December 3, 2017

Budgeting with Matrices

I've been trying to weave some big conversations about life and social justice into math class. I wanted to use matrices to analyze some complex data but it just wasn't falling into place nicely. Instead we did a fairly simple word problem about budgeting for young adults and had a decently nuanced discussion about it.

The point of using matrices is to organize data and reduce the repetitive nature of computation by having the calculator do the calculating. This was the first assignment where I allowed students to use a calculator so I had them do out all the computations for one person first to make sure they were using the calculator properly.

The Assignment. The Calculator Instructions.

They had a hard time with the "5 days a week, 3 miles each way" part, mostly due to a lack of reading that sentence. Many kids left the three values separate at first (income, meal spending and travel spending) which was fine because #4 has them go back and compare. So the front of this handout is a decently scaffolded matrix word problem. Standard fare. The back is where we move beyond matrices into real world application.

It turns out poor Sarah doesn't have enough money to cover rent, let alone groceries or fun. Jose has just enough for his essentials and Ana has what seems like plenty of money to spare. It was interesting to see what recommendations students made for each person to save money. Here's the first round of suggestions:

We discussed which of those options were reasonable. Is someone going to walk 3 miles to work? Moving or getting a different job makes more sense. I heard students talking about living at home or in the dorms though interestingly no one wrote that as a suggestion. Fixing the car (Jose) and buying a car (Ana) are good ideas in theory, but if someone is living in poverty and barely has enough money to pay rent they won't be able to save up the money needed to make a smart long term investment. 

In Massachusetts the minimum wage is $11 an hour, so there are lots of higher paying jobs out there, but these are kids who just graduated high school so they don't really have many better options. This is where we talked about why anyone would want to be like Sarah (going to school full time and in debt) when they could be like Ana (working full time). Yes, it looks like Ana is doing fine right now, but we haven't factored in any of her other expenses and she doesn't have much potential for getting a better job. We talked about how training (not necessarily college - we have several certificate programs at our school for automotive, child development, culinary and other trades) is necessary to get a well paying job.

This conversation gets us all the way through the beginning of #8 (who would you be and why). Students didn't write much for their "detailed budget" so the next class their opener was to list things people spend money on. Here are the results from my three classes:

We had a good discussion about what people absolutely need and what they feel like they need. I still didn't get great detailed budgets on the resubmitted assignments but I did get a bit more thought than the first round. I wonder what the cost/benefit would be for spending the time to have them research and make a complete budget. I think it would take more time than I want to spend in Algebra 2. All of the seniors go to a reality fair where they're given a salary (based on the job they choose) and have to go to a variety of booths to spend money - some are choice (pick your car) and some are random (get invited to a destination wedding, spend lots of money!). It's a cool day. 

Overall I'm happy with this assignment, it shows the benefit of using matrices to avoid repetitive computation, asks student to do some thinking about their own futures and hopefully develops a bit of empathy for people in poverty for the kids who aren't already keenly aware of how difficult choices can be when money is tight.

December 1, 2017

TMC18 Speaker Proposals

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 ( 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

November 13, 2017

Matrices - the Basics

I've been writing a lot of material lately. Why are matrix lessons so hard to find? I understand that they're plus standards in CCSS but even if you don't teach them in Algebra 2 someone should be teaching them somewhere...?? Anyway, it's been great in that I have total control over the difficulty level, language and content focus. Not so great in that it takes a ton of time. But since I put all this time into things, I'm going to share so that the time was worth it!

I was out (at ATMNE) for the first day of the unit so I made an introduction packet from my textbook and David's lovely post on matrix multiplication. The students who were willing to read and take some baby risks did great. The students who were scared to try informed me that they couldn't do it since I didn't teach them how. I need constant reminders that these are learned behaviors. Given enough time and consistency I can teach them new behaviors. But omg is it hard some days!

