## Desmos and LaTex Mashup

Over the past year, I have  created a lot of graphs in Desmos.  A collection of these can found here.  There you will find a wide range of graphs; some  are art related and some are only using sliders.  Also, some of the graphs are lab based and explorations.  These graphs have instructions for students to follow and some blank spots for their answers.  However, I have wanted students to work on paper, too, while working in Desmos.  So, I am now creating a Desmos Lab packet.  This lab packet will look very similar to the Desmos graph, allowing students to work through Desmos and record their work on paper.  Here are the steps if you would also like to create a worksheet/lab for students to follow.

1. 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.”
2. 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.
3. 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?
Next let a=4 and b=0.5. What is the height between the two blue lines?
Try more 'a' and 'b' values till you see a pattern.  See if you can write the height as function of b.
Your answer above probably has some trig functions involved.  See if you can write the equivalent algebraic expression.

$H_{eight}\left(x\right)=ax\tan\left(\arccos\left(x\right)\right)$
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.
$y=\left\{-ab\le x\le ab:f\left(ab\right)\right\}$
$y=\left\{-a\le x\le a:0\right\}$
Categories: blah-blah, desmos Tags: , ,

## What can I do for AMATYC?

Below is a reflection on my recent attendance of the 2012 AMATYC Conference, which I submitted to the NCMATYC fall newsletter.

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?”

Categories: blah-blah, presentations Tags: ,

## Standard Deviation of Sunset and Sunrise

Thanks to Kate Owens (@katemath) for sharing a great graph and saying the phrase “standard deviation” in a tweet. I was going to teach standard deviation to my Survey of Math class the next day, so this was perfect timing. The following is how I used the graph in class. Note that in description of the graph, the standard deviation is given.  This was not shown to the students until later. http://visual.ly/daylight-saving-time-explained

• How can you use the graph to figure out what day of the year has the longest night? shortest night?
• Where are the time changes? Does it really save daylight?
• What type of average (mean, median, mode, midrange) do you think is used in the graph?
• How can you use the graph to figure out which has a higher standard deviation, sunset or sunrise?
• Estimate the average sunrise time. What percent of the days have a sunrise earlier than than the average? later than the average.
• (I had two students pick two sun rises times. They were 5:20 and 6:40.)  How many sunsets occur during this time?
• Who would find the above sunrise interval useful? (One idea would be construction workers or people who work outside.)
• This graph is for Chicago, IL. Would it be similar for where we live, Hickory, NC?
• As you moved closer to the equator, would the standard deviation change? What about for a location in Alaska. How would their standard deviation for sunrises be?

I guided the class through those questions in a 50 minute class period.  I liked the activity because the data was in the form of a graph.  Usually, the exercises list out the data and have students calculate every thing. I plan on using the graph again and making a lab exercise about it.

Follow Matt Enlow (@CmonMattTHINK) on twitter and you see him post a #140digitnumber of the day. The 140 digit numbers follow some type of cool pattern. For example, one of his tweets were

Today’s #140digitnumber is the sum of the first 10^20 (one hundred quintillion) perfect sixth powers:

This one caught my eye because I will be teaching summations soon. So, I thought I would look at the sum of the perfect squares. In WolframAlpha can you type sum. I started using upper sums that were powers of 10 and then counting the number of digits in the sum. Do this enough times and you will notice that when the upper sum is $10^a$ the number of digits in the sum will be $3a$. Why does this happen?

First, the number of digits in a number can be found by taking the base ten log of the number and then the ceiling. For example, $5^{500}$ would have $ceil(500 \log_{10} (5))$ number of digits. Knowing this we can find out how many digits there are in sum of squares. Using the sum of squares formula $(1/6)n(n+1)(2n+1)$, take $ceil(\log_{10} (1/6)n(n+1)(2n+1)$ and now we can quickly find the number of digits in a sum. Note that for students this would be a good time to recap log properties.

Second, I went to desmos.com and created a slider for different $n$ values. I defined $n$ to be $n=b^a$ Here are a few $n$ values that create 140 digit numbers. $2^{155}, 3^{98}, 6^{60}, 9^{49}, 13^{42}$ Some neat patterns begin to show. $46^{23}, 47^{23}$ are the first pair of consecutives with the same exponent. $87^{24}, 88^{24}, 89^{24}$ are the first trio. My favorite so far is $62^{26}$ I stopped searching for $b=110$ and started search for $a=2$, that is some number $b$ squared that would make a 140 digit number. I found that $(2*10^{23})^2$ is one. Wow, this means that b value goes from 2 to at least $2*10^{23}$ One could create a program and find just how many $n=b^a$ values there are. I only found 25 of them. Also, what would be the largest amount of consecutive b values – that is $b^{a}, (b+1)^a, (b+2)^a, (b+3)^n...$

Overall, I think that these Twitter numbers will offer some great exploration in the classroom. I plan on showing the sum of squares in class and then have students explore sum of cubes, fours, fives, and sixes. Or a student can create their own function to sum. I will update later on how it goes. Below you can click on the desmos graph to see the summation I explained earlier.

