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.
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.
Answers
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.
Triangle y = –1.5x + 11
y = 1x + –4
y = 3x + 2 | Quadrilateral y = 1x + 10
y = –0.5x + –2
y = 1.75x + –11
y = –0.5x + 7 | Pentagon y = –^{2}/_{3}x + 8^{1}/_{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.