I was recently inspired by this tweet from Alex Overwijk:

I created my own set of logarithmic graphs - ten with base 10 (white paper) and ten with base 2, 3, 4 or 5 (green paper).

My students, in random groups on big whiteboards, had to determine the equation of the graph they chose. All groups started with base 10 graphs (many stayed with those for the whole activity) and checked their answers on Desmos (and with me). I had printed two sets so they had plenty of new graphs to choose from when they correctly identified their graph. I didn't give them a lot of hints as I circulated as I really wanted them to find strategies to help determine the equations. They were using what they could see about the graph (vertical asymptotes, for example) to help map points from the parent function to their graph. I was really impressed with their efforts. I did decide to write the base on the graphs that were not base 10 as I thought it was too much of a step up at that point.

Here is some of what I saw:

**Here** is the file. Thanks for the idea, Al!

My plan is to revisit these graphs when we solve logarithmic equations which will allow them to use algebraic skills to help find the equations without having to know/guess which point was mapped onto each new point.

Every year many of my grade 12 Advanced Functions students struggle with cofunction angle identities. Let me back up. They actually struggle with the entire second unit of trig. This includes proving identities and solving equations, but begins with cofunction angle identities. This is often where the confusion begins and it snowballs through the unit. I asked myself what I could do better, or differently, this time around so here is my attempt to help my students better understand the difference between related acute angles and cofunction angles.

I started by creating **this** investigation.

They worked in random groups on large whiteboards and definitely needed some coaching from me along the way. I feel that they finished the class with a strong understanding of the relationship between sine and cosine of complimentary angles.

We consolidated together and then recapped the following day with this:

I think that changing the acute angle to the vertical to α (not q) helped some see the difference between related acute angles and cofunction angles. It also helped me refer to what they had seen in the investigation - "Remember how alpha was the angle between the terminal arm and ...".

I wanted to build on what they had learned as we continued to move forward with compound angle identities so I decided to make a warm-up. I created a "game" that we have played for the last two days and we will play round 3 tomorrow. Each student got a small whiteboard and marker. On day 1, they had to write T for transformation, R for related acute angle or C for cofunction on their whiteboard. Each expression was on its own slide and students put up their whiteboards with each answer, which we discussed briefly. Here is the "game" for day 1:

They did well for the most part. Well enough that both they and I wanted to do another game the following day, so on to part 2. This time I asked for three things: T/R/C (same as day 1), quadrant and equivalent expression. Once again they each wrote their answers on a small whiteboard and held it up when they were ready. Here is the set of questions - students did not get to see the answers which are shown below:

Again, it went well.

This morning I shared the first two "games" with a colleague and she suggested making another one. We settled on giving an expression and a choice of potentially equivalent expressions, of which more than one could be correct. We made three of them and my colleague did them with one of her classes today and her feedback was that they really made the students think. She also pointed out how perfect these were as we head into proving identities.

My hope is that this daily practice will solidify their originally tenuous understanding of new concepts which will in turn give them more opportunities for success when they are working with identities and equations.

If anyone would like any of the files, please let me know.

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!