## Mathalicious Lessons for Developmental Math

**Background**

Developmental math at community colleges has been a hot topic in North Carolina. Within the past two years, the state has redesigned the curriculum so that basically the classes are broken up to modules and a student has to show mastery learning to move on to the next course. Furthermore across the state you will see many different versions, such as 4 week classes, 8 week class, emporium, seated, and online. The DMA (developmental math) range from DMA 010, which is operations with integers, to DMA 065 a prerequisite for pre-calc. Students that are non-stem need to have tested/completed out of DMA 010 through DMA 050. Soon, multiple measures, where students can use their high school GPA, can be used to test out DMA, compared to as of now a student must take a test for placement.

**My Experience**

Teaching this level of math to people whose range is 18ish-50ish has its challenges, as any type of math course. One challenge is that students have a lot of baggage from all of the previous math courses. For example, when teaching the concept of fractions, students have already seen this many times and their knowledge can be very fragmented. They might remember some type of tricks, what previous teachers did, or only how to enter fractions on a calculator. My task it re-piece all those ideas and also provide a “deeper conceptual understanding” of fractions. These classes are tough. In the past, I have tried to present the material in a way they might have never seen, like using cantor sets to learn fractions or like using lattice multiplication for decimals or like using expanded form in decimal and fractions or using Sam’s club photos to introduce the idea of variables, expressions, distributive property, yada yada yada. Students were engaged with these lessons but I think the approach was too mathy, especially for the liberal arts audience. I need to present the math in a way they have never seen before and in a way they can relate to their own lives.

**Mathalicious Lessons
**I have been aware of Mathalicious for a few years now but I have never actually used their lessons. (Actually, I stole a few of their ideas but I am now a paying member 😉 My original thought about Mathalicious was “Back when I was an 8th grade math teacher, those lesson would have been perfect.” That thought resonates with a lot I find on the internet: that is there seems to be plenty of resources directed towards a public school audience, some resources for the university audience, and very little for a community college audience. This distribution of resources makes sense to me since there are way more public schools and if you add in universities, then community colleges are sort of covered due to the intersection of public and university. With all of that said, I have decided that this summer I will try to implement Mathalicious Lessons with some community college flare.

**Matching up lessons to the course outline
**Here is the table of contents and the Mathalicious lessons that I will add.

- Algebraic Expressions
- Simplifying Algebraic Expressions Using Properties of Real Numbers
- Solving Equations Using Properties of Equality
- More about solving equations
- Formulas
- Problem Solving
- More About Problem Solving
- Solving Inequalities

- Ice Cubed – This lesson will cover formulas, geometric formulas, expressions, rates, and problem solving. This will also be a good lead into mixture problems. The lesson uses Lemonade, but being at a community college I change the drink to a “spiked lemonade” I might try to extend the lesson about what is the fastest way to cool down a six pack and what type of ice is best to keep a six pack cold.
- Viewmongous – I recently purchased a 55 inch TV so I have some resources for this one. (Mounting the TV in the corner of my room involved a good bit of math. The mount could only extend 13 inches from the wall and I had to calculate if the mount was too short.) This will be a good introduction to the geometric style of word problems that are in the text book.
- HI, BMI – This problem is in the textbook but only to re-write the formula. Student will practice solving equations and learn how to handle linear equations that have denominator.
- Calories In, Calories Out – My area of North Carolina is not known to be the healthiest. But this lesson will help student combining like terms and evaluating expressions. Also, is some conversions that students have to do, which is prerequisite skill of the class.
- Heart Rate – Having a wide age range for the class works well for this lesson. I have chosen to use this lesson towards the end since graphing is more of a focus in the next course.
- Not So Fast – Again, another great lesson because of the ages in my class. Also, the lesson focuses on Virginia, which our school is about 100 miles away from. I might try to extend the lesson to North Carolina, who has different pricing structure of speeding tickets. This lesson will also work for inequalities and domain restrictions.
- Text Me Later – This seems like a good lesson to have after Not So Fast. Yet, I don’t know if I will follow that order. I might do this lesson first because of the focus on percentages, ratios, and conversions. It could be a good refresher.

