Adding by a Form of 0: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on adding by a form of 0 (analogous to multiplying by a form of 1).

Part 1: Introduction.

Part 2: The Product and Quotient Rules from calculus.

Part 3: A formal mathematical proof from discrete mathematics regarding equality of sets.

Part 4: Further thoughts on adding by a form of 0 in the above proof.

My Favorite One-Liners: Part 122

Once in my probability class, a student asked a reasonable question — could I intuitively explain the difference between “uncorrelated” and “independent”? This is a very subtle question, as there are non-intuitive examples of random variables that are uncorrelated but are nevertheless dependent. For example, if X is a random variable uniformly distributed on \{-1,0,1\} and Y= X^2, then it’s straightforward to show that E(X) = 0 and E(XY) = E(X^3) = E(X) = 0, so that

\hbox{Cov}(X,Y) = E(XY) - E(X) E(Y) = 0

and hence X and Y are uncorrelated.

However, in most practical examples that come up in real life, “uncorrelated” and “independent” are synonymous, including the important special case of a bivariate normal distribution.

This was my expert answer to my student: it’s like the difference between “mostly dead” and “all dead.”

My Favorite One-Liners: Part 121

I’ll use this one-liner when I ask my students to do something that’s a little conventional but nevertheless within their grasp. For example, consider the following calculation using a half-angle trigonometric identity:

\cos \displaystyle \frac{5\pi}{8} = \cos \displaystyle \left( \frac{1}{2} \cdot \frac{5\pi}{4} \right)

= \displaystyle - \sqrt{ \frac{1 + \cos 5\pi/4}{2} }

= \displaystyle - \sqrt{ \frac{ 1 - \displaystyle \frac{\sqrt{2}}{2}}{2} }

= \displaystyle - \sqrt{ \frac{ ~~~ \displaystyle \frac{2-\sqrt{2}}{2} ~~~}{2} }

= \displaystyle - \sqrt{ \frac{2 - \sqrt{2}}{4}}

= \displaystyle - \frac{ \sqrt{2 - \sqrt{2}}}{\sqrt{4}}

= \displaystyle - \frac{ \sqrt{2 - \sqrt{2}}}{2}

That’s certainly a very complicated calculation, with plenty of predictable places where a student might make an inadvertent mistake.

In my experience, one somewhat surprising place that can trip up students seeing such a calculation for the first time is the very first step: changing \displaystyle \frac{5\pi}{8} into \displaystyle \frac{1}{2} \cdot \frac{5\pi}{4}. Upon reflection, perhaps this isn’t so surprising: students are very accustomed to taking a complicated expression like \displaystyle \frac{1}{2} \cdot \frac{5\pi}{4} and making it simpler. However, they aren’t often asked to take a simple expression like \displaystyle \frac{5\pi}{8} and make it more complicated.

So I try to make this explicitly clear to my students. A lot of times, we want to make a complicated expression simple. Sometimes, we have to go the other direction and make a simple expression more complicated. Students should be able to do both. And, to try to make this memorable for my students, I use my one-liner:

“In the words of the great philosopher, you gotta know when to hold ’em and know when to fold ’em.”

Yes, that’s an old song reference. My experience is that most students have heard the line before but unfortunately can’t identify the singer: the late, great Kenny Rogers.

Factorization of 2021

A line joining two infinitely small points

Been there, done that.

Source: Math With Bad Drawings, https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549/3862434577106673/

Usefulness of Mathematical Symbols in a Fight

Source: https://xkcd.com/2343/

Xmas Tree, Ymas Tree, Zmas Tree

I’m not gonna lie… I wish I had an ugly Christmas sweater with this theme.

Source: https://www.facebook.com/photo.php?fbid=10157861645601449&set=a.115338471448&type=3&theater

Is 8,675,309 prime?

This semester, to remind today’s college students of the greatness of the 1980s: I made my class answer the following question on an exam:

Jenny wants to find out if 8,675,309 is prime. In a few sentences, describe an efficient procedure she could use to answer this question.

