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Slide 2 Not a full entre, just some items to munch on. Slide 3 What is a Prime number? Colloquial Definition A number is prime if it is divisible only by one and itself. Is one prime? It seems sodivisible by onedivisible by itself Slide 4 What is a Prime number? Formal Definition A number is prime if it has exactly two whole number divisors. Is one prime? Noit only has one whole number divisor, i.e., itself Slide 5 Would Euclid pass the EOC in Geometry? Name these shapes? (Based on side lengths) Equilateral Triangle Isosceles Triangle Scalene Triangle Slide 6 Would Euclid pass the EOC in Geometry? Definitions Equilateral three equal sides Isosceles at least two equal sides Scalene no equal sides Is an equilateral triangle isosceles? Definitely yes! Slide 7 Would Euclid pass the EOC in Geometry? Definitions (from root words) Equilateral (from Latin) Aequi meaning even or level Latus meaning side Thus, equilateral means even sides, i.e. a triangle where all sides are equal in length Scalene (from Greek) (like Euclid himself) Skalns meaning stirred up or hoed up which implies uneven Thus, a scalene triangle has uneven sides or sides of unequal length Isosceles (from Greek) Isos meaning equal Skalns meaning uneven Thus, isosceles means equally uneven or equally different meaning two sides of the triangle are equally different from the third side. Skelos meaning leg (alternative...more common derivation) Using the more common etymology, isosceles means equal legs, by which Euclid meant exactly two congruent sides Slide 8 Would Euclid pass the EOC in Geometry? So, again, is an equilateral triangle isosceles? According to Euclid and the root wordsNO! Understanding this, remember the TEA chooses to use the at least two equal sides definition Proof (March 2007 TAKS Objective question 10 sample question 3) Which of the following must be true about an equilateral triangle and an isosceles triangle? Correct answer (C) An equilateral triangle is also isosceles. Slide 9 Can you name that geometric figure? Clearly, the naming rule for geometry is (number of sides prefix)-gon Or is it? Pentagon Five Sides Octagon Eight Sides Hexagon Six Sides Slide 10 Can you name that geometric figure? The name is derived from Latin instead of Greek, but generally its using the same rule. Or is it? (Technically, the suffix -lateral refers to sides whereas the suffix -gon refers to angles) Quadrilateral Four Sides Slide 11 Can you name that geometric figure? Shouldnt this be a trigon? Sounds wrong? Havent you ever heard of trigonometry? (trigon-ometry) Triangle Three Angles Slide 12 How we use ancient numeration daily Roman Numerals How people think we use Roman numerals today Counting Super Bowls Ozzy Osbournes Favorite Number The Most Interesting Number in the World I dont always teach Roman numerals, But, when I do, I prefer Slide 13 How we use ancient numeration daily Roman Numerals In reality, we use Roman numerals almost every day In most large numbers we write And we dont even think about it! Slide 14 How we use ancient numeration daily Roman Numerals Most people remember the basic symbols Most people remember the basic rules A symbol of smaller value to the left of a symbol of larger value is subtracted from the larger value. Symbols of smaller value to the right of a symbol of larger value are added together. Slide 15 How we use ancient numeration daily Roman Numerals Examples Most people dont remember the other rules Only symbols representing powers of 10 may be repeated and may only be repeated three times. Only symbols representing powers of 10 may be subtracted, may only be subtracted once, and may only be subtracted from the next two symbols of greater value. Slide 16 How we use ancient numeration daily Roman Numerals The most forgotten rule Each horizontal line written above a series of symbols represents multiplication by 1000. Examples Slide 17 How we use ancient numeration daily Roman Numerals The most forgotten rule Thus, the Romans lines naturally partitioned numerical strings into groupings of coefficients of powers of 1000. This is exactly the same way we use commas in numerical strings today. It is not a coincidence! Slide 18 How we use ancient numeration daily Babylonian Numeration Only used two symbols (cuneiform script) =1=10 Used groupings to create values up to 59 =6 =30 =58 Slide 19 How we use ancient numeration daily Babylonian Numeration Of greater interest is the fact that they used a base 60 positional system. This means they counted 0 59 before changing position In contrast, we use a base 10 positional system. This means we count 0 9 before changing position Slide 20 How we use ancient numeration daily Babylonian Numeration Base 10 Example Base 60 Example Slide 21 How we use ancient numeration daily Babylonian Numeration Base 60 Example (In cuneiform script) Slide 22 How we use ancient numeration daily Babylonian Numeration Base 60 seems too unusual? You use it daily How many seconds are in a minute? How many minutes are in an hour? 60 Slide 23 How we use ancient numeration daily Babylonian Numeration The Babylonians also loved multiples of 60 How many hours are in a day? How many degrees are in a rotation? How many days are in a year? 24 x60 360 6x60 360 6x60 Slide 24 The odds are you dont get odds Slide 25 What is probability? What are odds of winning? (odds in favor of an event) Slide 26 The odds are you dont get odds On a fair, six-sided die, what is the probability of rolling a 4? On a fair, six-sided die, what are the odds in favor of rolling a 4? Slide 27 The odds are you dont get odds An elementary school student knocks on your door and offers to sell you a raffle ticket to a school fundraiser. You buy one. The student now tries for the hard sale and asks you to buy a second ticket making the claim that buying the second ticket will: A) Double your chances of winning B) Double your probability of winning C) Double your odds of winning (odds in favor of winning) Which of these claims could the student have made without lying? Note that in a raffle the winning ticket is drawn from the pool of purchased tickets, i.e. a winning ticket will always be drawn. Slide 28 The odds are you dont get odds A) Double your chances of winning? This is true. B) Double your probability of winning? Hopefully, not true. C) Double your odds winning (odds in favor of winning)? This is true. You have one chance per ticket. So, 2 tickets = 2 chances. Slide 29 Now for the best part of snacking Junk Food!Junk Food! You know its not good for you but you just like it anyway! Slide 30 Why division is hard What is 25 5? Answer: 14 Huh? Do the division. 5 does not go into 2, but 5 goes into 5 onceso 25 5 = 1 remainder 20 But5 goes into 20 four timesso 25 5 = 1 then 4 = 14 WRONG! Slide 31 Why division is hard So 25 5 = 14? Still not convinced? Do the long division. 5 does not go into 2, but 5 goes into 5 once leaving us with 20 Now, we see 5 goes into 20 four times leaving us with no remainder Thusly, 25 5 = 14 WRONG! Slide 32 Why division is hard Still dont believe 25 5 = 14? Verify the answer with addition Next, add the ones First, add the fours Finally, add the results WRONG! Slide 33 How algebra lies Consider the following true statement Rewrite with equivalent values Factor the values on each side of the equation Add the same value to each side Rewrite each side with equivalent values Slide 34 How algebra lies Factor each side as a perfect square Take the square root of each side Let the square root and the 2 nd power cancel Slide 35 How algebra lies Add the same value to each side Simplify And, thusly, the universe explodes! WRONG! Slide 36 Why algebra didnt lie Remember this step Simplify the interiors This is true since What you have to remember is the square root property When you take the square root of both sides of an equation, you must introduce a on one side of the equation Slide 37 How to make trigonometry easy Simplify the following expression Obviously, we should cancel the ns and see the following WRONG! Slide 38 Why examples dont help in Calculus A student saw the following example in the text Evaluate the following one sided limit The student immediately recognized what the example was demonstrating and applied the knowledge they gained in the next problem. This is the actual students work Evaluate the following one sided limit WRONG! 5 Slide 39 Thanks for snacking! Richard Rupp Professor of Mathematics Del Mar College rrupp@delmar.edu