Systems of Equations Unit Plan

During this years MTBoS blogging initiative, I used the share the love prompt to capture some of the things I wanted to use and remember for teaching the quadratics unit. I am going to try to continue doing something similar to keep links and ideas for other units as well. I am really terrible about keeping all the activities I have used in the past or new things I want to try so this way I have an electronic record and I can share idea with other math people:

We are knee deep into a systems of equations unit in one of my Algebra classes. So far we’ve done the Systems of Equations Launch which I wrote about here. Which led into graphing systems. At the bottom of that post I wrote how I would adapt it next year by adding in some lines that never cross or end up being the same. Since I hadn’t done that with the launch, I used that idea for the warm-up the next day.

I projected a Desmos graph with a few different lines and told them that the battleship path was the red line. They were tasked with estimating mine placement for the other 4 lines.


This quickly brought up the “missing” orange line and the inability to lay a mine on the green path. We discussed possible numbers of solutions to a linear system then I had them sketch ideas for a system that could have two solutions.  Afterwards they did some more practice with graphing to find solutions. And ended by having the students create scenarios were you’d care about the intersection and then write up a problem which would fit that story. I collected them. Some will turn into warm ups or lagged review and some will end up  on quizzes or the end of unit assessment.  I’ve been working over the last few years to incorporate student generated problems. They seem to get excited about the possibility and its improved their problem writing because they want me to use theirs.

  • For the elimination method, I think I’m going to launch with a magical Ms. Micaela warm up. I can’t remember where I saw this, but I’ll try to update with credit as soon as I can find it. Basically, each student is asked to think of two numbers. Then I ask them to add the two numbers together and tell me the sum. And subtract the two numbers and tell me the difference. I will then “magically” tell them the two numbers they started with. I’ve done this before, and I’m always amazed how easy it is to impress high school students with my prediction abilities. After correctly predicting a few of the students numbers, one of two things happens. A student figures out what is happening or I ask the students to try and figure out what is happening. We write out x+y= # and x-y = * and this leads into elimination method. I might actually use the sticky notes like I did in the substitution method to show why we can combine the two equations. I’m still looking for other good elimination type activities, so feel free to share some below!
  • The meat of the unit is after they’ve seen the three methods. Why do we learn three? Which one is better? This is prime time for a math debate. I love having math debates.  They’ll pick teams and go for it. The first debate is usually informal. They debate with ideas they have already. The next debate is primed with examples. I pick out some systems for them, making sure I have a mix of problems that are suited for each method. In groups they have to solve each equation with all three methods. I provide some structure so that each person tries each method at least once. Then the small groups can discuss the pros and cons of each method. Then we go back to the whole class debate. Some team switching might occur here and all students usually are able to say “it depends” for the best method. But… I still have each team present their best case for why the method they are defending is awesome. The other teams then rebut by bringing up the draw backs. When all teams have had their say, we capture the strengths and weaknesses of each model into a visual for each student to keep. All future work, I don’t require them to solve any certain way, but leave a space for them to write why they chose their method. Come assessment time, I usually ask that each method be used at least once, but they are free to pick and chose when that happens.

Other things I want to remember to use this unit:

The MARs Formative Assessment Lesson on classifying solutions. 

Trashketball. I used to launch with this, but it will be a fun problem to use with the graphing method later in the unit too!

Drive or Fly? Lab. Another way I’ve launched systems before. I’m thinking this one might be a wrap up project or something we do in smaller pieces over the course of the unit for those days when we have a bit of time, short days, snow delay days, or sub days. I’m thinking I’ll introduce the first bit about go through the guess and talking about what is important and then have students finish the project when it works for them. (More so than many schools, our attendance patterns often leaves days when a few students are in vastly different places so have a challenge for those ahead will allow me to spend time catching others up).

Also, some other fun labs that I might try to fit in, either during this unit or as a lagged review later: Oreos by Christopher Danielson (would adapt slightly to have students figure out what they’d need to prove or disprove whether double stuff is real) and Stacking Ups by Andrew Stadel. If I do stacking cups, I’ll bring some in to do a “live” 3 Act if possible. My students tend to get more engaged that way.

  • Hopefully,  as I actually teach, I’ll be able to update the blog with links to what we do in class. (Specifically, I use math debates a lot, so I really want to do a post focused on that, but I also get too wrapped up in them to get pictures/notes to share). If you have any other awesome systems work I’m all ears!
  • Also, we will do systems of inequalities and non-linear systems as well. (Hinted at that in the Desmos warm up above) but this post is too long already so I’ll be capturing that in a future post!

