Thursday, 7 December 2017

Rate of Change Graphs

In grade 12 advanced functions (pre-calculus) we look at rates of change of functions. Our students have been working with rates of change since grade 9, but we now look at instantaneous rates of change, or slopes of tangents to a graph, along with rates of change of slopes of tangents. This can all be rather confusing to students so I decided to make a game to help my students build up their skills and vocabulary.

I started by placing all the desks in pairs, facing each other. Each pair had 2 small whiteboards and markers. I randomly assigned the pairs and had them decide who would be person A and person B.

Here was their challenge -


They jumped right in and were asking each other good questions. I paused them at one point and we talked about the fact that, as they couldn't ask about the actual graph, the person drawing should be able to come up with the shape of the graph but there could be horizontal and vertical translations. 


They soon discovered that the orange piles had more challenging graphs and actually chose those ones! Once they had each gone through a number of graphs, I suggested they could make their own. And, boy, did they! Here are some samples:





Along the way there were great conversations and many misconceptions came to light.

My colleague suggested having students create velocity-time and acceleration-time graphs based on the green and orange graphs (assuming they were position-time graphs). That produced even more misconceptions.

It was great to see them engaged and challenging each other in a really positive way.

Here are the graphs that I used - the first 6 were the green graphs and the last 6 were the orange graphs.


Friday, 1 December 2017

TMC 18

We are starting to gear up for TMC18, which will be at St. Ignatius High School in Cleveland, OH  (map is here) from July 19-22, 2018. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.


To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC18sessug). It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!


This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past nearly everyone who submitted on time was accepted, however, we cannot guarantee that will be the case. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 19 and 48 one hour sessions that will be either Thursday, July 19, Friday, July 20, or Saturday, July 21). That means we are looking for somewhere around 70 sessions for TMC18. We are requesting that if you are applying to speak for a 30 or 60 minute session that there are no more than 2 speakers and if you are applying for a morning session that there are no more than 3 speakers.


What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!


If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.


The deadline for submitting your TMC Speaker Proposal is January 15, 2018 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.


Thank you for your interest!
Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Muir, Jami Packer, David Sabol, Sam Shah, and Glenn Waddell

Thursday, 16 November 2017

Log Graphs

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.


Wednesday, 1 November 2017

Cofunction Angle Identities

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.

Thursday, 19 October 2017

Using Desmos Activity Builder in Unconventional Ways

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...
Annie Forest shares ways she uses Desmos with primary students here.






Tuesday, 26 September 2017

Thinking Classroom in MHF4U

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) = x² + 5x + 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 x³ + 5x² - 22x - 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 x³ + 6x² - 8x - 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. "


Wednesday, 2 August 2017

Be You, Not Whomever

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...