A New Take on Midyear Exam Review

Margaret teaches the highest level of Algebra 1 at our school, and I teach Pre-Calculus at the same level.  At the beginning of December, we both had conversations with our classes that began something like this:

I am not going to give you a review packet for your first semester exam…

**anxious/borderline angry looks exchanged around the classroom**

…instead, we’re going to do something better!  You’re going to write the exam review yourselves.

Wait, what, you’re asking?  Last spring before finals, Margaret and I decided that we were done with giving our upper level students an exam review packet or problem set and spending a couple of days in class going over it, as we had done in the past.  So, we decided to try something new, and we thought it was valuable enough that we both decided to try it again for the midyear exam that is coming up in a couple of weeks.  This year, iPads are making the workflow much easier for all involved.  

This is what the process looks like:

1)  We created a skeleton Google Doc with an outline of the topics to be covered on the exam.  We intentionally broke the topics up into a number of subsections equal to the number of students we each have.   We shared this document with our students specifically so that we could hold them accountable for their work via the revision history.

2)  On the day we introduced the project, students accessed the document via the Drive App on their iPads and claimed the topic they wanted.  Then, we gave them a day in class to work on researching their topic (going through notes, the text, old quizzes and tests, etc). and draft 2-4 questions that would help classmates review the topic.  We required that at least one question be conceptual and one question be computational.  

3)  Students entered their questions into the original Google Doc.  They used the built in equation editor to enter any equations.  If they wanted to use a graph or other image, they drew it in Notabilty or graphed it on Desmos on their iPad, took a screen shot, uploaded the screenshot to Drive, and then inserted the image into the review doc.  The limitations of the Drive app were such that most students did this step on the web-based version of Google Drive on a regular computer.

4)  Margaret and I reviewed the questions, made comments, and students used our comments to edit their questions into final versions.

Here’s a copy of my pre-calc review in its almost finalized version.  In order to write these questions, students needed to figure out what they need to know about a topic AND come up with questions whose solutions demonstrate that information.  This forced them to think a bit outside of the box, especially for some of the conceptual questions.  I think its so much better than completing a review packet!

5)  Once questions were finalized, students created solutions to each of their problems in Notability on their iPads.  These solutions are supposed to include a worked out solution to the problem with step by step explanations.  Students uploaded a screenshot of each solution to Google Drive and submitted a link to the image to me via a Google Form.

Directions for Creating a Solution

6)  Once all solutions were submitted, I made the response spreadsheet available to all students.  Each entry in the spreadsheet includes a verbal description of the topic, the number of the associated review problem, and a link to a solution.  When students are studying, they can go to this spreadsheet, find a topic they want to know more about, and then see a comprehensive explanation by one of their classmates.

7)  Once all the solutions are in, students are required to evaluate each other’s work, use the commenting feature on Google Images to share their evaluation.  The owner of each solution is then responsible for replying to the comment with an answer to a question or a resolution for an error.

Commenting Guidelines

This whole endeavor is still very much in the work in progress stage.  Stay tuned for a follow up post after exams about how we thought it went, how our students thought it went, and what we would do differently next time!

Teach them to fish

Two weeks ago, I had my pre-calculus students do an investigation about the characteristics of the graphs of power functions using the Desmos iPad app, which I wrote about in an earlier post.

Last Tuesday, they had a quiz that included the content of that investigation, and on Monday night, I was surprised and delighted to find an email from one of those 11th graders with the screen shot below attached and a question that said something to the effect of,

“I’ve been drawing all these graphs to practice and I’m a little unsure of why the graphs shown have this shape. Is it this reason?”


Possibly one of the best emails I’ve gotten lately because this boy was using a tool he had for a helpful purpose that I didn’t suggest. And you know what? That student got full points on the quiz.

And you can’t beat the colors!

Margaret’s love for Desmos is fully endorsed by the other half of this blog!

If you aren’t convinced, here’s some additional classroom action:

Yesterday in pre-calc, my students did an investigation about power functions.  They graphed groups of these functions and made generalizations about their characteristics.

Here’s what a group of power functions looks like on a graphing calculator:


Here’s what the same group of power functions looks like graphed on Desmos:


Which one makes it easier to tell that all the functions in this group are symmetric about the y-axis?  A picture is worth a thousand words.

Desmos Graphing App

Graphing calculators are good, but the Graphing Calculator app by Desmos is better.  Graphing calculators are expensive. (The Desmos app is free).   Graphing calculators require several clicks to change the ‘window’ of a graph.  (The Desmos app is interactive and I can zoom in with my fingers).  Graphing calculators can be finicky when entering complicated equations. (The Desmos app points out your error). 

I could go on an on about why I think Desmos makes a better graphing calculator for my students, but instead I’ll tell you about my Trigonometry students.

We are headed towards proving trigonometric identities, but before start simplifying them by had, I had them graph pairs of equations whose equations looked very different but graphs were identical.


Two equivalent equations graphed at the same time.

Imagine their surprise when only one graph showed up!  Several were concerned that they had done something wrong, but au contraire, they are equivalent equations!  We had a lot of fun entering pairs of equations and deciding whether or not they were equivalent.

My students have started calling these trigonometric identities “same, same, but different” because they look the same on the graph, but they have different forms when written.

When I did this same activity last year with graphing calculators, it took forever to enter the equations and then find their mistakes if they forgot parentheses.  The Desmos app saved us a lot of time and therefore allowed us to try lots of different pairs of equations.

Entering complicated equations is easy with this list of functions.

Entering complicated equations is easy with this list of functions.

You can change the x- and y-axis to radians!

You can change the x- and y-axis to radians too!

Desmos also has a web based calculator that is just as fun!