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.

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

]]>“Young man, in mathematics, you don’t understand things.

You just get used to them.” – John von Neuman

I am an instructor of mathematics.

I might not be a professor,

but with my might I profess, no more nor lesser

not like a passive polygon plastered across my back

a bumper sticker, allowing us to avoid eye contact

and not like a pierced cardioid tattooed “i less than three

math” out of sight, hidden underneath my sleeve

2MUCHTIMe time on my hands, too much has been achieved

A. Morgan, B. Russell, C. Gauss, D. Hilbert, E. Bell, F. Viete,

G.H. Hardy, I could go on naming them all day,

so mathematics boldly billboards across my chest

ignores sinusoidal fashion trends to project and express

a passion from my heart, a complex domain

a union of two parts, real and imagination, stain

countable Cartesian planes thirsty for a change

in position with respect to time, differentiate.

Sorry, off on a tangent, don’t want to complicate

or what’s that word, “remove one tenth”, decimate

a normal distribution, a significant sample of population

a standard deviation compressed by public comprehension

formulating sharp spikes out of any given Gaussian curve

to carve scars of G.P.A. and grade by grade serves

an affirmation of confirmation bias by such simple words

because everything you look for and all that you perceive,

has a way of proving what ever you believe.

“I’ve never been good at math”, “Letters are not math.”

“When I was in school, we didn’t do that type of math”,

“In all of my classes I have A’s, except for math.”

“I’ve been here for two years, trying to avoid math.”

These rationals expressions have a greatest common factor

that reduces down to an irrational thought of thereafter

“Math is hard.” Now, repeat with me class, “Math is hard.”

“Yes, math is hard.” now square that class, “Yes, math is hard!

Yes, math is hard! Thank God Almighty math is hard.”

Finding the expected value of math in any human’s life

the cost will always out weight, giving a negative fair price.

I mean, how is negative a squared is not the opposite of a squared

A false hypothesis for any conditional will be true. Who cares?

I care, but please listen, not because I’m a mathematician

I do not stand here on a soap box, a rectangular prism

My roots are at zero, neutral, with a compassionate grin

a parabola whose vertex is my chin, an understanding that begins

by knowing that math, a proper subset of life, battles with in

dancing to a divergent harmonic series, never to end.

Yes, math is hard. And so is life. “*You just get used to them*”. That’s the truth

no sugar coating, just 99.9 repeating percent absolute proof

But really, a victorious excuse “Math is hard!”. Where’s the logic?

Oh, here it is 2B V ~(2B), inclusive, exclusive how will you choose it?

]]>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.

]]>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.

]]>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.

]]>- 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.

]]>- Go to this website. http://jsfiddle.net/uRkkM/3/ The directions are in the lower right hand corner. All you have to do is drag “Equations List” to your bookmarks bar. Then, “From a Desmos Graph, it will open a new window containing text and latex for the expressions from the expressions list.”
- Now create a lab or some type of exploration in Desmos. When done click the “Equations List” and you will see all of the inputs from you Desmos graph that can be copied in LaTex.
- Finally, polish up your LaTex file. Now your students have a piece of paper that they can easily follow along with the Desmos graph.

There you go simple as that! I am excited to try this out with my students. Here is a link to what my looks like so far. Check it out and let me know if you have any ideas.

Here are is an example of those steps.

- The Desmos graph is here: https://www.desmos.com/calculator/yw9lovl86i
- The input after pressing Equations list looks like this. I copy it and paste it into a LaTex document
Semi Circle with radius 'a'. $f\left(x\right)=\sqrt{a^2-x^2}$ $a=4$ Below are two points on the semi-circle, such that the line connecting the two is parallel to the x-axis and is a factor 'b' of the diameter. $\left(ba,\space f\left(ba\right)\right),\space\left(-ba,\space f\left(-ab\right)\right)$ A factor "b" of the diameter. For example, b=0.5 means that the top blue line half of the diameter. b=0.2 would be two tenths of the diameter, or in other words, one-fifth. $b=0.8$ Begin by letting a=2 and b=0.5. What is the height between the two blue lines? Answer: Next let a=4 and b=0.5. What is the height between the two blue lines? Answer: Try more 'a' and 'b' values till you see a pattern. See if you can write the height as function of b. Answer: Your answer above probably has some trig functions involved. See if you can write the equivalent algebraic expression. Answer: $H_{eight}\left(x\right)=ax\tan\left(\arccos\left(x\right)\right)$ The equivalent algebraic expression $y=a\sqrt{1-x^2}$ $\left(b,\space H_{eight}\left(b\right)\right)$ Graphs of the two blue lines. $y=\left\{-ab\le x\le ab:f\left(ab\right)\right\}$ $y=\left\{-a\le x\le a:0\right\}$

When someone asks me, “Did you have a good conference?”, I often reply, “How could I not?”, and then continue to share all the wonderful events at the conference. To quickly sum up my AMATYC 2012 experience I got my math on, gave an Ignite talk (https://sites.google.com/site/2point718271827459045/) , got my math on some more, celebrated in the evenings, got my math on, and did my delegate duties. Feel free to look up my tweets from the conference for some extra inside scoop, #AMATYC12. Filling in all of the details would take too long, so I will only offer one short experience.

I attended three days straight of amazing session! In each one of those sessions, people would ask thoughtful follow up questions and offer excellent feedback. For example, I was in a talk that gave some ideas to spice up a trig course. The person that I was sitting next to mentioned to the presenter, “Just skip the ambiguous case stuff. Use the Law of Cosines for Angle Side Side and then apply the quadratic formula.” The presenter paused for a moment and replied exactly with what I was thinking, “I have never thought about it like that.” That 15 second comment was the golden nugget that I carried away from that session, inspiring me to create this graph of Angle Side Side, which can be seen here https://www.desmos.com/calculator/xubckxlnba, and write a blog entry about it http://t.co/aNGEHAoH.

What happened in that session was not an isolated event, and, as with every other conference I have attended, I have been overwhelmed with great ideas. Yet, trying to fit in all of those ideas into my currently overwhelmed scheduled just makes me feel more overwhelmed. Plus, my frugal instinct tells me not to let go of anything because I might find a time to use it. Yeah, right! That strategy has resulted in stacks of materials gathering dust on my shelf. Maybe there needs to be a session called “What to do with all that professional development you just had.” Am I the only one who walks away from an outstanding conference and only applies a small portion of it?

Perhaps there is a difference between an educator who attends conference and one who efficiently applies the conference. This reminds me of another AMATYC session I attended where the speaker had their students discuss the characteristics between an A student and a C student. The data in their survey showed that one perception of students is that doing homework is the main characteristic of an A student. As an educator, I know that being an A student is much more than just attending class and practicing a few exercises. But let’s make a switch and apply it towards professional development, like an AMATYC conference. What are the different characteristics of an A teacher versus a C teacher? Does a C teacher do professional development differently than an A teacher?

My first intention was not to give a talk at AMATYC or to be a NC delegate. After fulfilling those duties, I am thankful that I had a nudge from our current NCMATYC president to participate in those events. Being active at the conference helped challenge me professionally. Plus, just being around other people who are active in AMATYC is inspiring! Chat with them for a bit and you can see that they have many characteristics of an A teacher. I hope to carry this motivation forward with AMATYC and NCMATYC. Not only do I want to attend, I want to be involved. Not only do I want to participate at the conference, I want contribute throughout the year. Before going to Jacksonville, my perspective on joining AMATYC was “What could they do for me?”. I now wonder if this is a characteristic of a C teacher. I imagine that an A teacher would step up to the challenge and say, “What can I do for AMATYC?”

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