Category Archives: Technology

Desmos + Statistics = Happiness

Sunday – a quiet evening before President’s Day – checking out twitter – not looking for trouble – and then,

Wait..what’s this?  Standard Deviation?  It was my birthday this past Saturday, and the Desmos folks knew exactly what to get me as a present.  Abandon all plans, it’s time to play.  A lesson I picked up from Daren Starnes (of The Practice of Statistics fame) is a favorite of mine when looking at scatterplots.  In the past, Fathom had been the tool of choice, but now it’s time to fly with Desmos.  There are a few nuggets from AP Statistics here, and efforts to build conceptual understanding.

CORRELATION, LSRL’S AND STANDARD DEVIATION

Click the icon to the right to open a Desmos document, which contains a table of data from The Practice of Statistics.  In you are playing along at home, this data set comes from page 194 of TPS5e and shows the body mess and resting metabolic rate of 12 adult female subjects. One of the points is “moveable” – find the ghosted point, give it a drag, and observe the change in the LSRL (least-squares regression line) – explore changes and think about what it means to be an “influential” point.

Next, click the “Means” folder to activate it.  Here, we are given a vertical line and horizontal line, representing the means of the explanatory (x) and response (y) variables. Note the intersection of these lines.  Having AP students buy into the importance of the (x-bar, y-bar) point in regression beyond a memorized fact is tricky in this unit.  Drag the point, play, and hopefully we can develop the idea that this landmark point always lies on the LSRL.

Another “fact” from this unit which can easily wind up in the “just memorize it” bin is this formula which brings together slope, correlation, and standard deviation:

The formula is given on the exam, with b1 acting as the slope, so even memorizing it isn’t required, but we can develop a “feel” for the formula by looking at its components.

Click the “Means plus Std Devs” Folder and two new lines appear. we have moved one standard deviation in each direction for the x and y variables. Note that the intersection of these new lines is no longer on the LSRL. But it’s pretty close…seems like there is something going on here.

Ask students to play with the moveable point, and observe how close the rise comes to the intersection point. Can it ever reach the intersection? Can we ever over-shoot it? In the “Rise Over Run” folder, we can then verify that the slope of the LSRL can be found by taking a “rise” of one standard deviation of y, dividing by a “run” of one standard deviation of x, and multiplying by the correlation coefficient, r.


There’s other great stuff happening in the Desmos universe as well.

1.  This summer brings the 4th edition of Twitter Math Camp, to be held at Harvey Mudd College in California. I’m thrilled to have latched onto a team leading a morning session on Desmos. Consider coming out for the free PD event, and join myself, Michael Fenton, Jed Butler, and Glenn Waddell for what promise to be awesome mornings. To be honest, I feel the Ringo of this crew….

2. Can’t make it to the west coast this summer? Join me at the ISTE conference in Philadelphia, where I will present a learning session: “Rethink Math Class with the Desmos Graphing Calculator“. Bring your own device and join in the fun!

3. Are you new to the world of Desmos? Michael Fenton has organized an outstanding series of challenges, with 3 difficulty levels, to help you learn by doing. Try them out – they promise to get you think about how you and your students approach relationships.

4. Merry GIFSmos everybody!  The team at Desmos has developed GIFSmos to let you build your own animated gifs from Desmos files. EDIT – as Eli noted in the comments, credit for GIFSmos goes to Chris Lusto.  Thanks for being so awesome, Chris!

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Adding Distributions of Simulated Data

The current chapter on expected value and combining distributions in AP Statistics is one of my favorites for a number of reasons.  First, we have the opportunity to play games and analyze them…if you can’t make this fun, you are doing something wrong. Second, it often feels like the first time in the course we are doing some heavy lifting. Until now, we have discussed ideas like sampling, scatterplots and describing distributions – nothing really “new”, though we are certainly taking a much deeper dive.

The section on combining distributions contains a number of “major league” ideas; non-negotiable concepts which help build the engine for hypothesis testing later.  The activity I’m sharing today will focus on these facts:

  • The variance of the sum of independent, random variables is the sum of their variances.
  • The variance of the difference of independent, random variables is the sum of their variances.
  • The sum of normal variables is also normal.

First, we need to have student “buy in” that variances add. Then we have the strange second fact: how can it be that we ADD variances, when we are subtracting random variables? In this activity, we’ll look at large samples, and what happens when we add and subtract these samples. Since many students taking AP Stats have the SAT on their brain, and there is a natural need to add and subtract these variables, we have a meaningful context for exploration.

