Carl's Teaching Blog

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Category: Clog (Page 5 of 7)

CLOG: Moving towards symbolic representation

We have spent our time this past few weeks turning an interest in patterns into an interest in modeling. Each day we have done activities to get the students to think about number sense, and also about relationships and functions. Next we will focus on linear and quadratics simultaneously and look at both as examples of functions that we can use to describe how things grow and change. Teaching the two functions simultaneously will hopefully strengthen kids’ understanding of what functions are and what they could be more so than focusing immediately and solely on linear.

I taught my class twice, as per usual. The way our school is scheduled, we only see kids on Tuesdays-Thursday classes for 90 minutes each. The class has only met on 6 occasions (and one of those was taught by a sub who wasn’t given my subplans). At times it feels like we should be further along, but I want to remember to be patient.

Tuesday we looked at the penny circle on Desmos. We also opened up class with three visual patterns, one linear, one quadratic, and one exponential followed by a Contemplate then Calculate activity from This helped going into the penny circle because they were familiar with the types of equations, and also with counting things. The best part about Desmos activities are that it let’s the students work independently on tasks, and learn from each other. It is certainly because of these benefits that the universe conjured up a perfect storm of misplaced computer cart key and a soft lockdown drill kept me from actually using computers with my kids, and I ended up having a few kids walk through it with the smart board. Luckily I made a worksheet so I could see what kids are thinking about the activity, and the kids could still engage in this whole-class format. The students were able to see how the different equations look on a coordinate graph.

Following the activity we did an Interpreting Distance Time Graphs from the Shell Centre, which we did not have time to finish.

On Thursday we started with a visual pattern that worked out pretty well. This was the 5th one of these that I have done, but it was the first time I asked for symbolic equations for the patterns. I was surprised that students didn’t offer some kind of equation earlier, but I still wanted to wait until this point because we could start working with equations soon. To formally begin working with equations we looked at a modified version of Tina’s quilt squares, which has been a staple of my introduction to quadratics for years. Kids then worked on finding the number of grey squares and the number of white squares and eventually we had time to talk about the equations.

Clog: Getting ready for the next Episode

Today we finished up the first little “Episode” of my class, the goal of which was to  be able to identify relationships and functions as well as expose them to the terms ‘linear’ and ‘quadratic’ to describe their work with visual patterns. We had finished 8 different kinds of patterns by the time this class rolled around, including this one from youcubed. Since are now experts, I asked them to create their own patterns of blocks and put them on the board using Post-It notes, sort of a play on VNPS. Later on they were able to classify the patterns that they made as either Linear or Quadratic once they learned that that is a word used in math. The majority of the patterns they created were linear, which left me wondering if I should have exposed them to more non-linear patterns or if I just have a linear bunch of students.


Following this activity the students worked on a little reflection to wrap up this first Episode. I had this idea of breaking my class into little Episodes over winter break while my wife and I shamelessly barreled our way through Agents of Shield and The Blacklist on Netflix. Surely, some of the structures that the writers use to draw viewers into their stories could be adopted by teachers as they plan their units. My plan is to have these clear distinct units at the end of which is a little reflection, it’s roughly the “big idea” from the last 4 classes. Since the “big ideas” are the characters in my 2nd period show, the end of the Episode should highlight a new piece of information about one of the “big ideas.” The reflection on these things will help students keep track of all the important connections and representations. As we start future classes I plan to use the students reflections as part of a little introduction. This introduction could be mirror the “In the last episode…” or “Previously on Agents of Shield” announcements they have on TV which are followed by replays of key scenes from the series. So on the days where the concepts from this Episode stand to make particularly strong connections the ideas from the next Episode, I will try to adopt that by bringing back some of the more interesting reflections that student wrote.

What is the next Episode going to be about? Understanding! Specifically, getting to different forms of linear and quadratic equations with lots of understanding. We can look at difference tables, and break apart the meaning of the y-intercept and other variables in the equations, as well as looking at how these things reflect in the graph. I will work in some instructional activities and some problem strings in addition to counting circles as a way to help kids build some number sense. Since today’s post-it notes were a success, it seems like I will have to use more manipulatives, and try some Vertical Non Permanent Surface Problem Solving. If you have any ideas of things I should try, or other ways to make my class as addictive as a Netflix bing, let me know in the comments!

CLOG: First Class Of The Cycle

This Tuesday was the first math class I was teaching all year. This means Monday night I was wringing out my twitter feed like wet dish rag, trying to squeeze out every drop of teaching ideas into my class the next day. The turnover and structure of our school means that every marking period gives teachers the first day of school ‘blank page’ feeling.

My first day of school for this quarter needed to have some routines to build number sense, promote discourse, and get the kids prepared for an exploration of quadratics that put procedural knowledge in the backseat.  I settled on starting Tuesday with a counting circle for 10 minutes each class.  Sadie’s post from a few years ago seemed to be within six-degrees of separation of any post you can find on counting circles, and it pretty much convinced me to start my first class with this routine. As the cycle goes on I may try to work in a Problem String or some other routine.The idea is to start class each day with everyone working together and talking, as opposed to my usual Do Now-Review.

I like to think of this class as a unit in quadratics that puts procedures on the back burner. To plant the seed for some discourse around the quadratics I asked to get in groups and have a little conversation about two different kinds of visual patterns and to talk in small groups about what similarities and differences they saw in the two patterns.  The worksheet I used is below, and was a lightly modified version of one from last year.

Download (PDF, 50KB)

The worksheet didn’t really help promote the group work the way I would have wanted, in part because the task could have been more group optimized, and I could have really pushed the kids to stick to the group roles I had prepared. But the biggest problem was that I had only 5 students who showed up to class. Low attendance is a regular occurrence at our school, but this was basically a tutoring session. I had the whole class sit around the same table and instead of using chart paper had each group just write on a mini-whiteboard and slide the result down to the group at the other end of the table.

The two groups worked differently, one visual, one comparing the differences, so it was good. I asked them to describe their work to each other. Then on their paper, paraphrase what they saw in the other approach and how it connects to their own work. It showed that I will need to work on helping students describe their thinking in order to let everyone be successful this cycle.

The last thing we did in the class was talk about functions, and for this I dusted off a tried and true function lesson. The thing that this assignment needs is to have more challenging problems for the students who are able to finish the work quickly. If I could have an “important stuff” section of future assignments, and then “tough stuff” or “fun stuff” sections and ask student just to work on whatever they think is worth their time I would probably be more successful. I would also be just that much closer to Bowen and Darryl, whose idea I would be stealing.

For a first class it was pretty good. I hope to keep posting here throughout the cycle. The next post will be a talk about what ended up happening when I had to miss the second class of the cycle because of a meeting I had to attend.

Clog: “I don’t know, like a million?!?”

Over the weekend I was excited to attend my first baby class.  As a teacher, watching others teach triggers an unrealistic urge to by hypercritical. My wife is also a teacher so we left for lunch shocked at how much of the teacher’s time was spent talking at the unhelpful Powerpoint. She basically talked the whole time, constantly referring to the stuff we had to “get thorough”.  We had all sorts of pedagogical wisecracks about the experience while we ate at this greek restaurant that seemed a lot like Chipotle, and I thought about that lunch today in class.

Today’s class did not begin how I would have liked.  Unable to find a star wars themed Estimation180 kind of task, and unable to make one that would only appeal to fan boy trivia geeks(e.g. “Estimate the number of parsecs needed for the Millenium Falcon to complete the Kessel Run?”) Chipotle popped back in to my mind. The menu specifically.  Since we were talking about combinations and permutations, I thought let’s make an estimate of all of the things that are possible to order at Chipotle.  I gave them a menu that had all of the meals and proteins, and asked them to be specific about what they were taking into consideration.  I gave kids this menu that showed the meat and the menu choices.  To avoid over-scaffolding, I didn’t mention all of the sides, in hopes that the kids would think of the sides on their own.

Lots of kids immediately noticed that there would be more to it than the options listed, but they all seemed to shrink in the face of such a number.  I had a lot of exchanges where the kid would say ” Oh, that’s like a million?!?” as if they were comically startled to think of a number that big.  I would ask them to try and use the multiplication rule to take it into account.   Instead they would get overwhelemed and settled for 24 (four meals, 6 proteins), which would be the safe choice.

