Problem Solving, Review, and Extensions
Lesson 7 of 8
Objective: SWBAT apply what they know about exponential functions in the context of both review and new scenarios as they work on a problem set.
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.
- CME Project: Algebra 1 and Algebra 2 - Provides incredible ways to get students to think about relationships between mathematical ideas.
- Michael Sullivan's Algebra & Trigonometry - Clean layout and an incredible wealth of practice exercises and problems, with minimal frills.
- K. Elayn Martin Gay's Intermediate Algebra - for similar reasons to the Sullivan.
- Consortium for Foundation Mathematics: Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving - is organized as a series of rich, topical investigations, any one of which will have a place, depending on the course I'm teaching.
Opener: Two Sequences
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:
- Problem #1 is about auto depreciation, which we studied at the start of this unit. I want students to see what a textbook wording of a problem from this context might look like, and I also want them to show that they can interpret the parameters in a formula by writing sentences to answer parts (c) and (d).
- On problem #2, students analyze an outdated model for the growth in the number of cell phone users in the United States. It's a fun approach, because it makes kids feel like they "know more" than the invisible and un-approachable figure of the Author Of Their Math Textbook. Students are asked to their own research online, but if it's too inefficient to get computer access for just one problem, here's an easy link to use: http://www.infoplease.com/ipa/A0933563.html. Note that the given model stays pretty true to these values before diverging radically.
- Problem #3: The answer to part (c) of the rumor problem depends on the size of our school. Be ready to share that information as necessary, just in case some kids don't know. More importantly, I have not taught how to use logarithms to solve a problem like this. Instead, I show students how to make recursive calculations on their TI-83's and to count the steps. Just like I've noted in earlier lesson, I would take a different approach if this were full-fledged Pre-Calc course, but this problem shows how can meet students where they're at.
- Problems #4-8 give a quick treatment of compound interest. We'll spend a little more time on that in Unit 4, but it's also something that many kids have seen before. By placing these problems here, I seek to expose kids to this popular context and to require them to interpret parameters in a formula.
- For the compound interest problems and the graph problems (#9 and 10) - I left out any sort of "what do you notice?" question. I want that to be implied as students work through similar problems.
- Problems #11-13 are successively more open-ended, and refer back to the videos we watched at the start of class.