SIMULATING SAT MATH AND VERBAL SCORES

The printable classroom instructions for this activity are given at the bottom of this post.

photoTo begin, students use their graphing calculator to generate 200 simulated SAT math scores, using the “randnorm” feature on their TI calculators, and using the fact that section scores have an approximately normal distribution with mean 500 and standard deviation 110. Note – some older, non-silver edition TI-84’s won’t be happy with this, and a few students had to downgrade and use a sample of size 50 instead. There are a few issues with realism here: SAT section scores are always multiples of 10, which randnorm doesn’t “know”, and occasionally we will get a score below 200 or above 800, which are outside the possible range of scores. Also, there is a clear dependence on SAT section scores (higher math scores are associated with higher verbal scores, and vice versa), and here we are treating them independently.  But since our intent is to observe behavior of distributions, and not reach conclusions about actual SAT scores, we can live with this. In my class, no student questioned this as problematic.

Repeat the simulation in another column to simulate verbal scores. Then, for both columns, compute and record the sample mean and standard deviation. For my simulated data, we have the following:


It’s time to pause and make sure all students are clear on what we are simulating. We now have 200 students with paired data – the math and verbal score for each. Like most students, our simulated students would like to know their overall score, so adding math and verbal scores is natural. I help students write this command in a new column, then let them loose with the remaining instructions on both sides of the paper.

Students had little trouble finding the sum of the math and verbal scores, and computing the summary statistics. For my sample data, we have:

As students work through this, I want to make sure they are making connections to the notes they have already taken on combining distributions. I visited each student group (my students sit in groups of 4) to discuss their findings. Most groups could quickly identify that the means add, but what about those standard deviations? By now, if my students have taken good notes, they know that standard deviations don’t add, and that variances should. I leave groups with the task of verifying that the variances add.

Here’s the beautiful thing: students who immediately tell me that they “checked” the variances and verified the addition get the evil eye from me. In this simuation, students should find that the variances are “close” to adding, but not quite.  At the end of the acitivity, I ask students to conjecture why the addition is a “not quite” – even after I have beat into them that variances add.  There are two main reasons for this, and I was happy that a number of students sniffed these out.

  1. We are dealing with samples, not populations. There is inherent variability in the samples which causes the sample variances to not behave nicely.
  2. Variances add – but only if distributions are independent. Here, even though we created large random samples, there is still some small dependence. And while we don’t specifically cover the formula for dependent distributions in AP Stats, it’s worth discussing.

Next, it’s time to look at the differences.  Here’s students are asked to subtract math and verbal scores, compute the summary statistics, and compare the sum and differences. This was a nice way to go back and re-visit center, shape and spread.

photo

CENTER: Sums are centered around 1000, while differences are centered around zero.

SHAPE: Both distributions appear approximately normal.

SPREAD: The sum and difference distributions appear to have similar variability.

And this idea that the spread, and standard deviation, will be similar for both the sum and difference, can be also be explained by looking at the range of each population distribution.

  • For the sums, the max score is 1600 (800 M and 800 V), with a min of 400 (200 each)
  • For the differences, the max score is +600 (800M and 200 V), with a min of -600 (200M and 800 V).

Here, we can see that both distributiuons has the same range.

From start to finish, this exploration took about 30-40 minutes, and was worthwhile for verifying and developing understanding of the facts for combining distributions.  The student instructions and video notes students take beforehand are given below.  Enjoy!

Class Opener – Day 74 – PolyGraphs

My 9th graders have only about 2 weeks left with me before their final exam. Most of them will move on to Algebra 2 next semester, so my strategy with them has been 2-pronged: ensure we are produtive with new material and put them in a “happy place” to make a seamless tranisiton to Algebra 2. With a unit review today, and a pre-holiday-break quiz Monday, this was a perfect time to test-drive the new FREE PolyGraph activity from my friends at Desmos, along with the awesome work of Dan Meyer and Christopher Danielson. The Parabola activity sounded perfect for my class, though there are also activities featuring linear functions, rational functions and hexagons.

My freshmen have limited understanding of quadratic functions. While we have encountered some useful vocabulary regarding parabolas in my class (intercepts, vertex, domain and range), these students have not had a formal unit in graphing them yet. I was curious if students could transfer what they already knew to a new scenario. I was tempted to do a quick review of vocabulary before sending kids to the lab, but thought better of it. I want gut reactions.

In the activity, one student acts as the “picker” and chooses 1 parabola from a set of 16. The “guesser” then asks yes or no questions to help narrow down which parabola was chosen. “I don’t know” is also available as an option, if the question is not clear.

polygraphBetween games, students are given challenges to help guide their understanding of vocabualry and “good” leading questions.  I found these “intermission” questions to be extremely helpful, and noticed that the quality of the questions students asked improved after participating in them.

Some obeservations about my students in this activity, which we did for about 30 minutes.

  • Students didn’t have vocabulary to describe parabolas which “open up” (a>0) versus those which “open down”. The question “is it a smiley face or a frowny face?” was used by more than one student and led to some side discussions of what this meant.
  • Students also recognized that parabolas could have different widths, and describing the differences between these was more challenging. Questions like “Is it wide?” or “Is it narrow?” are helpful for identifying some extreme curves, but without a baseline for what a “regular” curve looks like, this leads to some confusion over which parabolas should be eliminated.
  • In the first round, few student mentioned the x or y-axis in their questions. Later, I noticed these became valuable tools for elimination.
  • Questions which attempted to use the vertex showed mixed success. “Is the vertex positive?” is unclear, but these attempts improved with more game plays. Similarly, attempts to describe domain or range often needed more work.
  • Students can be sneaky, and mine are no exception. Some students attempted to bypass mathematical conversation by asking “Is it in the top row?”.  Nice try – until they realize the parabolas are mixed up. Also, sometimes students were assigned to play against students sitting right next to them. Not ideal, but workable.

Here’s where I would go with this, if my next unit was on a formal discussion of quadratics: copy the student-developed phrases like “smiley face” nto a document. As we encounter those ideas formally in graphs, develop more math-specific language, match them up with the student descriptors, and improve the document. I want students to take ownership of their descriptions, and allow for their self-generated language. Hopefully, this builds richer connections to the vocabulary.

At the end of each class, I had students complete a Google Form evaluation. I appreciate the feedback from students who took this task seriously!

What did you enjoy most about this activity?

  • This activity was really really FUN!! I liked it because I was able to interact with my classmates. I had fun as well as learn.
  • It was interesting to see the language people used to describe the graphs.
  • i really liked that it was an interactive activity that we could do with our class mates. It really allowed us to think about math in a fun way!

How did this activity increase your understanding of parabolas?

  • I learned that a “smiley face” is positive and a “frowny face” is negative. these math terms will probably be useful.
  • I had to really think about the vocabulary and what I was saying while asking and answering questions
  • It forced me to think mathematically and use many math terms to figure out the answer.

Down the road, I think it could be fun to have one class code and invite students from a number of schools to join in. Knowing that their partner was somewhere in the room caused some goofy behavior, and I wonder how much more focus they would have if their partner was from a different school. In the end, I appreciate this activity because it is fun, forces students to think mathematically, and has clear entry points for class discussion after leaving the computers. Finally, can we have students use their already-existing Desmos accounts for logging in? I like that the data from all students is shared with me, and would be even better if their activity data is all in one place.  Awesome job team!

Class Opener – Day 53 – Don’t Take My Twitter Away!

Yesterday was an exciting day in the school-tech world, as a group of educators met at the White House and discuss education policy relating to technology – specifically the access and cost of broadband for students. A special daytime version of #edtechchat was held, and I had the opportunity to give my 2 cents on the opportunities technology allows for differentiation:

Thanks to Tom Murray and the #edtechchat crew for their sharing out from this exciting day.

But today brought my twitter enthusiasm to a screeching halt….overnight our tech department applied a new filter to the district network. Students and staff are now under the same network filet umbrella. The net result here is that TWITTER IS BLOCKED! {insert dramatic twist music}

Part of my morning then became e-mail back and forth between our tech folks and administrators, expressing my disappointment.  For now, I can access twitter through a “15 minute window”, which clearly is a non-solution. Through these “conversations”, the filtering of twitter for students has risen as an issue, so I have a few questions for my blog friends:

  1. Do students in your district have access to twitter during the school day?
  2. Have you experienced any discipline issues related to twitter?
  3. Why should students have twitter “unblocked”? Give me your best argument.

OK, so today was more of a rant…rather than an opener… Sometimes unpredictable episodes occur.

