Students will spend most of their time today working on a problem set. In this video, I share some of my thoughts about problem sets, their purpose, and what I think about when I make them. One of the most important ideas to note is that a problem set is not static: as teachers, it's our job to revise the sequence of problems we give to students, based on what they need to practice, what they've seen or have not, or even what they're studying in other classes as we seek interdisciplinary opportunities.
It will most likely to take two or three class periods for students to complete the problem set and then for us to review key ideas from this unit. As I note in the video, one purpose of a problem set is for students to review and apply in different contexts what they've seen over the last few weeks. I'm also likely to run another follow-up lesson during these couple days of review in which we investigate the parameters of exponential functions on graphing software like Desmos. I always teach that lesson to my Algebra 1 students, and it's written up in my Algebra 1 course here on BetterLesson.
Also in the video, I hold a pile of textbooks. Linked here are the books I flipped through while getting ideas for this problem set. I might also draw on activities from one of these for follow-up lessons, as necessary.
Today's opener in on the second slide of the lesson notes. Just like the problems that are printed on today's problem set handout, this one gives students a chance to review and apply what they know about arithmetic and geometric functions.
The phrasing of the problem also helps to prepare students for a brief investigation of compound interest: "In the geometric sequence, each term is 20% more than the term before it."
I post the problem as students arrive, and then I circulate to see how they're doing. I encourage them to discuss their answers with each other. After a few minutes, I post slide #3, which shows a graphical representation of this problem. I ask if any students thought about what the graphs would like, and students share their interpretations of what they see. I make sure not to spend too much time on this. I hope that students are comfortable with this problem; if I see that they are not, I'll make sure to address that 1-on-1 as students get to work.
The movie "Frozen" is popular right now, even for my high school students, who can be heard (some ironically, others not) singing it in the halls. So maybe you've heard the song "Let it Go," which includes the lyric "My soul is spiraling in frozen fractals all around."*
I do my best rendition of this line to get kids fired up, and then I ask (or sing, because it can fit in the song...see that?), "Wait, what is a fractal anyway?" Then I say that I'd like to show everyone a few short videos about fractals.
The first, Doodling in Math Class: Binary Trees, is one of Vi Hart's brilliant (please, look her up) tales of how she might spend her time bored in math class. She simultaneously gets kids on her side while obliterating us math teachers, takes us math teachers to task if we're the kind to not engage kids, and makes the math awesome.
The second, How to Draw The Sierpinski Triangle, is a little more dry, but it clarifies the idea of Sierpinski's famous fractal that was introduced in the first video.
I show both videos, and mostly rely on kids to see where we might go with this. If they have a lot of ideas, I open the floor to those. If no one's curiosity is peaked (which rarely happens), we can just move on. The key is simply to note how exponential growth might appear in both examples. At the end of the problem set, there are few prompts related to the ideas kids have seen here.
* What a nice little wordplay that was for the animators! The creation of fractals is a popular starting point in introductory computer graphics and animation, and that's exactly what these artists were tasked with creating, I presume, as they made the movie.
As kids watched the videos, I distributed today's Problem Set. When we finish the debrief from our viewing, it's time to get to work. I laid out some of my thinking about problem sets in the video at the start of this lesson. Once this one is in kids' hands, there's nothing too fancy today. Kids will work, alone or in groups, to do as much as they can. I'll circulate to troubleshoot, answer clarifying questions, or to join kids on extensions. Students will need computers to do some quick research as they finish problem #2, and they'll need graph paper, at least for #9 and #10, but many will want it for other problems as well.
For easy reference, each problem gets a slide on the lesson notes, so whenever we want to have a whole-class discussion, we can look at and mark up these problems on the board.
Here are a few notes on selected problems: