Carl's Teaching Blog

A place to talk about teaching and learning

Visual Patterns First: Putting Together A New Quadratics Module

This cycle we’re focusing on Quadratic Equations.  The topic of quadratic equations has been a bit of a murky area in our transfer school.  We don’t necessarily have a scope and sequence, and teach a bunch of modules, and it seemed to me we weren’t giving fair treatment to Quadratic functions.  I had one project “The Function Field Guide” that covered the topic in brief details, but thought we could create a new module so kids could understand all the cool things about these functions.  Creating this module has been going good so far, so I figured it would be good to blog about the broad strokes behind it’s construction and roll out.

Visual focus from the beginning

I wanted students to understand the purpose for the quadratic equations centra distinction from the function that most stduents think of when they think of equation (linear functions of the y=mx+b variety) so I started by showing this pattern.  From this the students sat and thought about the pattern of growth in the table, and how they coudl represent it.  on one hand we saw that it was not going up by a regular amount, that “It was going up by odds” and on the other hand some one looking at the fact that it was a square could see that the area of square would be x*x.  We talked about this a lot over the first day.  I thought it was too much because the kids seemed bored, but I’m not sure if the connection between the two methods was understood.  THe kids seemed unsatisfied by not being able to tie their “odd” noticing to something concrete.  So we kept doing visual patterns at the beginning of every class varying between quadratic and linear, and after defining the terms the kids now have an added task with each visual pattern, (is it quadratic, or is it linear, or something else?).

The visual pattern Do Now activities I used on the first six weeks of the cycle

The visual pattern Do Now activities I used in the first six weeks of the cycle

Fixing potholes along the way

‘Fixing the potholes’ is a term coined by the East Side Community School teachers talk of the MSRI event (around 56:00).  The idea is basically to fill in pot holes on the way to new destinations, not tear up and repave roads without making any forward progress.  If you find engaging problems to inspire kids to understand higher mathematics, the kids will realize what areas they need to work on.  Then we tie their curiosity to the need to repair their old understandings, and provide the space for that “pothole-filling” as we pave the road into more interesting content.   I much prefer this to traditional review: teaching it the same as before, but faster and with less support.

Instead of starting with an introduction on graphing, equations, linear functions I decided to work situations into the class where we do the review along the way. So when we started off looking at these patterns and classified the idea of a quadratic pattern, I get a chance to review linear functions for the students who may not have been signed up for my linear module and can benefit from seeing it in a new light.

Begin with vertex format

In the past when I’ve taught quadratics I’ve taught standard form and factored form first, because kids are often familiar with the operation of going back and forth between those two.  This year I am starting with the f(x) = a(x + h)2 + k form of function because I think these are the kinds of functions kids will generate on their own when looking at visual patterns.  If I see patterns like the ones above, I am not going to think, “oh, it’s like that square function, but we have to add two to it first” or “oh, it’s like that square function but with an extra thing on the side.  If kids are comfortable making and using these this form, they can quickly translate it to a graph, and we can start to make sense of the graph from there.

Building tent posts with reflective writing

So my way of thinking about this unit is kind of like building this series of mathematical tent posts, like a circus tent.  We get the main ones up, then all the little side posts and supports.  A the end of the unit, we’ll have created a place where kids can play around or build something cool.  The tent posts I’m planning will be the big thought for the week, and we’ll do it every Friday.  I’ll keep these big ideas on chart paper some where in the room to keep referring to through the week.  I’ll also ask kids to write about it, as well as create an example.

This tent post is pretty good so far.  The first big thought was what distinguishes a linear relationship from a quadratic relationship.  Last week was the shifts of the parabola and the connection to the vertex format.  Next week will probably involve either a further exploration of the parabola graph, the general function of the parabola, or maybe both.

Searching for the final project

My plan is to end up with a final modeling project. I have done a few things before, but I’m hoping I can find something better.  Since the last thing we will cover will involve the factored form and finding zeros as an important component.  Projectile motion makes a lot of sense, but that will involve a lot of physics.  I doubt I will have enough time to teach physics with understanding and I refuse to end this unit by telling kids “Here’s this function, don’t worry if you don’t understand it, just plug your numbers in.”

If you have any ideas for a good ending project, let me know in the comments!

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.

The view of San Francisco Bay from the steps of MSRI.

MSRI 2015 – A national math education conference focusing on developmental ed

This year there was a pretty spectacular conference in the hills of Berkeley California that brought a number of people who are involved in Developmental Math including Deborah Ball, Bill McCallum, Hyman Bass, but mixed among them were some lesser known names including Gregory Larnell, and a scene stealing group of high school math teachers from NYC.

The Mathematical Sciences Research Institute (MSRI) is a magical place in the hills above the University of Berkeley near the Lawrence Hall of Science.  It’s home to a group of academic researchers who work on their research from the inside of a beautiful complex, largely funded by MFA creator Jim Simons, which plays hosts to a number of national conferences with various focus on a yearly basis.  The annual Critical Issues in Math Education meeting was held in March and had the focus of “developmental mathematics at two- and four-year colleges and universities and the broader dynamic of mathematics remediation that occurs at all levels.”

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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.

Clog: “No, I just don’t get any of this, so I’ll just wait”

Seems like one of the side effects of my new emphasis on large scale problems is getting kids caught up when they come in late, weren’t paying attention, or otherwise find themselves lost.  I think tradition trains students to shut down when they are lost, as if the key to getting un-lost will be someone telling you what to do.  Like if you get lost in the choreography of a dance, you need someone to let you know what the next step was, or even to prompt you to look at the previous step, or just to show you the steps so you can follow it.  The “math-as-dance-step” metaphor breaks down once you want a student to do more of the creating themselves.  I mean, I don’t go to someone who is used to dancing ballet and say “Make a routine to this hip hop song” (unless of course I’m the producer of that Julia Stiles movie).  So when I want students to explore all the different ways of solving a complex problem, how is it that I let them know the steps?  Not the algorithmic steps to the specific math problem but the dance steps for thinking about and solving any kind of problem?

Today I tried to get students to work on a counting task from MARS and as they struggled, I wrote on the board the steps from Polya, about how to solve a math problem.  These steps had a lot of meaning for, but I don’t think the kids got it.  Abstracting the problem solving process is not a good thing to do when your kids just had a break down in their problem solving process.  Tomorrow I am going to roll out a whole bunch of problems that various people can have success with, and then afterwards ask them to share, and use that share out to abstract the steps required to solve a problem.  Hopefully if the problem comes from them, they will be more likely to apply it the next time they get to a place where it would have previously felt safe to just “wait.”

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