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# FInal Discussion - "Demystifying e" PLUS Skills Practice with Logarithms

Lesson 17 of 21

## Objective: This lesson provides students with the opportunity to analyze how they process reading difficult mathematical literature. The lesson also culminates with a checkpoint review of solving equations by using logarithms.

This is the first time that I have ever tried something like this! In the previous lesson, I sent the students home with a notes packet that continued our background lesson on e and the natural logarithm function. The critical piece (and justification for doing this) was that Day #1 of the “Demystifying e” lesson went extremely well! The students really bought into the in-depth nature of the mathematics, while the analogies of bacterial growth and money served to make the high level content easily accessible to all students. However, considering that I sent the students home with a lengthy notes page to look over, I was doubtful that it went as well when they were turned lose on their own.

Why did I do this?

I believe that it is important to expose students to a variety of learning tasks. In high school mathematics, rarely do we challenge the students to read and dissect mathematical literature. As I entered college, this was a major challenge to for me because I had never been exposed to it before. Even those individuals relatively fluent in mathematical skill can struggle through a detailed mathematical narrative or manuscript. Do I want to set the students up for failure? No! Do I want them to come back the following day with a desire to sort the details out and develop a list of tips and tricks for reading a mathematics resource? Yes!

How does this entry event look?

For this activity, I believe that it is extremely important to group the students in like ability groups for discussion. I have found that low students will become even more turned off to the mathematics if they are placed in an environment where they do not feel that they can openly express their lack of understanding with their peers. Instead, placing these students with other students who also struggle will help them realize that they are not alone… they are also OUTSTANDING at working together to find strategies that break down the paper into manageable chunks.

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To aid in the transition of our discussion, I hand the students a strip of paper featuring one of the following cartoons:

After allowing the students to read the cartoon, I ask them to find someone on the opposite side of the classroom who has a different cartoon than their own. When they find that person, I instruct them to explain their cartoon frame by frame to the other person. When they are done explaining, the other student is to re-explain the cartoon back to them AND include any mathematical references in the context of the storyline. (Look for video of this dialogue between students in the future!)

After this activity, I bring the students back together as a class and ask them if anyone had any difficulty understanding the storyline – the students always unanimously say something along the lines of “No, I wish we could do cartoons every day!”

Since there were no issues, I asked the students to make a list of things that were essential for them to understand the comic strip. During this time, I have each student make his or her own list, and do not allow any talking for about 3-4 minutes. I have found this promotes more creative and well thought-out answers from a wide variety of students. Next, I ask the students to share-out key points on their list so that I can create a class summary on the board.

Sample Student Responses: (your students may come up with additions to this list, but these are the most basic)

- We had to read the strips frame-by-frame
- Background knowledge was important to fully understand the problem
- A 5
^{th}grade student would not likely be able to understand the significance of the strip, not because it is above his reading level, but because of the level of the mathematics - The comic strip “concept” could be understood, even if the specific Trig/Geometry concepts had not been learned by the reader
- The setting for the comic strip was important to its understanding
- TV show/Football Game

- A 5

After making this list, I ask the students to draw comparisons between these strategies and the strategies that were helpful in breaking down the “Demystifying e” reading assignment. The students usually have very little difficult making the connections! I have listed sample responses from my class below:

- We had to read the strips frame-by-frame
- The paper we had to read was divided into sections that built upon themselves.
- We had to read one section to get to the “punch line” of another
- Background knowledge was important to fully understand the problem
- A 5
^{th}grade student would not likely be able to understand the significance of the strip, not because it is above his reading level, but because of the level of the mathematics - We had to know a little about exponential growth and logarithms.
- An Algebra I student would not likely be able to understand our reading assignment because they have not get had the depth of knowledge or experience with exponential functions.
- The comic strip “concept” could be understood, even if the specific Trig/Geometry concepts had not been learned by the reader
- In our homework assignment there were references to “limits” – a topic that we have not covered (and will not cover until our year #4 course). Although fully understanding the concept of a limit would have provided a more comprehensive look at e, it was not ESSENTIAL for us to understand what was going on in the explanation.
- The setting for the comic strip was important to its understanding
- TV show/Football Game
- Understanding that our reading was a “capstone” piece to our “Demystifying e” class lesson

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- UNIT 1: Culture Building Unit - Welcome to the New Year!
- UNIT 2: Exponential and Logarithmic Functions
- UNIT 3: Sequences and Series
- UNIT 4: Statistics: Something for Everyone
- UNIT 5: Review Lessons
- UNIT 6: Writing Prompts in Mathematics
- UNIT 7: Trig Tidbits
- UNIT 8: Functions, Problem Solving, and 21st Century Skills
- UNIT 9: Polynomials and Problem Solving
- UNIT 10: Probability
- UNIT 11: Imagine This! Imaginary and Complex Numbers
- UNIT 12: Let's Explore Radicals!

- LESSON 1: Introduction to Exponential Functions - Rational Exponents
- LESSON 2: Real Number Exponents
- LESSON 3: A Conceptual Introduction to Exponential Functions: Collaboration, Questioning, and Financial Advice!
- LESSON 4: Discovering Solutions to Exponential Equations
- LESSON 5: An "Exponentially" Rewarding Battle!
- LESSON 6: Productive Struggle - A Pre-Spring Break Challenge of Exponential Functions
- LESSON 7: Compositions, Inverses & Role Play: Extending Concepts to Exponential Functions
- LESSON 8: Exponential Functions and Logarithms - Connecting the Dots
- LESSON 9: Exponentials and Logarithms: LOUD and CLEAR!
- LESSON 10: A Wrap on Logarithmic dB's and An Extension to Logarithmic Properties
- LESSON 11: Classroom Journey: Exponential and Logarithmic Properties
- LESSON 12: Laws of Logarithms and Real Applications
- LESSON 13: Calculator Boot-Camp and Solving Exponential Equations
- LESSON 14: Logs, Loans, and Life Lessons!
- LESSON 15: "Demystifying e" Day #1
- LESSON 16: Logs, Loans, and Life Lessons Collaboration Day!
- LESSON 17: FInal Discussion - "Demystifying e" PLUS Skills Practice with Logarithms
- LESSON 18: Elephant Tracks!
- LESSON 19: End of Unit Differentiated Problem Set
- LESSON 20: Differentiated Problem Share Out - Unit Review
- LESSON 21: Unit Review