Perimeter and Area Models ID: 9837 By Holly Thompson Time required 30 minutes Activity Overview Students will look at data for the perimeter and area changes of a rectangle and triangle as their dimensions change. They will find that perimeter and area are modeled by linear and quadratic functions, respectively. Concepts Perimeter and area Linear and quadratic modeling Teacher Preparation This activity can be used in an Algebra 2 class or in a Precalculus class at any point in the class when linear and/or quadratic models are discussed. It could also be used in an Honors Algebra I class where students have done some work with area and perimeter. Students should already be familiar with the general shape of quadratic functions. The screenshots on pages 2 4 demonstrate expected student results. Refer to the screenshots on page 5 for a preview of the student TI-Nspire document (.tns file). To download the student.tns file and student worksheet, go to education.ti.com/exchange and enter 9837 in the quick search box. Classroom Management This activity is intended to be initially teacher-led, followed by students working individually, with a partner, or in small groups. You may use the following pages to present the material to the class and encourage discussion. Students will follow along using their handhelds. Be sure to cover all the material necessary for students total comprehension. The student worksheet Alg2Act18_PerimeterAreaModel_worksheet_EN is intended to guide students through the main ideas of the activity.. It also serves as a place for students to record their answers. Alternatively, you may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion. TI-Nspire Applications Calculator, Graphs & Geometry, Lists & Spreadsheet, Notes 2008 Texas Instruments Incorporated Page 1
Problem 1 A rectangle On page 1.2, students will drag point P to make the rectangle larger, changing its length and width, and therefore changing its perimeter and area as well. After they increase the size of the rectangle, they should view the data on page 1.3, displaying the values of the length, width, perimeter, and area of the rectangle that was collected as they moved point P. As a class, enter the formulas for perimeter and area of a rectangle and point out the formulas in the gray formula cells for Columns C and D (underneath the column title). They can then easily see how the perimeter and area are calculated. The scatter plot on page 1.4 shows the length vs. the perimeter of the rectangle. Discuss as a class which parent function would represent this data. You may wish to have students estimate a linear function for the data. Alternatively, have them graph the equation y = x and then transform the graph to fit the data. If you would like students to use this method, have them use the Text tool (MENU > Actions > Text) to display the equation y=x on the screen. Dragging the equation to the axes will display its graph, which can then be grabbed and dragged accordingly. 2008 Texas Instruments Incorporated Page 2
Yet another option is to have students perform a regression. Back on page 1.3, students can select MENU > Statistics > Stat Calculations > Linear Regression (mx+b). They should select length for the X List, and perimeter for the Y List. The regression equation can be saved as f1, and Column E (type e[ ]) can be designated as the 1st Result Column. Advancing again to page 1.4, students should press / + G to show the Entry Line, then to access f1(x), and then to graph the regression line. You may wish to have students clean up the regression equation by rounding. The regression equation is approximately y = 3x. A scatter plot showing length vs. area is shown on page 1.5. Again, discuss as a class which parent function would represent this data. As before, you can have students estimate a quadratic function for the data, transform the graph of y = x 2 to fit the data, or perform a quadratic regression. To perform a quadratic regression, return to page 1.3 and select MENU > Statistics > Stat Calculations > Quadratic Regression. Choose length for the X List and area for the Y List. Save the regression equation as f2, and set the first results column as g[ ]. On page 1.5, have students graph the regression equation as before and clean up the equation by rounding. The regression equation is approximately y = 0.5x 2. 2008 Texas Instruments Incorporated Page 3
Problem 2 A triangle For Problem 2, students will complete the same investigation using measurements from a triangle rather than a rectangle. This will enable them to see that perimeter is linear and area is quadratic, even if the shape of the figure is changed. If students are confident with how they completed the last problem, they could work on this problem in groups. Note: To ensure that each column will have the same number of entries (necessary for creating the scatter plot), the values for perimeter and area are collected from the diagram rather than calculated in the spreadsheet. On page 2.2, have students perform a linear regression to find the equation that models the scatter plot (base, perimeter). They should find regression equation is approximately y = 2.6x + 3. On page 2.2, students should perform a quadratic regression to find that an approximate model for the data (base, area) is y = 0.25x 2 + x. After students complete the above tasks, discuss as a class how the models can be useful and why perimeter is linear and area is quadratic. (Perimeter is one-dimensional and area is two-dimensional.) 2008 Texas Instruments Incorporated Page 4
Perimeter and Area Models ID: 9837 (Student)TI-Nspire File: Alg2Act18_PerimeterAreaModel_EN.tns 2008 Texas Instruments Incorporated Page 5