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Teaching Math: A Video Library, 9-12

Calculator-Based Labs

Content Standards
Statistics, Algebra Functions, Trigonometry
Process Standards
Reasoning, Problem Solving, Communication, Connections
School: North Country Union High School, 1,200 students
Location: Newport, Vermont
Teacher: Jean McKenny
Years Teaching: 18
Students in Classroom: 25
Grade: 11 – 12
Calculator–Based Labs
Video Overview
Student groups learn how they will conduct two experiments using the Calculator–Based Laboratory (CBL)^TM—a temperature probe and a motion detector attached to graphing calculators. In each experiment, the calculator graphs collected data and the students write an equation for the graph. After the first experiment, the class reconvenes and groups discuss their results. Then students conduct the second experiment.

The Class Challenge
Students conduct the following experiments:

Use a temperature probe attached to a graphing calculator to collect data on temperature change in a cup of hot water.
Use a motion detector attached to a graphing calculator to collect data on the gravitational acceleration of a falling ball.

Course and Curriculum
Prior to this elective Probability and Statistics course, students took Algebra 1 and Geometry, and most students took Discrete Mathematics. While taking this course, which focuses on data analysis, some students also took Algebra 2, Precalculus, or Calculus. In this lesson, students applied prior knowledge about functions, such as curve fitting and exponential and quadratic functions, and data analysis. Although students had previously used the graphing calculator for data analysis, this was the first time they used real data and the Calculator-Based Laboratory (CBL)^TM. While the CBL^TM can also create an equation, Ms. McKenny asked her students to develop their own equations based on the statistical plots and the data from their calculators. The following day, groups presented their work and discussed their data and conclusions.
Ms. McKenny assessed students by examining their equations and group work and by probing student thinking through questioning. She created these groups based on the amount of equipment available. For other lessons, she organizes groups randomly, unless there is a broad range in student ability and prior mathematics experience, in which case she creates heterogeneous groups.

A Pre–Viewing Exploration for Teacher Workshops
Solving the Coffee Cup Problem
Objective: To collect data that represents a natural phenomenon and then find a regression equation that best describes the data.
Materials: Graphing calculator with an attachable probe system or graphing calculator, a stop watch, and a scientific thermometer; boiling hot water, and a cup at room temperature for each small group.
Activity
Solve this problem in small groups.
If you are using a probe system, pour the hot water into the cups and set the probes to record the temperature every 10 seconds over a period of 6 minutes. The system records data points (time, temperature) and graphs the function.
If you are using a scientific thermometer, pour the hot water into the cups and record the temperature every 10 seconds over 6 minutes. Then, enter the ordered pairs (time, temperature) into the calculator and graph the function.
Explore the time dependence of the temperature predicted by Newton's Law of Cooling.

Newton's Law of Cooling
The difference between the temperature of an object and the temperature of its surroundings decreases or increases exponentially with time.
T(t)–T(s)=[T(o)–T(s)]e^–kt
where T(t)=the temperature of the object at time t
T(s)=the temperature of the surroundings
T(o)=initial temperature of the object
k=proportionality constant

What is the initial temperature?
What is the final temperature?
What is the importance of the initial temperature difference (T(o)–T(s)) ?
Under what conditions will the water temperature decrease or increase?

Exploring Content

Describe the function. Does it increase? decrease? Is the increase or decrease constant? Is it a curve? a line?
What will happen to the curve when the water temperature reaches room temperature?
Use the regression equation function on the calculator to find the line that fits best. Graph the function you think best matches the data on the graph.
What equation best fits your original graph?
How does the initial temperature difference between the hot water and the room temperature change the water temperature?

Exploring Teaching Issues

Do you think the use of probe system technology in mathematics classes is important? Why or why not?

Exploration Answers
Activity

Answers will vary depending on the individual experiment. The boiling point of water is 212 degrees Fahrenheit (100 Celsius).
Answers will vary depending on the individual experiment, but the temperature should be between the boiling point and the room temperature.
The larger the initial difference between T(o) and T(s), the faster the initial temperature falls.
The temperature decreases if the initial temperature is higher than room temperature and increases if the initial temperature is lower than room temperature.

Exploring Content

The function decreases at an uneven rate. At first, the temperature drops quickly, and then it drops more slowly. The graph of the function is a curve.
The curve will flatten into a horizontal line.
Answers will vary depending upon what portion of the graph you selected to fit.
If the cup is cold, the temperature decreases faster; if the cup is hot, the temperature decreases slower.

Topics for Discussion
How does this technology allow students to learn mathematical modeling?

How would you encourage students to think and talk about connections between the two experiments?
Describe how the graph patterns reflect the two modeling situations. What patterns in scatterplots suggest a linear function model? an exponential function model? a power function model?
How could students conduct and discuss the same modeling situations if they did not know these functions?
How did Ms. McKenny use the graphs to assess students' understanding and the accuracy of their equations?
What questions would you ask to assess the students' understanding of using functions to model data?
How have you incorporated technology into some of your lessons?
The graphing calculator can collect and display data and find precise equations to fit the data. Which tasks would you have the calculator perform in one of your lessons? Which tasks would you have your students perform? If students use the calculator to perform all three tasks, what instructional sacrifices are made and what opportunities are created?
How can you ensure that technology is used as a tool for learning mathematics rather than just for fun?
Discuss other modeling activities that include curve fitting.

What would you plan for the next day to continue the students' exploration of curve fitting?
How can you find and test algebraic models that fit patterns in data?
How might this lesson be taught without the data–gathering technology shown?

How would the teacher–student and the student–student interactions be different without the technology?
What are some ways students could gather their own scientific data?
What other materials, such as a ruler or string, could be used to create a phenomenon that represents a linear model?
Is student access to technology important? Why or why not?