use a counter example to prove Tomas is incorrect.
Rachel casually holds a water balloon out of her 2 story window, and it falls to the ground when she lets go. She predicts this will happen every time. What type of reasoning is Rachel using?
Cliff sees a Chevy rusting in a field, and concludes that all Chevys end up rusting in fields. What type of reasoning is Cliff using?
Tyrone throws a ball into the air as far as he can. He really chucks it up there. Because it went out of sight, he figures it’s gone and walks away. The ball hits him in the head 2 seconds later as he’s walking. What kind of reasoning should Tyrone have used?
Kelsey predicts that Alabama will win another National Championship. She bases this prediction on the fact that Alabama has won 13 championships –way more than any other school, and the fact that Alabama is the current National Champion. What kind of awesome reasoning is Kelsey using?
Write the converse for this statement: “If Tennessee wins at football, then it’s a miracle.”
Give a counter-example for this statement: “If √81 = x, then x = 9”

42. Biconditionals
**If both a conditional, and its converse are true, they are combined into a Biconditional
**We use the words “if and only if” to combine a conditional and its converse
Conditional: If two angles have the same measure, then they are congruent (true)
Converse: If two angles are congruent, then they have the same measure (true)
Biconditional: Two angles have the same measure if and only if the angles are congruent (true)
**NOTE: A Biconditional sentence does NOT START WITH IF! It DOES NOT have the word THEN in it!
**We abbreviate If and only If as IFF

43. Symbology

44. Example Write this as a biconditional:
Perpendicular lines are two lines that intersect to form right angles
Two lines are perpendicular IFF they intersect to form right angles
A right angle is an angle whose measure is 90 degrees
An angle is a right angle IFF its measure is 90 degrees

45. Law of Detachment
Suppose a mechanic knows that generally speaking, if a car has a dead battery, the car won’t start. The mechanic begins working on Tom’s car, and determines that the battery is dead.
What does the mechanic conclude?
Now suppose the mechanic begins working on another car, and finds that the car won’t start. Can the mechanic conclude that the car has a dead battery?

46. What we Learn
In the first example, we have a hypothesis and a true conclusion applied to an example –car batteries and cars that won’t start
We apply this hypothesis to a new situation (different car) and we can draw the same conclusion –car won’t start
We CANNOT apply the conclusion (car won’t start) to a new situation, and assume the hypothesis is the same (dead battery)

47. Law of Detachment
If you are given a true conditional, then you can apply that to a new conditional statement that has the same qualities/properties, and draw the same true conclusion
Example: If three points are on the same line, then they are collinear
New conditional hypothesis: Points A,B,C are on the same line
By the Law of Detachment, you can conclude that points A,B,C are collinear
Conditional Statement, If Points A,B,C are on the same line, then they are collinear

48. Example
If M is a midpoint of a segment, then it divides the segment into two congruent segments (true statement)
Given: M is the midpoint of segment AB –Now we have a specific example. We can turn this into a conditional statement:
If M is the midpoint of segment AB, then it divides segment AB into two congruent segments

49. Example
Example: If it is snowing, then the temperature is less than or equal to 32 degrees. This is a true statement.
Given: It is 20 degrees outside
You conclude that the temperature is equal to, or less than 32 degrees
Is this correct? Why or why not?

50. Example Given: If a road is icy, then driving conditions are hazardous
Given: Driving conditions are hazardous
Is it possible to use the Law of Detachment to draw a conclusion?

51. Law of Syllogism
The Law of Syllogism allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement
Or as Mr. Bass calls it “This is called Cutting Out the Middle”

52. Example
Given: If a number is prime, then it does not have repeated factors
Given: If a number does not have repeated factors, then it is not a perfect square
By the Law of Syllogism: If a number is prime, then it is not a perfect square
---Hey diddle diddle, cut out the middle ;)

53. Example If you live in Little Rock, then you live in Arkansas
If you live in Arkansas, then you live in the 25th state to enter the union
By the Law of Syllogism: If you live in Little Rock, then you live in the 25th state to enter the union

54. Inverse
An “Inverse” is when you negate (make the opposite) both the hypothesis and conclusion of a conditional statement
Conditional: If two angles are congruent, then their measure is equal
Inverse: If two angles are not congruent, then their measure is not equal
The easiest way to do this is to add the word “Not” or “don’t”

55. Contrapositive
First, switch the hypothesis and conclusion. What do we call this?
Converse
Then negate both the Hypothesis and Conclusion. What do we call this?
Inverse
Therefore, a contrapositive is when you take the Inverse of the Converse

56. Example Conditional: If you live in Toronto, then you live in Canada
Converse: if you live in Canada, then you live in Toronto
What is the Inverse?
If you don’t live in Toronto, then you don’t live in Canada
What is the Contrapositive?
Contrapositive (inverse of the converse): If you don’t live in Canada, then you don’t live in Toronto.

57. Symbology Statement Example Symbolic Form You Read it Conditional
If an angle is a straight angle, then its measure is 180
p --> q
If p, then q
Converse
If the measure of an angle is 1800, then it is a straight angle
q-->p
If q, then p
Biconditional
An angle is a straight angle IFF its measure is 1800
P<-->q
P If and only If q
Negation (of p)
An angle is not a straight angle ~p
Not p
Inverse
If an angle is not a straight angle, then its measure is not 180
~p --> ~q
If not p, then not q
Contrapositive
If an angle’s measure is not 180, then it is not a straight angle
~q -->~p
If not q, then not p

58. Assignment
Page

59. Bellringer If you are happy, then you have joy. Write the converse
Write the Inverse
Write the Bi-conditional
Write the Contrapositive
Hunter has joy. Based on the Law of Detachment, can you draw a conclusion?
If you are happy, then you have joy. If you have joy, then you smile. Using the Law of Syllogism, write the conditional statement based on these two conditional statements.

60. Unit 4 Quiz 3 Definitions Match the Definition If-Then statement
General to Specific Logic
The part after If
A statement which uses IFF
A statement which negates a conditional statement
The part after Then
Specific to General Logic
Switch the hypothesis and conclusion
To make “not” or “don’t”
A statement which negates the Converse
Conditional statement
Converse statement
Inverse statement
Bi-Conditional statement
Contrapositive statement
Inductive Reasoning
Deductive Reasoning
Negation
Hypothesis
Conclusion

61. Unit 4 Final Extra Credit 2 Points each show all work
Paige analyzes this picture, and
concludes that X is 110 degrees
What kind of reasoning did Paige use?
Is Paige right?
What is X?
Blake and Hannah are arguing. Blake says the sum of the first 100 odd numbers is greater than the sum of the first 100 even numbers. Hannah says he has it backwards.
Who is right?
What are the sums for each?
125 2 3 4
X+ 15 7 8 5