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: • Have red hair • Have Black or Brown hair • Have Blonde hair
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.
Standards • SPI’s taught in Unit 4: • SPI 3108.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. • SPI 3108.2.1 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 3108.4.4 Analyze different types and formats of proofs. • SPI 3108.4.12 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 3108.1.1 Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely in mathematical reasoning. • CLE 3108.1.2 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 3108.1.3 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 3108.1.4 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 3108.1.5 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 3108.1.7 Use technologies appropriately to develop understanding of abstract mathematical ideas, to facilitate problem solving, and to produce accurate and reliable models. • CLE3108.2.1 Establish the relationships between the real numbers and geometry; explore the importance of irrational numbers to geometry. • 3108.2.3 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: • 3108.1.1 Check solutions after making reasonable estimates in appropriate units of quantities encountered in contextual situations. • 3108.1.6 Use inductive reasoning to write conjectures and/or conditional statements. • 3108.1.13 Use proofs to further develop and deepen the understanding of the study of geometry (e.g. two-column, paragraph, flow, indirect, coordinate). • 3108.1.14 Identify and explain the necessity of postulates, theorems, and corollaries in a mathematical system. • 3108.2.1 Analyze properties and aspects of pi (e.g. classical methods of approximating pi, irrational numbers, Buffon’s needle, use of dynamic geometry software). • 3108.2.2 Approximate pi from a table of values for the circumference and diameter of circles using various methods (e.g. line of best fit). • 3108.4.2 Compare and contrast inductive reasoning and deductive reasoning for making predictions and valid conclusions based on contextual situations. • 3108.4.15 Identify, write, and interpret conditional and bi-conditional statements along with the converse, inverse, and contra-positive of a conditional statement. • 3108.4.16 Analyze and create truth tables to evaluate conjunctions, disjunctions, conditionals, inverses, contra-positives, and bi-conditionals. • 3108.4.17 Use the Law of Detachment, Law of Syllogism, conditional statements, and bi-conditional statements to draw conclusions. • 3108.4.18 Use counterexamples, when appropriate, to disprove a statement. • 3108.4.37 Identify similar figures and use ratios and proportions to solve mathematical and real-world problems (e.g., Golden Ratio).
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?
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.
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…?,?
Summing Odd Numbers • What if you wanted to add the first four odd numbers? • In other words, you want to add 1+3+5+7 • Well, you could easily add those numbers and get 16 • But what if you wanted to add the first 83 odd numbers? (1+3+5+7+9+11…etc…) • Would you sit there and type 1+3+5+7+9+11… etc. on your calculator? • Hopefully, you would try to find a pattern
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? What’s special about each of these numbers? It would be 83 squared, or 6889
Summing Even Numbers 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
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?
Summing All Numbers 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
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?
Assignment • Page 85 1-30 (Skip #2-5)
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?
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
Fibonacci Sequence • The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filiusBonaccio, "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
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?
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.
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 1.618 – 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
Fibonacci and The Golden Ratio • 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
Fibonacci In Nature • Romanesque Broccoli, Conch Shell, Pine Cone • http://mathbits.com/MathBits/PPT/Fibonacci-Faces2007.html • http://www.intmath.com/numbers/math-of-beauty.php • http://www.facialbeauty.org/divineproportion.html
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.
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.
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?
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
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
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
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.”
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”
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 • What is the counterexample? • Mr. Bass lives in Clarksville (Montgomery County) and he’s at HCHS
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
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”
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. • What is the truth value of this statement? • This is false. Rectangles have four sides, but they aren’t all square
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 • What is the truth value of this statement? • If it is false, what is the counterexample?
Examples • If x = 2, then |x| = 2 • True? • What is the converse? • If |x| = 2, then x = 2 • True? • Hint: Absolute value problems always have a counterexample (the negative)
Examples • If x = 4, then x2 = 16 • True? • What is the converse? • If x2 = 16 then x = 4 • True? • Hint: “square” problems always have a counterexample (the negative)
Assignment • Page 93 5-19 • Worksheet 2-1 • Worksheet 2-2
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
Assignment • Page 110 6-17
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.
Unit 4 Quiz 3 Match the Definition Definitions 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
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 5 7 8