To determine what students had learned from the packet as well as see what details they would attend to, we did a WODB the next class. After my last post I got some more feedback and decided on this set. Kids found reasons for all four in every class! I was surprised that the non-square matrix didn't get the most votes since it's the one that jumps out to me.

Then we practiced matrix multiplication by comparing matrix multiplication to the multiplication of numbers. For some students this sequence flowed really nicely. We did a problem from the homework and someone asked if we always had to bring down the matrix on the left. Cue a couple problems that follow a*b vs. b*a. Lovely counterexample of the commutative property for matrix multiplication. The next pair of problems has students multiply by the identity. For students who were paying attention they caught on to the pattern quickly and realized that this matrix was special. Students who weren't paying attention got extra practice multiplying matrices. I let them in on the 'secret' that mathematicians are lazy! They look for patterns to make their work easier and we should be like mathematicians as we work. It's a tricky balance to structure a class so that students are always looking for patterns but to make sure that they aren't just assuming there's a pattern because there always is one. That was one benefit of starting with testing the commutative property - in that case the thing to notice was the lack of a pattern! Finally we multiplied some inverses. Another benefit to the pattern finding structure is that students compared work with their group when they weren't seeing a pattern, they found mistakes faster than in a random set of practice problems. I chose my inverses very carefully which allowed students to recognize two aspects of inverses (opposite signs and switching a with d). We finished off with some homework on all the operations they've learned so far.

A few students asked during all of that rather tedious matrix multiplication practice what the point of these things was, when would anyone ever use this?? So I told them about graphic design using a matrix to represent all of the points of a figure and operations with other matrices to transform them on the screen. I could've sworn I saw a video on them using this process for the fur in Monsters Inc. at the Pixar exhibit at the Boston Museum of Science but I can't find the video online. I could, however, make up a problem set to demonstrate the process by 'animating' a triangle. This might have been my best lesson all year! It was last block before a long weekend but yet students found this lesson very approachable and got some self checking practice applying matrix operations. Bonus- I enjoy activities that blur the (non-existent) lines between algebra and geometry.

Most students finished their transformations with time to spare (in an 80 minute period) so I had another practice sheet ready to start in class and finish for homework. My colleague and I like my textbook's problems with variables and expressions as entries in matrices. Some of the entries are self checking, others make simple equations and students get some practice solving one variable equations (which, yes, my algebra 2 students do still need). However, no one on the internet has these lovely problems, including our Kuta Software. So I searched class zone (I still don't understand the structure of that site but I'm learning, kinda) and compiled all the ones they wrote on an additional equation practice sheet.

Next class we'll do some other applications of matrices and row games to review the basics before heading into systems.

What else do you have for matrices? Where on the internet are these lessons hiding?

November 11, 2017

Explore Math Project and DREAM

Last week was both the end of the quarter and ATMNE. I got the chance to see Tracy Zager talk (both her keynote and her workshop) and reflect on how first quarter went. Tracy's keynote was about mathematical inquiry. One method of allowing kids to ask the questions is to provide students time to play with math. Her daughter wondered why we DEAR (drop everything and read) but never DREAM (drop everything and math). After the talk, some of us who teach high school were discussing what this might look like at our level. I suggested that my version of Explore Math is one option, after all one of the columns is titled play!

This year's version of Explore Math asks students to explore something from one column per quarter and then share briefly with the class. I assigned it in all three sections of Algebra 2 as well as my honors precalculus. I've really struggled with my Algebra 2 classes and part of it is that they don't enjoy math, at all. They see every aspect of the subject as painful and it is taking every teacher tactic in my bag to convince them that math might be interesting to think about. I hoped that they would find something to grab their interest when completing this project. However, students continued to resist - I'm too shy to present, none of those 17+ options appeal to me, I don't know how to [fill in the blank]. When the first person presented on a mathematician they realized that I wasn't asking for a giant research project and almost all decided to also look up a mathematician, but there wasn't much exploration of math happening. Don't get me wrong, it was awesome to see my diverse students represented in the diverse mathematicians they chose but most didn't even look up the kind of math they studied. Maybe step one needed to be seeing that someone like them did math before they could be convinced to take another tentative step toward enjoying math? Here are the statistics from the first round:

