Summary
In this lesson, students will discover how to evaluate logarithms and use that knowledge to write equations in logarithmic and exponential forms. Students will learn how to solve exponential equations with and without using logarithms and then apply their knowledge to solve real-world problems involving exponential equations. This is the first lesson in the "All About That Base" lesson duo.
Essential Question(s)
How do we use logarithms to solve exponential equations?
Snapshot
Engage
Students observe a candy bar as it is cut in half repeatedly, modeling exponential decay. Students see that the remaining amount of candy bar is approaching zero, the asymptote.
Explore
Students discover how to evaluate logarithms.
Explain
Students complete guided notes with the class and formalize their understanding of solving exponential equations.
Extend
Students apply what they have learned to solve real-world problems and complete a Gallery Walk. Students then summarize the process of solving exponential equations.
Evaluate
Students demonstrate how to solve exponential equations using logarithms.
Instructional Formats
The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.
Materials
Lesson Slides (attached)
What Power? handout (attached; one per pair; printed front only)
Guided Notes handout (attached; one per student; printed front only)
Guided Notes (Teacher Guide and Model Notes) (attached; for teacher use)
Scenario Cards document (attached; one per class; printed front only)
Scenario Cards (Sample Responses) (attached; for teacher use)
Scientific calculators
Candy bar (one per class)
Knife
Cutting board
Pencils
Paper
Tape or other adhesive to hang student work on the wall
Exit Ticket handout (optional; attached; one half-sheet per student; printed front only)
Engage
15 Minute(s)
Introduce the lesson using the attached Lesson Slides. Display slide 3 to share the lesson’s essential question with students. Go to slide 4 to share the lesson’s learning objectives. Review each of these with students to the extent you feel necessary.
Go to slide 5 and ask students to share what they think will happen when a candy bar is cut in half repeatedly. After students have shared their ideas, place the candy bar on a cutting board and use a knife to cut it in half. Set one half aside and cut the remaining half in half. Repeat and encourage the class to continue sharing predictions. As you continue to cut the remaining piece of candy bar in half again and again, ask students if their ideas have changed while watching this process.
Stop when you have reached the point where the remaining piece of candy bar is too small to cut in half safely. Ask students to imagine you have the ability to continue cutting the candy bar piece in half for days, years, and even centuries into the future. Again, ask students to share their predictions about what will happen.
After sufficient class discussion, connect this activity to exponential decay. Take the concept of "someday" not having any candy bar remaining and connect it to the horizontal asymptote of an exponential function: As time moves toward infinity, the remaining amount of candy bar approaches zero—but it never equals zero.
Explore
10 Minute(s)
Ask students to find partners or assign student pairs. Display slide 6 and pass out the attached What Power? handout to each pair of students.
Have students use questions 1 and 2, which have been completed for them, to help them answer the remaining questions. Inform students that the WhatPower function is "just some function" and that they should treat this handout like a puzzle or a riddle.
Once students have completed the handout, transition through slides 7–8 so they can check their work.
Explain
20 Minute(s)
Display slide 9 and explain to students that the WhatPower function is actually the logarithm function. Have students write and evaluate the two expressions on slide 9. Students can write these expressions on the back of the What Power? handout or on a piece of scratch paper. After a few minutes, ask for volunteers to share how they arrived at their answers.
Go to slide 10 and pass out the attached Guided Notes handout to each student.
Before working through the front side of the Guided Notes, have students flip to the back of the handout and copy the table shown on the slide. Have students work with their partners to complete the table. Again, ask for volunteers to share their thinking. Be sure that the class understands the equivalent forms of logarithmic and exponential equations before moving on.
Go to slide 11 and have students return to the front side of their Guided Notes. Complete the handout as a class.
Have students add their completed Guided Notes to their math notebooks if that is a classroom norm.
Extend
40 Minute(s)
Have students form groups of 3–4 or assign groups. Introduce students to the Gallery Walk strategy and give them an overview of this activity:
Work with your group to solve the problem on your given scenario card.
Create a mini-poster to display your work.
Split your group into presenters and analysts.
Presenters: You stay with the mini-poster and answer questions about your scenario.
Analysts: You go to each mini-poster and analyze your peers’ work.
Switch roles and repeat.
Summarize the process of solving exponential equations.
Display slide 12 and pass out one scenario card to each group. Ask students to work together to solve the given problem before creating a mini-poster to show their work.
Go to slide 13 and review the expectations for the mini-poster.
After groups complete their mini-posters, go to slide 14. Have groups divide in half and decide who will be presenters and who will be analysts. Groups of three can have one presenter and two analysts or vice versa. Assure students it does not matter who presents or analyzes first, since they are to switch roles later in the activity.
Guide presenters to the "presentation gallery" and have them hang their mini-posters. After reminding students of their roles’ responsibilities, have them begin the Gallery Walk.
Once students have had enough time to visit 3–4 mini-posters, go to slide 15. Ask students to pause and switch roles. Again remind students of their roles’ responsibilities and have them begin the Gallery Walk.
After giving this group of analysts about the same amount of time to visit mini-posters, go to slide 16 and ask all students to go back and sit with their groups. Have students get out a piece of notebook paper and write a short paragraph explaining the general process of solving exponential equations.
Evaluate
5 Minute(s)
Display slide 17 and use the Exit Ticket strategy to assess what students have learned individually. Using the Exit Ticket handout or the piece of notebook paper with the Gallery Walk paragraph, have students write and solve the two exponential equations on the slide.
After you have collected students’ responses, use their feedback to determine if remediation is needed or if they are ready for the next lesson: "All About That Base, Part 2."
Resources
Eureka Math. (2014). Algebra II module 3, topic B, lesson 8: The "WhatPower" function [Handout]. NYS Common Core Mathematics Curriculum. https://www.engageny.org/resource/algebra-ii-module-3-topic-b-lesson-8/file/108756
K20 Center. (n.d.). Gallery Walk / Carousel. Strategies. https://learn.k20center.ou.edu/strategy/118
K20 Center. (n.d.). Bell Ringers and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081
