I love how Desmos Activity Builder has given students the opportunity to discover many concepts in mathematics at their own pace. A well designed activity will get them to predict, test and validate their ideas, helping misconceptions come to the surface along the way. That light bulb moment when you hear your students exclaim "Oh, I get it!" is amazing. The activities on **teacher.desmos.com** are all fantastic, however I thought I would share a few less conventional ways of using Activity Builder.

**#1.**

I was helping create a test recently and wanted to include some "student" work for my students to analyze. To accomplish this I create an activity with a graph screen and then a sketch screen. Here was *f*(*x*):

And here is "Martha's" graph of the reciprocal of *f*(*x*):

Using the sketch feature to create work for students to discuss is quick and easy. It really helped me see what relationships they understood.

**#2.**

If students are creating their own graphs you can collect them into one activity to allow you to discuss or show them off more efficiently.

If you add a graph screen to a new activity you can paste the URl into the first line of the graph screen and that entire graph page will be loaded.
Paste a link like this: https://www.desmos.com/calculator/sr04cmo3vk as shown below.

You can then preview the activity to see each graph in turn.

**#3.**

Although you can make them part of a larger activity, both Card Sorts and Marbleslides can be stand-alone activities. These are options under the Labs tab. (You may have to turn this option on - I'm not sure if this is still required.)

You could create a card sort as a warm-up or exit ticket. Assuming all students have access to technology, they can complete one in a very short amount of time and you get really quick feedback (see green/red below).

Marbleslide challenges can be used at all levels of graphing and are delightful! Sean Sweeney has posted 36 Marbleslide challenges **here**. I will stop on that note so that you can go try them out yourself. This is the one that I am currently working on, from Set 14:

From the #MTBoS...

My advanced functions classes (grade 12 - similar to pre-calculus) are doing really well so far. I am mixing up VNPS (vertical non-permanent surfaces - i.e. whiteboards) with some direct instruction and a lot of explorations with Desmos. **Here** is the link to the Desmos activity I created to introduce increasing/decreasing intervals. Using the pause button was great to help focus everyone's attention as common misconceptions came to light.

Yesterday, we started investigating the remainder theorem. In their random groups, they divided f(*x*) = *x*²

+ 5*x* + 6 by *x* + 2, then by *x* + 3, then by *x* - 1. They also looked at f(-2), f(-3) and f(1). Then I asked what they noticed. Some really didn't notice much so I asked them to divide f(x) by (*x*+4) and find f(-4). They saw that the remainder from the division was the same as the value they calculated and that the value they were calculating was zero only if the divisor was a factor of f(x). (Note to future self: this was too scaffolded; fix for next year.)

So today we started with this: Find a factor of *x*³ + 5*x*² - 22*x* - 56. They were in new random groups of 3 and clearly did not make the connection to what they had started yesterday. Here is some of what I saw - lots to talk about!

We discussed our objective here - to factor this polynomial, which would allow us to sketch it. I asked something like "If something is a factor, what do we know?" The light bulb went on and they ran with that, finding at first one factor, then the remaining two using a variety of methods. Seeing that some were trying all integer values of *x* in f(*x*), I created a new question for them: Factor *x*³ + 6*x*² - 8*x* - 7. Those who had found all the factors to the previous question by systematically trying all integers starting at 0 soon got tired and asked if there was a better way (that was the point of the question). I suggested they look at the constant terms in their factors and the original polynomial. They were remarkably quick at putting those pieces together.

So, soon groups knew that only needed to try ±

1 and ±

7. They made more mistakes along the way (see below!), but there was progress. They found that they could not determine the other factors using the factor theorem, but had to divide by the first factor they had found. They understood the process and had ownership of it, having tried many paths that didn't take them where they wanted to go before figure out what would work consistently.

We didn't get through very many examples, but I firmly believe that it's better to work through one or two examples in depth, allowing students to find the pitfalls along the way and find their way out of them, than to spoon feed students a multitude of examples.

It's been ridiculously hot here the past few days so I feel somewhat incoherent and am not sure what point I was trying to make with this post anymore... I guess the takeaway I see is that using VNPS and VRG to let students explore and make mistakes is really powerful. I am really trying to get my students to do the thinking, not fall back on memorizing an algorithm, despite the fact that many think that is the best way to learn (ack!). I am trying to convince them that __understanding__ the mathematics will take them so much further and be far more beneficial to them in the long run. That making attempts, some of which will fail, will prepare them for other times when they will struggle and not know where to start when solving a problem.

