Geometry is a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; the study of properties of given elements that remain invariant under specified transformations.
Basically what that means is geometry is the study of the laws that govern the patterns and elements of mathematics.
Definition from Merriam-Webster Online Dictionary.
An explorer wishes to cross a barren desert that requires 6 days to cross. He can only carry his equipment and clothing and nothing else. If one man can only carry enough food for 4 days, what is the fewest number of men traveling on this exploration?
Explorer and 2 men carrying food.
Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument .
So deductive reasoning either states laws and/or conditional statements that can be written in if…then form.
There are two laws that govern deductive reasoning .
If the logical argument follows one of those laws, then it is said to be valid , or true.
A farmer has a fox, goose and a bag of grain, and one boat to cross a stream, which is only big enough to take one of the three across with him at a time. If left alone together, the fox would eat the goose and the goose would eat the grain. How can the farmer get all three across the stream?
Goose ----- Fox & Grain Goose Trip 1 All ----- ----- Goose Trip 4 Fox & Grain ----- Goose Grain Trip 3 Fox Goose Grain Fox Trip 2 Leave New Take Back Leave Behind Take Over
Sam, Maria, Tim, and Julie are all skilled at the video game Alien Invaders. Julie scores consistently lower than Tim. Sam is better than Maria, but Maria is better than Tim. Who is the better player, Julie or Maria?
Robert is shopping in a large department store with many floors. He enters the store on the middle floor from a skyway, and immediately goes to the credit department. After making sure his credit is good, he goes up three floors to the housewares department. Then he goes down five floors to the children’s department. Then he goes up six floors to the TV department. Finally, he goes down ten floors to the main entrance of the store, which is on the first floor, and leaves to go to another store down the street. How many floors does the department store have?
Deductive v Inductive Reasoning
Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a proof.
This is often called a logical argument .
Inductive reasoning uses patterns of a sample population to predict the behavior of the entire population
This involves making conjectures based on observations of the sample population to describe the entire population.
Examples of Deductive and Inductive Reasoning
Andrea knows that Robin is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. Therefore, Andrea reasons inductively that Todd is older than Robin based on past observations.
Andrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. Andrea reasons deductively that Todd is older that Robin based on accepted statements.
Lesson 1.1 – Logic
Lesson 1.2 Patterns
Lesson 1.2 Objectives
Describe and predict patterns found in sequences, figures, and word problems. (L4.1.1)
Define inductive reasoning.
Identify counterexamples for various conjectures.
Inductive Reasoning is the process in which one looks for patterns in samples and makes conjectures of how the pattern will work for the entire population.
A conjecture is an unproven statement based on observations.
A conjecture is math’s version of a hypothesis , or educated guess.
The education comes from the observation.
Using Inductive Reasoning
Much of the reasoning in Geometry consists of three stages
Look for a Pattern . Look at examples and organize any ideas of a pattern into a diagram or table.
Make a Conjecture . Use the examples to try to identify what step was taken to get from element to element in the pattern.
Verify the Conjecture . Use logical reasoning to verify the conjecture is true for all cases.
Example 1.5 A man starts a chain letter. He sends the letter to two people and asks each of them to send copies to two additional people. These recipients in turn are asked to send copies to two additional people each. Assuming no duplication, how many people will have received copies of the letter after the twentieth mailing? What pattern was being formed with the mailings?
Find the pattern and predict the next figure.
Find the pattern and predict the next number.
1, 4, 16, 64,…
Multiply by 4
-5, -2, 4, 13,…
+3, +6, +9, +12
1, 1, 2, 3, 5, 8,…
Add the previous two numbers in the list
1, 2, 4, 7, 11, 16, 22,…
+1, +2, +3, + 4, +5, +6, +7
In order to keep the spectators out of the line of flight, the Air Force arranged the seats for an air show in a “V” shape. Kevin, who loves airplanes, arrived very early and was given the front seat. There were three seats in the second row, and those were filled very quickly. The third row had five seats, which were given to the next five people who came. The following row had seven seats; in fact, this pattern continued all the way back, each row having two more seats than the previous row. The first twenty rows were filled. How many people attended the air show?
