Calculator-Based Lessons Help Examine Real-World Data
Students may determine equations based on graphs and regression models.
By Donna Iadipaolo
Probes and sensors connected to graphing calculators allow students to learn mathematical modeling because these devices gather data for a real-world experiment. A teacher may encourage students to think and talk about connections between a cooling experiment and a ball-dropping experiment, for instance, by discussing how equations can be determined from the data in each case. Many of the National Council of Teachers of Mathematics (NCTM) Standards may also be covered by such endeavors.
Students may recognize the cooling experiment as exponential because the graph curves and decreases quickly before gradually leveling off. They may recognize that the ball-dropping experiment is parabolic because it contains a maximum and curves outward to a minimum. In contrast, scatterplots that suggest a linear function form a line and depict a proportional relationship between the independent and dependent variables. Exponential functions are often confused with power functions. While an exponential function’s general form is y=a^x, a power function’s general form is y=x^a. If a student is not familiar with which graph to use for a particular function ,a correct regression model could be chosen by figuring out which form the data fits best. The NCTM standards require students to “understand patterns, relationships, and functions.” One of the ways they can meet this standard is by learning to identify which graphs to use to perform particular functions.
The issues that I would raise to assess the students’ understanding of using functions to model the data are:
- Describe their process for formulating the equation.
- Describe the general form for a linear, exponential, quadratic, and power function.
- Determine the equation for each graph generated from the data.
Personally, I have incorporated technology into some of my online lessons this past year by having students use the Apple spreadsheet program “Numbers” to create a scatterplot and use regression models. Furthermore, I would have students collect and display the data and find precise equations to fit the data. The instructional sacrifices made in having students use the calculator to generate the equation initially does not allow students to perform calculations based on certain data points to determine equations. By having students explore the regression models before allowing them to use a graphing calculator, you can help ensure that students know which function to use for a particular problem. Use of technology and application of various regression models also fits into the “Representation” NCTM standards. Specifically, in this work, students “create and use representations to organize, record, and communicate mathematical ideas.”
Applying Real-World Data
One can ensure that technology is used as a tool for learning mathematics, rather just for fun, by having students apply real-world data. Other modeling activities could include curve fitting. For instance, live experiments collecting data could also ensure that technology is used as a tool for learning mathematics, such as collecting data on students walking at a constant rate versus accelerating. Students could also research and input available data, such as looking at population growth for a certain region over time and then modeling the best equation for that data. Certain sports data could also relate such as studying data between the number of minutes a sports player plays and the amount of points that s/he scores. Applying mathematical aspects to the real world further connects to the NCTM standards because students “use representation to model and interpret physical, social, and mathematical phenomena.”
More Graphing Calculator Lessons
Students are shown how to use a graphing calculator to solve various algebraic problems. There is a focus on polynomials.
Students use graphing calculators to explore and record data and communicate their results and analysis.
Students use a motion sensor and graphing calculator to collect data on a free-falling object. Students examine velocity and distance in relation to time and determine how these variables relate to acceleration.
Students use electronic probes and graphing calculators to collect and process data. They then analyze what a bouncing ball would do in various situations.