# Latest Updates to My Geometry Course

My geometry course… funny. It isn’t really my geometry course anymore. Little by little it’s becoming more of a curated set of links that make me wish I still had a classroom. That’s what community will do for you and the innovative math community more than takes care of people. I’ve noticed some excellent learning experiences that my geometry course has been missing. So, I’m adding them. All three are Desmos activities.

Parallelograms in the Coordinate Plane by Jared – This one asks some very interesting questions and has a nice format in which to do it. Due to the algebraic nature of proving this parallel, parallelograms are a nice target for a Desmos activity.

Transformation Polygraph by Julia Finneyfrock (@jfinneyfrock) – Seriously. Check this activity out and then daydream about all the outstanding vocabulary the students would be forced to use while competing with each other.

Working with Dilations by Mr. Rothe – This one is particularly handy because my unit on dilations and scale factor wasn’t my favorite. This is a very robust activity (36 slides), so time might become an issue, but depending on how you run it, I feel like there’s a ton of value embedded in the different learning activities.

If you give any of these a try, please let me know how they go.

# 4 Desmos Activities I’m adding to My Geometry Course

It’s been about 14 months since I last taught Geometry, but that hasn’t kept me from keeping a keen eye out for high-quality activities to add to My Geometry Course.

Here are the goods that I’ve found looking through the searchable offerings on Desmos.

Ryan Brown offers a nice introduction to polygons using a do-it-yourself Polygraph activity. This strikes me as a nice kick-off to an early unit in Geometry. Serves as a good vocab review. For high schoolers, it would likely serve as a nice low-entry point to Polygons and wandering around listening would give a nice window into what your students could possibly missing as you are about to embark on a new unit.

Kate Nowak offers a nice algebraic introduction at circles in her Activity-Builder offering.  I like this because the common core circles standards tend to be a little more algebraic than some of the other units, so we put it toward the end of the year as we are ramping up the students for their next challenge (which is often Algebra II). Using specific vocabulary like “proportional” and having the student drag points around the graph sets a good tone for what students can expect in a second semester circles unit.

Pizza Delivery by Scott Miller is a very interesting take on reflections. I am actually bummed that I’m not teaching Geometry any more because I’d like to see what this one looks like. You’re going to want to look at this one.

Mathy Cathy offers a nice introduction to reflections as well. I really like the questions she chooses and the tasks seem approachable as reflections are explored for the first time.

# The Struggle for New Math Instructional Technology

This article by Jack Smith IV (mic.com) says what I’m thinking better than me. Matt McFarland (Washington Post) has a similar argument most focused on the price (which mysteriously hasn’t followed the trends of any other piece of personal technology by staying quite high).

The argument: Texas Instruments has some kind of a stranglehold on secondary mathematics here in the United States and it certainly isn’t that it is the best tool for the job.

The arguments that I most often hear from teachers in support of sticking with TI are that it is the tools that have been in place so long that they are the easiest for teachers to teach. It’s just easier to have the students all using a single tool that the teachers are really familiar with.

In an article written in The Atlantic, Alexis C. Madrigal argues that there is very little need for an update in the technology if there hasn’t been a corresponding update in the math content requiring the technology. “After all, the material hasn’t changed (much), so if the calculators were good enough for us 10 or 15 years ago, they are still good enough to solve the math problems.”

And also that these tool are the tools that are allowed on the SAT or the ACT. For some teachers that I talk to, this is kind of a big deal. And there is a reasonable logic to it. Those tests are pretty important (especially in Michigan, where school accountability for math is connected to student outcomes on the SAT). So, why confuse a student’s brain with a variety of tools that, when the rubber hits the road, they won’t be able to use?

It’s worth noting, however, that College Board (the designers and publishers of the SAT as well as the Advanced Placement exams) have a particular target audience in mind. In states where the SAT isn’t the state-mandated accountability measure (meaning every single high school junior statewide will take the test), which students are likely to take a test designed by College Board? Students who are college bound and/or enrolled in advanced placement classes. For many, many of these students TI Calculators have “worked out fine”. Also, they’ve likely been the only tool made consistently available. And it’s better than graphing all of your parabolas by hand.

But, have you ever watched a group of students in an algebra class for the second time trying to remember those key sequences and explore images on those tiny, granulated screens? And by now, they know there are technologies out there that are easier to use.

I am certainly not accusing College Board of being deliberately prejudiced. I am not accusing TI of being deliberately prejudiced. I’m simply accusing them of having goals that are different than “100% of math students will learn math at the highest possible level”.

But that’s my goal. And it should be the goal of all of our classroom teachers. And the question should be whether or not TI calculators fit within that goal.

Well, in Michigan, there might be a couple of potential cracks in the shell through which the light of better technologies (like Desmos and Geogebra) could potentially shine through to teachers previously unwilling to explore them.

The Redesigned SAT, which launches in March 2016, has a math portion that includes two sections. There is a significant portion that prohibits calculators completely. Then, there is a section that allows calculators, but the College Board includes questions “for which calculators would be a deterrent to efficiency.” D’ya catch that? That’s our window. This allows for an entirely different message than simply “TI’s are allowed on the SAT, so we’ll use them.”

You could reasonably guess that AT LEAST half of that test should be able to be done with no calculator at all. This could potentially leave a huge hole in one of the most stubborn supports for the TI-status-quo. Now, we are potentially keeping these clunky, expensive devices as the primary tool because of possibly 25 questions that they’ll use it on for a single testing sequence. That’s a much tougher sell.

The other main issue (teacher familiarity) is is a matter of exposure. The teacher prep courses could support this for our preservice teachers. (In 2004, when I was a preservice teacher at WMU, I had one “instructional technology in math” course that has a TI-89 or Voyage200 as a required purchase. Luckily my girlfriend (now my wife) had one.)

Instructional technology in math should be way, way more than a semester-long how-to on a single TI device. Desmos and Geogebra and other available technologies that are free for use in the classroom provide valuable opportunities to give students different experiences with problem-solving in mathematics. But teachers, like all people, are going to stick with what they are familiar with.

And most of these young math teachers (most born in the early 90’s), we raised in the classroom as math learners with TI calculators. Given no other experiences, they aren’t going to have just their familiarity to build from. Especially when, for most of them, the TIs “worked out fine.”

As for the teachers already in the classroom, that’s part of the work of math-minded instructional techs… like me. I’ve got two sessions coming up that are focused on Desmos as a tool to engage math learners.

The SAT did it’s part and I intend to do mine. Our students deserve better than TI or nothing.

# Exploring Reflections with Desmos Activity Builder

Geometric transformations take up a good chunk of the first quarter of Geometry (at least the way that I had it sequenced). The tricky part of teaching transformations in Geometry is the delicate balance between the non-algebraic techniques and understandings and their algebraic counterparts.

For example, consider the two following statements.

Two images are reflections if they are congruent, equidistant from a single line of reflection and oriented perpendicularly from said line of reflection.

The reflection of A(x, y) is A'(-x, y) if the line of reflection is the y-axis and A'(x, y’) if the line of reflection is the x-axis.

As a geometry teacher, am I to prefer one over the other? In my experience, they both present challenges.Students are often a little more enthusiastic about the first (which is why I start there), but can often be more precise with the second. The second requires the figure to have vertices with coordinates and the line of reflection be an axis. The first requires the figure be drawn (and fairly accurately, at that.)

So, in my attempt to learn how to use Desmos Activity Builder, I wanted to produce something useful. So, I made a Desmos Activity to bridge the transition from a visual, non-algebraic understanding of reflections to an algebraic one.

Full disclosure: This activity presupposes that students are familiar with reflections in a visual sense. It isn’t intended to be an introduction to reflections for students who are brand new to the topic.

Feedback, questions… all welcome.

# Have you seen Desmos Polygraph?

Most of us are  old enough to remember the game Guess Who, right? It looked like this:

It went like this: You and an opponent each had a board in front of you that looked like this. Then you had a card with a face on it. So did your partner. By asking only questions that could be answered by “yes” or “no”, you needed to eliminate all the faces that were not on your opponent’s card in fewer turns than your opponent.

So you might ask “Does your person have glasses?” or “Is your person wearing a hat?” Based on the answered, you could flip some faces down because you knew they couldn’t be correct.

Okay. Now apply that to math. Imagine instead of human faces, it’s parabolas.

They also have one for lines, hexagons, rational functions, quadrilaterals (both basic and advanced).

Still yes or no questions. Students still have to determine which pictures to eliminate based on the answer to the yes or no questions. Set-up is incredibly easy.

As if that wasn’t cool enough, if you can think of a set of pictures you’d like to see, you can create your own Polygraph activity.

I thought it would be cool if there was one for systems of linear functions.

If you haven’t taken the time to explore Desmos yet, I’d say it’s about time.

# If haven’t met Desmos yet…

I would like to use this post as a shout-out… an atta way… an unpaid advertisement, if you will.

There are a lot of tech tool providers out there. Not all of them want to work with you. Not all of them want to hear your feedback. Not all of them provide their products openly for free on lots and lots of different platforms. (In fact, I sat in a focus group with an instructional tech developer once. The producct in question was being sold to districts for \$10,000 per building. When we all picked our jaws up off the floor they said that they set the price high on purpose. If too many people bought it, they didn’t think they would be able to properly support everyone who was using it.)

Desmos is not any of those things. I’ve never gotten responses from instructional tech providers and developers quite like I’ve gotten from the folks at Desmos. I have two anecdotes that will help illustrate this.

1. I had an idea and made this. When I finish a post, I send out a tweet. The folks at Desmos responded to my tweet.

I want to make sure you clicked both my “made this” and their “few possible ideas.” If you didn’t do that, do it now. Do you see the difference? Do you see what happened there? They took the little bit of an idea that I had and added to it stuff that I didn’t even know how to make Desmos do. The tweets that followed were an exchange between them and me that helped me learn how to do that stuff.

That was a unique event. That had literally never happened to me before. The developer of the tool reached out to my feeble attempt to use their tool and personally improved it and instructed me all the while giving me the credit?

At least it was unique until it happened again. Last week I had an idea. This idea. And I tweeted out the post. And then…

Once again, be sure that you check out their idea compared to mine. Mine was a nice start (I hope you see the progress I’m making learning how to use the tool). There’s was expert level. The folks at Desmos are eager to build on the ideas of the educators who are trying to put their free tool into play in the classroom.

If you haven’t gotten a chance to try some Desmos work? Maybe consider redesigning a lesson and then, just maybe, they’ll have a few ideas for you, too.

# Desmos-Enhanced: The remodeled Pencil Sharpener Problem

Lately, I’ve found it tremendously enjoyable to revisit some of my favorite homemade problems and use Desmos to model them.

I decided to remodel the Pencil Sharpener Problem this time. If you’re not familiar, go check it out. Here’s how it goes.

Three boys are held after class for detention. I told they have to stay for a half-hour, but they can leave earlier if they can grind down 100 pencils by hand in less than a half-hour.

So, the three of them decide to take the me up on my offer and begin cranking the pencil sharpeners as fast as they can. Each of their top speeds is recorded on video. If we assume that they keep their top speed up the whole time and don’t slow down, then how long with their detention last?

This problem has created some fantastic student work. Enough so that it is almost tempting to force pencil and paper work.

However, I couldn’t resist the temptation to create a Desmos worksheet for it.
Besides, by now, Pencil Sharpener Problem is ready for an extension. How’s this:

It seems safe to assume that the boys will tire as they crank the pencil sharpeners over and over and over. How about we say that the each subsequent pencil takes 5% longer than the previous pencil. So, if first pencil took 60 seconds, the second one took 63 seconds, and the third one would take 66.15 seconds, and so on.

Would it still make sense for the boys to grind away pencils? Or should they just sit quietly for 30 minutes?