Lesson 1.2 – Patterns
Lesson 1.3 Day 1: Conditional Statements
Lesson 1.3 Objectives
Write conditional statements. (L4.2.1)
Write the inverse, converse, and contrapositive of a conditional statement. (L4.2.2)
Create a negation of a statement, including “there exists” and “all” statements. (L4.2.3)
Write a biconditional statement. (L4.2.4)
Utilize symbolic form of if-then, inverse, converse, contrapositive, negation, and biconditionals. (L4.3.1)
Apply the laws of detachment and syllogism. (L4.3.2)
Identify a counterexample.
A conditional statement is any statement that is written, or can be written, in the if-then form.
This is a logical statement that contains two parts
If today is Tuesday , then tomorrow is Wednesday .
The hypothesis of a conditional statement is the portion that has, or can be written, with the word if in front.
When asked to identify the hypothesis , you do not include the word if .
If today is Tuesday , then tomorrow is Wednesday .
The conclusion of a conditional statement is the portion that has, or can be written with, the phrase then in front of it.
Again, do not include the word then when asked to identify the conclusion .
If today is Tuesday , then tomorrow is Wednesday .
Write the statements in if-then form.
Today is Monday. Tomorrow is Tuesday.
If today is Monday, then tomorrow is Tuesday.
Today is sunny. It is warm outside.
If today is sunny, then it is warm outside.
It is snowing outside. It is cold.
If it is snowing outside, then it is cold.
The converse of a conditional statement is formed by switching the hypothesis and conclusion.
If tomorrow is Wednesday , If today is Tuesday , then tomorrow is Wednesday . then today is Tuesday
The negation is the opposite of the original statement.
Make the statement negative of what it was.
Use phrases like
Not , no, un, never, can’t, will not, nor, wouldn’t, etc.
Today is Tuesday . Today is not Tuesday .
Write the negation of the following statements.
It is sunny outside.
It is not sunny outside.
I am not happy.
I am happy.
My dog is black.
My dog is not black.
The inverse is found by negating the hypothesis and the conclusion.
Notice the order remains the same!
If today is not Tuesday , If today is Tuesday , then tomorrow is Wednesday . then tomorrow is not Wednesday .
The contrapositive is formed by switching the order and making both negative.
If tomorrow is not Wednesday , If today is Tuesday , then tomorrow is Wednesday . If today is not Tuesday , then tomorrow is not Wednesday . then today is not Tuesday .
Write the converse , inverse , and contrapositive of the following statements.
If you get a 60% in the class, then you will pass.
Converse – If you pass the class, then you get a 60%.
Inverse – If you do not get a 60% in the class, then you will not pass.
Contrapositive – If you do not pass the class, then you did not get a 60%.
If there is snow on the ground, then the flowers are not in bloom.
Converse – If the flowers are not in bloom, then there is snow on the ground.
Inverse – If there is no snow on the ground, then the flowers are in bloom.
Contrapositive – If the flowers are in bloom, then there is no snow on the ground.
A biconditional statement is a statement that is written, or can be written, with the phrase if and only if .
If and only if can be written shorthand by iff .
Writing a biconditional is equivalent to writing a conditional and its converse .
All definitions are biconditional statements .
Write the conditional statement as a biconditional statement .
If the ceiling fan runs, then the light switch is on.
The ceiling fan runs if and only if the light switch is on.
If you scored a touchdown, then the ball crossed the goal line.
You scored a touchdown if and only if the ball crossed the goal line.
If the heat is on, then it is cold outside.
The heat is on if and only if it is cold outside.
