My Favorite One-Liners: Part 78

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

One of my pet peeves is students who don’t give me a reasonable amount of time to grade their final exams and compute their grade for the semester. Usually, “reasonable” means “tomorrow morning.” (However, sometimes I have to give multiple final exams on the same day, and there just isn’t enough time in the day to compute grades for all of my classes in that amount of time.)

So, to head this off, I’ll announce to my students when they can expect me to finish grading the finals so that it’s safe to ask for their grade for the semester; usually the answer is “9:00 tomorrow morning.” And, to make sure that no one bugs me before then, I’ll give the following playful admonition:

Anyone who asks me for their grade before 9:00 tomorrow morning gets an automatic F in the course.

Was There a Pi Day on 3/14/1592?

In honor of Pi Day, here’s a bonus edition of Mean Green Math from two years ago.

Mean Green Math

March 14, 2015 has been labeled the Pi Day of the Century because of the way this day is abbreviated, at least in America: 3/14/15.

I was recently asked an interesting question: did any of our ancestors observe Pi Day about 400 years ago on 3/14/1592? The answer is, I highly doubt it.

My first thought was that $latex pi$ may not have been known to that many decimal places in 1592. However, a quick check on Wikipedia (see also here), as well as the book “$latex pi$ Unleashed,” verifies that my initial thought was wrong. In China, 7 places of accuracy were obtained by the 5th century. By the 14th century, $latex pi$ was known to 13 decimal places in India. In the 15th century, $latex pi$ was calculated to 16 decimal places in Persia.

It’s highly doubtful that the mathematicians in these ancient cultures actually talked to…

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The Perfect Geometrical Christmas Present

Behold, a three-dimensional calendar on the faces of a rhombic dodecahedron:

Source: https://www.wired.com/2016/10/gorgeous-12-sided-wooden-calendar-geometry-nerd-life/

Faculty Office Hours

Kudos to Arizona State University for making this public service announcement.

Pizza Hut Pi Day Challenge (Part 6)

On March 14, 2016, Pizza Hut held a online math competition in honor of Pi Day, offering three questions posed by Princeton mathematician John H. Conway. As luck would have it, years ago, I had actually heard of the first question before from a colleague who had heard it from Conway himself:

I’m thinking of a ten-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?

I really like this problem because it’s looks really tough but only requires knowledge of elementary-school arithmetic. So far in this series, I described why the solution must have one of the following four forms:

O , 4 O 2 , 5 8 O , 6 O 0,

O , 6 O 2 , 5 8 O , 4 O 0,

O , 2 O 6 , 5 4 O , 8 O 0.

O , 8 O 6 , 5 4 O , 2 O 0.

where O is one of 1, 3, 7, and 9. (The last digit is 0 and not an odd number.) There are 96 possible answers left.

Step 8. The number formed by the first eight digits must be a multiple of 8. By the divisibility rules, this means that the number formed by the sixth, seventh, and eighth digits must be a multiple of 8.

For the first form, that means that 8O6 must be a multiple of 8. We can directly test this:

816/8 = 102: a multiple of 8.

836/8 = 104.5: not a multiple of 8.

876/8 = 109.5: not a multiple of 8.

896/8 = 112: a multiple of 8.

For the second form, that means that 8O4 must be a multiple of 8. This is impossible. Let O = 2n+1. Then

8O4 = 800 + 10 \times O + 4

= 800 + 10(2n+1) + 4

= 800 + 20n + 14

= 2(400 + 10n + 7).

We see that 7 is odd, and therefore 400 + 10n + 7 is not a multiple of 2. Therefore, 2(400 + 10n + 7) is not a multiple of 4 (let alone 8).

For the third form, that means that 4O8 must be a multiple of 8. This is also impossible. Let O = 2n+1. Then

4O8 = 400 + 10 \times O + 8

= 400 + 10(2n+1) + 8

= 400 + 20n + 18

= 2(200 + 10n + 9).

We see that 9 is odd, and therefore 200 + 10n + 9 is not a multiple of 2. Therefore, 2(200 + 10n + 9) is not a multiple of 4 (let alone 8).

For the fourth form, that means that 4O2 must be a multiple of 8. We can directly test this:

412/8 = 51.5: not a multiple of 8.

432/8 = 54: a multiple of 8.

472/8 = 59: a multiple of 8.

492/8 = 61.5: not a multiple of 8.