I'll end this with part of an e-mail I received from a student who graduated last year - I can't tell you how much it meant to me: "I have to say, I think your calculus class has been the most useful so far. The problem solving skills I learned in that class have taught me to set up equations and approach problems from a different point of view (in multiple classes... especially chemistry)! The self learning technique also helped because that's pretty much all I do now before I go to a lecture. "

In the spirit of Carl's keynote at TMC17, I thought I would pass along my best advice to all those who are taking on new challenges and have half a dozen #1TMCThing: don't let who you are get lost as you try to implement others' great ideas.

It took me a long time to figure this out - I can't be you so I need to make your idea work for me. And sometimes that means it just won't. After TMC15 I so wanted to jump on the High 5 bandwagon (giving every student a high 5 on the way into class), but I just couldn't. Merely thinking about it made me cringe, despite all the great things everyone was saying about it. I have also wanted to be more like <insert teacher's name here> and it took me some time to realize that I can't teach like them because I'm not them. So be sure to sift through all the great ideas you have collected and find the ones that you can actually put into action. Make them yours, adapt them as needed, and make them great. If you have found an activity that you think you can implement well with your students, work through it and tweak it so that it represents what your students need. Cultivate your own style while stretching yourself to be better, always. Be you, not whomever.

I really hope this doesn't sound preachy. Not sure whether to hit Publish, but #justpushsend and all...
This is a thoroughly unhelpful blog post except for those of you who were in my Desmos class at Exeter. It's just a whole bunch of links...

Desmos Bank
blog.desmos.com

Sunday:
Monday:

A while back, Pam Wilson shared an old linear matching activity. It had students match up a graph, two points, a slope and two forms of a linear equation to form a set. I really liked it, but it used old calculator screen captures for the graphs. I cleaned it up and ran it with my students. It went well, but I learned that it works better if each type of card (graph, slope, equation, etc.) is printed on one colour of paper so that students have a complete set when they have one card of each colour.

**Here** is the .docx file and **here** is the .pdf file. I'd love to hear if you use it and how we could make it better!
I ended my first full unit teaching in a Thinking Classroom (you can read about the beginning of this journey for me **here**) in my calculus classes just before the March break. I surveyed my students and had mixed feelings about the results. Many suggested 20 minutes of notes at the beginning of class, yet many of these same students also said that they never looked at the notes I post on Google classroom each day. I e-mailed Alex Overwijk to ask for advice, saying that I felt like I was doing something wrong, or at least not entirely right. His response went along the lines of "they don't like being uncomfortable, they don't like having to struggle and you aren't doing anything wrong". I think that what led to me feeling as I did had a lot to do with the very skill-based nature of the unit. They were taking derivatives and taking more derivatives and then they took some more derivatives. This led to groups taking turns doing questions. The questions were not challenging enough to require them to work together as a group to solve them. I'm not sure how to change that for the let's-learn-how-to-take-derivatives unit, but I will ponder that some more before we get there again next year.

This past week we have been working on the elements of curve sketching. Using the first and second derivatives to help determine intervals of increase and decrease, local maximum and minimum points, intervals of concavity, points of inflection, etc. I have continued to use visible random groups and they have continued to work on the VNPS (whiteboard/chalkboard) for almost the entire class each day. I have tried to be more intentional about what I do work through with them - mostly at the beginning of the class. The questions have been more interesting and more challenging for them. I am really pleased with their efforts. I am finding that they are putting all the pieces together more easily and that I am also thinking more deeply about the material. And it's fun! At least twice this week the bell rang at the end of my afternoon class without anyone in the room being aware that it was the end of class. They didn't want to stop. It's incredible how much fantastic work they are producing and how well they can explain it all to me. I am not quite sure how much gushing is appropriate, but my students are awesome. I snapped this picture of some of them this morning and it makes me happy and proud to look at it.

**Here** is the progression I used for the week (apologies - I got lazy and didn't include all the answers). They did not all get through every question yesterday and today, but I believe that they all have a solid grasp of the material. I continue to post filled-in notes at the end of each day should they wish to review the work or try any of the questions on their own.
I was going to blog about the last few days in my calculus class, including "Leibniz Day", but I just can't. I found out at the end of the day today that a student I taught all of last year passed away over the weekend. Sometimes teaching is really hard.

**Here** is the docx version of what I did and **here** is the PDF version.