 I wanted to show them all of the other possibilities that it seemed most people were scared to explore. At the board I walked through the rest of the possible meal options, one at a time.  “What are the choices for beans? Brown and black?” Ok, that’s 2 more, so multiply by 2″ in as engaging a manner as I could.  At the end we multiplied it out and got something like 516,094, allowing kids to have two kinds of meet, any of the salsas, and also getting an optional extra tortilla on the side.

Yes, I am fully aware that this sounds like I’m defending a teacher led call and response.  I felt the full irony of me doing pretty much what our birthing class instructor was doing over the weekend. At that moment, with the do now almost over I genuinely wanted to see what we could up with, it’s hard to turn that off. At the same time, kids were watching me do math and sort of cheering along.  The argument could be made that this diversion was not really valuable.

What I think makes this valuable is that I am making explicit the process that one has to go through in order to both think through a problem, and really justify their thinking.  This process is important, and now I can refer back to this component of the lessons when I want to explain to students how to think through and justify their reasoning with similar problems, and I can assess this I provide an opportunity for them to do a similar type of counting on their own in the future.

Clog: “I’m not asking for a right answer, I’m asking ‘How Do You Know'” #HDYK

So today I uttered my favorite new acronym in class while explaining what groups should be doing.Group work has become the new bane of my existence because I both realize it’s importance, and also recognize the fact that I have not done it well. …yet.  My new favorite acronym, HDYK  (How Do You Know), is hopefully going to help with that, along with new roles and what I learned employing those roles today.

For a number of reasons, I tried to get kids to focus on the process of group work, not necessarily the outcomes.  These were the roles that I suggested in order to get students to focus on listening to each other:

  • Involver – Your job is to make sure that all of your group members are involved in the task.  Ask questions to other members to make sure they have their ideas included.
  • Task master – Keep track of the task that needs to be done and the amount of time required to do it.  Remind people of the work the group needs to finish if it seems people are off task.  Let people know when they are ahead of schedule as well.
  • Summarize and Share – Keep track of everyone’s thoughts and prepare to share all of the thinking with the whole class.  Also be prepared to let others in the group know what is going on if anyone gets lost.

This was a pretty interesting switch on group work (which I totally stole from someone else, but I can’t remember who) and it delivered some success.  It seemed nice to see people flocking to certain roles.  Perhaps next time I will find ways to immerse kids more intensively in these roles.  One thing I may do with would be to have little conferences with the different roles, like ask the Involver to give me a report on the groups functioning or pull out the Summarizers to meet together and share ideas.

The task for this class was also one which I hoped to hear a lot of “How Do You Know” from kids. Each HDYK  was hoping to get kids to justify their response to the following question which could hopefully get us to start thinking about permutations, combinations and other ways of counting:

Dimoni is going to make a new restaurant.  He promised it would be fancy, gluten free, paleo-friendly, vegan, low-carb, high-fiber, seasonally appropriate, locally grown, and tasty.  This left him with the following ingredients.

Sweet Potatoes, Radishes, Carrots, Onions

…and some other stuff.  Given that he has such few ingredients he wanted to make as many dishes as he could using all of these ingredients.  He planned to say he could make 50 different combinations of these items.

Is it possible to make 50 different arrangements of these items?  Work together to detail how to figure out all of the possibilities.

In how many different ways could you describe an arrangement?

When the students were in groups I realized how this could appear like a “What’s the answer” kind of task.  Many kids would come over and say “I got 12, is it right?” To which I would say “How do you know it’s 12?”  Students were supposed to come up with some way to justify that they have whatever answer they have.  Many instead focused on asking me whether or not I could justify their answer as “right.”  After some direction most of the groups got to a place where they felt they could justify the choice and many were saying things like “Yes, our answer is 12, because he shouldn’t be trying to act like radishes and onions is a different arrangement than onions and radishes.”  In the future I think I would replace the word “possible” with “reasonable” to suggest that kids should have a reason for what they say.

It seemed a little more conceptual than some kids were able to latch on to, and with so many people expecting me to guide them, it was really easy for kids to get left out.  I probably should have done a Next time the task will need to be really clearly laid out to show that we are emphasizing the HDYK and not the answer.  My role as facilitator has to be reduced too, perhaps the involver could be the person who is allowed to ask me questions, and can only ask me questions when they have heard from everyone in the group.


Next time I see this group will be on Friday and we will spend that time comparing all of the responses from each of the different groups.

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