# Mathematics: Establishing Trigonometric Identities

Title: Establishing Trigonometric Identities
Discipline(s) or Field(s): Mathematics
Authors: Kavita Bhatia, Jinbo Lu, LaVerne Harrison, University of Wisconsin – Marshfield/Wood County
Submission Date: January 1, 2011

Executive Summary

Our goal was to help students to understand the process of establishing identities and to carry out the process effectively.

In establishing an identity, students have to apply a series of logical steps, while always keeping their objective in mind. There is no template problem, no recipe to follow. The steps for establishing an identity are unique to that identity. Our focus was on teaching students what constitutes the proof of an identity and how to present the proof in logical steps.

In designing the lesson we tried to link students’ prior knowledge about algebra, in particular algebraic equations and rational equations, with trigonometric functions. To emphasize this we began the lecture by first establishing an algebraic identity, followed by the replacement of the present variable with a trigonometric function to obtain a trigonometric identity. Other examples of identities were then proved on the board with each step being provided by a different student and the underlying algebra underscored by the instructor. Finally the students were given a worksheet with different types of identities to work on as the instructor observed and answered questions.

There were six identities on the worksheet. Most students attempted 4 ~ 5 and on average completed the proofs of about 4 identities correctly, using varied approaches. Furthermore, the students displayed enthusiasm and confidence in carrying out their work. This carried over to their performance in the final exam: more students attempted and successfully established the identities compared to previous years, when a majority of the students simply skipped these problems.

The Lesson:

The Study:

# Mathematics: Credit Cards as an Application of Exponential Equations

Title: The Mathematics of Credit Cards (as an application of exponential equations)
Discipline(s) or Field(s): Mathematics
Authors: Kirthi Premadasa, Paul Martin, Clare Hemenway, University of Wisconsin-Marathon County
Submission Date: December 14, 2008

Goals of the lesson.

a) Students will learn how to make basic interest calculations, when calculating the monthly payments and the monthly interest charges on credit card balances.

b) Students will learn what “amortization” means and will learn to fill in an amortization table.

c) Students will use logarithms to solve the exponential equations encountered in the calculations of the payment periods for different payment methods.

d) Students will develop an insight to the advantages and the disadvantages of the different payment methods.

Final Report:

# Mathematics: Arc Length of a Curve as an Application of Integration

Topic: Arc length of a curve as an Application of Integration
Discipline(s) or Field(s): Mathematics
Authors: Kavita Bhatia, University of Wisconsin-Marshfield/Wood County & Kirthi Premadasa, UW-Marathon County
Submission Date: March 30, 2010

Lesson Goals:

a) Students will learn how to make a manual calculation of a Riemann sum for the arc length of a givensample curve using a few subdivisions.

b) Students will use the knowledge obtained through the Riemann Sum ‐> Integrations models that they have seen before, to “discover” an integration formula for the arc length of any continuous curve.

c) Students will use the formula that they “discovered”, together with the integration techniques, taught in the course to evaluate the actual arc length of the curve.

d) Students will understand the underlying theme behind all the Riemann sums that they have encountered.

# Mathematics: Intro to Partial Derivatives in a Business Calculus Course

Title: Introduction to Partial Derivatives in a Business Calculus Course
Discipline(s) or Field(s): Mathematics
Authors: Erick Hofacker, Ioana Ghenciu, Don Leake, Alexandru Tupan, University of Wisconsin-River Falls
Submission Date: March 2, 2009

Executive Summary

This introductory lesson to partial derivatives to a class of business and social science majors focuses on conceptual understanding in several different ways. It opens with a couple of questions on car loans aimed at assessing the experience and intuition of the class concerning changes in multivariable functions. Then with the help of a computer applet borrowed from MIT the lesson introduces the concept of partial derivatives through its geometrical meaning. TI-89 calculators provide a way for students to easily compute partial derivatives algebraically for a simple polynomial function. Through these two technological tools students explore the relationship between the 3-D graph of a two-variable function and its partials. The 75 minute lesson ends with a couple of partial derivative applications from the fields of business and economics.

The lesson is based on a laboratory/guided discovery approach. Technology is used as a tool for exploration. The learning activities were ordered to achieve understanding first geometrically, then algebraically, and finally through application. Lower-level computational skills were placed in support of higher-level conceptual understanding. Some later questions were directed toward giving students the opportunity to discover connections with previously-learned material. The application portion of the lesson is designed to help students see connections between the mathematics curriculum and other disciplines.

This lesson study reinforced the notion that discovery learning, supported by technology that helps students visualize and compute, is very helpful in the introduction of a conceptually difficult topic such as partial derivatives. The lesson also highlighted the importance of constant and immediate assessment in the classroom. The gulf between an instructor’s perception of student understanding and what is actually the case can be tremendously broad, especially toward the end of a long semester. A third revelation is that usually simpler is better. It is preferable to focus on understanding a few concepts well in the classroom. Finally, the importance of personal contact, student-to-student or student-to-teacher, cannot be overemphasized. While working in a computer lab, the information is right there in the face of the student on the computer screen. In a lengthy classroom or lecture hall, it is far too easy for the weaker student to disengage. In addition every learning environment needs to provide a way for instructors to get within every student’s “sphere of learning.”. Students that are not easily accessed in the classroom, whether in the back of a long classroom or against the wall in a computer lab are in danger of being lost.

Below are links to the lesson materials used to teach it.

Lab Activity

Clicker Questions

Below are links related to the study of the lesson.

Exam Questions

Example of Student Work from Lab Activity

Sample Homework Questions

Rubric for scoring Exam Questions

Example of Student work from Exam Questions

# Mathematics: Simple and Compound Interest in Investment Contexts

Title: Simple and Compound Interest in Investment Contexts
Discipline(s) or Field(s): Mathematics
Authors: Kathryn Ernie, Laurel Langford, and Erick Hofacker, University of Wisconsin-River Falls
Submission Date: March 2, 2009

Executive Summary

Our main goal for this lesson is for students to understand the difference between simple interest and compound annual interest. Prerequisite to understanding these concepts is the understanding of the mathematics concepts of rate (interest rate) and percents. A related goal is the recognition of the additive nature of simple interest providing a linear rate of growth (additive sequence) and the multiplicative nature of compound interest providing an exponential rate of growth (geometric sequence). Included in our goals is the ability to represent these relationships in numeric, tabular, and graphical forms.

Part of the rationale for this project defined in the fall of 2007 was the recent home foreclosures problem in the U.S. (indicating that individuals did not understand the mathematics perhaps of home loan agreements). Unfortunately, the impact of the foreclosure crises was felt even more strongly a year later during our lesson study with the failure of numerous financial institutions and major losses in the stock market.

The recent national interest in financial literacy as it relates to citizens understanding rates, percents, investment, interest earned, and growth relate directly to this lesson study. This first lesson on the mathematics of financial literacy is on simple interest earned in contrast to compound interest earned annually.

The investment context first introduced was the additive application of simple interest. Students represented an investment in numeric and tabular form and extended the data by working in small groups using a calculator. This data was also analyzed using its graphical form. The compound interest earned (exponential rate of growth) was studied in the same fashion by small groups of students. Students made longer term predictions as to which form of investment would be best over time. Excel was used to investigate further the impact of longer term investments in contrast to each other. These activities were at an appropriate level and resulted in students analyzing differences between the two types of interest earned both numerically and graphically. By the end of the lesson, students readily recognized the type of interest earned directly from only a graphical representation.

Below are links to the lesson plan materials used to teach it.

Handouts for Activities
All 3 handouts are linked here.

Excel Examples

Excel examples from the lesson with graphs.

Bonds and Prices
Link to current source of bonds.

CDs and Pricing
Link to current source of CDs.

Below are links related to the study of the lesson.

Field Notes
Observation notes on the lesson.

Student Responses
Sample student work and summary

Assignment and Solutions

# Mathematics: Calculating the Distance Between a Point and a Line

TItle: Calculating the Distance Between a Point and a Line – by Hand and Using TI-89/TI-Voyage 200 Calculator Technology
Discipline(s) or Field(s): Mathematics
Authors: Theresa Adsit, James Meyer, Gary Wardall, University of Wiscnsin-Green Bay
Submission Date: December 12, 2007

Executive Summary

The lesson topic is the distance between a point and a line using an algebraic approach and a calculator based approach to problem solving.

Learning Goals: The immediate academic learning goals of this lesson were to develop students’ understanding of the derivation of the point to line distance formula and to develop the ability to apply the point to line distance formula to solve problems. The ongoing academic learning goals of this lesson were to develop the ability to use the calculator to build structures to solve problems involving systems of equations, to develop a greater understanding of the similarities between calculators and other forms of technology, and to further develop strategies for solving multi-step problems.

Instructional Design: The lesson was divided into five steps. The first step was instructor led and involved the determination of the shortest distance between a specific point and a specific line using the techniques of algebra and paper and pencil. The second step mimicked the first but rather than using paper and pencil the instructor and students used either a TI-89 or TI-Voyage 200 calculator. During the third step of the lesson the instructor and students then developed the point to line distance formula for any point and any line using the TI-89 or TI-Voyage 200 calculator. The fourth step of the lesson involved the students verifying the formula by using the developed formula along with the point and the line from parts one and two to determine if the developed formula did indeed yield the same results as their previous calculations. Finally, in step five the students worked collaboratively and then independently on an assignment related to the lesson.

