Math Labs. Activity 1: Rectangles and Rectangular Prisms Using Coordinates. Procedure

Transcription

1 Math Labs Activity 1: Rectangles and Rectangular Prisms Using Coordinates Problem Statement Use the Cartesian coordinate system to draw rectangle ABCD. Use an x-y-z coordinate system to draw a rectangular prism. Verify their properties using slope and distance. Equipment Graph paper Isometric paper Straight edge Graphing calculator Procedure 1 Draw the x- and y-axes on a sheet of graph paper. Use each unit as one. 2 Use your graphing calculator to generate random integers to use as coordinates for two of the vertices of a rectangle. Your calculator requires that you specify the lower and upper boundaries. You can generate more than one integer by specifying the number of integers you desire. l You need two integers, one for the x-coordinate and one for the y-coordinate. Use 4 and 4 as the boundaries. Press M. Arrow over to PRB. Select 5 for randint(. Enter 4, 4, 2. Press E. Plot the point on your graph paper. Label the point A. l Repeat to generate a second vertex. Use 5 and 5 as boundaries. Plot the point and label it B. 3 Draw AB. Let AB be one side of rectangle ABCD. Calculate the slope of AB. 4 Based on the properties of a rectangle and the slopes of parallel and perpendiular lines, find each slope. CD BC AD 5 Use the slopes for BC and AD and a rise/run movement on your graph to find possible coordinates for points C and D. Plot your points and complete the drawing of rectangle ABCD. Math Labs 249

2 6 Draw the x-, y-, and z-axes on isometric paper. Use each unit as one. y z x 7 Use your graphing calculator to generate three random integers to use as dimensions of your rectangular prism. Use 3 and 12 as the boundaries and 3 for the number of integers. 8 Use the random integer function to generate an ordered triple that will be one of the vertices of your rectangular prism. Use 0 and 5 as the boundaries. Plot the point on the x-y-z coordinate system and label it S. Two faces of the prism are parallel to the x-y plane, two are parallel to the x-z plane, and two are parallel to the y-z plane. The coordinates of points in a plane parallel to the y-z plane have equal x elements, but the y and z elements differ. 9 What element of ordered triples in a plane parallel to the x-y plane are equal? 10 What element of ordered triples in a plane parallel to the x-z plane are equal? The face STUV is parallel to the y-z plane. Using point S from Step 8 and the dimensions from Step 7 you can write the ordered triples of the vertices of the face. Suppose point S is (4, 1, 5) and the dimensions are (3, 8, 4). The other vertices of the face STUV are: (4, 1, 9) Add the z of the dimensions to the z element. (4, 9, 5) Add the y of the dimensions to the y element. (4, 9, 9) Add the y to the y element and add the z to the z element. 11 Use your point S and your dimensions to name vertices STUV of your figure. Plot the ordered triples and label them S, T, U, and V. What is the dimension of this face? 250 Chapter 4 Linear Equations

3 12 The face that is opposite of STUV is also parallel to the y-z plane. Use the x of the dimensions to determine the x element for the vertices of this face. 13 Write the ordered triples for the face in Step 12. Plot the points and label them W, X, Y, and Z. 14 What are the dimensions of faces that are parallel to the x-y plane. Name the faces. 15 What are the dimensions of faces that are parallel to the x-z plane. Name the faces. Activity 2: Measuring in Inches and Centimeters The relationship that converts measurements in inches (i) to centimeters (C) is C 2.54i. Problem Statement You will measure several lengths in inches and centimeters and plot corresponding pairs of measurements on a graph. You will interpret the graph to verify the value of the coefficient Equipment Calculator Tape measure marked in inches and centimeters Sheet of paper Procedure 1 Measure and record the width of a sheet of paper in inches and 2 Measure and record the length of a sheet of paper in inches and 3 Measure and record the width of the classroom door in inches and 4 Measure and record the height of the classroom door in inches and 5 Measure and record the width of the teacher s desk in inches and 6 Measure and record the length of the teacher s desk in inches and Math Labs 251

