## 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!

## Multiple Representations of Patterns

I have all my classes doing a Visual Pattern from Fawn as a warm up once a week. With my Algebra 1 students I decided to extend the assignment to create posters showing how the patterns relate to the table, words, graph, and general formula.

I did a similar assignment last year for exponential growth and decay when we got to that section, but I figured I should start with linear patterns. When we do the project again for exponential patterns hopefully the connections will be even stronger.

The process:

The warm up was a simple linear pattern from Visual Patterns.

I always have them fill in a table and attempt to come up with a rule. After they did this, I asked one student to share the results with the whole class using the projector. While she was explaining I had another student plot the table points on a graph on the board.

We had a quick discussion on where the pattern “up 2” showed up in the four representations (drawn pattern, table, algebraic expression, graph) and had presenting students mark the 2s on the examples. Then I asked where the plus one in the rule came from. The student at the front said it was the one red star in every pattern. The one drawing said: “Oh, if I extend this back, its right here!” (pointing to the y-intercept). I asked them to find it in the table. (It wasn’t there….So add it! Where would to show up?)

I then handed each student a new pattern and asked them to make a table, graph, rule, and color code the connections between the representations. When they were confident, I had them make post sized versions to hang around the room.

A few examples:

Things I noticed: Some students drew in a Stage 0. I asked why, and for the most part these students thought of stage 0 as taking away the pattern part of stage 1. I asked the others why they didn’t draw a stage 0. The response was generally, “Is there a Stage 0? I see the start value here in 1…” I want to dig deeper into this difference.

Note On Slope: We haven’t talked slope yet. Students saw the pattern as change in number of squares or dots exclusively since n was going by 1s, and I didn’t push them on it yet although I did start those discussions one on one with the students as they worked. My plan is to give them Step 1, 3, and 6 of a pattern and ask them to come up with a rule. As well as to introduce slope/rate of change with a similar task (and non-integer changes) and have the discussion on rates of changes of 1.5 vs 3/2; and where those show up in the different representations. Then have them go back and update the current posters and with creating new examples with differing x changes as well. I’m not sure if I should have done this during this task, waited to do a connection of representations until we had already talked about this, or if this do it now and revisit is best.  The students have definitely benefited from looking at the different representations, and there ability to come up with pattern rules has been improved but part of me is still worrying I should have done the Step 1,3,6 thing at the same time…maybe giving that to my students who needed more of a challenge for their original poster and then had a class discussion about how they were the same/different.

## Time Distance Graphs

Algebra 1A is reviewing parts of a graph, writing stories from graphs, and graphing based on situations. I ran across a really fun idea from Mr. Orr where he had students create motion videos, create an answer key (i.e. a nice graph of the scenario) and then gallery walk and try to create graphs based on each pairs video. I knew I wanted to do it with my class. I am not one to one, or even close to that, though, so I adjusted the activity to fit. Instead, we all had to share the one iPad I did have.

Each student got a set of steps and a place to sketch what they thought the steps would look like on a graph. I didn’t give them much direction here. As they finished, I pulled students out into the hall to film there series of steps. I asked them to think about the distance they traveled over the course of doing the activity and re-sketch the graph if doing it changed their minds. (So in the worksheet below, most students had two sketches for #1).

After the videos and sketches, we talked about what a time distance graph shows. I projected a completed graph and students try to come up with a story. Each student then graphed there own graph on a full sheet of graph paper. I checked over them for accuracy, asking students to explain areas I wasn’t sure about and re-draw until we both agreed. They stapled a blank piece of paper on top and wrote the letter of their story on the outside.

Overnight, I collected all the different clips and arranged them by letter. I also taped up the graphs around the room, with only the letter showing. We opened the next day by watching all the clips as a class. Each student sketched what they thought each graph would look like and labeled it with the correct letter. After the clips, students walked around the room to check their answers. I also asked them to write down any “Notices or Wonders,” especially if the graphs didn’t match they had drawn.

Here is a sample video and solution:

We got back together as a class and took a poll on which graphs had matched or not. We re-watched the most troubling clips and discussed what made them harder to draw. We also talked about how the graphs might not mach exactly, but still be considered correct. Without having actual measurements on the ground, “fast walking” might always be steeper than “slow walking,” but by how much would vary.

Then, students were given a reflection/exit ticket. Instead of creating graphs, they were given a graph and had to create the story.

The whole thing took about 1 class period, but split into two days. (The end of one class and the first half of another. I have 50 minute classes, but they are small. With a larger class it might have taken longer, or with more iPads it could have been shorter as we wouldn’t have had to watch all the videos together before the walk. Mr. Orr had the iPad videos sitting around the room, one video at each poster.)

It was a fun and active way to introduce motion graphs. I will definitely be using it again. Hopefully next time with more video devices.

## Crazy Taxis and Trig Ratios

Short class Wednesday had me on two different sides of the teaching spectrum today. Algebra 1A and 1B did Mr. Orr’s 3 Act Taxi problem complete with great discussions and hard working students. Geometry A and B had a makeup work day because no one was caught up and it’s crunch time. I had avoiding letting the Wednesday be a make up day all semester, but I cracked today.

Note: My kids think taxis are really expensive or Mr. Orr thinks they are really cheap. I can’t tell because the only time I’ve ridden in one in recent memory was in Korea and the exchange rate doesn’t lend itself to great comparisons.

