Math Art

We are doing state testing today, so my room will be a quiet place. Thought I’d share some math-y art instead:

Posting regularly is harder than I’d imagined it would be. Maybe it wasn’t the best idea to start at the end of the year in the midst of all the craziness that happens. I’m teaching over the summer, but a reduced load… and ‘m planning on setting some goals for more regular posting next year. I’d love to do a 180, but I’m scared to commit 🙂 Hopefully TMC will be a good push into really jumping into the MTBoS.

Although we still have just over 3 weeks of school left, I had my advisory class do a bit of self reflection of the year. The next weeks are filled with testing and racing the clock to finish, so I wanted to capture their thoughts now as well give them a little boost to show them how far they’ve come.

I typed out a few prompts and had them finish the sentence, then I typed the response side into Wordle to come up with the main ideas. Reading the individual responses was more powerful though, so I wanted to throw up a couple of those ideas as well:

We are an alternative public high school, so some students choose to come and others are required to attend. Some are here for a few weeks, others years. They are all in such different spaces, but they are all really cool people. I’ve had a great year and hope we can finish strong together.

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.

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….