Equivalent Statements If the conditional statement is true, then the contrapositive is also true. Therefore they are equivalent statements! If the converse is true, then the inverse is also true. Therefore they are equivalent statements! Take the converse and make both parts negative. Take the original conditional statement and make both parts negative. Switch the hypothesis with the conclusion. Written just as it shows in the problem. If ~q, then ~p If ~p, then ~q If q, then p If p, then q Contrapositive Inverse Converse Conditional
For extreme negation , such as: All, Everyone, Nothing, Nobody, etc
It is sufficient enough to show that at least one item can negate the statement.
All… (or Everybody…) Somebody There exists one that… Nobody There exist one that… Nothing There exists one that does not… Everyone There exists one that does not… All Negation Phrase
Write the negation for the following statements.
All dogs are black.
There exists one dog that is not black.
Nobody likes tomatoes.
There exists one person who likes tomatoes.
Somebody is going to get in trouble for toilet papering the school.
Everybody is going to get in trouble for toilet papering the school.
Lesson 1.3A Homework
Lesson 1.3 Day 1 – Conditional Statements
Lesson 1.3 Day 2: Symbolic Notation
Symbolic Conditional Statements
To represent the hypothesis symbolically, we use the letter p.
We are applying algebra to logic by representing entire phrases using the letter p.
To represent the conclusion , we use the letter q.
To represent the phrase if…then , we use an arrow, .
To represent the phrase if and only if , we use a two headed arrow, .
Example of Symbolic Representation
If today is Tuesday, then tomorrow is Wednesday.
p: Today is Tuesday
q: Tomorrow is Wednesday
We read it to say “If p then q.”
Recall that negation makes the statement “negative.”
That is done by inserting the words not, nor, or, neither, etc.
The symbol is much like a negative sign but slightly altered…
Use the statements to construct the propositions.
p: It stays warm for a week. q: The apple trees will bloom.
If it stays warm for a week, then the apple trees will bloom.
It does not stay warm for a week.
~ p ~ q
If it does not stay warm for a week, then the apple trees will not bloom.
~ q ~ p
If the apple trees will not bloom, then it does not stay warm for a week.
If the apple trees bloom, then it stays warm for a week.
It stays warm if and only if the apple trees bloom.
Law of Detachment
If p q is a true conditional statement and p is true, then q is true.
It should be stated to you that p q is true.
Then it will describe that p happened.
So you can assume that q is going to happen also.
This law is best recognized when you are told that the hypothesis of the conditional statement happened first.
Law of Syllogism
If p q and q r are true conditional statements, then p r is true.
This is like combining two conditional statements into one conditional statement.
The new conditional statement is found by taking the hypothesis of the first conditional and using the conclusion of the second.
This law is best recognized when multiple conditional statements are given to you and they share alike phrases.
Are the following arguments valid? If so, do they use the Law of Detachment or Law of Syllogism ?
Scott knows that if he misses football practice the day before the game, then he will not be a starting player in the game. Scott misses practice on Thursday so he concludes that he will not be able to start in Friday’s game.
Valid - Law of Detachment
If it is Friday, then I am going to the movies. If I go to the movies, then I will get popcorn. Since today is Friday, then I will get popcorn.
Valid – Law of Syllogism
If it is Thanksgiving, then I will eat too much. If I eat too much, then I will get sick. I got sick so it must be Thanksgiving.
Invalid – Argument is out of order to use Law of Syllogism
A counterexample is one example that shows a conjecture is false.
Therefore to prove a conjecture is true, it must be true for all cases.
Conjecture: Every month has at least 30 days. Counterexample: February has 28 (or 29).
To find a counterexample , use the following method
Assume that the hypothesis is TRUE .
Find any example that would make the conclusion FALSE .
For a biconditional statement , you must prove that both the original conditional statement has no counterexamples and that its converse has no counterexamples.
If either of them have a counterexample, then the whole thing is FALSE .
Find a counterexample for the following statements.
If it is a bird, then it can fly.
Ostrich, Penguin, Emu, Cassowary, Rhea, Kiwi, and the Inaccessible Island Rail
Write a truth table of the connectives and their negations (L4.2.2)
if and only if
What is a Truth Table?
A truth table displays the relationships between truth-values of propositions.
Truth tables are especially useful in determining the truth-values of complex propositions constructed from simpler propositions.
Building a Truth Table?
Every truth table is constructed to verify the validity of every possible outcome of the individual proposition.
So, all truth tables should begin construction in a similar fashion:
Create two columns for p and q .
Even if they are not used that way in the proposition.
Fill the columns for p and q with every possible combination of outcomes.
ie. Both true, both false, only one is true.
Add extra columns for any negation of p and q .
These columns should contain truth-values that are opposite of their original columns.
Add extra columns for any intermediate propositions that are used in the final proposition.
The last column should be the final proposition.
“If…Then” Truth Table
Recall that “ If p , then q . ” can be denoted as:
If Mr. Lent wins $1,000,000, then he will give you $100,000.
An “if…then” statement will be false when p is TRUE and q is FALSE, and will be true for all other cases. T F T T p q q p p : Mr. Lent wins $1,000,000. q : He will give you $100,000. F F T F F T T T p q q p
“And” Truth Table
An “and” statement is written as “ p and q .” and can be denoted as:
Mr. Lent wins $1,000,000 and he will give you $100,000.
An “and” statement will be true when BOTH p is TRUE and q is TRUE, and will be false for all other cases. T F F F p q q p p : Mr. Lent wins $1,000,000. q : He will give you $100,000. F F T F F T T T p q q p
“Or” Truth Table
An “or” statement is written as “ p or q .” and can be denoted as:
Mr. Lent wins $1,000,000 or he will give you $100,000.
An “or” statement will be false when BOTH p is FALSE and q is FALSE, and will be true for all other cases. T T T F p q q p p : Mr. Lent wins $1,000,000. q : He will give you $100,000. F F T F F T T T p q q p
Truth Table Involving Negation
Remember to add an extra column for the negated proposition.
p ~ q
Mr. Lent wins $1,000,000 and he will not give you $100,000.
Remember, an “and” statement will be true when BOTH p is TRUE and q is TRUE, and will be false for all other cases. F T F T F T F F p ~ q q p p : Mr. Lent wins $1,000,000. q : He will give you $100,000. ~ q F F T F F T T T p ~ q q p ~ q p ~ q q p
“If and Only If” Truth Table
Recall a biconditional is written “ p if and only if q .” and is denoted as:
Mr. Lent wins $1,000,000 if and only if he will give you $100,000
A “if and only if” statement will be true when p and q are BOTH TRUE and when p and q are BOTH FALSE, and will be false for all other cases. T F F T p q q p p : Mr. Lent wins $1,000,000. q : He will give you $100,000. F F T F F T T T p q q p
Construct a truth table for:
(p q) (p q)
T T T F T F F F T F F T “ OR” “ AND” “ IF…THEN” Q: Remember when is an “if…then” statement is false? A: When the FIRST proposition is TRUE and the SECOND proposition is FALSE.
(p q) (p q)
q p F F T F F T T T
(p q) (p q)
q p F F T F F T T T
(p q) (p q)
p q q p F F T F F T T T
(p q) (p q)
p q p q q p p : Mr. Lent wins $1,000,000. q : He will give you $100,000.
Construct a truth table for:
(p q) ( ~ p q)
F F T T T F T T “ IF…THEN” “ IF…THEN” T T T F Q: Remember when is an “if…then” statement is false? A: When the FIRST proposition is TRUE and the SECOND proposition is FALSE. “ OR” T T T T p : Mr. Lent wins $1,000,000. q : He will give you $100,000. (p q) ( ~ p q) q p F F T F F T T T (p q) ( ~ p q) q p F F T F F T T T (p q) ( ~ p q) ~ p q p F F T F F T T T (p q) ( ~ p q) p q ~ p q p F F T F F T T T (p q) ( ~ p q) ~ p q p q ~ p q p
Lesson 1.4 Homework
Lesson 1.4 – Truth Tables
Lesson 1.5 Logical v Statistical Arguments Using Necessary or Sufficient Conditions
Lesson 1.5 Objectives
Distinguish between statistical and logical arguments. (L4.2.2)
Differentiate between a necessary and a sufficient condition for an argument. (L4.3.3)
You are playing a game of Old Maid. The game is played by drawing cards from your opponent’s hand to make a matching pair from your hand. The loser is the person left holding the Old Maid card after all pairs have been made.
Imagine you are left with 5 cards in your hand and one of them is the Old Maid. Describe the chances of your opponent drawing a card from your hand and leaving you with the Old Maid.
Are the odds more in your favor or your opponent’s in terms of having the Old Maid after that one draw?
Statistical v Logical Argument
A statistical argument is a way to come to a conclusion involving the use of data, numbers, odds, probabilities, percentages, etc.
The chances of you keeping the Old Maid is 80%.
A logical argument a way to come to a conclusion by using other valid statements such as laws, definitions, postulates, and theorems.
This typically does not involve numbers or data.
The chances of you keeping the Old Maid is more in your favor because you keep more cards than your opponent draws.
Use the given situation to make a statistical and a logical argument.
You are rolling a number cube (dice) with the numbers 1-6 on it. What is the chance of getting an even number versus an odd.
Statistical : ½, 50%, 1:2
Logical : Same as getting an odd because there are the same number of each type.
Drawing from a deck that has 10 black cards and 5 red cards, do you think the next card will be red?
Statistical : 1 / 3 , 33%, 1:3
Logical : More likely to get a black card since there is more of them.
You flip a coin 10 times and 8 times you get a head. Do you think you will get a head on the next flip?
Statistical : ½, , 50%, 1:2
Logical : Same as getting a tails because there are only two possible outcomes with each flip.
A necessary condition of a statement must be satisfied for the statement itself to be true.
Having gasoline in my car is a necessary condition for my car to start.
If we say “ x is a necessary condition for y,” we mean if we don’t have x, then we won’t have y.
Or put differently, without x, you won’t have y.
This means that x must happen in order for y to happen, but it does not mean that having x guarantee s that y will happen.
Example : There is gas in the car but the battery is dead.
A sufficient condition of a statement is one that, if satisfied , will make the statement true.
Rain pouring from the sky is a sufficient condition for the ground to be wet.
If we say “ x is a sufficient condition for y,” then we mean if we have x, then we know y must follow.
In other words, if we have x we can guarantee we have y.
Example : If it is raining, then we can guarantee that the ground will be wet.
Necessary v Sufficient
Remember, necessary conditions are must haves .
So you have to think, can the conclusion happen without the condition?
If it the conclusion cannot happen, then it must be a necessary condition .
If the conclusion can happen without the condition, then it must not be necessary !
And to recap, a sufficient condition is a condition that guarantees the conclusion.
The conclusion may happen without it, but…
IF the condition occurs, the conclusion MUST happen.
It is a way to make the outcome happen, but it is not the only way .
It is possible for a condition to be necessary and sufficient. Example: Getting credit for Geometry is a necessary and sufficient condition for graduation!
Decide the best statement to complete the sentence.
Having oxygen in the earth's atmosphere is a (necessary/sufficient) condition for human life .
Earning a total of 95% in this class is a (necessary/sufficient) condition for earning a final grade of A.
Pouring a gallon of freezing water on my sleeping sister is a (necessary/sufficient) condition to wake her up.
Being at least 16 years of age is a (necessary/sufficient) condition for being able to obtain a driver’s license in Michigan.
Deliver the opening arguments of a proof by contradiction. (L4.3.2)
Review with Algebra
Is the following true or false?
5 = 5
What about if we added 3 to both sides?
3 + 5 = 3 + 5
True 8 = 8
What if we now subtracted 6 from both sides?
8 – 6 = 8 – 6
True 2 = 2
What if we now multiply both sides by 8?