61% mathematician
0% didn't do
2 recent articles, 2 recent topics, 1 unsolved problem

Alg 2 (3 sections):
64% mathematician
32% didn't do
1 problem

59% mathematician
36% didn't do
1 grad level math

59% mathematician
41% didn't do

I'm thankful I made this a year long project so that now for the remaining three quarters students will have to choose other categories. It seems wrong to be forcing my students to play, but after many years of learning to resent math it's going to take a strong shove toward playfulness to get them to consider it as something that they could engage with independently. (These results have a little bit to do with the project being homework, but I provided time in class on several occasions to work on this or other make up work.)

Tracy asserted that allowing kids to ask the questions is 1) intellectually honest, 2) a good way to teach and 3) important for equity and access. I have been trying harder to make the structure of class transparent to students. For example, I've shared with them the 'secret' that mathematicians are lazy! So they look for patterns to make their work easier. Most lessons involve us playing with problems to test some ideas and then generalizing our results, but when students aren't driving the questions it doesn't feel like playing. A recent lesson on matrices flowed beautifully for some students but others were so caught up in the drudgery of multiplying matrices (because integer operations require significant brainpower) that they weren't seeing the overarching ideas on their own. Turns out it's hard to recognize the significance of the identity matrix when you forget whether 5*0 is 5 or 0. So we're working our way toward 1) and 2) with some students feeling like mathematicians each day. What about 3)?
"The person who poses the question is the person who frames the debate."
When Tracy said this sentence I stopped, wrote it down, tweeted it out and only partially heard her next several sentence as I grappled with this huge revelation. Politics can be decided based on who poses the question and how they frame it. How do I give my students this power? How do I make sure that my students recognize this so they can question the premise of others questions? And, honestly, how do I even consider doing any of this when I have two new preps this year? I look back on the things I did last year and regret how many of them I've let drop this year. And then I remember that I only had two preps last year and I'd taught both of them at least 3 times before. While I should cut myself some slack I'm not going to give up entirely. Up next is solving systems using matrices with technology which sounds like a good time to mix in some messy data, hopefully I can find some worthy data sets for students to play with. They can ask the questions. They can judge others' questions. We can do some aspect of this important work each class and hope that by the end of the year students see their relationship with mathematics with a bit more positivity.

October 31, 2017

Matrices WODB

I've managed to avoid teaching Algebra 2 for all but one year of my career (prior to this one). Between Algebra 1 and Precalculus I am familiar with most of the topics in the course, but matrices don't appear in either of those courses in our sequence. I forgot how much familiarity with content makes a difference when prepping! Pulling from our textbook and David's lovely post on matrix multiplication I made a packet to introduce matrices and their basic operations while I'm out. I wanted to start the next class with a Which One Doesn't Belong? but I couldn't find one. So I thought I'd make one. I started out thinking about dimensions- the top right is the only one without a dimension of 3, the bottom left is the only one without a dimension of 2. Then I thought we could play with the numbers to get something interesting.

But when I was chatting with my colleague we were struggling to come up with what other reasons we could set up. I know that when Christopher first started making these he asked people for four characteristics and then made each shape using just 3 of the four (so each one wouldn't belong due to the characteristic it was missing). We tried to think what other characteristics we could include and decided to use the scalar multiplication and equation solving from the packet to build other aspects. I also made the sum of the digits equal 10 for three of the four just for fun.

But when I tweeted this set I still wasn't getting the kind of responses I was hoping for. And I am confident this is because I'm not familiar with matrices. I'm going to have to spend some time familiarizing myself with inverses before I can teach them. By the end of the unit I'll have an entirely different idea about what important characteristics of matrices are. So until I get there, help me out? Describe a matrix using four characteristics and let's see if we can build a good WODB together!