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.

Mathematics: Establishing Trigonometric Identities – The Study (Final Report)

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:

  1. Students will learn how to make basic interest calculations, when calculating the monthly payments and the monthly interest charges on credit card balances.
  2. Students will learn what “amortization” means and will learn to fill in an amortization table.
  3. Students will use logarithms to solve the exponential equations encountered in the calculations of the payment periods for different payment methods.
  4. Students will develop an insight to the advantages and the disadvantages of the different payment methods.

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:

  1. 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.
  2. 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.
  3. 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.
  4. Students will understand the underlying theme behind all the Riemann sums that they have encountered.

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.

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.

Basketball Example
Warm-Up to Related Rates Worksheet
In-Class Related Rates Examples
Rales Rates Worksheet

Below are links related to the study of the lesson.

Observation Forrm
Student Survey
Related Rates Homework Spring 2006
Related Rates Homework Fall 2006
Student Data