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Math Snacks

Math Snacks. Not a full entrée, j ust some items to munch on. 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 so…divisible by one…divisible by itself. What is a Prime number?. Formal Definition

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Math Snacks

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  1. Math Snacks Not a full entrée, just some items to munch on.

  2. 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 so…divisible by one…divisible by itself

  3. What is a Prime number? • Formal Definition • A number is prime if it has exactly two whole number divisors. • Is one prime? • No…it only has one whole number divisor, i.e., itself

  4. Would Euclid pass the EOC in Geometry? • Name these shapes? (Based on side lengths) Equilateral Triangle Isosceles Triangle Scalene Triangle

  5. 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!

  6. 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) • Skalēnós – 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” • Skalēnós – 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

  7. Would Euclid pass the EOC in Geometry? • So, again, is an equilateral triangle isosceles? • According to Euclid and the root words…NO! • 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.

  8. Can you name that geometric figure? Clearly, the naming rule for geometry is (number of sides prefix)-gon Or is it? Hexagon Six Sides Pentagon Five Sides Octagon Eight Sides

  9. Can you name that geometric figure? The name is derived from Latin instead of Greek, but generally it’s using the same rule. Or is it? (Technically, the suffix -lateral refers to sides whereas the suffix -gon refers to angles) Quadrilateral Four Sides

  10. Can you name that geometric figure? Shouldn’t this be a trigon? Sounds wrong? Haven’t you ever heard of trigonometry? (trigon-ometry) Triangle Three Angles

  11. How we use ancient numeration daily Roman Numerals How people think we use Roman numerals today… The Most Interesting Number in the World Ozzy Osbourne’s Favorite Number Counting Super Bowls “I don’t always teach Roman numerals, III But, when I do, I prefer…”

  12. 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 don’t even think about it!

  13. 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.

  14. How we use ancient numeration daily Roman Numerals Examples Most people don’t 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.

  15. 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

  16. 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!

  17. 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

  18. 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

  19. How we use ancient numeration daily Babylonian Numeration Base 10 Example Base 60 Example

  20. How we use ancient numeration daily Babylonian Numeration Base 60 Example (In cuneiform script)

  21. 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 60

  22. How we use ancient numeration daily Babylonian Numeration The Babylonians also loved multiples of 60 How many days are in a year? How many hours are in a day? How many degrees are in a rotation? 24 ⅖x60 ≈360 6x60 360 6x60

  23. The odds are you don’t get odds • You’re in a casino about to play a dice game. Would you rather play a game where the probability of winning is or where the odds of winning are 1:5? • Most people say the 1:5 game is better • Most people just got fooled!

  24. The odds are you don’t get odds • What is probability? • What are odds of winning? (odds in favor of an event)

  25. The odds are you don’t 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?

  26. The odds are you don’t 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.

  27. The odds are you don’t 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. So, 2 tickets = 2 chances. You have one chance per ticket.

  28. Now for the best part of snacking JunkFood! You know it’s not good for you …but you just like it anyway!

  29. 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 once…so… • 25 ÷ 5 = 1 remainder 20 • But…5 goes into 20 four times…so… • 25 ÷ 5 = 1 then 4 = 14 WRONG!

  30. 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!

  31. Why division is hard • Still don’t believe 25 ÷ 5 = 14? • Verify the answer with addition Finally, add the results First, add the fours Next, add the ones WRONG!

  32. 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

  33. How algebra lies • Factor each side as a perfect square • Take the square root of each side • Let the square root and the 2ndpower cancel

  34. How algebra lies • Add the same value to each side • Simplify And, thusly, the universe explodes! WRONG!

  35. Why algebra didn’t 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

  36. How to make trigonometry easy • Simplify the following expression • Obviously, we should cancel the “n’s” and see the following… WRONG!

  37. Why examples don’t 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 student’s work • Evaluate the following one sided limit WRONG! 5

  38. Thanks for snacking! Richard Rupp Professor of Mathematics Del Mar College rrupp@delmar.edu

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