##
* *Reflection: Developing a Conceptual Understanding
Chain Emails - Practice with a Geometric Sequence - Section 2: Investigation

One group of students had an interesting discussion in class today about Question #3. They calculated how much money Bill would have to spend if 8,000,000 people received the email coupon for ice cream. Although they forgot to add in the previous days, they had a rich discussion about whether or not this could be the right answer because they amount of money he would spend seemed so large (over 39 million dollars). I let them talk and discuss for quite a while until they seemed to be at an impasse. When we did discuss their thoughts, I used their thinking of "too big" to talk about exponential growth. Often times I find students don't have a conceptual understanding of what it means to continually make a number larger by multiplying it by 10. I think this idea of how much money Bill would have to spend was a good way for them to understand just how big things can get when you repeatedly multiply by 10. We also looked at how much he would spend at earlier days to think about whether our operation (multiplying) made sense. Eventually, these students were able to come to a consensus about how expensive this proposition would be!

*"That answer seems too big" A Discussion about Exponential Growth*

*Developing a Conceptual Understanding: "That answer seems too big" A Discussion about Exponential Growth*

# Chain Emails - Practice with a Geometric Sequence

Lesson 4 of 10

## Objective: SWBAT solidify their understanding of geometric sequences. SWBAT generate tables, graphs, recursive equations, and explicit equations to represent a geometric sequence.

#### Opening

*5 min*

I start class today by reminding students that we have been working on arithmetic and geometric sequences for the last few days. In the last lesson, they learned the defining features of an arithmetic sequence and how to identify such a sequence from a graph, table, recursive, and explicit equation. I let them know that today we will be looking at another problem situation and using these representations to help us make sense of the problem.

We start by just reading through the problem in Don't Break the Chain together. I ask for a volunteer to go to the board and draw a diagram that would show the first two days of Bill's sent e-mails. I want to make sure students are understanding the email chain and are clear about how the growth of the chain occurs.

#### Resources

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#### Investigation

*30 min*

Next, I let students get to work in small groups on Questions 1 and 2. If some groups finish early, they can move on to Question 3. I emphasize with students that for Question 2, I would like them to represent their work with a table, graph, recursive function and explicit function.

Things I watch for as students work:

- Many of my students will have trouble scaling the graph. I encourage them to think about how many dots they can represent and might give them a hint to scale on the y-axis by 1000. I let them know it's ok if they cannot show the full week of the spread of the emails.

- If there's time, I might show students how to use a web based graphing app like desmos or plot.ly to make their graphs. This makes working with this scale a little bit easier, but still requires students to think about the large numbers they need to represent.

- If students have trouble writing the recursive function, rather than showing them how to do it, I refer them back to our previous lessons. They should have a recursive function written from the second lesson of this unit where the pattern was multiplying by 2. They should be able to see how to tweak this function to fit this new situation. I might also ask them to write the recursive pattern in words first, rather than going straight to function notation so they have a sense of what it means for a pattern to be recursive in the first place.

- Many of my students will struggle with the explicit equation. I'll ask them to do the best they can working across the table and thinking about how they can write the Out values as relatives of the previous values (8 x 10 x 10). If they cannot get the explicit equation, I let them know that we will work on it as a group in the Discussion section of the lesson.

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#### Discussion

*20 min*

At the start of today's discussion I have students share out their table, graph, recursive, and explicit equations. I follow the Breaking the Chain PowerPoint to make sure students understand the sequence and that I hit all of the main points. A lot of my focus in today's discussion is helping students to recognize how they can determine if a sequence is geometric by looking at its graph, table, recursive and explicit function.

We will spend a fair amount of time during the discussion working through the explicit formula. I anticipate that most of my students will have trouble writing it at this stage in the unit. We work with a table with two additional columns (included in PowerPoint) so students can see that 80 can be written as 8 x 10 and then 800 can be written as 8 x 10 x 10 and so on. Next, I try to elicit from students that the number of 10s they need to multiply by is one less than the number of days that have gone by. I also refer students to a previous geometric sequence to see how we wrote that one to see if they can find some clues to writing the current explicit equation.

#### Resources

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#### Closing

*5 min*

Today's closing activity asks students to reflect on the similarities and differences between arithmetic and geometric sequences. The reflection questions are included in the Break the Chain PowerPoint. Students respond in writing to the following prompt:

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Don't Break the Chain is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

http://www.mathematicsvisionproject.org/secondary-1-mathematics.html

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- UNIT 1: Introduction to Algebra: Focus on Problem Solving
- UNIT 2: Multiple Representations: Situations, Tables, Graphs, and Equations
- UNIT 3: Systems of Equations and Inequalities
- UNIT 4: Quadratics!
- UNIT 5: Data and Statistics
- UNIT 6: Arithmetic & Geometric Sequences
- UNIT 7: Functions

- LESSON 1: Exploring an Arithmetic Sequence - Dots
- LESSON 2: Exploring a Geometric Sequence - Different Dots
- LESSON 3: Push Ups - Practice with an Arithmetic Sequence
- LESSON 4: Chain Emails - Practice with a Geometric Sequence
- LESSON 5: Decreasing Arithmetic Sequences
- LESSON 6: Decreasing Geometric Sequences
- LESSON 7: Arithmetic Means
- LESSON 8: Geometric Means
- LESSON 9: Arithmetic & Geometric Sequences: Putting It All Together
- LESSON 10: Sequences Assessment