Purpose
The Animation Tutor CD is a curriculum development project that was created through a grant from the National Science Foundation. It contains eight modules that encourage and support visual thinking through manipulation and animation to help people reason about mathematics. The mathematical content in these modules is typically taught in high school but, because students often struggle with this material, the modules can later be used in college or at home as a review of content that many students never learned.
Each module provides instruction on a particular situation, such as population growth, rather than on a particular mathematical procedure such as exponential functions. One advantage is that users can more easily compare and contrast different mathematical content such as linear and exponential models of population growth. A second advantage is that focusing on situations emphasizes the application of mathematics. A third advantage is that the modules can also be used in other courses that include the topics. A course on biology could use the Population Growth module, a course on finance could use the Personal Finance module, a course on chemistry could use the Chemical Kinetics module, and a course on physics could use the Catch Up module on acceleration.
Many of the problems initially require estimates to improve estimation skills and provide motivation for learning analytic techniques for calculating the answers. Animation of students’ answers (such as the time to fill a tank or complete a round trip) provides visual feedback on the accuracy of their estimates and calculations. Some problems also include graphs so students can attempt to infer what kind of functional relations link the variables.
The following provides a brief overview of the eight modules. A more extensive summary of their content follows the overview.
Overview of Animation Tutor Modules
- Dimensional Thinking
Demonstrates through object manipulation that proportional reasoning applies to perimeter but does not apply to areas and volumes. Generalizes ideas to irregular shapes. - Chemical Kinetics
Presents two concepts that are fundamental to calculus (area under a curve and a tangent to a curve) within a scientific context. Introduces key ideas of chemical kinetics to provide the context. - Mixtures
Demonstrates through object manipulation how the balances and interest rates on two credit cards determine payments of annual interest. Generalizes ideas to mixtures of ores. - Personal Finance
Emphasizes the distinction between simple and compound interest as examples of linear and exponential growth. Application to both earning and owing interest. - Population Growth
Requires estimating and adjusting parameters to improve goodness-of-fit of linear and exponential models of population growth. Investigates extrapolation to predict future growth. - Average Speed
Uses average speed as an example of a weighted average problem. Multiple representations illustrate the constraint that average speed can not exceed twice the slower speed. - Catch Up
Includes problems in which one person catches another by traveling at a constant speed or a constant acceleration. Applies ideas to determining a safe driving distance. - Task Completion
Illustrates how the same equation applies across a variety of problems that differ in story content. Shows different methods for solving the same problem. - Leaky Tanks
Demonstrates how a simple (no leak) solution can be adapted to solve more complex problems in which there is a bottom leak or a side leak. The side leak solution combines the no-leak and bottom-leak solutions.
Dimensional Thinking
Brian Greer, Bob Hoffman, and Stephen Reed
The Dimensional Thinking module begins with the question of whether a 12-inch pizza for $6.99 or a 20-inch pizza for $12.99 is the better buy. Dividing price by diameter results in the incorrect answer that the smaller pizza is the better buy. The purpose of this and other problems is to correct the overuse of proportional reasoning to situations in which it does not apply. The module demonstrates that doubling the side of a square or the diameter of the circle doubles its perimeter but does not double its area. Students manipulate geometric objects to learn that doubling a side of a square or the diameter of a circle increases area by a factor of four. Doubling the side of a cube or the diameter of a sphere increases volume by a factor of eight. Applications to standard and irregular forms show both correct (perimeter) and incorrect (area, volume) uses of proportional reasoning.
Chemical Kinetics
Kathy Tyner, Stephen Reed, and Susan Phares
The Chemical Kinetics module uses the context of chemical reactions to discuss two concepts that are important in calculus – area under a curve and the tangent to a curve. Students estimate the proportion of a curve that exceeds a critical value of kinetic energy to estimate the proportion of atoms that will form molecules. They make these predictions for kinetic energy distributions at low- and high temperatures before viewing simulations. The concepts of average and instantaneous rates of change are illustrated for a curve showing the half-life of aspirin. The instruction shows that it is possible to find an instantaneous slope (represented by a tangent) that has the same value as the average slope (represented by a secant) for different intervals of the curve.
Mixtures
Stephen Reed and Bob Hoffman
The Mixtures module begins with calculating annual interest on two credit cards by using arithmetic. Annual interest can be constructed by physically adding the interest owed on one card to the interest owed on the other card. The same solution method works for solving how much iron can be extracted from a mixture of low-grade and high-grade iron ore. The algebra problems require calculating the amount of money owed on a credit card or the amount of iron ore. The height of stacks of money or bars of iron ore can be manipulated to show how changing the value of a variable influences other values in the equation.
Personal Finance
Bob Hoffman and Stephen Reed
The Personal Finance module compares investing money at simple and compound rates of interest to illustrate the difference between linear and exponential growth. Students calculate how their money grows over a five-year period for simple- and compound rates of interest. They also use algebra to determine how much money they would earn by partitioning their savings between a certificate of deposit and a savings account. The mathematics of investing are also applied to borrowing money, such as calculating monthly payments on a $200,000 mortgage at 6.5% interest over 15 years and the same loan at 7% interest over 30 years. Graphs of the remaining principle for loans and the amount earned for investments depict how the amount of money changes over time.
