Logarithms in Science & Engineering

Grade Level:

/ 11th /

Subject:

Algebra 2 AA

Teacher:

Jeff Dauwey

Prepared By:

/ Safa Herfat

Analyze Learners

Overview & Purpose

This lesson plan will review logarithms and exponential equations by applying the mathematical conceptsto real world applications in science and engineering. The goal of the lesson is to apply algebra 2 concepts to topics learned in the chemistry classroom.The importance of logarithms and exponential functions will be tied to local companies like P&G and GE Aircraft Engines. The majority of the lesson will be implemented in the Algebra 2 classroom but certain portions involving chemicals and their lab notebooks will be implemented in the chemistry classroom. Tomorrow's lesson will include a brief introduction discussing how a sound level meter works, then getting into brief applications in engineering, and the students will work on an M&M activity that shows exponential decay. They will gather data, plot their data, and find a linear regression equation for their data using a graphing calculator. The lesson will then be continued in the chemistry class, where they will be given mystery pKa's, which they will have to convert to pH using a logarithmic equation, test a few labeled solutions with litmus papers to find their pH's and then match the "mystery pKa" with the corresponding solution. /

Education Standards Addressed

MATH
Patterns, Functions, and Algebra Standard

Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

Data and Analysis and Probability Standard

Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators.
Design and perform a statistical experiment, simulation or study; collect and interpret data; and use descriptive statistics to communicate and support predictions and conclusions.
Connect statistical techniques to applications in workplace and consumer situations.

Mathematical Processes Standard

Apply mathematical knowledge and skills routinely in other content areas and practical situations.
Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.

SCIENCE
Scientific Inquiry

Participate in and apply the processes of scientific investigation to create models and to design, conduct, evaluate and communicate the results of these investigations.
Make appropriate choices when designing and participating in scientific investigations by using cognitive and manipulative skills when collecting data and formulating conclusions from the data.

Select Goals and Objectives /

Teacher Guide

Student Guide

Objectives

(Specify skills/information that will be learned.) /

At the end of the 11th grade unit, students will be able to understand various real world applications for logarithms.
At the end of the 11th grade unit, will be able to perform computations involving logarithms and exponents.
At the end of the 11th grade unit, will be able to solve for variables in real world equations involving exponents and logarithms.

STEM Objectives:

Exponential population in the Cincinnati area
Logarithms are used by researchers every day in academia and industry for modeling and simulation purposes
GE aircraft engines
Modeling & analysis
Materials research
P&G
pH very important in body care
shampoos, toothpaste
UC biomechanics and biomaterials research which impacts people in need of orthopaedic reconstruction
Elderly
High school, college, and pro athletes

Materials Needed

Paper
Pencil
Graph paper
Sound level meter
Graphing Calculator
M&Ms
Litmus Paper
Vinegar
Water
Baking Soda Solution
Ammonia or NaOH

Select Instructional Strategies –

Information

(Give and/or demonstrate necessary information) / This lesson includes a direct activity and an inquiry based activity that students will have to figure out for themselves.
Utilize Technology / A sound level meter will be used in the introduction.
Graphing calculators can be used to find the best fit linear regression equation. /

Other Resources

(e.g. Web, books, etc.)

Require Learner Participation

/ Catch: How do sound level meters work?

Smack ruler on desk to measure sound level
Record sound level while talking
Measure headphones from a student’s ipod and talk about hearing loss
Talk about other loud events and their relative sound level ranges
The standard sound level meter is more correctly called an exponentially averaging sound level meter
Input pressure converted to voltage by one circuit and then passed through a logarithmic circuit to give a readout in decibels (dB). This is 20 times the base 10 logarithm of the ratio of a given sound pressure to the reference sound pressure.

The reference pressure is set by International agreement to be 20 micropascals for airborne sound. It follows that the decibel is in a sense not a unit, it is simply a dimensionless ratio—in this case the ratio of two pressures.

Informing of objectives:
Students will be informed of the objectives listed above. The main objective of this lesson is apply exponential equations and logarithms to chemistry and engineering.
Prerequisite Learning:
Students should have completed the chapter in the Algebra 2 text on logarithms and exponential functions.
Presenting Material:
Several applications will be briefly reviewed.

Exponential growth and decay
Population growth
Medical Dosages
Investment
Half-life of radioisotopes
Data Analysis - modeling using exponential functions
Logarithms
Sound level equation
pH equation
using natural logs to solve for exponential variables
give example: tissue biomechanics

Require Learner Participation (Cont.)

Activity

(Describe the independent activity to reinforce this lesson) / Activity #1: M&M Exponential Decay
(in Algebra 2 classroom)
Have students form groups of 3 or 4. Give each team a zip lock bag with 50 M&M’s inside, the procedure handout and the handout on finding a mathematical model for an exponential equation. Students will gather data and be challenged to come up with a formula that models their data results using the TI-83 and the 2nd handout.
Providing Feedback:
Groups will present and compare their findings to the other groups. This activity can lead into a discussion about real-life situations that can be modeled exponentially, e.g. population growth.
Activity #2: Mystery pKA
(in Chemistry classroom)
Display pKa to pH conversion equation on the board. Place the following solutions in 5 different beakers:

Concentrated HCl
Vinegar
Water
Baking Soda Solution
Ammonia or NaOH

Students are to form groups of 3 or 4. Give each group 5 strips of litmus paper. Students are supposed to figure out how to match each mystery pKa to the proper solution.
Providing Feedback:
Prior to starting, the students will write a protocol and have it checked by the instructor. The instructor will monitor each group’s progress and provide feedback. / Student Guide
Activity #1:

Students will follow procedure given on handout
Students will record their data and findings on a given worksheet
Directions on how to find a mathematical model for an exponential equation also given on another handout.

Activity #2:

Being an inquiry based activity, students will be given materials and a logarithmic equation and are expected to figure out the procedure on their own.

Evaluate (Assessment)

(Steps to check for student understanding) / The same assessment will be given at the beginning and end of the lesson. The assessment should not take longer than 15 minutes to complete. The file name for the assessment is Lesson 3 Assessment.doc. If students are still having trouble grasping the concepts following the lesson, these concepts will be reviewed by the instructor. /

Additional Notes