Major Findings about Student Learning: The students with the assistance of the instructor were able to build the appropriate structures using either a TI-89 or TI-Voyage 200 calculator to solve a problem involving systems of equations and to derive a formula involving systems of equations. The students were collaboratively and individually able to apply the developed formula to other problems in the assignment. Students questioned each other and the instructor more often during the collaborative work period than during the instructor led portion of the lesson. Some students did have an underlying misunderstanding of the benefits of a formula.

# Mathematics: Rolle’s Theorem and the Mean Value Theorem

Title: Rolle’s Theorem and the Mean Value Theorem
Discipline(s) or Field(s): Mathematics, Statistics, Computer Science
Authors: Joy Becker, Petre Ghenciu, Matt Horak, Helen Schroeder, University of Wisconsin-Stout
Submission Date: April 1, 2008

Executive Summary

The topic of the lesson is Rolle`s Theorem and the Mean Value Theorem.

Learning Goals.

1. Students will understand the meaning of Rolle`s Theorem and the Mean Value Theorem, including why each hypothesis is necessary.
2. Students will complete problems and applications using Rolle`s Theorem and the Mean Value Theorem.
3. Students will appreciate the discovery process of developing mathematics and have a better understanding of the construction and proof of mathematical theorems.

Lesson Design. The lesson was designed in order to emphasize the discovery process and the role of proof in mathematics. The first major piece of the lesson is an activity that asks students, in several steps, to draw graphs of functions satisfy various hypotheses. The last graph that students were asked to draw is impossible to draw, because any graph satisfying all of the required conditions would violate Rolle`s Theorem. Rolle`s Theorem is introduced in this way. A second activity involving graphs related to the Mean Value Theorem is used to introduce or study the Mean Value Theorem. These graphing exercises are intended to help students discover for themselves the two theorems and help them to appreciate the discovery process in mathematics.

The second major part of the lesson is to work problems involving the theorems to better understand how the theorems are used and apply in practice. The variety of problems is intended to emphasize different aspects of the theorems, including why the hypotheses are necessary and how to apply the theorems to modeling applications and more abstract settings.

The final part of the lesson is to prove the Mean Value Theorem assuming Rolle`s Theorem. This portion of the lesson is expected to be difficult for students, so ample time should be allotted for question and discussion.

Major Findings. During the first round of the lesson, we learned that students seem to catch on quickly that the second graphing exercise is almost identical to the first and that therefore the last graph is impossible to draw. This seemed to cause a significant reduction in their engagement with the lesson. However, when this activity was changed for the second round, the decrease in performance on certain quiz and homework problems suggests that the repetition may actually have served its purpose of emphasizing the hypotheses present in the two theorems.

Below are links to the lesson outline and the materials required for the lesson.  Included with the materials are all of the in-class examples used as well as quiz, homework, and exam problems.

Lesson Outline

# Statistics: Interpretation of Confidence Intervals

Title: Interpretation of Confidence Levels
Discipline(s) or Field(s): Mathematics, Statistics
Authors: Jeff Baggett, Brooke Fridley, David Reineke, University of Wisconsin – La Crosse
Submission Date: June 9, 2006

Report Excerpts:

Our group decided to address the topic of confidence intervals and the interpretation and use of confidence intervals, in particular focusing on the interpretation of confidence intervals (i.e. what intervals say and what they do not say). We chose this topic after having noticed students having difficulty with the interpretation and use of the confidence interval and not so much the computation of the confidence interval.

Based on the results of the pre and post quiz, the misconception regarding the interpretation of a confidence interval by applying it to individuals rather than means seems to have been adequately addressed by the lesson.  However, students still do not have a clear understanding of the interpretation of a confidence interval for the mean as it relates to the subtle difference between probability and confidence.

Full Report Below:

# Mathematics: Related Rates in Calculus

Title: The “Perfect” Related Rates Lesson: A lesson study in calculus
Discipline(s) or Field(s): Mathematics
Authors: Joy Becker, Christopher Bendel, Petre Ghenciu, Laura Schmidt, Radi Teleb, University of Wisconsin – Stout
Submission Date: February 28, 2007

Executive Summary

The lesson topic is related rates in Calculus I or Calculus & Analytic Geometry I. Related rates problems tend to be difficult for students since they are generally word problems that require setting up equations before solving. This topic is important as one common example of an application of derivatives.

