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!

Pythagorean Theorem and the Distance Formula: Live 3 Act

I was reading twitter when I found Mr. Orr’s 3 Act Task Corner to Corner task. I had just taught the Pythagorean Theorem the day before and the distance formula was on tap for the day. I had a giant thing of string from Algebra’s battleship task, so I thought….why not recreate the scenario in class.

I taped a piece of string from one corner on the floor to the kitty corner  one on the ceiling.  I have tables and a relatively small class ~15 students so I was able to push tables to the sides for the day. When I greeted the class at the door, I asked them to watch their heads. That got a few chuckles until they saw the giant string. Instead of giggles, I got excited chatter. Many were variations on”What is Ms Micaela up to now?”,  but many students were asking each other math-y questions as well. The bell hadn’t rung, I was still greeting in the hall, and already I overheard the question I wanted.

When class did start, one student asked the why question, so I asked them right back. “Why do you think I put this here?” and hinted I’d already heard some great math talk. We quickly settled on “How long is the string?” I told them I’d hung it right before class (in fact a few students were trickling in as I finished taping) and I didn’t measure it as I was doing it. In fact, the string was still attached to the yarn ball, but that I would give them other information if they wanted it and I knew it.

But first, I had them each guess/estimate the length. Afterwards I held up a meter stick and asked if anyone wanted to re-estimate. One student did.

I didn’t have nice pictures set up like Mr. Orr since it was last minute, but I did give them the height, width, and length of the room. One asked for the diagonal length and I honestly answered that I hadn’t measured that, but ensured them they could figure it out. (I though about having the students measure the room dimensions, but I also liked the double Pythagorean and visual understanding needed if the diagonal wasn’t known so I didn’t.)

Each time I gave a piece of information (height, width, length) at least one student would shout “I want to change my guess!” So we started collecting all the changes in estimates on the board.

They turned to their table partners and started in on trying to solve. One table quickly figured out that Pythagorean Theorem would work, but didn’t “see” the problem in 3-D quite right. Other groups saw the problem, but didn’t jump to Pythagorean theorem. I had them conference will a different group and then go back to their seats and try to solve. I also asked a group to come up and draw their thinking on the board.

They were excited and engaged and worked hard to explain their thinking to their tables.

It was a lot of fun. I know its not feasible for everyone, so I’m glad people like Mr. Orr make the videos, but I loved having the actual string in the room.   Instead of revealing the answer. We cut down the string and measured right there in class. It was so satisfying for them! We had a quick discussion on why our answers might have been a bit off the actual answer even if the math was done correctly.

Afterwards, we started in on the progression Mr. Orr shows here for the distance formula. He links to the Desmos files at the end. I adapted a bit and stuck them on a Power Point if that is easier: distance-formula-lead-up

I have always taught slope as as “change in x” over “change in y” and used the delta symbols, so we did the same thing in the distance formula. I have found this reduces sign errors and makes kids think about what it means. A few students who had seen the formula before asked it is related to “the y2 y1 thingy” so I added the traditional formula up as well and we discussed how they said the same thing. They seemed to leave feeling confident and I even overheard two students debating which way (distance or Pythagorean theorem) was better on the way to their next class. That is a teaching win.

Systems of Equations Launch

After wrapping up our linear functions unit, the students had one day off. When they returned to class on Thursday, each table had been turned into a mini command center. Big sheets of graphing paper were stuck down to the table and an assortment of string, scissors, tape, rulers and three colored dots were at each table.

When students were sorted into teams, they were handed the mission sheet:

I told them they could only use the supplies on the table and at the end of the activity, I’d need a report on where the mines should be laid in the form of coordinates.  And they were off! (Side note, the original question had more information, basically telling them how to solve, so I just erased it which is way the type is a bit crazy. I’ll type up a nicer version for next time with the additions I add at the end of this post).

After we stopped, I took the coordinates and posted each teams on the board. No groups had the exact same answers. They wanted to know if they “won” but I told them they’d have to wait.

We had quick whole groups and table group discussions on the activity itself. I told them the new unit was called “Systems of Equations” and asked for feedback on what they thought that meant. We also discussed the idea of solutions to a single line (where are all the possible places the mines could have been laid), and whether one mine would have been sufficient (if the system was all four together) and finally grouped the equations into three different systems, with the battleship equation being in each system along with one enemy ship.

I told them that the next day they were going to have to prove whether or not their mission was successful. In preparation, they has the last few remaining minutes of class to grab a pencil (which was not an initial supply) and add any information to their work that might help that cause or that they wanted to remember. Some wrote the three systems, some added points, most added labels.

The next day the groups came back to their work from the day before. I told them they could use any strategies they could think of to try and prove the solutions. The most common were tables for each line and plugging in the solutions to see if they were true. The answer to one of the systems consists of nice integers. The other two are fractions, one of which was easier to estimate with a graph (involving halves) and other which was very unlikely to have been chosen perfectly.

Students narrowed down intervals and we had a good debate. Mines might not have to have a direct hit? But what if we wanted to be sure? This was a great quick launch. Graphing is awesome. We didn’t have to discuss it as a method, the kids figured out that it was, but now they wanted more ways to find an answer. I told them we’d learn two more methods over the course of the unit and then they’d come back to find precise coordinates for the mines at the end (Score, one assessment question written!)

Changes/Ideas for next time: Add a fourth enemy ship that does not intersect the battle ship. (Maybe another that will, but too far out to see on the graph. Or one that ends up on the battleship path.) Six enemies might be too many, but each group could have a different subset of the enemies and we could come together as a class to discuss each.

Another idea might be to provide each student with partial information. Give them some time to look at theirs and then find classmates to form a team to that would have all the needed information.

