## Algebra

### Farey Fraction Visual Patterns

The Farey Sequence, $$F_{n}$$ is the list of all simplified fractions between $$\dfrac{0}{1}$$ and $$\dfrac{1}{1}$$ with denominator less than or equal to $$n$$. For example,

$$\large F_{5}=\dfrac{0}{1},\dfrac{1}{5},\dfrac{1}{4},\dfrac{1}{3},\dfrac{2}{5},\dfrac{1}{2},\dfrac{3}{5},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{1}{1}$$

### $$k^x-x^k$$ Slider and $$x^y=y^x$$ graph

Use the slider to change the value of k, and to see dynamically how the graph of $$h(x)=k^x-x^k$$ changes. Negative values have $$x^k>k^x$$. Which value of k gives a graph that is never negative? Why?

Here is a very pretty graph of $$x^y=y^x$$ with areas where $$x^y < y^x$$ shaded in green and areas where $$x^y > y^x$$ shaded in purple.

I am working on an article about some problems related to these equations, including The Biggest Product Problem. There’s a lot of interesting stuff here!

### Last Chance U and Alternatives to Remedial Algebra

Note: Spoiler free, and you don’t have to have watched the series to read the post.

I just finished watching Last Chance U, a new documentary series screening on Netflix, which follows the 2015 football season of the East Mississippi Lions — an often championship junior college team full of students who are good enough to play Division I football, but who, due to academic, behavioral, or other issues, have had not yet been able to do so successfully. The series focuses especially on a few of the players and on two adults from the college: the dominant and often angry football coach and the loving and often exasperated guidance counselor.

Some of the drama in the series is the expected “Will they win the game?” sports drama and some of it involves complicated questions about ethics, race, and justice, but there is also a surprising amount of drama that involves questions like, “Will he go to math class? Will he pass his English class?” Miss Wagner, the counselor, gets up-to-the-minute attendance reports on the students (once even during a class), as well as results of all their graded assignments. She regularly computes their likely grade point averages, based on their work so far.

One of the classes that she discusses the most is algebra; most of the discussions focus on the number of absences students have, but there is one shot from the class itself. This scene could be titled, “Generic U.S. Math Class.” The teacher is at the board. The students are in desks. The students face the board. The teacher is the only one who talks. He explains about substituting zero to find the intercept and how you only need two points to find the equation of the line. The class looks boring. I wouldn’t want to go to it, especially on a day that also includes lots of running, tackling/being tackled, and getting yelled at for hours in football practice.

The math class serves the purpose that math classes unfortunately often serve: to help the students learn to do things they really don’t want to do in the name of responsibility and delayed gratification (the football practices also often have this emphasis). With the English class there’s a hint that one of the short stories was actually interesting to the students, but with math, the audience can simply assume that there’s nothing interesting about the content; the students should go to class because going to class will help them achieve their goal of getting a high enough GPA to play football at a D1 school.

After seeing the the clip from the algebra class, I started thinking about potential alternatives to remedial algebra for the students in the film (who are very different from the population I teach). Below are some quantitative reasoning and liberal arts mathematics topics that I’d think about trying if I taught at EMCC and had the freedom to experiment (beyond having students compute their GPA’s themselves!).

Financial Mathematics: Many of the students want to get to the NFL in order to make a lot of money. They can study topics like interest, credit, mortgages, and use spreadsheets, all in the context of how they would manage their money when they have it. They’ll acquire many useful skills, whether or not the NFL works out, and they can also cover many algebra topics.

Sports Data:  Students can investigate how football rankings are determined and the biases inherent in different methods. They can look at the probabilities that teams who win the coin toss will win in overtime and alternative methods for conducting overtime. They can look at ways to construct season schedules. They can look at algebra concepts through graphs of real data; for example, by using software to graph height vs. weight for linebackers and then for running backs and link the features of the graphs to the features of the players.

Science of Sports:   Perhaps a science class can be linked to the math class. Students can attach sensors to themselves during practice to study many kinds of motion (including the algebra of such motion).  They can study linear equations involving momentum or take pictures of the trajectory of a ball and fit an equation to it.   They can look at current research about concussions and maybe even contribute to it.

Social Justice Data:  Several of the students come from communities where many of their peers are in jail or are dead. Perhaps in conjuction with a history, literature, or social science class, they can gain a deeper understand the roots of some of the issues and the strengths in their communities. They can study data on incarceration rates, educational funding, the results of slavery, literacy, and other topics.

Mathematics of Games:   The students in the series are competitive, and some might enjoy a class where the mathematics originates from games. They can study the mathematics of a wide variety of games such as NIM, Mancala, dominoes, poker, black jack, MasterMind, SET, etc. I don’t know whether the gambling topics would be acceptable, but I did check and Mississippi has casinos (but not a state lottery).

All of these topics can be taught at a variety of levels of mathematics, including accessible levels that support basic skills development. Pairing the course with one or more other courses can create learning communities, which, along with the active learning strategies, makes it easier to create a sense of belonging, which can contribute toward students wanting to come to class and wanting to succeed in class. Clearly most students at the school are not football players, but judging from the series, there are many other students who are interested in football and might find the proposed courses interesting, engaging, and relevant to their future goals.

Remedial courses, especially in mathematics, are currently getting attention as a significant cause of students’ not completing college.   The EMCC website and catalog show that the math program at EMCC is quite traditional; its remedial math classes are the same classes offered at colleges and universities across the country.  The issues with remedial algebra are not primarily about EMCC and they’re not primarily about football players, but the small part of Last Chance U that’s about an algebra class was an interesting way for me to reflect on some larger issues about these courses.

### The Handshake Problem

The handshake problem is an old chestnut — if everyone in the room shook hands with everyone else, how many handshakes would there be? Then generalize. What I have to add to teaching the problem is a handout (doc version, pdf version) with different (fictional, but based on reality) students’ strategies for solving the problem. This handout is good for homework after students have worked on the problems themselves and listened to their classmates’ strategies. Read more >>