Quadratics: Sharing the Love MTBoS Blogging Initiative

Explore the MTBoS prompt for the week is to share the love and the resources from other great bloggers. I wanted to do that by collecting some of the blog posts that will help inform the next unit I’ll be teaching (added, bonus, I’ll be able to find them easily when the planning begins in earnest). Algebra 1 starts semester 2 off right with a unit on quadratics. It is usually one of the more challenging ones for my students, but I have grown to really enjoy teaching it both here and in my third year math class which also has a quadratics unit with a bit more depth.

First: I want to remember the my students created assessment questions on the topic last year. They were great. I want to use the student generated questions this year and have the students create more of their own.

Second: I loved the series of Headache/Aspirin posters from Dan Meyer. His posts and all the comments given ideas to launch many topics one of which happens to be factoring trinomials!

Third: Jennifer Fairbanks wrote a blog post for last year’s Explore MTBoS in which she shared a quiz question for the quadratics unit. It allows students to have some choice while still getting them to practice multiple methods and understand the strengths and weaknesses.

Fourth: Lisa Henry has put together some practice for students on sketching graphs from zeros and other important points. I do an activity like every year and I never save it so I start from scratch. Lisa is kind enough to share hers so is already made for me!

Fifth: One thing I’m really excited about is the Marble Roll Lab from Mary Bourassa. She did it with her students and used a linear model so we did that earlier in the year. But, like she mentions, the actual relationship is quadratic so I’m going to have my class revisit and try to model with a quadratic to see if they get better results. Either way, we’ll talk about reasons why they get better (or if they get worse, reasons that could occur as well).

There are many more awesome quadratic ideas out there so I’ll certainly be adding to the list as I get more into the planning, but I’m excited to teach the unit. I’ll certainly be using algebra tiles for polynomial operations, factoring and completing the square for example.

If any of you have some favorite activities/lessons/launches/questions…. please share a link below!



And you can play along or keep track of other math bloggers here.

Adapting for Discourse

In one of my classes (Bridge to College) there are sets of developed lesson plans that address the standards for the course. Because of agreements with the community colleges in our state, I do have to stick fairly closely to the outline provided. Luckily, the lessons themselves are pretty well thought out and have a huge focus on the practice standards and the idea of more than one right way to solve problems approach that I like to employ in my other classes.

I do, however, take the opportunity to make small additions or adjustments to further the opportunity for student discourse.

For example, we are working on a unit dealing with measurement and proportional reasoning. The seemingly obligatory scale drawing project comes near the end. I printed off a picture, cut it into one inch squares and asked the students to each recreate a square on an 8 inch square. At the end, the pieces would be put together to get one giant image. They didn’t know what they were creating in the beginning.

My small changes: No much in the way of guidance. Each student tried there own method. Halfway through class, I had them find someone who was using a different method to scale up and have a quick chat. After they finished their first squares, they each did another one and had to use a different method. Most did the one from their partner chat, but one or two thought of a totally new way to approach. 


After they are assembled into the wall art, the lesson plan lays out the following questions to consider:


1. Look at the finished product and evaluate the display. Did the lines match up? Which part looks the best? Which piece would have been the easiest to recreate? The hardest? Why?

2. What is the relationship of the perimeter and area between your original square and the square you created? What is the relationship of the perimeter and area of the original square to the final class project?

3. If we did the project using 4” x 4” squares how would that have affected the perimeter and area?

My additions:

First will be in the form of a debate:

  1. Which method works best for scaling? (Two or three teams depending on how many methods they settle on).

Second will be a group think: 

  1. What makes a good strategy? (precision, speed, etc)
  2. After we come up the list of attributes, they’ll be asked to rank them in importance and have a quick share out why they think so. I’ll pose the debate question again to see if any minds have changed. I’m expecting an ‘it depends’ as a final answer, but a well thought out, justified, and detailed it depends. 

Nothing too exciting or revolutionary but by using the debate structure, students spend time doing all the important steps of good discourse: listening, responding, justifying, prompting others, etc. It’ll add a bit of time onto the project, but time well spent. A scale factor project is just not that interesting all on its own. But really thinking about how to choose a strategy and when and why other choices might be better/worse can then be applied in more complex problem situation later.

The squares are starting to go up (This is the point the students figured out what they were creating. I did give them all random inside bits to start with):



What are some of your favorite strategies for increasing meaningful student discourse?

Edited to add the finished product: img_0750