Also could change how the information was given. I liked the practice of graphing from standard form, but some information could have given by coordinates or an initial sighting (one point) and speed (slope) depending on what review/practice is needed for the given group on students.

**Edited to Add: The context could easily change to Zombie Attack. Or trying to drop aid packages along routes. Treasure hunt. Or any other more positive situations. I kept it as battleships this year, but might adjust depending on students/issues in the class as well.**

Measuring Angles (Plus a Little Dancing)

Geometry A is working through a unit on definitions and introduction to proof, with a good dose of reviewing key ideas that may or may not have been learned in middle school math. Our topic today was recognizing: vertical angles, linear pairs, corresponding, alternate interior, alternate exterior, and same-side interior and how to use that information to start building proof.

I put up a set of lines and a transversal and had student call out angles that fit the above terms. Then I had them all choose a set of taped lines that were spread around the room. We went through each location again, trying to find as may pairs that fit said relationship. After students were comfortable with each, I hooked up “Dance, Dance, Transversal.” I have no idea who I first saw the idea for “Dance Dance Transversal” from or I’d give credit, but there are lots out in the inter-webs of math land, so if you want more, Google returns many hits. We played a few rounds to warm up. Basically, I had a PowerPoint set up to cycle through the terms (I limited it to vertical, AIA, AEA, and corresponding) and students had to jump to the correct location. We had music going and in later rounds, judges. We crowned the class champion and I let them know they’d have to dance on the unit exam.

Then, each student was tasked with getting a protractor and measuring the eight angles formed by the transversal. I had purposefully used both parallel and non-parallel set ups (About 2 parallel for every non-parallel). After measuring, they walked around and had to write down at least two notices and two wonders. (Examples “I noticed all the vertical angles were the same.” “I won
der why some gro
ups have only two different angle measurements and others 4.”   “I noticed an error, or what I think must be an error in set B.”   “I’m wondering if my tape was messed up, because it doesn’t look the same as most of them.”)

We were running short on time, so we shared out a few of the Noticings and Wonderings as a class and we’ll pick back up there tomorrow. The plan is to agree upon, prove, and then practice with the key ideas.

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.

Spaghetti Challenge

I wanted to start the year with a project. I value discussion, team work, and creating and I wanted students to know this from day one. I threw around some grand ambitions, but settled on the spaghetti/marshmallow challenge mostly because the supplies were cheap and easy to jump right into. I was worried that students would have done it already or be bored, but I was definitely wrong. Students were thrilled. My favorite comment of the day “You aren’t tricking us right? I hope math class will be good all year and not just today.”

I had students fill out reflection forms and I got all fives, with the exception was of two fours. (1-5 scale). All groups begged to stay longer after class. Two groups worked through lunch hour, and one group tried to stay into their next class.  I’d call that a good opening. I also got pages of notes of individual students learning styles, ability to work together, communication skills, etc.

One interesting note was the responses to the question: What worked well? Students  tended to choose their own contribution to the tower. For example for one group, one said “adding something to pull back the wobbly parts” (her idea) and the other: “making a triangle at the bottom to make sure its even/balanced.” (again, said students idea). Their tower was successful at 26.5″ and there was no way to tell tell which one made the difference, if both were needed, or something else entirely. On the flip side, the “What would you change” was usually more supply or group oriented. “Be neater” “Use material smarter” “Be less wobbly”. This may be normal, but we started doing community circles this year and the students had been really critical about themselves earlier in the day, so I was excited to see them proud of their contributions.

In most of the groups they started out two people working on totally different ideas for same tower and it slowly turned into better team dynamics. A few groups hit it off right from the start and only would do work if both agreed. A few assigned tasks and then came together at the end.

We had some good discussions. What assumptions can we make? How long does it have to stand to count? (The absolute highest was 30 inches. it stood alone for about 1 or 2 seconds and fell. Another at 26.5 inches stood until the group karate chopped it down at the end of class). Is height really the most important part? What are we learning?  “We can go higher!”

Also, I forgot string was a material, so they only got tape, spaghetti and a marshmallow. Three separate groups came back later to tell me a brilliant new idea that “Would for sure be the highest.”  One of the the more inventive ideas was to make a long thin strand and tape the marshmallow to the top of the spaghetti and the ceiling so the long, thin strand didn’t have to actually hold weight.

Puzzling Minds

Keeping with our problem solving short Wednesdays, we did math station puzzle rotation today. Certain students really dislike when I assign one problem that takes the whole 40 minute block, so I thought I’d give out a few shorter activities today.

First station: Arrange all five pieces into a square. All the students found the 4 piece version and more than half tried to get away with sticking the little left over square on top. Most students were eventually successful with the five piece square although I know a little hint giving was done. I like this puzzle since every student walked in and said “This is easy!” and then quite quickly, “This is impossible!” After finishing, they’d sketch it in a notebook for me to check off.

<— All the pieces

A tricky Student —>

Second Station: Logic Puzzle Fun. Students could choose any of the varieties of logic puzzles I put out and solve it. They have seen each one of the choices at least once during the year.

Third Station: Four 4s. Basically, using exactly four 4s and any mathematical operation, create expressions to represent all the integers 1-100. This was the biggest hit. Students were whining and complaining at the beginning, but as soon as they got into it, they really got into it. I had multiple students try to skip their next class and stay in to do four 4s. I told them they could take them with them, but most replied “But I want you to know I did it myself.” I told them I trusted them, but still something about doing the problem while I’m in the room was important to them, so I told them it could be something they pull out as pre-class work or end of class work if they ever finish early. I feel so corny saying it, but it was so fun to watch them work in this one. They’d light up when they found a ‘hard’ number or used a funny operation. I got to introduce a whole lot of math vocabulary that we never get around it using. They loved this so much, I should have spent the whole hour on it. I had figured it would be the least liked activity. I was very wrong.