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# A Wrap on Logarithmic dB's and An Extension to Logarithmic Properties

Lesson 10 of 21

## Objective: Students will share out their findings from the logarithmic study of sound, as well as discover, analyze, and prove logarithmic properties.

I’m going to shoot straight with you here… this is not the most smooth transition! Going from “share out time / presentations” to the next activity is far from seamless. In a perfect world, the presentations in the prior section can be completed at the end of the previous day. However, since my classes are only 45 minutes long, I was unable to perform the dB’s investigation AND allow time to share out – so I was forced to start this lesson with this piece.

If you try something that serves as a smooth transition between presentations and homework collection time, please let me know! It is not the end of the world, and the students in my class adapt well, but good teachers always strive for continuity!

As I collect the homework, I take a quick snap shot or two of a 2-3 students papers… in particular, I focus on #13 and #14. After all homework has been collected, I display these three pictures on my projector screen and begin a discussion with the students about what they notice. I work hard to focus my conversations on building a desire to see if “the sum of the logs is ALWAYS equal to the log of the products” – and how we can ensure this is the case in ALL situations. I emphatically work to draw out of the students that we could test specific numbers FOREVER and still not be entirely certain… but we can use mathematical proof to make our claim!

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#### "Cut strips" Proof Activity

*10 min*

*Prior to the activity:*

*1) **Print, copy, and cut the resource attached in this lesson*

*2) **Scramble the strips on each page and place them in an individual envelope*

*3) **Write on the board the statement the class is setting out to prove based on the discussion. This is also found as the bottom strip on the page*

Begin by handing each student an envelope. Leave the students answers’ up from #13 on the document camera or projector screen. I usually start be telling the students that inside this envelope is EVERYTHING they need to prove the statement we have made, as well as extend the findings of homework #13 to all cases. (What a powerful envelope, huh?!)

I have the students work to organize the strips in the proper order and provide mathematical reasons for their claims. This serves as a wonderful spiral review of the N-RN standards covered in the opening section of this unit.

What does the teacher do during this time? Rotate the room and help individual students. Upper level students will be particularly troubled by what to write in the “reasons” portion. Although they will understand the concept, they will need support in writing their thoughts in mathematical notation. On the other hand, other students may choose to write their reasons more informal in nature. I allow this to happen because what is important to me is the thought process, not the formality of the responses. (MP1: Make sense of problems and preserve in solving them.)

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#### Homework Worksheet

*10 min*

Circulate the homework worksheet attached to this lesson. The worksheet asks the students to duplicate the activity we just completed – minus having the crutch of arranging the strips. The students have developed an important skill though completing the previous task, and they have built upon their knowledge of exponentiation and logarithms. Mathematical proof is an intimidating thing for many students, but I have found that easing them in provides more of them access to the experience.

Would I take time to do mathematical proof every day?

*No. *

Do I believe that supplementing this unit with a small lesson on mathematical proof enhances the learning experience for my students and helps them to make connections?

*Absolutely!*

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- UNIT 3: Sequences and Series
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- UNIT 5: Review Lessons
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- UNIT 10: Probability
- UNIT 11: Imagine This! Imaginary and Complex Numbers
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- 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
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- LESSON 17: FInal Discussion - "Demystifying e" PLUS Skills Practice with Logarithms
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- LESSON 19: End of Unit Differentiated Problem Set
- LESSON 20: Differentiated Problem Share Out - Unit Review
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