Decreasing Geometric Sequences
Lesson 6 of 10
Objective: SWBAT compare and contrast an arithmetic and geometric sequence. SWBAT represent a geometric sequence with a constant ratio between 0 and 1 using multiple representations.
In the previous lesson, students explored an arithmetic sequence that had a negative constant difference. In today's class, they will look at a geometric sequence that decreases in the out column. I start class today by reminding students of the work they did in the previous class. Then I ask them if they think it's possible to have a geometric sequence where the numbers in the Out column of the table would decrease instead of increase. Some students may need a reminder that in a geometric sequence, the previous term is always multiplied by the same number. I let students share out their thoughts on this idea and ask them HOW a geometric sequence might decrease.
Next, we read through today's task, Chew on This together. Students can jump right into to problems 1 through 5. They will likely recognize the first two questions from previous work and should not struggle too much to write an explicit and recursive function representing this situation. In Questions 3 through 5, students explore a similar situation that has a common ratio rather than a common difference. I watch for students who have trouble writing the explicit function because it starts at 1, rather than at 0 in the In column. When to use x-1 versus x as an exponent is often a struggle for my students.
The real work for this task comes in Question #6, so I want to make sure students get to it with plenty of time to work on it. I think there is an opportunity in Question 6, depending on the level of students in the class, to think through allowing students to productively struggle with the problem before lending a guiding hand. Most of my students will approach Question #6 by finding 60% of 100,000 and then subtracting that amount from the original amount of candy. I watch out for students who then think that same 60% is subtracted each time and forget that they should subtract 60% from the new amount of candy. I might encourage students to create a three column table here, so they can see how much candy is subtracted each day but also have a separate column to keep track of how much candy is left (which is what the problem asks for). Why do I let students go through all of this tedious work without just showing them they can multiply by .4 each time? I think the "long" way is accessible to most students, it makes sense to them. I would rather use this understanding they develop as a jumping off point for repeated multiplication, rather than just "show" them the correct way to use repeated multiplication.
In the discussion section of today's lesson, I'll ask students to share their work for this problem. Some may have been able to get to multiplying by .4, so I'll look for those students to share out their work later. I might encourage students to think about a "shortcut" by asking them something like, "What if you thought about how much candy remained after each day instead of how much you should subtract?" How might that change the math you're doing?"
Many of my students really struggle with this problem, so depending on how much time I have, I might consider pushing the discussion to the start of the next class, rather than trying to fit it in today.
I want the discussion on Questions 1 through 5 to go fairly quickly so we can really dig into Questions 6 through 7. I do spend some time on the 1 through 5 section asking students to compare how much more candy Augustus would have with the second scenario than with the first. I might ask them how much candy he would have on day 30 of each plan. The difference between the two answers is quite striking so I want to spend some time having students think about why. This is a good place for students to get the idea about how much bigger numbers get when they are repeatedly multiplied.
Next, we'll take a look at how students handled Questions #6 - 8. I start by having someone who did the problem using subtraction come up and share their work. I want to emphasize to students that this is NOT the wrong way, it works perfectly well and makes sense to most students. We can talk about whether or not we think we might be looking at a sequence here and I elicit ideas from students about how we might figure that out. At this point, I might have another student come up who used repeated multiplication and have him/her share out the work. We spend some time thinking about how multiplying by .4 achieves the same result as this multiplying by .6 and subtracting. I might ask students why they think this works? We also talk about how giving away 60% of something is the same as keeping 40% of it. I might give another example to illustrate to students why this works.
Once we have a working table for this problem and students can see the common ratio of .4, I ask students to write a recursive and explicit function. Most students will readily come up with the recursive, but many students will struggle with the explicit. My students are still learning that the common ratio becomes the base of the exponent and they struggle to write the exponent itself. We also spend some time talking about the starting value and what that means in this problem. Where should it go in the explicit function?
As in the previous lesson, I want students to begin to notice certain characteristics of geometric sequences that they can identify in different representations. I end class today by asking them to complete the identifying a geometric sequence table so they have a guide for how to identify geometric sequences in future work.
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