Learning Goals:  There are two immediate goals for this lesson: 1) Students will understand that related rates problems are applications of implicit differentiation and 2) Students will be able to translate, compile, model, and solve a related rates problem and interpret the meaning of the answer. A longer-term goal is that students’ problem-solving and critical thinking skills will be improved.

Lesson Design: The lesson is designed to span two class days. On the first day, students start by working through an introductory worksheet, which extends what they have previously learned to introduce the concept of related rates. Since word problems are often a stumbling block for students, the lesson includes an overview of problem-solving strategies, somewhat specific to related rates, although they can be generalized. A warm-up worksheet reviews necessary material and gives students a chance to set up equations, an essential part of the problem-solving process. On the second day of the lesson, the instructor works through two examples with the class to model the problem-solving process, and students are given a chance to solve problems on their own or in small groups. The examples and worksheet problems were chosen to show students a variety of different types of related rates problems, starting with more straightforward problems and ending with more difficult problems.

Major Findings about Student Learning:  In terms of our specific lesson goals, by looking at the data we collected, the first two were achieved by most students: 1) Students will understand that related rates problems are applications of implicit differentiation and 2) Students will be able to translate, compile, model, and solve a related rates problem and interpret the meaning of the answer. Since the third goal, “Students’ problem-solving and critical thinking skills will be improved,” is more general, there will need to be a series of lesson studies in order for it to be assessed properly. Is this the “perfect” lesson? The answer is probably no. However, the planned activities did visibly increase student engagement and responsiveness. The lesson developed will help instructors to assemble an excellent lesson, depending on the classroom settings and other institutional factors.

Below are links related to the lesson plan and the materials used to teach it.

Below are links related to the study of the lesson.

# Mathematics: Exploring Difficulties with Combining Rational Expressions

Title: Lesson Study in Exploring Difficulties with Combining Rational Expressions
Discipline(s) or Field(s): Mathematics
Authors: Laura Schmidt and Diane Christie, University of Wisconsin-Stout
Submission Date: February 28, 2007

Executive Summary

Learning Goals. The overall learning goal is to have students be able to add and subtract rational expressions. Students will first combine expressions with common denominators, then find a common denominator to combine expressions with unlike denominators. Long-term goals not directly assessed by the lesson are to ease anxiety when dealing with fractions and to have students realize the connection between adding/subtracting rational numbers and adding/subtracting rational expressions.

Lesson Design. The lesson reviewed addition and subtraction of fractions, demonstrated addition and subtraction of simple rational expressions, and worked up to difficult examples. The lesson began with three examples of rational numbers, one with common denominators and two with un-like denominators, followed by rational expressions with common denominators. Examples of rational expressions with un-like denominators started out simple and increased in difficulty level. The number of expressions to be added increased along with the difficulty in the factorization of the denominators. The examples were chosen so that the answers could be rewritten in reduced forms at the end to remind students to check that final step in their answers. Due to the anxiety that this lesson has caused in the past, hard examples were presented by the end so that students could be exposed to more difficult problems.

Major findings about student learning. The findings showed that even though students were successful at the beginning problems in the homework, they were intimidated by the “difficult look” of the later homework problems and simply did not attempt them. This was evident in the analysis of the homework where the amount of incomplete problems drastically increased at a certain problem when the difficulty level was higher. In the revised lesson, more difficult examples were used, and it was stressed that the steps remain the same even though it looked much harder than previous examples. Several days later when the students had to use the lesson to solve equations involving rational expressions their confidence level was greater and the majority of students got the correct answers.

Below are links to the lesson plan and the materials used to teach it.

Rational Expressions: Addition and Subtraction Lesson Plan

Teacher’s Lesson Examples

# Mathematics: Rate of Change in Context

Topic: Rate of Change in Context: a Lesson Study in Calculus at the University of Wisconsin at River Falls
Discipline(s) or Field(s): Mathematics
Authors: Laurel Langford, Alexandru Tupan, Ioana Ghenciu, Don Leake, University of Wisconsin – River Falls
Submission Date: May 5, 2007

Executive Summary

Our goal is for students to better understand rate of change in context, including the skills of moving flexibly between algebraic and graphical representations and analyzing behaviors given information about the rate of change. In this lesson, students practice these skills in concrete examples using average rate of change, as a preparation for doing similar work with derivatives. These activities are at an appropriate level, with some review, and some critical thinking work, and they prompt valuable discussion among students about rates of change.

A video clip showing part of the lesson as first taught by a member of the Calculus: Rate of Change lesson study group.

Below are links to materials used to teach the lesson.

