This example presents a series of explorations based on two
linked representations of linear functions. The grades 9–12 section on
the Problem Solving Standard includes an episode describing how a
teacher engaged her students in problem solving and reasoning with tasks
such as those presented in this example.
The following links to an updated version of the e-example.
Relationships of Lines and Parabolas (applet)
This two-part e-example allows for students to explore the
effect of coefficients and constants within linear and quadratic functions.
Options include the ability to the change the forms of equations and trace the
The e-example below contains the original applet which conforms more closely to the pointers in the book.
||Exploring Linear Functions: Representational Relationships
Technology allows the linking of multiple representations of
mathematical situations and the exploration of the relationships that
- How do changes in the values of m and b in the linear function f(x) = mx + b affect the graph? In the applet below, use the sliders to adjust m and b in f(x) = mx + b. How is the graph affected by changes in m? By changes in b?
- What are the effects on the graph of f(x) = mx + b if the values of m and b are changed simultaneously? Click on the Connect Sliders button. Adjust the sliders again. What happens to the equation and to the graph? Summarize your findings.
- What are the effects of changing m and b simultaneously for different lines? Uncheck the Connect Sliders button, and specify a different initial line. Then click on Connect Sliders again. Describe what happens. Compare your results to the results you obtained in task 2.
- Click on the Show Trace button. Predict starting values for m and b that, with the sliders connected, would result in the lines intersecting in—
Try out your values. Describe what happens.
If m and b are simultaneously varied by the same amount, a family of lines is generated.
- Express the equation f(x) = mx + b so that it represents the
family of lines. (Hint: Consider using k to represent the amount m and b
- Use your equation to explain why all the lines meet at a common point and why the x-value at that common point is always –1.
How to Use the Interactive Figure
This interactive figure
allows the user to manipulate both the points and the line to observe the effects
on the sum of residuals shown at the bottom of the figure.
Modify Line: Checking this
box allows the user to modify the placement of the line in two ways: (a)
by dragging the line, the user rotates the line around a the y-intercept,
thus changing the slope of the line without changing the y-intercept;
and (b) by dragging the point at the y-intercept, the user translates
the line without changing its slope. To disable this capability, click on the
checked box to clear the check.
Modify Points: Checking this
box allows the user to modify the placement of the points. Once the box is checked,
the user can change the position of any red point by clicking on that point
and dragging it to a different position.
Squares: This button allows
the user to choose the square of the vertical distance between each point and
the line, or "distance squared," as the mode of calculation for the equation
at the bottom of the figure.
Absolute Value: This button
allows the user to choose the "absolute value" of the vertical distance between
each point and the line as the mode of calculation for the equation at the bottom
of the figure.
Shortest Distance: This button
allows the user to choose the perpendicular distance, or "shortest distance,"
between each point and the line as the mode of calculation for the equation
at the bottom of the figure.
Dynamic graphical representations can suggest mathematical relationships that may be surprising to students and teachers, and exploring such relationships can help students gain a deeper understanding of important concepts. With technology, representations can be manipulated rapidly and in ways that would not be practical using paper and a pencil.
The purpose of this task is for students to better understand linear functions by exploring the relationship between symbolic and graphical representations. The first task draws students' attention to the roles played by the parameters in a linear function and engages them in observing, describing, and comparing relationships among mathematical objects. They can also begin to link representations.
In addition to exploring what happens when b and m are adjusted separately, students can use the Connect Sliders option to explore what happens when they simultaneously vary b and m by the same amount. In trying to understand why any pair of lines seems to intersect at a common point, students will find that it is useful to develop a general symbolic representation of the family of lines whose slopes and intercepts differ by the same amount. They can then prove that all such lines will pass through the point (–1, b – m).
The episode on pages 338–40 in the Problem Solving section for grades 9–12 describes how this situation developed in an instructional setting. See that discussion for further details.
Take Time to Reflect
- What do you think are students' greatest conceptual difficulties in representing graphical relationships with symbols?
- What questions would you ask students in order to better understand their thinking about the relationships they see?
- What probing questions could you ask to help students expand their understanding of graphical and symbolic representations and the relationship between them?
- How might you help students see the need to explain more formally the phenomena they are observing?
The preceding activities were used in a research study conducted by Rose Mary Zbiek and colleagues at the University of Iowa during the CAS-Intensive Mathematics project (NSF award number ESI 96-18029).