Geometric Series Formula, Too
Lesson 7 of 11
Objective: SWBAT derive the formula for the sum of a finite geometric series.
Set the Stage
I begin this class with a review of the "rules of deriving" that we formulated as a class at the close of yesterday's lesson. Instead of just telling my students what we wrote, I ask for volunteers to come up and write the rules on the board, one rule per person. I ask if there are any questions then explain that today they will get to help derive a new formula.
I tell them that I've received a mailer offering a grand prize of either $100,000 lump sum or $5000 per month for life and can't decide which offer I would choose if I won. I explain that I've checked with my bank and they'll give me 1.4% interest payment per year if I put the whole $100,000 in savings. What should I do? I ask my students to work in teams of two or three (they get to choose their partners for this activity) and help me figure out which deal is better. (MP1, MP2)
Initially there are teams that want to just go with one option or the other without really considering the mathematics, but I try to help them focus their thinking by asking directed questions like "How do you know this is the better option? Can you prove it to the class?" or "What evidence do you have that this is the better option?" As they work there are questions that arise from individual teams that I answer for the whole class like "How long do you think you'll live?" I answer that one with "I plan to be around at least another 20 years or so, so you can use 20 as your target if you want." I walk around offering encouragement and support as needed, and also looking for teams with particularly interesting approaches to answering the question.
After several minutes or when all the teams have finished, I ask certain teams to share their work with the rest of the class. (MP3, MP8) I try to select a team that has simply calculated each term of the $100,000 investment over the 20 years and the total from the $5000 per month without interest for 20 years. I also select a team that chose to try to multiply $100,000 by the interest and then by 20, because this demonstrates a misunderstanding of compounded interest. I can address this fairly easily by asking my students to calculate how much I'd have at the end of the first year (including interest), then emphasizing that the result is the amount I start with the second year. When we're done, the class should recognize that I would have approximately $101,420 if I took the $100,000 and put it into savings to get interest and I would have $120,000 if I took the $5000 per month option and didn't receive any interest. This problem and the work involved are intended to set my students up for figuring out a better way to answer questions like this.
Put it into Action
I begin this section by reminding my students that this work all goes in their notebook so they have it as a reference later! I then tell them the story of Gauss as I heard it when I was a student. Here is a link to an interesting article about the veracity of the story: Gauss Story. I give them an opportunity to try to find Gauss' shortcut then walk them through the math if needed. (MP1, MP8) Someone always asks if the story is true and I respond that I think it very well could be because the mathematics isn't really that complicated. I also explain the mathematical vocabulary for this kind of problem is finding the "sum of a geometric series" and invite discussion and/or questions.
I have my students work independently on this activity to give them a chance to flex their mathematical muscles and build their mathematical confidence. (MP1, MP2, MP4) I distribute the sum of geometric series handout and ask my students to read through the directions. After answering any questions, I tell my students they will have almost 30 minutes to work this out and that if they come up with a solution before that time, they should be sure to check and see if it works. Rather than telling them how to check their work, I try to let them decide what to do and ask leading questions if necessary like "How does that prove your answer correct?". As they're working I walk around giving encouragement and support. For students who are struggling I offer hints like "Have you tried writing a recursive formula for the sum?", or "How do you think Gauss would tackle this problem?" This is a difficult challenge for some students but as I discuss in my Formula Too video, it's a great opportunity to help students build perseverance and learn to make connections to previous work. As students complete the challenge I ask them to consider ways to check their work and to demonstrate to the class that their formula works. (MP6)
When everyone is done, I randomly select students to share their formulas by writing them on the board. Since several students can write at the same time, it provides a measure or anonymity for students less certain about their answer. I ask the class to critique the formulas on the board collectively, looking for similarities and differences. (MP3) When we've reviewed and refined to a single formula, I tell my students to try the formula on my prize problem from the beginning of class to see if it works.
I wrap up this section of the lesson by celebrating success! I tell my students that they have derived a formula for the sum of a geometric series...and that once done it will be theirs forever.
Wrap it Up
To wrap up this lesson I give each student a notecard and ask them to reflect on the process they used to derive the formula today. I tell them to respond to the three questions below which I selected to focus my students' thinking on the process instead of just the result. (MP3) When they finish the notecard I remind them to include the new formula in their notes for future use.
- What strategy did you do first to try to answer the question and how well did it work?
- What changes to your strategy did you make? If you didn't make any changes did your strategy work?
- Would you use a similar strategy to answer other mathematical questions and why or why not?