# 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:

### Hashtag Math

Last year as a test review, I asked my students to categorize problems. Beyond knowing the mechanics, I want my students to know the when and why. Thus I asked them to tell me what “big theme” each problem represented. With a set of review problems, I began to #hashtag the problems. Two examples are shown below:

I have no scientific research showing that this helps my students make connections and categorize concepts, but they certainly remembered and enjoyed hashtagging math problems! This is a technique I plan on continuing and is worth considering beyond just review. At minimum, hashtagging in class was unique, so it stood out as something special. Their favorite was #factoredform, so I wonder how long until its trending?

### 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.

### An Explanation of Conditional Expectations… Part Two

In my first (found here) of a three part series on statistical analysis, I discussed how data can inform decisions within countless industries. My research in human capital, and specifically education, provides for various usages of data in decision making. Specifically, available data can be used for predictions of when, where, and which students may have trouble, determinants of parental satisfaction, admission decisions (at both secondary and collegiate levels), and student achievement both academically, on standardized tests, and in future wages.

This post is intended to shed light on the science of data analysis, specifically conditional expectations. The mathematical approach to conditional expectation is based on heavy statistical concepts, some of which are not appropriate for this venue. That being said, my intention is to provide an accessible explanation of the techniques used in statistical analysis. For a mathematical approach, I recommend the seminal text Econometric Analysis, by William Greene, 2002; or Econometric Analysis of Cross Section and Panel Data, by Jeffrey Wooldridge.

Econometrics, and essentially any data analysis, is based on determining a prediction. In statistical terms, this is called an expectation. A *conditional expectation* is a prediction based on available information.

For instance, consider guessing the height of a random human being. The average human height is 5 foot 6 inches, so, this would be a logical starting point for our prediction. But if we know more information about this random person, we can improve our expectation. Specifically, if we knew the person was male, we would want to change our expectation conditional on that fact. The average height of an adult man is 5 foot 9.5 inches, so that would be our new prediction given some information.

If we also knew that the person weighed 240 pounds, we may want to increase our expected height. Note here that there is no causal assumption, just a change in our expectation, given some piece of information. This is correlation: the taller a person is the more, on average, we expect that person to weigh. We may predict 6 foot 1 inch for our random male weighing 240 pounds. Data analysis can help us with our predictions, given some imperfect information.

Next, consider that we also knew this random person’s SAT score was 1200. We probably would not consider changing our expectation of height. If we had data on a sample of people with the variables height, sex, weight, and SAT score, some variables may be good predictors of height and be statistically significant while others, SAT score in this example, would not be statistically significant and would not persuade us to change our expectation.

Multiple Regression Analysis essentially considers a sample of data and determines the predictive success of each variable. Using the information from a statistical data program (even excel can do this reasonably well), we can arrive at a predictive equation for the most logical expectation conditional on the information we have at our disposal.

The data will find the coefficients- the “B’s”- and also determine the likelihood that each “B” is a significant predictor. In this case, I suspect B3 would not be statistically significant.

In part three, I will discuss an example of student achievement data from a nation-wide sample and how conditional expectations can be used to inform decisions in many fields.

### 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)