8 2 = 8 2
True 16 = 16
And now if we divide both sides by 4?
16 4 = 16 4
True 4 = 4
What do you observe happened throughout all this manipulation?
As long as we performed the same operation on BOTH sides of the equal sign we created another true, or equivalent, statement.
What is a Proof?
A mathematical proof is a sequence of justified conclusions used to prove the validity of a statement, or conjecture.
It is more than just showing your work!
You must state a reason why each step was done.
The reasons why are typically found by stating a…
Another Theorem, or
A mathematical proof shows that a conclusion is true for ALL cases.
These are used to create theorems , which are true statements created as a result of other true statements.
That is the proof process!
Definition of a Postulate
A postulate is a rule that is accepted without a proof.
They may also be called an axiom .
Postulates are used together to prove other rules that we call theorems .
An algebraic proof involves solving Algebra equations by providing a reason for each step along the way.
Again, it is more than just showing your work!
You must now state a law or property of Algebra to show why each step was done.
Algebraic Properties of Equality Addition Property If a = b, then a + c = b + c. When you add the same number to both sides during solving. APOE Subtraction Property If a = b, then a - c = b - c. When you subtract the same number to both sides during solving. SPOE Multiplication Property If a = b, then a c = b c. When you multiply the same number to both sides during solving. MPOE Division Property If a = b, then a c = b c. When you divide the same number to both sides during solving. DPOE Distributive Property a(b + c) = ab + ac When you multiply a number next to parentheses by everything inside. Distribute Combine Like Terms ax + bx = (a + b)x ALL OF THIS WILL BE DONE TO ONE SIDE OF THE EQUATION ONLY! CLT Substitution Property If a = b, then a + b + c = b + b + c. When you plug a number in for a variable in an equation. SUB If a = 6, then find a + 5. 6 + 5 = 11 Abbreviation Helpful Hint Example Definition Property
Recipe for a Proof
If 5x – 18 = 3x + 2, then x = 10.
Always rewrite the problem first. And the reason why is to state the GIVEN problem. And you should know when to stop because it will be th EXACT statement/conclusion you are trying to show is true. Reasons Statements 1. 5x – 18 = 3x + 2 Reasons Statements 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements 2. 2x – 18 = 2 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements 2. SPOE 2. 2x – 18 = 2 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements 3. 2x = 20 2. SPOE 2. 2x – 18 = 2 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements 3. APOE 3. 2x = 20 2. SPOE 2. 2x – 18 = 2 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements 4. x = 10 3. APOE 3. 2x = 20 2. SPOE 2. 2x – 18 = 2 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements 4. DPOE 4. x = 10 3. APOE 3. 2x = 20 2. SPOE 2. 2x – 18 = 2 1. Given 1. 5x – 18 = 3x + 2 Reasons Statements
A proof by contradiction proves the given conjecture true by attempting to prove the opposite is true.
The point is: They both can’t be true at the same time!
The first step in a proof by contradiction is to assume the desired conclusion is not correct.
So rewrite the problem by taking the negation of the conclusion only .
And then try to prove it as we have done before.
Example of a Proof by Contradiction The solution to x + 8 = 17 is not 10.
If this way is wrong, then the original conjecture must be true! 1. Given 1. The solution to x + 8 = 17 is 10. Reasons Statements 2. 10 + 8 = 17 1. Given 1. The solution to x + 8 = 17 is 10. Reasons Statements 2. SUB 2. 10 + 8 = 17 1. Given 1. The solution to x + 8 = 17 is 10. Reasons Statements 3. C LT 3. 18 = 17 2. SUB 2. 10 + 8 = 17 1. Given 1. The solution to x + 8 = 17 is 10. Reasons Statements 3. C LT 3. 18 = 17 (Contradiction) 2. SUB 2. 10 + 8 = 17 1. Given 1. The solution to x + 8 = 17 is 10. Reasons Statements
Write the first step in constructing a proof by contradiction for the following: