Assessing Problem-Solving with Desmos

Throughout my teaching career, I’ve always told students that teaching them to be nimble problem-solvers and good critical thinkers is high on my priority list.  However, when it comes time to test the students, I have struggled to find an appropriate way to genuinely and fairly test problem-solving.  When I include a novel problem that requires some creativity, my students perceive it to be an unfair, “trick” question.   Of course, I emphasize problem solving in collaborative projects, but 30% of my course grades are usually from tests.  If problem-solving is really important to me, I ought to find a way incorporate it into my major assessments.

On a recent regular- track Pre-Calculus test, I think I’ve made some good progress.  The first 80% of the test was fairly traditional and non-calculator.  I think I tapped into some higher-order thinking skills, but not much problem-solving.  When the students had completed this section, they turned it in and got the following two problems.

The first problem requires the students to imagine how to complete an even function.  In class, we identified even functions and proved that certain functions were even, but I never asked the students to do this.  Also, we practiced how to reflect a function over the x-axis, but we had never considered how to reflect over the y-axis.  The second problem asks the students to consider the absolute value as a transformation.  This is also something that I might have considered in an Honors section of Pre-Calculus, but we hadn’t talked about in this class.

While the students were working, I walked the class and watched them explore.  As I had hoped, the students were trying many different things.  A few students needed some encouragement to try many varied examples and that is not surprising given how little they have explored mathematics in their high school careers.   On the second problem, there were a few students who had used the function from first example and said, “Mr. Hinkley, the transformation doesn’t do anything.”  With some encouragement they tried a wider variety of functions.

Overall, I was very impressed with the results.  They spent quite a bit of time working on these two problems and were largely successful.  I got some very detailed explanations of how the absolute value transformation worked.  Rather than following a repetitive process like transforming a simple function, the students were engaged in real mathematics.

My hope is to follow this format for all the rest of the tests this year and see if the students are able to progress in their skills.

Student Writing about Math Technology

The last essay I assigned in a math class was when I was a graduate student at the University of Illinois teaching a summer math class called Numeracy.  A star of the football team (I had several football players in the class…they had their own extra large desks) wrote a paper outlining how he would have to subtract taxes, agent’s fees, and the cost of owning several different houses from his multi-million dollar NFL contract.  Did I mention that this was the last paper I had assigned?

As I planned how I was going to integrate technology in my AP Calculus class, in particular how I was going use both Desmos and the graphing calculator, I decided to ask my students to weigh in on the Calculator Wars.   I told them that I would select the top two and publish on them on my blog.  (This was before Dan Meyer gave my blog a shout out and sent my subscribers through the roof!)

So, here’s the Technology Writing Assignment.

And here are the top two papers.  I should note that I brought in a colleague from the English Department and asked her to read my top six with the names hidden.  So the fact that my son’s essay is one of the finalist isn’t complete nepotism.  Also notice that the two essays take a pretty different stand, so I avoided the trap of only choosing the papers that I agree with.

Jackson Vail’s Calculator Essay

Ezra Hinkley Math Essay

Apologies to Aristarchus

2300 years ago the Greek astronomer Aristarchus estimated the size of the moon relative to earth during a Lunar Eclipse.  In 2000 A.D., Peter Alway, a college astronomy professor, showed how his class made a similar calculation using pencil and paper.

Now, Desmos makes a similar calculation remarkably easy.  Using my son, Ezra’s photo (taken with just a digital camera), my Pre-Calculus class is able to embed the photo into Desmos and mess with circles until they line up.  Here’s what the final version should look like.

Convincing them how to deal with the Penumbra/Umbra calculation is a little more complicated.  It’s too bad the last step requires a “Well based on astronomy, we know that…”.  However, I don’t think my students have ever had a more relevant application of finding the equations of circles.

Here’s the assignment:  Lunar Eclipse Prob Set

Here’s the beautiful photo taken by my son, Ezra: 

Teach Them to Fish

I have been working on a major overhaul of my AB Calculus curriculum this summer.  In particular, I’ve been thinking about the best way to integrate technology as I write the problem sets. (Here are a couple of examples of this summer’s work: 3. Continuity Problems   4. Derivative Rules Problems  Contact me if you’d like all that I have completed or solutions.)  My vision is to be largely textbook-free this year in order to allow the students more opportunities to investigate calculus without the strictures of a textbook. A corollary benefit is the reduction on the lower back pain associated with carrying 35# backpacks home every day.

I have found as I write these problems that I am frequently writing at the end of every one: “Verify with Desmos.”  For years, I have struggled to fully integrate the graphing calculator into my classroom.  At Medomak Valley, we have TI-nspire calculators in the classroom, but students are not required to purchase their own and given the socio-economic status of our district, it is an unreasonable expectation for many students. I believe that Desmos is a potentially revolutionary technology in mathematics education.  In the past, I have had my students use the graphing calculator to verify that they have found a line tangent to a function only once, usually in class.  The process of entering the function, the line, then zooming is just too cumbersome to do every time.  However, Desmos is quick, easy and intuitive.  The graph you see is so rewarding… it jumps out at you and says:  “You’ve done it correctly!”.

Seeing a visual demonstration of the correctness of your analytic solution certainly doesn’t warrant the use of the word “revolutionary”.  What is revolutionary is the idea that students can become comfortable enough with Desmos this year that they will see it as a tool to investigate and explore mathematics!  In the past I have always built teaching modules to foster student exploration.  Years ago, they were problem sets based on the TI-83 calculator – maybe graphing a string of vertical transformations of a quadratic function.  More recently, I have built modules on Geometer’s Sketch Pad or GeoGebra for students to investigate a specific mathematical property.  The difference now is that I want to get students to the point where they will simply open Desmos and build their own modeling tool from scratch.

To take this out of the realm of the abstract, here’s a specific example from my Derivative Rules Problem Set (See link above).  The last problem asks students to solve the classic AB Calculus problem:  Find the shortest distance between a point and curve.  (At this point in my curriculum, we are solving this by finding the normal line, not by minimizing the distance.)  Here’s the problem: 

The first three parts are very traditional.  I encourage those interested to follow my instructions to build their own investigation on Desmos.  Here’s the finished product if you’re short on time.  This dynamic graph is simple, elegant and beautiful and connects many disparate concepts.  As a mathematician, I have always prospered happily in the “Analytic” realm of our three-legged stool of math education:  Analytical, Graphical, Numerical.  I never needed to see math visually to understand it, but I recognize that I do understand it at a deeper level through this visualization.  For example, here’s something I noticed: the point that models distance mimics f(x) when a is small and becomes linear when a is large.  Obvious from the equation for distance, but I wouldn’t have thought of it without seeing the graph.

In my title, I referenced the proverb about fish and fishing.  When I build a tool on Desmos, GSP or a TI-83, I have given them a fish.  When they can open Desmos on their own and build their own investigation, I have taught them to fish.

Calculus with Clay

Once my Medomak Valley High School AP Calculus students had completed their AP Exam, I went in search of a genuine interdisciplinary project that would apply what they had learned in Calculus, but also enrich their understanding.  Plus it had to be fun!  After a discussion with Medomak’s pottery teacher, Brooke Holland, we decided that having the students make a pot on the pottery wheel and then find the volume of their pot (interior and exterior volume) using calculus would be an excellent culminating project.

Here is the assignment: AP Calculus Final Project

The first day in the pottery studio was fascinating as an educator.  I watched another teacher in action, I observed my predominantly left-brained students in a completely novel and somewhat unnatural environment, and I participated as a student as well, joining them in learning how to throw a pot.

As educators, one of our mantras is that students need to be willing to fail in order to have genuine learning.  Isn’t sitting at a pottery wheel for the first time a perfect example of this?  Here’s the evidence of our first failures….I mean attempts:

The final products were a bit better.

After the bowls were fired in the kiln, it was time to begin measuring the volumes in two different ways.

First, each student took a photograph of their bowl and embedded the photo into Desmos , a revolutionary on-line graphing calculator and matched the Desmos scale to the measured scale of their photograph.  They then added functions (with sliders) to approximate the cross-sectional curves on their bowls.  The students were creative with their choices for function types:

• sine and inverse sine
• linear
• exponential
• logistic
• quadratic

Once they had their best fit for a model, they calculated the exterior volume using a combination of the volume by revolution formulas from calculus.  Some students used the discs and washers method, others used cylindrical shells.  Since the models for the pots were so wacky, all students appropriately used the TI-nspire graphing calculator to compute the values of the integrals.

To estimate the interior volume, students used calipers to either estimate the thickness of the pot at a set intervals or to measure the diameter of the interior of the bowl.  Again, Desmos was used to find a model for the interior and calculate the volume.

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Finally the students dunked their pots into a water tank to calculate both the interior and exterior volumes.  This was harder than I thought it was going to be.

Here’s a video showing our fun!

For a class of mostly seniors, they put quite a bit of work pulling all the details together.

In the end, I couldn’t pass up the opportunity to take some math-art photos.

Joseph Powell

Aaron Smeltzer

Megan Reed

JP Lobley

Gavin Felch