**Final Thoughts
**I am excited to try these all of these lessons and a bit scared because of how the lessons differ from the text book and the assessment. At my school, students have to score at least an 80% on the final exam to exit the course. Though the lessons don’t mimic the final exam, I feel that students will be prepared. A lot of what I read, shows that people who take developmental math, mostly need to work on their attitudes toward math and being in a “lower level course”. The students needs to have a positive experience and a sense that they belong, as shown in this NY Times article, “Who Gets to Graduate?”. Overall, my thought is to hook students in the classroom with the Mathalicious lessons, which will hopefully motivate students to work on the more mathy stuff outside of the classroom, and to build a collaborative environment around the Mathalicious lessons so that students have a sense of belonging.

The class starts next Tuesday. I will try to blog about experience.

## AMATYC 2013 Presentation

Here are the slides for my AMATYC 2013 Presentation. The slides include links to all of the Desmos graphs.

## Math Is Hard

Here is the poem that I will be presenting at the Ignite session for AMATYC 2013 There are 50 slides that go with poem that I will try to upload later.

*You just get used to them*”. That’s the truth

## Linear Regression by the Median of Slopes

I have been trying to come up with a lesson about linear regression that involves more than pushing a few buttons, like on the TI-8ish, or using sliders in Desmos. I tried to search the web for lessons of other people but I could not find what I was looking for. Then I came across a method of finding the line of best fit called Theil–Sen estimator. Here is the method.

As defined by Theil (1950), the Theil–Sen estimator of a set of two-dimensional points (

x,_{i}y) is the median_{i}mof the slopes(y−_{j}y)/(_{i}x−_{j}x) determined by all pairs of sample points. Sen (1968) extended this definition to handle the case in which two samples have the same_{i}x-coordinate. In Sen’s definition, one takes the median of the slopes defined only from pairs of points having distinctx-coordinates.Once the slope

mhas been determined, one may determine a line through the sample points by setting they-interceptbto be the median of the valuesy−_{i}mx._{i}^{[8]}As Sen observed, this estimator is the value that makes the Kendall tau rank correlation coefficient comparing the sample data valuesywith their estimated values_{i}mx+_{i}bbecome approximately zero.

I really like this idea because it reinforces a lot of procedures of linear equations. Here is how I might do the lesson. A link to the entire Desmos graph is here.

First give the students data and have them plot it with Desmos. This data is the annual gross ticket sales (in 100’s of millions) where x=0 for 1995. Using the table feature in Desmos is great.

Next I would have students find the First Order Differences and plot these on the same graph. We would have a discussion about what these values mean and also talk about how these are approximately constant so a linear model would be a good fit.

Next we would begin finding the median slopes. We might begin by asking how many different slopes could be found between 17 points. Obviously, we would not find them all so we would assign a certain amount for each student to find. Then we would gather up all of those slopes and plot them in Desmos. This should be a great visual example to see the outliers of slopes within the data. (For this example, I only found 10 different slopes. Also, note that the first oder differences could be used as slope values. Those slope values are for consecutive points.)

Then we can discuss what “average”, (mean, median, mode, midrange), we should use to find the “average slope”. In Desmos, finding the median slope is easy. Click on the top line, then hide it. Click on the bottom line, then hide it. Click on the new top line, hide it. Click on the new bottom line then hide it. Continue doing so and this will result in the median slope. Here is a picture of the final two.

We can also plot that median slope with the first order differences. This could bring up a good discussion about do we really need to find other slopes or could we just use the first order differences to find the “median slope”

Next we can go back to the table and find the median y-intercept. In the Desmos table, we will make a column of values that is the expression y-(median slope)x. We can also plot those points to show what the y-intercept would be for each data point. Here is that graph.

Now that we have all of those different y-intercepts we can use a slider to estimate the median y-intercept value. We could also throw the values into a spread sheet if we wanted, but I think the slider will be good enough. I made the slider have a lower bound of 4 and an upper bound of 5. The b value ended up being 4.554.

Finally, we are ready to plot the line of “median fit.” using the equation y = (median slope)x + (median y-int)

For only using ten different slopes, I would say that the line looks pretty good. However, the data did a have a strong correlation to begin with. I have not compared the “median line” to line of least-squares because I think that would be a good follow up. I think this method goes into the heart of regression. Students get to see how many different lines are used to find the best line. Student review stats concepts and how outliers impact different averages. Students are creating a lot of evidence for their model, instead of just relying on the “r-value”.