## Absolute Values

Algebra B is working on absolute value equations. We started off using Dan Meyer’s “How Old” game where students see a picture of a celebrity and have to guess the age. I made sure to throw in a picture of my daughter at the end. It is amazing how excited the students get when I bring a picture or talk about her. Plus, it was a gimme answer. Who doesn’t like to be exactly right at least once. After about 10 people, ages are revealed and the best guesser is announced. It was a small class, but we had a good split of students who calculated their guess score by using absolute values and those that kept signs and even a few that explained the idea of the calculation correctly but still had negatives in the answer. We looked at two student’s calculations on the projector to pick a winner. The lower number actually was farther off (used negatives) so we talked about ow we wanted to determine how far off each was without caring if they were low or higher. It was quick, about 5-10 minutes for the whole thing and a good introduction to the idea of absolute value.

I introduced the mathematical notation and then asked students to come up with examples of when the sign didn’t matter (use absolute values) and when the sign would matter (not absolute values.) Then I threw up a quick number line on the board and had students place sticky notes where they should belong.

We then moved on to graphing. I had all the students predict what they thought the graph would look like, graph the parent function, and then discuss why they looked how they did. As time allowed, student then started to graph transformations of the parent graph, mostly by using tables and plugging in values. A few caught on that the transformations work like all the others we’ve done so far and skipped some of the plugging in steps. (All this work is still in the student’s notebook, and I didn’t take pictures as I was walking around working with students, but I’ll get it up soon.)

Lastly, we started looking at the graphs to find solutions to absolute value equations. I like starting graphically for two reasons. It reinforces the skill of find solutions from a graph (especially the multiple solution part) and it highlights the need for careful thought when solving algebraically since there are two solutions.

Our last few minutes together, I put up a couple absolute value equations which could easily be solved by sight and had students think of both solutions. We’ll follow up tomorrow with more involved equations.

**Last semester’s Algebra B class did a more involved graphic introduction which I really liked. We graphed the parent function and then in different sets 3-4 transformations of the equations on the same graph in different colors. I liked that graphing activity better, but it was also the first major transformation example. I did radical graphs and transformation first this time, so I skipped the longer discovery this go round. I think I might bring it back next time. Having 4 graphs together gives a better comparison. and allows for finding solutions to lots of equations at once.

## Introduction to Systems of Equations

My Algebra 1A students are just starting the systems of equations unit, and what better way than a TrashketBall competition. I’ve seen a few people use Trashketball as a fun review game, but Mr. Orr shared an awesome three day lesson on his blog. I borrowed Day 2 and 3, adjusted them a bit and made it an introduction to systems of equations.

Students played 4 rounds of 1 minute each to find their average make rate per minute, converted that to makes per second and came up with an equation using that rate. Then they graphed their lines. With a partner, students discussed what their graphs meant and compared and contrasted the lines. (All start at 0, but all different rates etc). A few pairs shared out insights.

We then graphed all the lines on a projected Desmos graph. I asked who would win a basketball competition and how they knew. Then I asked how to make the game more fair. Eventually they settled on giving the slower rate a head start. They jumped back with their partner and overlaid their original graphs using patty paper on one graph. They had to decide who should get the advantage and how many balls that advantage should be. One group wanted to give the ‘better’ player a late start (moving the x-intercept instead of the y) which I said was fine. They graphed the new line on the same graph as the line that wasn’t changing.

Up to this point we had just thought about how to make the game more fair. I asked them to make observations about the new set of lines. The first observation was that they intersected. Another student pointed out the the originals intersected too, but that the intersection was no longer at (0,0). We ran with that and talked about what the intersection would mean and why did we want to move it away from the origin. That intersection point became new time length for each partner to compete against each other. I had them write down the intersection point and decide what each value meant (time of game, # of makes expected)

Almost all of these games were quicker than the original one minute, but one generous competitor gave a bigger advantage had an almost 2 minute game. New partnerships were formed and they played one more time.

Closing as a whole group discussion about the game. We had a few ties and most were fairly close to the predicted make values. A few were outliers so we talked about those as well. I then introduced the term “systems of equations” and “solution to a system of equations” and the students contented them to the context of the trashketball game.

Tomorrow, we’ll move into solving systems graphically more formally, with lots of in class practice time. Then we’ll move into pictorial puzzles to lead into the substitution method. Elimination/Combination will round out the introduction to systems – still looking for the best hook for that one.

I really encourage you to try out Trashketball. Every single student was engaged and doing lots of great math. I will definitely be refining it for next year, maybe even extending as a unit long project. I did also promise a trashketball performance task on the unit final….

## Flying Higher

While one class was racing cars yesterday, another was flying drones. I liked the gather data and predict aspect of the car lab, so I created a similar format for the drone lab.

Students were given a drone and the task: Determine the maximum height the drone can reach. Find out how distance from the controller affects this number. Use the data to make predictions about other distances and heights.

We spent about 15 minutes outside flying the drones and gathering data. Then the students came back in and Desmos to create a scatter plot and line of fit for the data. We then had a quick group discuss/recap to talk about the model and why or why not the line would make a good prediction for other trials.

Take Aways: Everyone wanted to fly, so most groups switched off controllers. Our class theory was that the controller may have actually had a bigger effect on drone height than distance. Also, we used “paces” to measure. Some groups were much better at doing these uniformly. Either way, both of these introductions of error were good talking points.

More Notes: The students are getting good at explaining their thinking orally, especially when prompted or questioned about flaws. When they write about their thinking, they still need a lot of work. Even when I say, write what you just said to me, the paper version never makes as much sense. We need to work on this!

Notes: Our drone’s battery life is very short (~6 minutes) and charge time is long. I keep one spare with me, but I also let the students know this. They have to be able to get the data they need in a limited amount of flight time.