Lesson Description and Materials
The lesson as taught by the lesson study team

Revised Lesson
The lesson as revised in response to our observations

Sample student work
This file contains scanned portions of student work from the lesson

# Mathematics: Statistical Inference of Means

Title: It depends on what “mean” means: a lesson study on sampling distributions
Discipline(s) or Field(s): Statistics
Authors: Abdulaziz Elfessi, Heather Hulett, Dan Nordman, University of Wisconsin – La Crosse
Submission Date: August 29, 2006

Executive Summary

In this activity, we hope to help students differentiate and explain three statistical terms at the heart of statistical inference: the mean of a population, the mean of a sample of observations, and the mean of the sample means.

Past experience indicates that term “mean” can be very confusing for students in an Elementary Statistics class, especially when the same word choice may be applied in all three situations above, with different meanings in each case. Understanding the differences, as well as the connection, between the three types of means above is important for the most basic hypothesis tests in statistics: testing if the population mean equals a certain value by looking at just one random sample. The idea that data from a small sample can be used to estimate the mean of an entire population, which cannot be obtained directly, is critical to statistical applications in many, many fields.

The specific learning goals of the lesson are as follows:

• Students will practice applying statistical techniques to data collected from samples.
• Students will see and explain sample variability and how sample size decreases the variability of sample means.
• Students will see graphically that the “typical/central/mean/expected” value of a sample mean is the same as the population mean.

In this lesson students will take random samples of different sizes and calculate their averages. They will then put their averages on Post-It notes and place them in the correct spot on the chalkboard to make histograms that will represent the sampling distributions. By comparing their sample means, the mean of the histogram (the mean of all the samples), and the population mean (which will be revealed at the right moment), they will hopefully get a fuller appreciation of the three different uses of the word “mean”.

The activity was successful in several ways. Students enjoyed the short exercise in drawing random samples and were surprised by some of the sample means obtained. As the histograms were formed, they saw clearly how the variability decreased as sample size increased. Finally, they got to see how most sample means gave close approximations to the true population mean.

# Computer Science: Discovering Inheritance through a Popular Video Game

Topic: Discovering Inheritance through a Popular Video Game
Department(s) or Field(s): Computer Science, Mathematics, Statistics
Authors: Terry Mason, Diane Christie, Radi Teleb, Bruce Johnson, University of Wisconsin – Stout
Submission Date: Spring 2007

Executive Summary – The lesson topic is inheritance in Computer Science 1 (CS1) courses. Inheritance is a powerful tool which is generally not fully understood by beginning students in computer science. They may understand the mechanics of making inheritance work, but do not always comprehend the utility and power of it. A deeper understanding of the topic is a learning goal that all teachers strive for in their students. This topic has a broad application as the introduction to programming is a course that is taught by many instructors in colleges and high schools throughout the world.

Learning Goals – The goal of this lesson is to illustrate the power and utility of inheritance as a tool in computer science with the graphics and engagement experienced by students playing video games. The lesson is designed using a familiar Mario game implemented in Java. The students were engaged in the project by first playing the game to identify the sprite objects. This set up a class discussion on how these objects are organized into an inheritance hierarchy through shared characteristics and functionality. The students complete the project by using inheritance to complete the functionality of the game.

Lesson Evaluations – The results of surveys and quizzes compare the results of one section of students that completed the older inheritance laboratory with two sections of students that completed the new video game based laboratory. Student engagement in the new laboratory ranked close to exciting versus a ranking between marginally interesting and interesting for the older lesson. Student surveys show that students believe that the new lesson was exciting and it increased their understanding of inheritance hierarchies, the power of inheritance, and the usefulness of the lab. Student grades on a quiz administered four days after the laboratories show that student scored slightly higher after completing the new lesson compared to students completing the older lesson.

Observations and Exit Interviews – The lessons were observed by members of the lesson study team. Students showed a high level of engagement in the game and identifying the objects for missing functionality. They expressed a sense of accomplishment in extending the functionality of the game. In addition they showed a sense of accomplishment. Two different groups shouted “Yes!” when their new code provided the expected functionality of the game. In addition, students were engaged enough in the lesson to spend extra time to further investigate the code.

Links to materials used to teach the lesson and data generated by the study.

Handout of Lesson for Students
The handout provided to students to complete the lesson.

Evaluation of the Inheritance Lesson based on Mario
Evaluation of the results for the quiz and survey administered to students participating in the new laboratory compared to students completing the older laboratory.

Quiz Administered 4 Days after Lesson
The quiz given to all students after completing their lessons.

Survey of Student Engagement
Survey given to students at the completion of laboratory lesson for both the new and old lesson.