One other thought would be to have student’s create an error region for the model. This might help them understand ideas of interpolation and extrapolation. Plus, it might allow us to discuss standard deviation, too. In the graph below I graphed {median slope(x) + 1.15(median y-int)} and {median slope(x) – 1.15(median y-int)} to create a 15% above and 15% region. I could have found the standard deviation of the median b value and done three standard deviations above and below.

The more I explore this concept the more it seems like it turning more into a statistical analysis. I need to determine if that is the route I want to go on since the class I am developing this for is “Math Modeling” course.

I hope all of this gives you some ideas about linear regression. I have not designed the lab sheet that will go with this yet. I would love to hear feed back if you have any.

## Your Homework: Play Temple Run

Summer is here. Yippee for me! This means that I can get back to blogging some of ideas. So, here is my next idea, The Math of Temple Run.

Have your ever played Temple Run? Probably, yes, given the game’s popularity. If not, here is information about the game. http://www.imangistudios.com The game is free to play. Did you catch that – FREE! But what is better than being free is that Temple Run has a lot math problems waiting to be explored. Here are some that I will be trying out.

- How fast is the person running? The game keeps track of how far you run in meters. This means that you can time how long the person runs and then calculate the speed.
- Is the person’s speed possible? I won’t spoil it for you but you need to see how fast the person runs.
- The person runs faster as the game progresses. Students can make a chart of distance and time because the game flashes up the distances as you travel. Does the increase of speed follow a quadratic pattern or is it more like a piece-wise linear?
- Next have students play the game and record 15 rounds of data. Here is a link to the table of data that I made. https://www.desmos.com/calculator/barud9egsn
The data looks to have a strong linear correlation, which allows us to explore rates of change. What does the rate of change mean for the blue dots and what does the rate of change mean for the red dots? Are certain games better than others? Is a better game based on the distance? Is a better game based on the amount of coins? Is a better game related to the number of coins compared to the score?

- In Desmos we can use a slider to create a line of best fit. The ones I made were y=(coins/dist)x and y=(score/dist)x I will probably discuss with students why we can make the initial value of zero. (Actually, one of the goals in the game is to go 1,000 meters and get zero coins.) Next, with the slider values we can either make the numerator or denominator equal to 1 and adjust the other slider accordingly. Once they have the line of best fit, we can talk about what it means for the data points to be above/below the line of best fit.
- Remember that each student will be playing the game, hopefully. So, this means we will have many different graphs. This will allows us to talk about how you can look at a graph and say, “That is good player. This is a so-so player. Etc.” or to be able to look at graph and point out who has had more experience playing the game.
- How is the score calculated? After students record the amount of coins, distance, and score they will have some data to try and figure out how the score is calculated. The game does not tell how the score is calculated. All you see is a running total in the upper right corner. Wikipedia does give a formula for the score. http://en.wikipedia.org/wiki/Temple_Run However, that formula was not working for me. I might doing some type of error. But this is good because it gives students the opportunity to figure it out. Plus, the formula opens up the door to the floor function and ordinal numbers.
- I might throw in Game Theory at the end. Not quite sure.

Plenty of stuff to work on there. I have not made the worksheets yet but I will update the post soon. If you could, please play the game 15 times and record your data into this Desmos graph https://www.desmos.com/calculator/b2bycwrenn. Save your work and then post it here in the comments or on twitter @LukeSelfwalker. I will gather up the data and post it later. Thanks for your help.

## My New Part-time Dream Job

I really don’t know how else to put this – I am now part of Team Desmos! For now, most of my time is spent running the Desmos twitter feed. I remember three months ago, a person asked me, “Luke, what would be your dream job?” I responded, “To teach at a community college and to work for Desmos.” Now, all of the sudden, poof, it happened. But how did a community college instructor who lives on the opposite coast from where Desmos is located get this chance of a lifetime? Good question. The following post gives a short description of how all of this happened.

