Group students into pairs, and give each pair a small white board or
other means to display their work to the larger group. The students may
be seated near computers or have laptops in their pairs in anticipation
of a later portion of the lesson.
Display the overhead, Think of a Graph.
Ask, "Think about a graph. Along the horizontal axis is the length of a
side of a square. Along the vertical axis is the perimeter of a square.
With your partner, sketch what you think the graph would look like. You
have two minutes to sketch something and hold it up for everyone to
see." Students may begin to plot points. Encourage them to think about
the shape of the graph without plotting points.
Group the graphs that students sketch according to their
general shapes. If all of the shapes are the same, ask, "Why do you
think that the graph would look like that?" If there are different
shapes, ask students to explain what other groups might have been
thinking. This part of the lesson is to initiate thought about graphs
and relationships so that students do not merely manipulate the applet
without thinking about the relationships in what they see. It is not a
time to press for exact answers; that will come later. For now,
students should put aside these graphs, but they will return to them.
Students will now use the Square Graphs
applet for exploration. The applet may be used in a whole-group setting
or by pairs or small groups of students in a lab setting, depending on
the availability of technology.
Note that if students use this tool to create a trace and then
resize the graph, the trace may not appear as it should. Although the
straight line traces (Graphs 1, 2, and 5) will appear correctly if the
grid is resized, the other traces (Graphs 3, 4, and 6) will not. If
students wish to resize the grid, they should first hide all points and
clear the graph of all traces by clicking the X
in the lower right corner; then, they should resize the grid and redraw
each of the traces. This will ensure that they do not consider
Say to students, "This applet will show you relationships among
several measures related to squares. Let’s try one together. To show a
portion of Graph 1, click on Show Graph 1.
Drag either point C or D to change the size of the square." Wait while
students do this, and monitor their use of the applet. Provide help as
necessary. Discuss the results with students by asking the following
- Describe what you see. What shape is this graph? [The points seem to form a straight line segment.]
- What relationship between two measures of a square might
explain the graph? [Students may see how this graph relates to what
they just sketched, perimeter as a function of side length.]
- Write an equation that describes the graph. [P(s) = 4s, where s is the side length. Students may first write P(s) = s + s + s + s.]
- Explain how you know this equation is correct. [The graph
appears to contain points (2, 8) and (4, 16). A line through these
points would have a slope of 4 and would pass through the origin.]
Have students return to their earlier work from the Think of a Graph
activity. Ask, "How do the results of the applet activity compare with
the graph you sketched? Would you change any of the graphs you created
during the Think of a Graph activity? Why or why not" Allow students to
share their thoughts.
Distribute the Graphing What
activity sheet. Note that the class has already answered the questions
for Graph 1. Have students answer the questions as a whole class and
record the results. [The independent variable is side length; the dependent variable is perimeter; the function family is linear; the rule is P(s) = 4s; and, the rationale is that the perimeter is the total distance of four side lengths.]
Assign one of the remaining five graphs to each pair of
students. This ensures that all graphs are discussed by at least one
group, in case there is not enough time for all groups to discuss each
graph. Each pair starts with its assigned graph and then moves through
the other graphs in order. For example, a pair that starts with Graph 5
would then consider Graphs 6, 2, 3 and 4, in that order. For each
graph, the pair must do the following:
- Identify the independent and dependent variables;
- Determine the function family to which each graph belongs; and,
- Write and explain a symbolic rule for the function.
Students should record their results on the Graph Chart.
(You may wish to have students use a large version of this chart that
can be hung on the wall; or, have students use a version of this chart
on a transparency sheet, so their results can be displayed on the
Depending on time, and after students have an opportunity to
consider at least two of the five graphs, have the students return to a
whole-group setting. One student originally assigned to a particular
graph should share their pair’s results. Other students are responsible
for questioning the student/pair for their reasoning as well as for
clarity. When students answer the three questions about the six graphs,
the results can be recorded in an overhead version of the Graph Chart. Alternatively, you can wait until the entire discussion is completed and display the Graph Chart Results.
When the information for all of the functions has been
discussed, the class should compare and contrast the results. Give
pairs a chance to discuss the following question before students share
their ideas with the whole class: "Why does it make sense that the
relationships between some of these measures are linear while others
are not?" [Students might draw on units of measure. Linear functions
arise when both measures are expressed in the same unit (e.g., both
in cm, rather than one in cm2 and the other in cm).
Quadratic relationships and square root relationships arise when one
measure is expressed in a particular unit but the other measure is
expressed in that unit squared.]
- Arcavi, Abraham, and Nurit Hadas. "Computer Mediated Learning: An Example of an Approach." International Journal of Computers for Mathematical Learning, 5 (January 2000): 25‑45.
- Foletta, Gina M., and David B. Leep. "Isoperimetric Quadrilaterals: Mathematical Reasoning with Technology." Mathematics Teacher, 93 (February 2000): 144‑147.
Questions for Students
1. Why does it make sense that the relationships between some of these measures are linear while others are not?
[Students might draw on units of measure. Linear functions arise when
both measures are expressed in the same unit (e.g., both in cm).
Quadratic relationships and square root relationships arise when one
measure is expressed in a unit but the other measure is expressed in
that unit squared (e.g., one in cm2 and the other in cm.]
- What ideas about diagonal, perimeter, and area did students use to talk about the graphs?
- What ideas about inverse functions emerged as students distinguished between the graphs?
- Did you find it necessary to make adjustments while teaching
the lesson? If so, what adjustments, and were these adjustments