December 26, 2017

Snowflakes (Geometry Review)

My contained special ed class has been following the 8th grade IM curriculum. I decided last week that I would give them a midterm exam on units 1-3 in January which meant we'd need to start reviewing some of the ideas from the geometry units. Folding snowflakes the class before winter break seemed like a good place to start!

I begin each class by projecting an image and asking "How many?" To preview the lesson that day we counted with snowflakes:


(I found this on google images but the source is great - it includes instructions for these and more!)
Students counted snowflakes, holes, points, stars, hearts and snowmen (can you see them in the center snowflake? They're wearing top hats!). They used specifics like first, second and third snowflake as well as small, medium and large holes. They reasoned about totals and made comparisons and wrote equations. Have I mentioned that I love this routine for the rich mathematics and language it invites? Because I do love this routine.

After counting everything we could, it was time to start making our own! I googled something like geometry snowflakes and came across a pdf with an FBI copyright warning which I ignored for my class but I won't post it here. Instead you can have the version with a million ads, sorry. The instructions are the same and that link includes the helpful hint to make sure the smallest triangle is on the outside at the end. I'm glad I tried to make one before doing it with the class or we would have had some asymmetrical snowflakes round 1!

Here's where the geometry review came in. As I walked around the room checking everyone had folded correctly, we discussed a vocab word for each step. Step 1: line segment, Step 2: midpoint, Step 3: vertex, Step 4: name that triangle!, Step 5: congruent. I was impressed with how close to congruent most students angles were. Often when I've done origami in class before students struggle to line things up precisely but this set of instructions was quite accessible.

Once everyone had their folds done it was time to cut. I was surprised to find most of them had never done this before and were hesitant to make a cut. I’m glad I had made a few samples the day before and could fold them back up to show how I cut. Once they saw an example they hesitantly made a cut. It was cool to see that several students unfolded after their first cut to see what happened, then refolded and cut some more. The reactions of “did that cut do what I expected?” were great. Students were definitely invested and expressed enthusiasm or dismay about their creations.

After they finished cutting I asked them to notice and wonder about their own result. Then we gathered all the samples together to discuss notices and wonderings. They were able to name the base shape as a pentagon, describe methods to create stars on the inside and outside, and how to make a tree (one not pictured in the close ups had a tree in the snowflake, the small decorated trees were a happy accident when the para cut too much and her snowflake turned into 5 parts!). Then we discussed aesthetics - we seemed to like the ones with more paper cut out. Kids were eager to cut more of their first one or try again. Everyone who didn’t know where to start cutting the first time was eager to try something new out after our discussion!


December 24, 2017

Representation in Novels

Yesterday was my first day of vacation. I slept for a much needed ten hours (I refuse to get the flu or any of the many other illnesses students brought into my classroom last week) and then flipped through my kindle to find a book to read. I found Not Your Sidekick and the description said something about teens and superpowers. Sounds like a perfect first day of vacation read. Pretty early on I find out the main character is bisexual, but this isn’t a coming out story. The book isn’t about her sexuality, she just happens to be a part of the LGBTQ community. Later on the author mentions her brown skin, but it’s not a book about struggling to fit in as a person of color. In fact, slowly across the two books (yes I read two books yesterday, it was awesome. Since the third book in the trilogy isn’t out yet I’m writing rather than reading today.) we find out that three of the four main characters have brown skin. We also find out that one is trans and another has two moms. These details all come up as they are relevant. The book is about super powers and conspiracy and dating. The characters are whole people. It’s not “an LGBTQ book” but instead is a futuristic fantasy book with representation. However, I’m sure that when I first downloaded it I chose it because it was in the LGBTQ category of some book list. Books like this are hard to come by. I’m so glad I found it again and finally read it because our GSA is currently fundraising so we can donate books with LGBTQ representation to the school and city libraries. What are some books you’ve read that rock the rainbow?

p.s. Rock the Rainbow is our new GSA slogan on our awesome wristbands and keychains:

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 (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

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?