Get Perpendicular with Geoboards!
Lesson 7 of 20
Objective: SWBAT use the concept of slope to determine if two lines are perpendicular, or they simply intersecting.
During today's Warm up students set up an x- and y-axis with 2 rubber bands stretched across the center of a Geoboard. It is helpful to use colored rubber bands, but I also have students use an erasable marker to label x and y on the board itself to identify each axis.
As they work on the Warmup, I first want students to think about the definition of parallel lines and the geometric reason that the lines are parallel. Most of my students will say that 2 lines are parallel because they never intersect, which is an understanding based on the definition. The geometric reason the lines never intersect is because they have the same slope.
The second concept I want students to think about is the difference between intersecting lines and lines that are perpendicular. Again, the slope measurement provides critical data in making this determination.
After the warm up, the students need to remove the rubber bands from the Geoboard except for the x- and y-axis. The x- and y-axis are needed in the next activity.
Teachers Note: Materials needed for the Warm-up and the Geoboard Activity
1. Geoboard per student or per pair
2. Rubberbands- about 10 per student or per pair if supplies are limited.
3. Expo marker with eraser or towel to wipe the marker off of the Geoboard.
In this Geoboard Activity I have students form 6 lines on a Geoboard set up to model a coordinate plane. I have students work in pairs on one board for this activity. I have students set up with table partners that are of the same level (homogeneous) as much as possible. In this activity, there are two sets of lines that have the same slope and are therefore parallel. The slopes are such that the two sets of lines are perpendicular to each other. I want my students to first visualize the parallel lines using the concept of slope, then to reason about the idea that some lines are perpendicular to each other.
I have posted the Geoboards of two students (student 1 and student 2) as they work on this activity. I often find it is helpful to demonstrate the x- and y-axis and to model how to form the first line on the Geoboard. The idea that the Geoboard can be used as a coordinate system is not always obvious to students:
Once rubber bands representing the x and y axis are placed in the center of the board, the points on a standard Geoboard form a coordinate system that goes from -2 to 2 along the x-axis and the y-axis.
Explaining the grid thoroughly at the beginning will clear up any confusion to form the lines correctly.
At the end of the lesson, students should be able to explain verbally and in writing that 2 lines are parallel if they have the same slope, and that 2 lines are perpendicular if the slopes meet two conditions:
1. the slopes are opposite signs
2. the slopes are reciprocals of each other
So, two lines are perpendicular if the slopes of each line are opposite reciprocals of each other. I used to teach this idea as a single statement, but I find it helps my students to break it down into two testable criteria. Students should also be able recognize that when two lines have different slopes that are not opposite reciprocals, then the lines only intersect.
After students complete the task on their Geoboards, I plan for them to compare the 6 lines and describe the relationships among them. I ask my students to write the equation of each line. They can use point-slope form or slope-intercept form for this task. The slope of each line is found by counting vertical change over horizontal change using the slope triangle, and the y intercept is found by observing where the line crosses the y axis. Each point on the line is determined by the coordinates from -2 to +2 for both x and y.
During this lesson, we often end up discussing the idea that a vertical and horizontal line on the same graph will always intersect at a 90 degree angle, and therefore will be perpendicular. Throughout the lesson, I emphasize the difference between the definition of parallel and perpendicular lines compared to the reasons based on the relationship of the slope of the lines.
As a followup, I have students create their own design. Then, as a formative assessment, I ask my students to complete the steps of the investigation using their own design. This task helps me to assess their skill at using slope to describe the relationship between two or more lines.
I bring closure to this lesson by taking students from the simple concrete example of using the Geoboard to a real world design problem, Parking Lots! I present two different types of parking lot line designs. The two most common parking lot line designs are rectangular (Perpendicular Parking Lines) and diagonal (Parking Lot Design -- Angled). I provide students with an image showing the dimensions of parking spaces for standard motor vehicles.
We begin the Parking Lot Lines Problem by discussing whether one style or another would be better if a student were building a parking lot for a retail store that he/she owned. I plan for students to refer to space, difficulty or ease of parking, how attractive each looks, and the effect on cost. I want students to consider the pros and cons of different designs and to make a decision based on an analysis that involves mathematical criteria.
I expect the closure to take about 15 minutes.