Population Growth
Stephen Reed, Bob Hoffman, and Diane Short
The Population Growth module compares linear and exponential models of population growth by computing a (least squares) goodness-of-fit index between actual and predicted data. Students adjust the y-intercept and slope to try to find the best-fitting linear model of population growth in the United States during the 19th century. They then compare this model to an exponential model of the same data. Students next make predictions about the accuracy of extrapolating the best-fitting linear and exponential models to predict population increases in the 20th century. The distinction between linear and exponential growth is illustrated by asking when a tank of bacteria will be ¼ filled if the bacteria double every minute beginning at 11:00 am and fill the tank at noon. An animation and simultaneous graphing of the increase in population provides feedback about the explosiveness of exponential growth. The instruction concludes by applying polynomial functions to world population growth.
Average Speed
Stephen Reed, Jeff Sale, and Susan Phares
The Average Speed module provides instruction about weighted averages by showing that average speed is a weighted average of the amount of time spent traveling at the different speeds. Students begin by estimating the average speed of a round trip for a speed of 60 mph on the initial trip and 30 mph on the return trip. Animated feedback reveals that 45 mph is incorrect because a car traveling at this speed in both directions returns sooner than the other car. Students continue to receive animated feedback as they practice estimating average speeds for different initial and return speeds. The counterintuitive idea that the average speed can never exceed twice the slower speed is explored graphically as the asymptote of a function, conceptually as total distance divided by total time, and algebraically as a derivation based on the weighted average formula.
Catch Up
Stephen Reed and Bob Hoffman
The Catch Up module contains problems in which one person catches another person by traveling either at a constant speed or a constant rate of acceleration. The problems require a variety of mathematical skills including estimating answers, selecting an appropriate equation or mathematical relation, solving single and simultaneous equations, matching graphs to situations, providing conceptual explanations, and planning solutions. The bridge problem shows how a challenging mathematical problem can be easily solved by combining a spatial simulation with proportional reasoning. Determining safe driving distances provides an important context for spatial reasoning about deceleration.
Task Completion
Stephen Reed, Susan Phares, and Jeff Sale
The Task Completion module provides instruction on solving algebra word problems by showing how the same equation applies to different situations. The situations require finding the time it takes for either two pipes to fill a tank, two workers to complete a task, or two people to meet by traveling toward each other. The key to solving these problems is to convert completion times into rates that specify how much of the task is completed during each unit of time. Students also learn how to solve a more complex version of these problems in which two pipes begin filling the tank at different times. In addition to showing solutions based on a single algebraic equation, the instruction shows solutions that decompose a problem into its parts.
Leaky Tanks
Stephen Reed, Susan Phares, and Jeff Sale
The Leaky Tanks module continues with instruction on more complex variations of the tank-filling problems by introducing problems in which there is a loss of liquid from the tank as it fills. Students are asked and receive feedback on how they would modify the simpler equation to solve these more complex problems. The solution to the bottom-leak problem requires reducing the rate of fill by subtracting the rate of loss. Calculating how long it takes to fill a tank with a side leak can be accomplished by breaking the problem into two parts. The fill time below the leak is found by using the equation for a no-leak problem and the fill time above the leak is found by using the equation for a bottom-leak problem.
Instructions for Using the Modules
We designed the modules to be independent of each other to offer maximum flexibility in their use. There are none-the-less some principles that should help in sequencing. The first four modules typically do not require the construction of equations for solving algebra word problems. These problems are more concerned with applying and understanding general formulas. The last four modules do require the construction of equations for problems that are more typical algebra word problems. In addition, it can be helpful to think of the modules as four pairs in which the first module in each pair helps prepare students for the second module. Using proportions in Dimensional Thinking provides preparation for using proportions in Chemical Kinetics, comparing linear and exponential functions in Personal Finance provides preparation for changing the parameters of these functions in Population Growth, thinking about speed in Average Speed provides preparation for thinking about acceleration in Catch Up, and solving tank problems in Task Completion provides preparation for solving leaky tank problems in Leaky Tanks.
The modules can be used by individual students, by small groups of students, or by the instructor for a class demonstration. The problems vary in difficulty and the more difficult problems should be more accessible to small groups of students than to individual students. An instructor may also want to engage the entire class in the most difficult problems by projecting the animation at the front of the classroom. The bridge problem and the safe-driving-distance problems in the Catch Up module are examples of challenging problems that might benefit from a class presentation.
All modules have the same navigational aids in the upper right corner of each screen that allow users to return to the previous screen, advance to the next screen, return to the table of contents, or quit the program. Users can advance to any topic in the instruction by clicking on that topic in the table of contents. If you wish to return and replay prior animations, you should first return to the table of contents and click on a topic. Animations are typically reset at the beginning of each topic and may not work properly if you back up by using the back arrows.
We appreciate suggestions for improving the modules. Discovered problems or general suggestions for improvement should be sent to Steve Reed at
