Linear Alignment

• Lesson
6-8,9-12
1

This lesson allows students to explore linear equations and the effects of changing the slope and y-intercept on a line. It gives students exposure to y = mx + b, and can be used as an introduction to the topic. Using graphing calculators, students are challenged to overlap lines onto the sides of polygons. To achieve this goal, students change slopes and y-intercepts of lines, noting observations about behavior as they work. As students change the y-intercept of a line, they see it raise or lower the line. As students change the slope, they see it affect the steepness of the line.

Begin this lesson by demonstrating the activity. You may choose to use the coordinates for the triangle given in Question 1 on the Linear Alignment activity sheet or use a different set of points so students can later work through Question 1 on their own.

 Linear Alignment Activity Sheet

Students will first need to enter the given coordinates into lists on the graphing calculator, with x-coordinates in one list and y-coordinates in a second list. Note that the first coordinate is repeated at the end of the list. This is necessary for the triangle to be graphed properly. Using the statistical plot function, create a line plot of the two lists. This should draw a triangle in the graph window.

Note: The standard window for graphing calculators goes from –10 to +10 on both the x- and y-axes. This creates a rectangular view in most graphing calculators. To minimize the complexity of knowledge in using the graphing calculator needed for this activity, this is the view that was used to create the activity sheet. For a square view, set x to go from –15 to +15. This helps reinforce the behavior of slope. You may also choose to adapt and change the limits to create other polygons.

Lead a class discussion about the characteristics of the line segments that make up the polygon shown in the window. These characteristics may include the direction of a line (increasing or decreasing from left to right) and whether some lines are higher or lower than others. If students are not yet familiar with the equation y = mx + b, introduce it as the slope-intercept form of a linear equation. As the name implies, the equation tells you the slope and y-intercept of a line. Do not discuss what these terms mean — students will discover the meanings during the activity.

To help students discover the behavior of y = mx + b, start by graphing y = 1x + 1. This line and the triangle that you created in the step above should be displayed in the same window. Have students compare the line to the sides of the polygon. Ask them if they think they can "move" the line so that it overlaps any of the sides of the polygon. Instruct them that to move the line, they will have to change the values of m and b. To aid the discussion, ask students some of the following questions:

• Is the line parallel with any of the sides of the polygon?
[Yes, it is parallel to one side.]
• What is the difference between the line you graphed and the parallel polygon side?
[One line is higher than the other.]
• Go back to the equation of the line and change the coefficient of x to 2. Then look at the graph again — how did increasing this number affect the graph of the line? Is it now parallel to a polygon side?
[The line is now steeper, and it is no longer parallel to any of the sides.]
• What if you change the constant instead? Try graphing y = 1x + 2. How did increasing this number affect the graph of the line?
[The line moves higher on the graph.]

Ask students what numbers they think they will need to use to adjust the line so that it overlaps one of the polygon sides. This would be a good chance to have students make predictions about different kinds of numbers, such as positive and negative integers, fractions, and decimals. Discuss their possible effects on the slope and y-intercept of the line. Remind students that the only numbers they will be changing are m and b.

Distribute the activity sheet. Read through the instructions at the top of the first page, and clarify what students are expected to do. Arrange students in pairs or groups so they can share observations and ideas, and ensure that each group has a graphing calculator. Allow students to begin work on the activity, helping them as needed.

There are 3 sets of coordinates given on the activity sheet. The activity can be done in many variations. Here are a few suggestions:

• Have students all work on the same polygon, starting with the triangle. Have a class discussion after all groups finish a polygon, and then allow students to tackle the next polygon.
• Mix up groups, assigning each group a different polygon. Have representatives share their observations about the different polygons afterwards.
• For students with a stronger understanding of slope, have them graph the pentagon first.

These answers are the exact answers. However, this lesson is designed as an exploration of linear equations, so exact answers are not necessary. Look at students' answers individually to determine if the lines overlap an acceptable portion of the sides of the polygons. You may choose to have a discussion with the class about what makes an answer acceptable. There can be many answers that are all equally good. To reinforce the concept of m = rise/run, you can have students give all answers using fractions.

 Triangley = –1.5x + 11 y = 1x + –4 y = 3x + 2 Quadrilateraly = 1x + 10 y = –0.5x + –2 y = 1.75x + –11 y = –0.5x + 7 Pentagony = –2/3x + 81/3 y = 0.5x + 6 y = –2x + –9 y = 0x + –3 or just y = 3 y = 1.5x + –9

Summary

To finish the lesson, lead students in a discussion of their observations. Ask students to compare the tables of x- and y-values, graphs, and linear equations. This will help students to make a connection between the many representations of linear functions.