Amazingly, it turns out that 8,675,309 is a prime number, though I seriously doubt that Tommy Tutone had this fact in mind when he wrote the classic 80s song. To my great disappointment, nobody noticed (or at least admitted to noticing) the cultural significance of this number on the exam.

Naturally, I didn’t expect my students to actually determine this on a timed exam, and I put the following elaboration on the exam:

Although Jenny has a calculator, answering this question would take more than 80 minutes. So don’t try to find out if it’s prime or not! Instead, describe a procedure for answering the question and provide enough details so that Jenny could follow your directions. Since Jenny will need a lot of time, your procedure should be efficient, or as quick as possible (even if it takes hours).

Your answer should include directions for making a certain large list of prime numbers. Be sure to describe the boundaries of this list and how this list can be made efficiently. Hint: We described an algorithm for making such lists of prime numbers in class. (Again, do not actually construct this list.)

I thought it was reasonable to expect them to describe a process for making this determination on a timed exam.  Cultural allusions aside, I thought this was a good way of checking that they conceptually understood certain facts about prime numbers that we had discussed in class:

  • First, to check if 8,675,309 is prime, it suffices to check if any of positive prime numbers less than or equal to \sqrt{8,675,309} \approx 2,945.387\dots are factors of 8,675,309.
  • To make this list of prime numbers, the sieve of Eratosthenes can be employed. Notice that \sqrt{2,945} \approx 54.271\dots, and the largest prime number less than this number is 53. Therefore, to make this list of prime numbers, one could write down the numbers between 2 and 2,945 and then eliminate the nontrivial multiples of the prime numbers 2, 3, 5, 7, 11, \dots 53.
  • If none of the resulting prime numbers are factors of 8,675,309, then we can conclude that 8,675,309 is prime.

I was happy that most of my class got this answer either entirely correct or mostly correct… and I was also glad that nobody suggested the efficient one-sentence procedure “Google Is 8,675,309 prime?.”

Engaging students: Dot product

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Haley Higginbotham. Her topic, from Precalculus: computing a dot product.

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For the dot product of vectors, there are lots of word problems regarding physics that you could do that students would find more interesting than word problems self-contained in math. For example, you could say that “you are trying to hit your teacher with a water balloon. Your first try had a certain velocity and distance in front of the teacher, and your second try had a certain velocity and distance behind the teacher. In order to hit the teacher, you will need half the angle between the vectors to hit the teacher. Figure out what angle and velocity you would need to hit the teacher with a water balloon.” This could also turn into an activity, where the students get to test their guesses to see if they can get close enough. There would be need to be something they could use to accurately catapult their water balloon, but that’s a different problem entirely.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

The dot product (and vectors in general) can be seen in physics, calculus 3, linear algebra, vector calculus, numerical analysis, and a bunch of other upper level math and science courses. Of course, not all students are going to be taking upper level math and science courses. However, out of the students going into STEM majors, they most assuredly will see the dot product and by seeing how vectors work earlier in their math careers, they will be more comfortable manipulating something they have already seen before. Also, the dot product and vectors are very useful as a tool to use in upper levels of math and in many different applications of engineering and computer science. In the game design, the dot product can be used to help engineer objects movements in the game work more realistically as a single unit and in relation to other objects.

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E1. How can technology be used?

Geogebra is a great site to use since it has a tool https://www.geogebra.org/m/PGHaDjmD that will visually show you how the dot product works. It’s awesome because you get multiple different representations side by side, so that students who understand at different levels can all get something from this visual, interactive program. They can see how changing the position of the vectors changes the dot product and how it relates to the angle between the two vectors. Also, students will most likely be more engaged with this activity than just doing a bunch of examples with no real concept of how all of these pieces relate together which is not good in terms of promoting conceptual understanding. I think you could also use Desmos as an activity builder to make something similar to the above tool if students find the tool confusing to either use or look at.

References:
https://hackernoon.com/applications-of-the-vector-dot-product-for-game-programming-12443ac91f16

 

 

Happy Pythagoras Day!

Happy Pythagoras Day! Today is 12/16/20 (or 16/12/20 in other parts of the world), and 12^2 + 16^2 = 20^2.

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