# Mathematics

### NCTM Presentation: Integrating Authentic Economics Applications into the Math Classroom

These are the resources that will be presented on Saturday at 11:00 (room 108 in BCEC).

Handout: Economic Applications NCTM

Slides: NCTM economic applications

Extra Resources:

### An Election: Heads vs. Tails

Yesterday during a tangent in my Algebra 2 class I realized how little understanding my students had about the voting process as well as how predictions are made and how, when, and why networks announce winners without all of the votes counted.

Today, I ran an activity that pitted Heads vs. Tails. Each student received a penny and flipped the coin to determine their vote: Heads or Tails. The class consisted of 19 students.

In our first election, we strictly counted votes- a popular election. Tails beat heads 10-9.

In our second election (everyone re-flipped their coin), the girls announced first: tails 6 and heads 5. I asked our students, what do we EXPECT to happen with the remaining 8 votes? Assuming fair coins, we expect 4 for heads and 4 for tails. So if we had to bet, would we bet that heads would win or tails? Since tails already had a 1 vote lead, tails was most likely to win. In fact, the scenario most likely to result in a win for heads was 5 heads and 3 tails. This was not a probability class, but with some basic concepts we could have given percentages to each outcome.

Back to this specific election: With only the girls reporting, the vote was 6 for tails and 5 for heads. I asked one boy to announce his result. He flipped tails, so the vote count was 7 for tails and 5 for heads. I declared, “I am ready to call the election for tails!” The students protested; there were 7 students still left to vote. Could heads have come back and won? Yes, but this was unlikely since 5 of the 7 remaining flips would need to be heads. We counted the votes and tails won 10-9 (4 of the 7 remaining flips went to heads). I did not have a specific percentage in mind when I “called” the election (the real chance when I declared tails victorious was around 75%, but I wanted to make a point). We discussed how, when, why (and the implications) newspapers and television stations “call elections.”

In our third election, I told the class that the front row (5 students) always voted heads and were going to vote for heads again. With 14 students left to flip, I said that I had enough information to call the election and was sure heads would win. Again, with the likelihood around 75% that heads would win, there was a decent chance tails could come back, but again, I was making a point. We then discussed how this activity mirrored the Electoral College with states like California consistently voting democrat.

Finally (and I did not have time to do this election), I was going to weight each row differently. The first row would vote and their winner would receive 4 votes, the winner of the second row would receive 3 votes, the 3^{rd} row two votes and the back row one vote. I was going to have each row announce in order 1-4, then redo this system with the back row announcing first.

Please leave a comment if you have any ideas on improving this exercise or need any help implementing it in your class.

### Education: Skills or Signals?

It’s that time of year when almost 100 million Americans go back to school. Yes, approximately 25% of the US population is enrolled at least part-time as a student at some level of kindergarten through graduate programs. My question is whether these students are learning skills applicable outside of the classroom or are they just earning credentials for their resume?

The easy answer is both, but I would argue that after initial universal skills are learned, education acts as a signal of ability to comprehend and analyze. It is important to distinguish between learning a skill, such as the ability to problem solve, versus signaling that you have the ability to problem solve. Before going too deep, an example should clarify the difference.

Consider high school math, in Algebra or Geometry a student may learn the skill of deductive reasoning and problem solving. Then a few years later in Calculus, the students signal that they have the requisite skills needed to comprehend Calculus concepts. Note that this signal (the ability to understand Calculus) may not be an applicable “life skill,” but if you ask a college admissions officer they would tell you it is quite valuable to signal to colleges that you are able to complete a Calculus course.

What skills will be acquired in schools this fall that will be applicable? Obviously basic reading, writing, and arithmetic are necessary skills learned in the lower grade levels. Once these skills are mastered, the second level of thinking comes into play. That is, using factual knowledge to solve problems. Finally third level thinking, including synthesizing (or combining) knowledge from various sources to derive original thoughts and conclusions. The last two “levels of thinking” are not necessarily taught on their own, yet students acquire these skills through high school and these attributes are the base argument for a “liberal arts education” promoted by colleges.