Have each group of students prepare an explanation of their results and observations on poster board. Their explanations should include the equations they used and a description of how the slope and y-intercept affect the lines in the graph. Have students present this information to the class. While this will likely result in many groups making similar presentations, students will benefit from seeing that other groups came to conclusions similar to their own.

As an alternative summary activity, create a coordinate poster and place several lines on it. Have students write descriptions of the individual lines and descriptions comparing lines. Place the descriptions near the lines they describe. Some ideas are listed below:

• If m is larger, then the line is steeper.
• Negative slopes make y decrease as x increases.
• Positive slopes make y increase as x increases.
• Positive y-intercepts make the line cross the y-axis above the origin.
• Negative y-intercepts make the line cross the y-axis below the origin.

Assessments

1. Have students write out on index cards what they do not understand fully or questions they would like to explore further. For example, a student may want to discuss how a positive or negative slope affects the graph. Collect the index cards and randomly distribute them to other students. Have each student address the question or comment on his/her index card.
2. Have students create a list of coordinate points for their own polygon and have other students create functions to overlap them.
3. Provide an overhead with several lines drawn on the coordinate plane. Have students explain what they know about the slopes and y-intercepts.
4. Present students with envelopes filled with pieces of paper that have a linear equation, a graph, or a table of values on them. Ask students to match each equation with the appropriate graph and table.

Extensions

1. Have students explore other representations of linear equations. One approach is to give students 2 equations and ask them to create a real-world situation that represents the equations. For example, if given the equation y = 5x + 10, students may suggest the equation represents a situation in which t-shirts cost $5 each with a$10 initial fee. For a fundraiser, the equation y = 3x + 20 could represent earning $3 for each candy bar plus a$20 donation.
2. Introduce students to m = rise/run, and have them calculate the actual slope of each side of a polygon. Then, have students compare these answers to the answers they found during the activity.
3. Introduce the concepts of horizontal, parallel, and perpendicular lines as they are represented by linear equations. In the activity, the quadrilateral has 2 parallel sides and the pentagon has 2 perpendicular sides and 1 horizontal line. You could use these examples to begin an investigation.
4. Ask students to bring in pictures in which lines are clearly visible. Have them draw a grid over the image and find the linear equations for those lines.

Questions for Students

1. If a line is not overlapping with any of the polygon sides, how could I change the equation so it overlaps?

[You can change the line in two ways: You can be adjust its steepness by changing the coefficient of x, or you can shift it up or down by changing the constant.]

2. How does changing the slope affect the graph of the line?

[Larger values make the line steeper, while smaller values make the line less steep.]

3. What do you think will happen to the graph if I use a negative integer for the slope?

[Negative integers will make the graph go down from left to right; that is, the y-values will decrease as the x-values increase.]

4. How does changing the y-intercept affect the graph of the line?

[Larger positive values move the line up, while negative values move the line down. These changes do not affect the steepness of the graph.]

5. Are there other equations that will work? What is the range of values that can be used for the slope and y-intercept?

[Answers will vary. Since this is an estimation activity, there can be several answers that are equally good. Allow students to explore a range of values, rather than anticipating a set answer.]

6. Why are some equations different but still create a similar graph?

[Equations that are fairly close and differ only by tenths or hundredths will create similar lines. If students zoom in on the graphs, they will be able to see the differences in the graphs.]

7. Without graphing the lines, how will the graph of y = 3x + 2 compare to the graph of y = 5x + 2? How do you know you're right?

[The second graph will be steeper because the slope is greater.]

8. Without graphing the lines, how will the graph of y = 3x + 2 compare to the graph of y = 3x – 2? How do you know you're right?

[The first graph will cross the y-axis above the origin because the y-intercept is positive. The second graph will intersect the y-axis below the origin because the y-intercept is negative.]

9. Without graphing the lines, how will the graph of y = 3x + 2 compare to the graph of y = 1/3x + 2? How do you know you're right?

[In the first graph, when x increases by 1, y increases by 3. In the second graph, when x increases by 3, y increases by 1. This can be determined by looking at the coefficients of x in both equations, which is the slopes of the lines.]

Teacher Reflection

• Did students make the anticipated observations about slope and y-intercepts? How could you facilitate student learning of these concepts?
• How did your lesson address auditory, tactile, and visual learning styles?
• Was your students’ level of involvement high or low? What could you change to make more students more enthusiastic?
• How did the technology work in your class environment? Did it create behavioral management issues? If so, how did you address them?
• How do you feel about your level of expertise in leading a lesson with technology? Were you adequately prepared to address questions relating to the technology? What are some resources you can research to learn how to use technology better?

Learning Objectives

Students will:

• Create a polygon on a graphing calculator using the LIST function
• Graph linear equations to overlap lines on the sides of the polygon
• Record observations about how different values affect the slope and the y-intercept
• Draw conclusions about the behavior of m and b in the equation y = mx + b