Mistakes are Great

Earlier in January, I wrote about the Collaborative Exam Review that my pre-calculus classes (a total of 43 students) wrote using a variety of tools from Google Drive.  Today was exam day, so the project is done, and have decided that what I liked best about is was that it was their responsibility and it wasn’t perfect.

In preparing the review, students had to take responsibility for knowing what they needed to know (so they needed to become experts on their topic) and they had to take responsibility for knowing it well enough to explain it coherently (via their solution image) to others.

Even so, kids made mistakes.  However, because of the culture of responsibility for the material that already existed in this review, the mistakes actually made the comments and discussion on the review problems/solutions that I required in the week leading up to the exam even better.  Students evaluated peer solutions and, when they had questions or found mistakes, used Google Drive’s commenting features to offer constructive criticisms or additions to the material was already there.  Lots of different students started getting in on the discussions, going back and forth about what was right and what was wrong.  At the end of the day, I think this helped them demonstrate their mastery of the material better than simply completing a review packet.  After all, you can’t tell someone else they did something wrong if you don’t know how to do it yourself!


Overheard from a Chemistry Teacher

I was in the main office the other day and I heard a chemistry teacher singing the praises of his Apple TV.  When reviewing homework (lots of math in chemistry), he asks students to mirror their iPads.  If a student has a question, he asks that student to mirror her iPad so the class can look at her work.  They work together to find the mistake and correct the problem.  Then they move on to the next one.  He said the students love sharing their work with the class and helping each other find mistakes … it’s like a puzzle (for the whole class to work on together).


Cold Season = Increased Absences

It’s cold season in Massachusetts (I’m even writing this while surrounded by cough drops and tissues).  Cold season means my students miss class more often and come back to school overwhelmed by make-up work.

In our 1:1 environment, I write out my examples in Notability and my students take their notes in Notability too.  When a student is absent, he/she typically asks a classmate to share the notes from class before coming back to school.  While this doesn’t make up for the missed class, when absent students come to see me the next, they have already seen the notes and are prepared to ask questions other than the dreaded “did I miss anything?

Note: Students typically “share” their notes with each other by sending their Notability file via email to another student.

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!

Pooling our Data

Last Monday in Geometry, it was time for my students to ‘discover’ that when two parallel lines are cut by a transversal, corresponding angles are congruent, etc.  I think its pretty typical for geometry teachers to have students construct parallel lines and a transversal and measure pairs of angles to determine special relationships.  Here’s the activity as suggested by a geometry textbook I have:

I think this is a good activity, and I’ve done it before, but I’ve never been entirely satisfied with it.  First of all, invariably someone starts shouting out the conclusions before everyone is done measuring, and students start making assumptions about their own results, and the magic is lost.  Second of all, I feel like the teacher usually ends up saying something like, “and every time you draw parallel lines and a transversal, alternate interior angles will be congruent.”  I don’t want my students to believe everything I say without evaluating its reasonableness first.  They need more evidence than that.  I am trying to teach my geometry students to adopt an attitude of skepticism; they shouldn’t be drawing conclusions unless they can explain them.

In an attempt to improve this activity this year, I had students work with partners, but I imposed a strict ban on speaking while they were working, so that no pair’s work would be influenced by that of others.  Some students asked me if they could assume certain results would be the case, and I reminded them of the dangers of assuming.  I had them submit all of their measurements via a Google Form, and then we looked at the pooled results as a class to see if we could establish patterns.  I did this with a class of students that need a lot of direction, and it did not exactly go smoothly: some students measured the wrong angles, some students measured the right angles incorrectly, some students entered their data in the wrong place, but we got it done eventually, and I was still happy I did it.


1) It was a struggle.  I wouldn’t confirm or deny anything my students were thinking and saying for most of the activity, so it forced some actual thinking.

2) Our examples and data were student created.  They weren’t perfect, but they were real.  Not all angles in life are 60 degrees, and sometimes that makes them hard to measure, but you might still have to measure them.

3) There were mistakes.  Students didn’t only see their results, but they saw those of their classmates.  When something didn’t look right, we could have a conversation about whether someone made a mistake, or whether it was actually the case that the pattern people thought they were seeing wasn’t a pattern at all.

I liked this activity, and I liked how my students’ iPads made it relatively easy to execute.  But it got me thinking about something I’m starting to not like about my students having iPads, but that’s a topic for another post…