So, when does education stop being skill based and start becoming a signaling device? My conclusion is that this balance transfer occurs in the midst of the traditional high school years. The American high school curriculum (including calendar and schedule), include many requirements that act as hurdles. Also, many advanced graduate programs fail to provide students with necessary skills. These degrees simply act as acknowledgement that a student has read the core literature of a field and does not enable students to go out into the world ready to produce original ideas.

Below are two graphs that I believe represent traditional educational paths for American students and the subsequent returns of skill development from that education. The first graph represents a student whose returns to schooling are negative, that is each year fewer skills are acquired. Unfortunately, I believe this is the path of the majority of American students. The second graph represents a student that take a career orientated path and ends their education in a trade school or college or graduate program that gives the student real skills to be applied outside of the classroom.

If we want education to have meaning for our students, we should gear curriculum to achieve the second path shown above. Ideally, we could create a skill development graph that is a horizontal line where students are constantly acquiring applicable skills. This may seem far off, but if we want our future citizens to be equipped to face the challenges of an ever-changing economy and job market, we need to be sure that they are prepared with applicable skills, not a diploma that recognizes that they jumped through the appropriate hoops.

### Polar Coordinate Investigation Using Wolfram Alpha (help needed see bottom)

In my summer school PreCalculus class today we continued our discussion of Polar Coordinates. I began the class by asking the students to draw the graph of:

r = 2sin(θ)+2 by hand. Their gaph should have looked like this:

We discussed the process of translating between rectangular coordinates and polar coordinates when a student asked “can we just transform the entire function?” This led to a geeky, yet exciting, hour discussion and investigation.

First, we simplified the problem by dropping the “+2” from r = 2sin(θ)+2 (don’t worry we’ll go back to it…). So we algebraically converted the equation r = 2sin(θ) into rectangular coordinates (x and y). After substituting the conversions in (y=rsin(θ) and r^{2 }=x^{2}+y^{2}), the algebra cleans up nicely by completing the square and a circle with center (0,1) and radius of 1 emerges. After half the class had already plugged the polar coordinate equation into their calculator, we all agreed that this process worked for r=2sinθ, but what about r = 2sin(θ)+2?

The algebra was messy at best (plus I erased the final x,y equation before I was able to plug it into wolfram alpha), but here wass our equation:

Here’s the input into Wolfram Alpha: graph y=2sin^2(aCos(x/(sqrt(x^2+y^2))))+2sin(aCos((x/(sqrt(x^2+y^2))))

Anyway, here was the graph Wolfram Alpha spit out:

Where’s the bottom!!!!

So, in our equation, to find the angle of the polar coordinate in terms of x and y, we had to use inverse cosine, whose range is only zero to pi. In order to get the bottom, we needed angles from pi to 2pi. To find that, we had to shift our cosine by pi; that is, we had to add pi to each of the angles:

Wolfram Alpha input: graph y=2sin^2(aCos(x/(sqrt(x^2+y^2)))+pi)+2sin(aCos(x/(sqrt(x^2+y^2)))+pi)

Wolfram Alpha output:

But how do we get both? A simple google search (remember this is the middle of class and my computer’s monitor is projected to the entire class, so a little bit of a gamble… good thing Wolfram Alpha isn’t a subsidary of GoDaddy!) landed me on this blog: Gower’s. Somewhere in the comments, I found that a simple comma would allow my to graph both equations together, so i input the following into Wolfram Alpha:

graph y=2sin^2(aCos(x/(sqrt(x^2+y^2))))+2sin(aCos(x/(sqrt(x^2+y^2)))), y=2sin^2(aCos(x/(sqrt(x^2+y^2)))+pi)+2sin(aCos(x/(sqrt(x^2+y^2)))+pi)

Wolfram Alpha thought for awhile, and returned nothing.

PLEASE HELP US if you know how to get wolfram alpha to graph two intense equations. (note: I can graph two simple graphs: input: graph y=2x,y=x^2)