Exploring the Slopes of Parallel and Perpendicular Lines

Transcription

1 Lesson Summary: Students will be asked to discover the relationship between the slopes of both parallel and perpendicular lines. The students will use the knowledge discovered and Cabri to find missing coordinates of a parallelogram and the missing coordinate of a rectangle. Keywords: Parallel, Perpendicular, Slope, Cartesian coordinate system NCTM Standard: 1. Approximate and interpret rates of change from graphical and numerical data (p.305). 2. Analyze relationships using the Cartesian Coordinate System (p.313). Learning Objectives: 1. Students will be able to discover relationship of slopes between parallel and perpendicular lines. 2. Students will be able to find the fourth coordinate of a parallelogram and a rectangle given three coordinates, using their knowledge of parallel and perpendicular lines. Materials Needed: 1. Cabri II Geometry Software 2. Pencil, Paper 3. Lab Handout Procedure/Review: 1. Attention Grabber: Begin by asking the students, What are some instances where parallel lines are used in everyday life? What about perpendicular lines? 2. Students can be grouped in teams of two or, if enough computers are available, they may work individually. 3. Assessment will be based on the students completion of the lab and the extension worksheet. 4. A review of slope and equivalent fractions may be necessary prior to instruction.

2 Team Members: File Name: Lab Goals: 1. To find the relationship of the slopes of parallel and perpendicular lines. 2. To apply their knowledge of parallels and perpendiculars to find missing vertices of parallelograms and rectangles. Investigation: 1. Draw line r. Then draw parallel line t. (Line tool, parallel tool) 2. Find the slope of line r and line t. (Slope tool) What do you notice about the slopes of parallel lines? 3. Scroll down screen, and draw line w. Then draw perpendicular line h. (Line tool, perpendicular tool)

3 4. Find the slope of line w and line h. What do you notice about the slopes of perpendicular lines? 5. Now try multiplying the slope of line w and the slope of line h. (Calculate tool) What do you notice? Drag the result to the top of the screen and label the result product of slopes. 6. Grab line w and move it to a different location. What is the product of slopes? Once again, drag line w to another location. Is there a pattern? Definition: When two numbers product is negative one, the two numbers are called negative reciprocals. For example, 4/3 and 3/4 are negative reciprocals. Definition: The slope of a line containing two points with coordinates (x 1, y 1 ) and (x 2, y 2 ), is given by the formula: y 2 - y 1

4 Extension #1: M= x 2 - x 1 A parallelogram is a four-sided polygon whose opposite sides are parallel. Given A(1,1), B(6,2), C(2,4), find the coordinates of a point D so that A, B, C, and D form a parallelogram. This problem has three possible solutions for the coordinates of point D. Here is a walkthrough of one possible solution. 1. Create a new page in Cabri, Show Axes and Define Grid. (show axes and define grid tools) 2. Plot points A, B, and C. (point tool) 3. One possible solution can be found by drawing AB and BC. (segment tool) 4. Draw a line parallel to AB through point C. 5. Draw a line parallel to BC through point A. (parallel line tool) (parallel line tool) 6. Label the intersection point D. (point tool) 7. Find the coordinates of point D. (equation and coordinates tool) Find the other two possible coordinates of point D. Extension #2: A rectangle is a four-sided polygon whose opposite sides are parallel and whose adjacent sides are perpendicular. Given A(3,1), B(6,-2), and C(7,5), find the coordinates of point D so that A, B, C, and D forms a rectangle. How many possible solutions are there for the coordinates of point D? Why?

5 Solutions Investigation: -2. Parallel lines have the same slope. 4. Answers may vary 5. Perpendicular lines have slopes product is negative one (negative reciprocal slopes). 6. The product of the slopes remain Extension #1: Extension #2:

6