In other words, we’re down to

O , 4 O 2 , 5 8 1 , 6 O 0,

O , 4 O 2 , 5 8 9 , 6 O 0,

O , 8 O 6 , 5 4 3 , 2 O 0,

O , 8 O 6 , 5 4 7 , 2 O 0.

For each of these, there are 3! = 6 ways of choosing the remaining odd digits. Since there are four forms, there are 4 x 6= 24 possible answers left.

In tomorrow’s post, I’ll cut this number down to 10.

A request to the athletic department

Even though I’ve had nothing but good professional relationships with the athletic department at my own university, I still think this is really funny.

Source: http://imgur.com/vJWkqhe

Shuffling cards

I really enjoyed this article concerning the mathematics of shuffling a deck of playing cards, justifying the claim that no two properly shuffled decks of cards have ever been the same: http://www.matthewweathers.com/year2006/shuffling_cards.htm

100,000 page views

I’m taking a one-day break from my usual posts on mathematics and mathematics education to note a symbolic milestone: meangreenmath.com has had more than 100,000 total page views since its inception in June 2013. Many thanks to the followers of this blog, and I hope that you’ll continue to find this blog to be a useful resource to you.

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Twenty most viewed posts or series (written by me):

  1. All I want to be is a high school teacher. Why do I have to take Real Analysis?
  2. Analog clocks
  3. Another poorly written word problem
  4. Arithmetic and geometric series
  5. Common Core, subtraction, and the open number line
  6. Exponential growth and decay
  7. Finger trick for multiplying by 9
  8. Full lesson plan: magic squares
  9. Full lesson plan: Platonic solids
  10. Fun with dimensional analysis
  11. Importance of the base case in a proof by induction
  12. Infraction
  13. Inverse Functions
  14. My “history” of solving cubic, quartic and quintic equations
  15. My Mathematical Magic Show
  16. Square roots and logarithms without a calculator
  17. Student misconceptions about PEMDAS
  18. Taylor series without calculus
  19. Was there a Pi Day on 3/14/1592?
  20. What Happens if the Explanatory and Response Variables Are Sorted Independently?

 

Twenty most viewed posts (guest presenters):

  1. Engaging students: Classifying polygons
  2. Engaging students: Congruence
  3. Engaging students: Deriving the distance formula
  4. Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries
  5. Engaging students: Distinguishing between inductive and deductive reasoning
  6. Engaging students: Factoring quadratic polynomials
  7. Engaging students: Finding x- and y-intercepts
  8. Engaging students: Laws of Exponents
  9. Engaging students: Multiplying binomials
  10. Engaging students: Order of operations
  11. Engaging students: Pascal’s triangle
  12. Engaging students: Right-triangle trigonometry
  13. Engaging students: Solving linear systems of equations by either substitution or graphing
  14. Engaging students: Solving linear systems of equations with matrices
  15. Engaging students: Solving one-step and two-step inequalities
  16. Engaging students: Solving quadratic equations
  17. Engaging students: Square roots
  18. Engaging students: Translation, rotation, and reflection of figures
  19. Engaging students: Using right-triangle trigonometry
  20. Engaging students: Volume and surface area of pyramids and cones

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If I’m still here at that time, I’ll make a summary post like this again when this blog has over 200,000 page views.

Danica McKellar from NOVA

I really enjoyed this:

Why Reluctant Students Still Should Learn Math

I love this quote from Math With Bad Drawings about why students should learn math:

In every walk of life, humans need to reason. So of course, they can learn these intellectual skills in other places. You don’t need math. But gosh, does math make it easier!

You can learn to taxonomize in biology, by considering the classification of organisms. But your taxonomies will never be perfect, because life doesn’t fit into neat little boxes. (I’m looking at you, protists.)

Life doesn’t… but math does.

Or you can learn to dissect arguments in civics. But emotions will flare. It’ll be tough to agree on premises. And even if you do, words like “justice,” “freedom,” and “common good” are subject to fuzzy interpretations and subtle misunderstandings. All words are like that: a little vague, tricky to pin down.

Except in math.

Logic shows up everywhere. But in math, it’s the whole game. Math isolates the operations of logic and reason so that we can master them.

In short: math is the playground of reason.

I recommend the entire article: https://mathwithbaddrawings.com/2016/06/08/a-quadratic-of-solace-or-maybe-math-class-has-a-purpose-question-mark/