Piecewise Functions and Restrictions on Desmos

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I love checking my blog stats. Above are just some of the many search terms which cause people to end up here on the blog. You search, I listen. Armed with Camtasia (provided by my friend and barbecue savant Jason Valade from TechSmith) here is a tutorial I hope you find helpful as you start your school year. Resolve to make Desmos part of your classroom culture this year, then check out the Desmos File Cabinet of graphs to get you started.  Also, check out classroom strategies for using Desmos to explore function inequalities in the second video below.

DOMAIN RESTRICTIONS AND PIECEWISE FUNCTIONS

 

INVESTIGATING INEQUALITIES USING DESMOS

4 Tools I am Pumped to Try

Another great week with Discovery Education is over, and I was happy to share some fun math resources to the Fellows at the Siemens STEM Institute. But being a team lead doesn’t mean that I’m not learning as well – here are 4 great online resources I had a chance to try out this week, and look forward to using in my classes to increase engagement.

Kahoot – create fun quizzes for your class with this tool. You broadcast the questions one at a time, and students earn points based on their speed on correct answers. Keeps class-wide leaderboards. Kahoot

ThingLink – annote pictures with information, and share your works digitally. Provide information about key aspects, or external links to share additional information. Here is my first attempt, using a sampling distribution example as the context.  Click the link to try it out!

http://www.thinglink.com/scene/556526980820893698

Plickers – load the applet, print out the pre-made response cards, and prepare for a quick, engaging formative multiple-choice assessment.  By scanning the room with the app, you pick up student responses, which can then be linked to students if you choose.

Plickers

Answer Garden – billing itself as a “minimalistic feedback tool”, Answer Garden allows students to share briefs thoughts on a topic you choose, then gathers the results in a word wall.  Here is how I used this as a potential class bell-ringer in a talk I gave earlier this week:

Prompt

Students can them respond on a laptop or smart phone, with multiple responses allowed. How could these responses then allow us to start a class discussion?

Responses

Please Rip Apart My Flipped Videos!

It’s the Most Wonderful Time of the Year!

-Andy Williams

It’s the beginning of August. I’m sitting with 50 fantastic educators from across the country. I’m at Silver Spring at Discovery Education Headquarters.  This must mean it’s another year of the Siemens STEM Institute! During the week, you can follow the goings-on through the Institute Website, or visit the site often as we populate it with resources and come back often!

Wall


LodgeTonight’s Keynote speaker is Dr. Lodge McCammon, follow him on Twitter, a pioneer in not only using video, sound and motion to enhance educational practices, including classroom “flipping”, but also encouraging low-cost, simple-to-implement solutions. Check out Lodge’s YouTube Channel and enjoy his work, especially his series of 50 States Songs.

Today Lodge shared one of his catchy songs, this one featuring planetary motion, moons and their orbits. His kinesthetic lecture technique challenges us to apply movements with meaning by finding simple movements to represent otherwise complex topics.

I have written about Lodge’s talks here on the blog before, and invite you to go back and check out his powerful educational message. My first go-around with Flipping was also documented here on the blog, along with some mid-year reflections and tools.

So, where do I go from here?

Well, to be honest….I made lots of videos last year and some of them suck. Some I really, really like, and were on-target for my classroom expectations. Others…not so much. My goal this year is to review all of my videos and assess their effectiveness, then edit, re-shoot, re-format…whatever it will take to make my videos work best for kids.  You can check out my many algebra and stats videos on my YouTube Channel.

SO, I NEED YOU YOUR HELP TO RIP APART MY VIDEOS!!!

Below are 3 examples of videos I used last year to “flip”instruction. Each of them was made using a different device or format, and while there are parts of each that I think are effective, none of them are perfect.  I invite you to leave comments about any or all of my videos here, or in the YouTube comments.

Video 1: Completing the Square. Made using Doceri.

PROS: Content is clean. Mostly on point. After assigning this video, was able to quickly dive into problems the next day.

CONS: Probably too long. Tried to cover too much in one video.

Video 2:  Random Variables. Made using SMART Notebook

PROS: These are not easy example problems. Students can go back and “rewind” to think about processes.

CONS: These are a lot of rules in this section. The different format types may make it difficult to identify the “big ideas”.

Video 3: Samping Distributions. Recorded live in my dining room.

PROS: After some initial discomort, I find I have the most fun doing these live videos, and that perhaps they do a better job in engaging students with the ideas.

CONS: Am I clear? Are the visuals good enough?

So I invite you to rip apart my videos, provide guidance, comments, and share your success stories with flipping!