4 7 Graph your data. Graph the measurement in centimeters on the vertical axis. Graph the measurement in inches on the horizontal axis. Caution: Study the range of the data. Then choose the scales for the x- and y-axes. Make certain all the data will fit on the graph. 8 Draw an unbroken line that best connects the six points on your graph. Is the graph a straight line? If a point is not on the straight line, double-check your measurements and graph for that point. 9 Choose any two points on the graphed line such as A and B in the drawing shown here. These points need not include the points you plotted to draw the graph. Based on the values of these points, subtract the smaller centimeter value from the larger centimeter value. The result is the difference in centimeter values for the two points. Label this on the graph as cm. A Measurement in centimeters cm B slope of line = cm in. in. Measurement in inches 10 For the same two points, and in the same order, find the difference in inch values. Label this on the graph as in. 11 Divide the cm value by the in. value. This is the slope of the graphed line and is the value of m in the slope-intercept form of a linear equation y mx b. Compare your calculated slope to the value 2.54 in the equation 1 (cm) (in.). 12 For your graphed line, what is the y-intercept? Does this value make sense? 252 Chapter 4 Linear Equations

5 Activity 3: The Equation of Lines Problem Statement Examine the equation of lines to discover the slope-intercept form of a linear equation. Use this equation to generalize characteristics of slanted, horizontal, and vertical lines, as well as the ordered pairs of intercepts. Equipment TI-Nspire technology Procedure 1 At the home screen of the TI-Nspire handheld open a new document. Select Add Graphs & Geometry. To view the grid on the coordinate graph, select Menu. View. Show Grid. 2 Hide the entry along the bottom of the screen as you will use this space later. Select Menu. View. Hide Entry Line. 3 To place the first point, select Menu. Points & Lines. Point. Use the arrow buttons to move the pencil to place a point in Quadrant III. Press enter to actually place the point on the grid. Name the ordered pair of the point you graphed. 4 Place a second point on the y-axis above the origin. Because you want this point to be on the y-axis, choose Point On from the Points & Lines menu. Move your cursor to place a point on the y-axis above the origin. Be sure that the y-axis is highlighted when you place this point. Name the ordered pair of the point you graphed. 5 Select Menu. Points & Lines. Line to draw the line that contains both points you graphed. Use the arrow buttons to place your cursor over one point. When it begins to blink, press to select the point. Move your cursor to the second point and make sure the point is blinking before you press to select that point. Math Labs 253

6 6 Use Measurement. Slope to calculate the slope of the line. Place the cursor on the line so that it is blinking. Click the line to measure its slope and move the resulting number to the lower right corner of the screen. 7 Press to put away the measurement tool. You can label the number as slope. Double click on the measurement to get a cursor. Scroll to the beginning of the number and type slope=. Press to get out of the text box. 8 Use Menu. Action. Coordinates and Equations to name the ordered pair of the y-intercept. Click on the blinking point on the y-axis to get the coordinates of this point. Move the resulting ordered pair to the lower middle part of the screen. 9 Use Menu. Action. Coordinates and Equations to calculate the equation of the line graphed. Click on the blinking line and then move the resulting equation to the lower right corner of the screen. Press to put away the measurement tool. 10 Compare the information you placed on the lower part of the screen. a. Where in the equation is the slope given? b. Where in the equation is the y-intercept given? 254 Chapter 4 Linear Equations

7 11 Click and hold the cursor over the point in Quadrant III. Once you have a closed hand, you can move this point around the screen. Use the left arrow to move it to the left. Notice how the slope and equation updates based on the new point. Move that same point to the right and notice the changes. Move it up and move it down. Each time, notice the changes to the slope and the equation. Does this confirm your answer to Step 10 part a? 12 Move the cursor until you have a negative slope. Describe the line. 13 Move the cursor until you have of slope equal to zero or very close to zero. Describe the line. 14 Move the cursor until the second point is also on the y-axis. What is the slope? Describe the line. 15 Move the point off the y-axis and press to release the point. Click and hold the cursor over the point in on the y-axis. Once you have a closed hand, you can move this point. Because it was constructed as a point on the y-axis, the point will only move along this line. Notice the changes to the ordered pair and the equation. Does this confirm your answer to Step 10 part b? 16 When is the constant term in the equation positive? When is the constant term in the equation negative? 17 Use what you discovered about the slope and the y-intercept of a line to write an equation of a line that has the slope and y-intercept characteristics given. a. slope 5 5; y-intercept (0, 22) b. slope 5 0.4; y-intercept (0, 4) c. slope 5 m; y-intercept (0, b) Math Labs 255