I first came across the Desmos calculator back in late 2011. Honestly, the first attraction that hooked me in was the price, which was free. So, I began messing around with it and I quickly saw how easy it was to use. If I had a question, Eli, who is the CEO, would respond to my emails. (I remember thinking how cool it was that the CEO of the product I was using was helping me.) Then as I used Desmos in the classroom, by using an overhead projector, I saw how much more my students were engaged. Also, Desmos allowed us to explore math concepts in a way I could have not done earlier. The combination of customer service and student enjoyment motivated me to see just how many different ways could I apply Desmos to my teaching.

Some graphs I made were art. Some graphs I made were math demonstrations. Some graphs were for my own enjoyment. So, yeah, no need to hide from the truth, Desmos became one of my hobbies. A collection of my graphs can be seen here. When ever I finished a graph, I would share it on Twitter. Once in awhile an educator would respond and most of the time Desmos would give a RT (retweet). This gave me nice warm math fuzzies. I love to share math and knowing that people across the world were seeing my math influenced me to begin promoting Desmos when ever I could.

I did three different presentations (NCCTM, NCMATYC, and SOCOMATYC) and all of them had a bit of Desmos with the topic I was discussing. Each time I would ask the audience if they had heard of Desmos and only a few hands would go up. Then as I would introduce them to the slider feature, the choice of colors, the projector mode, and so on, each one would be amazed. The look on their faces reminded me when I first began.

Another place I was sharing Desmos work was on a facebook page . I run a facebook page for the students at the school I work. Like I said, I like to share and talk about math and this is just another venue. So, on this page I would post Desmos graphs that my students would create and Desmos would see and share them as well. Students loved to showcase their work and I loved to share it.

Overall, the main theme in my story is sharing math. Social media removes the physical distance and makes sharing easy. Sharing is a common part of life and Desmos has integrated that into their calculator. I was able to share my graphs with two clicks, copying and pasting a URL. Students can do the same and, even better, they want to. Also, students can print out their graph and share a hard copy, too. They did this once to make a coloring book for Kindergartners. And, yes, I print mine out, too, and I have them hanging in my office. So, just how did I get to join Desmos? I would say by sharing what I love to do. I know there are thousands of people who do the same and that is why I feel so lucky to have been chosen.

## Flip the First Day

Ah, the first Day. As a student or as a teacher we all know about the first day. Usually this means going over rules, syllabus, and other guidelines. I began to think about the first day and thought, “If I want my students to come to class prepared, then why on the first day of classes do I stand in front of them telling them every thing? Shouldn’t the first day be what models how the semester is going to be? Isn’t there some thing about how first impressions are the most important?” So, after marinating on that thought this is what I decided to do on the first day.

- First I passed out an index card. Before hand, I wrote the numbers 1-30 on the back of the cards and shuffled them up. I had students write down basic information about them and some other fun stuff, such as “Write down one lie and one truth about yourself but don’t tell me which is which.”
- Next I told students that now you will get in groups of three such that the sum of the numbers of the back of your cards is divisible by three. (We quickly went over the divisibility rule of three. Students were able to form groups pretty quickly. I would say about two minutes.)
- Next I told them to get into groups of four, such that if I were to pick any three members of the group their sum is not divisible by three. (This took a bit longer about four minutes.)
- Finally, I told them to get into groups of five such that if I pick any three members of the group their sum in not divisible by three. (I gave them about two minutes before telling them that to form such a group would be impossible.)

During this exercise, students were using Socrative (socrative.com check it out) to share what their group numbers were. I have the overhead projecting socrative and this allowed me to point out some patterns, such as “Will three consecutive numbers be divisible by three? Can three prime numbers be divisible by three.” I did not tell them why you can always find three number whose sum is three from a group of five numbers. Again, trying to model that some ideas take time. Hopefully, some students will come to class saying, “I figured it out.”

Once students were back in their seat, around 35 minutes of class had passed. I told them, “Notice how I did not start the class by going over the syllabus and me telling you every thing. This is a model for how this class will be run. I expect you to come to class prepared and ready with questions.” Yada, yada, yada, the speech went on a bit more. I will see how many actually go and look at the syllabus and everything else in Blackboard. Regardless, I am happy with how I set the learning environment on day one.