## Trick with 27

I finished watching another fantastic video from Numberphile.  The title is perfect “Beautiful Card Trick”.  After you watch it I think you will agree.

The video has some wonderful math and I cannot wait to try out different versions of the trick.  Here are some examples that come to my mind.

1. If you have a class of 27 students or more, then you could have each student put their name on a card, making a deck of 27 names.  I think this could be a good first day activity.
2. Do one of those activities where you use four fours to make the numbers 1-27.  Put each one a card, making a deck of 27.  Then do the trick.
3. Have students type in “random city” into WolframAlpha.  A lot of math could be done with each city.  Then collect 27 cities and do the trick.
4. This would be a great introduction into bases. The trick could then be followed with base four and other bases.
5. Have students create a function and list out 27 inputs-outputs in coordinate form.  Do the trick with those 27 coordinate points.

Overall, there are this card trick is simple and it can easily be modified, probably into at least 27 different ways.  Hopefully, you find at least one way to share the trick.

Bonus video! This video goes beyond only 27 cards. You will see that (the number of cards) = (number of piles) ^ (number of times shuffled).

Categories: blah-blah Tags: , ,

## What is tangent?

Trig functions have a lot of acronyms. Is this good or bad, well I don’t know. What I do know is this: tangent is much more than just sine over cosine or opposite over adjacent. However, the text books I teach from usually focus on that. So, I thought I would make an investigation that allows students to see that tangent can also be seen as length of the tangent line to the unit circle at $(0,1)$ Wikipedia gives some good pictures. Click here to see the article.

Note the location of tangent and secant

I also like in the article when it writes about “Slope Definitions”.

“Tangent combines the rise and run” meaning that Tangent takes the angle of the line segment and tells its slope; or alternatively, tells the vertical rise when the line segment’s horizontal run is 1.

So, next time you talk about the trig functions be sure to expand on the idea of tangent. Here is a Desmos graph I made that you can use. Click on the image to see the graph.

Categories: desmos Tags: , ,

## Angle Side Side

Recently, I was at the 2012 AMATYC conference and heard many great ideas for my classroom.  One idea was this,

Don’t use Law of Sines for Angle Side Side, just use Law of Cosines and then apply the quadratic formula.

What a revelation that was!  My students never seem to get the ambiguous case.  Maybe it is the way I try to explain it. Overall, it seems like a bunch of hoops to them.  But by using the quadratic formula students refresh their memory of the quadratic formula and also apply some basic trig.  So, let’s look at an example.

Let the angle be $20.5^{\circ}$ the adjacent side be 31 and the opposite side be 12. Going straight to the Law of Cosines we get the following:

$12^2 = b^2+31^2-2*12\cos(20.5^{\circ})b$ now set the equation equal to zero and we get
$0=b^2-2*12\cos(20.5^{\circ})b + (31^2-12^2)$ now find the values for the quad formula
$a=1, b=2*12\cos(20.5^{\circ}), c= 817$.  Since the discriminant is greater than zero there will be two solutions or in other words, two lengths for the third missing side. Once you have the two sides, then use the Law of Sines to find the angles. Voila, you are done!

I have created a Desmos graph that can be used to further model the idea of Angle Side Side. (See below) In the graph, you can move the opposite side, which is the ‘c slider”, to try and connect with the missing side.  Through careful observation you will notice that the value of the ‘c slider’ that makes a triangle actually is the +/- discriminant/(2a) part of the quadratic formula!  Having students discover this correlates to the idea how the quad formula is the vertex +/- distance to x-intercepts. In this case, the quadratic formula is the distance to the height +/- distance to opposite side. So when the discriminant is imaginary this means there is not a distance to the opposite side. If the discriminant is zero, then the opposite side is actually the height, altitude, of the triangle. Finally, when the discriminant is greater than zero there are two distances to the opposite. I think this is a great bridge back to x-intercepts and reinforces the importance of the quadratic formula.

Check out the desmos graph by clicking on the below picture. It is the Angle Side Side tutorial I mentioned above. Let me know what you think. Will you still use the Law of Sines?

Categories: desmos