Math Snacks

Math SnacksNot a full entre,just some items to munch on.1What is a Prime number?Colloquial DefinitionA number is prime if it is divisible only by one and itself.Is one prime?It seems sodivisible by onedivisible by itselfWhat is a Prime number?Formal DefinitionA number is prime if it has exactly two whole number divisors.Is one prime?Noit only has one whole number divisor, i.e., itselfWould Euclid pass the EOC in Geometry?Name these shapes? (Based on side lengths)

Equilateral TriangleIsosceles TriangleScalene TriangleWould Euclid pass the EOC in Geometry?DefinitionsEquilateral three equal sidesIsosceles at least two equal sidesScalene no equal sides

Is an equilateral triangle isosceles?Definitely yes!

Would Euclid pass the EOC in Geometry?Definitions (from root words)Equilateral (from Latin)Aequi meaning even or levelLatus meaning sideThus, 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 unevenThus, a scalene triangle has uneven sides or sides of unequal length

Isosceles (from Greek) Isos meaning equalSkalns meaning unevenThus, 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 sidesWould 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 definitionProof (March 2007 TAKS Objective question 10 sample question 3) Which of the following must be true about an equilateral triangleand an isosceles triangle?Correct answer (C) An equilateral triangle is also isosceles.

Can you name that geometric figure?Clearly, the naming rule for geometry is(number of sides prefix)-gon

Or is it?PentagonFive SidesOctagonEight SidesHexagonSix SidesCan 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)QuadrilateralFour SidesCan you name that geometric figure?Shouldnt this be a trigon?Sounds wrong?Havent you ever heard of trigonometry?(trigon-ometry)TriangleThree AnglesHow we use ancient numeration dailyRoman NumeralsHow people think we use Roman numerals today

CountingSuper BowlsOzzy OsbournesFavorite NumberThe Most Interesting Number in the WorldI dont always teach Roman numerals,But, when I do,I prefer

IIIHow we use ancient numeration dailyRoman NumeralsIn reality, we use Roman numeralsalmost every day

In most large numbers we write

And we dont even think about it!How we use ancient numeration dailyRoman NumeralsMost people remember the basic symbols

Most people remember the basic rulesA symbol of smaller value to the left of a symbol of largervalue is subtracted from the larger value.Symbols of smaller value to the right of a symbolof larger value are added together.

How we use ancient numeration dailyRoman NumeralsExamples

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.

How we use ancient numeration dailyRoman NumeralsThe most forgotten rule

Each horizontal line written above a series of symbolsrepresents multiplication by 1000. Examples

How we use ancient numeration dailyRoman NumeralsThe 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 commasin numerical strings today.

It is not a coincidence!

How we use ancient numeration dailyBabylonian NumerationOnly used two symbols (cuneiform script) =1=10Used groupings to create values up to 59

=6 =30 =58

How we use ancient numeration dailyBabylonian NumerationOf greater interest is the fact that they used abase 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

How we use ancient numeration dailyBabylonian NumerationBase 10 Example

Base 60 Example

How we use ancient numeration dailyBabylonian NumerationBase 60 Example (In cuneiform script)

How we use ancient numeration dailyBabylonian NumerationBase 60 seems too unusual? You use it dailyHow many seconds are in a minute?

How many minutes are in an hour?

6060

How we use ancient numeration dailyBabylonian NumerationThe 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?

24x603606x603606x60The odds are you dont get oddsThe odds are you dont get oddsWhat is probability?

What are odds of winning? (odds in favor of an event)

The odds are you dont get oddsOn 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?

The odds are you dont get oddsAn 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 winningB) Double your probability of winningC) 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.The odds are you dont get oddsA) 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.Now for the best part of snacking Junk Food!

You know its not good for youbut you just like it anyway!Why division is hardWhat is 25 5?Answer: 14Huh?Do the division.5 does not go into 2, but 5 goes into 5 onceso25 5 = 1 remainder 20But5 goes into 20 four timesso25 5 = 1 then 4 = 14WRONG!Why division is hardSo 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 = 14WRONG!Why division is hardStill dont believe 25 5 = 14?Verify the answer with addition

Next, add theonesFirst,add thefoursFinally, add theresultsWRONG!How algebra liesConsider 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

How algebra liesFactor each side as a perfect square

Take the square root of each side

Let the square root and the 2nd power cancel

How algebra liesAdd the same value to each side

Simplify

And, thusly, the universe explodes!

WRONG!Why algebra didnt lieRemember 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

How to make trigonometry easySimplify the following expression

Obviously, we should cancel the ns and see the following

WRONG!

Why examples dont help in CalculusA student saw the following example in the textEvaluate 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 workEvaluate the following one sided limitWRONG!

5Thanks for snacking!Richard RuppProfessor of MathematicsDel Mar Collegerrupp@delmar.edu