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All in the Family

  • Lesson
9-12
1
AlgebraGeometry
Gina M. Foletta
Location: unknown

Students use a dynamic geometry applet to conjecture about the relationships between characteristics of a square: side length, diagonal length, perimeter, and area. Graphs are used to represent functional relationships between two characteristics, such as diagonal length as a function of perimeter. This lesson helps students deepen their understanding of basic functions (e.g., linear, quadratic, square root) and their knowledge of the measures of a square.

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.

1110 overhead  Think of a Graph Overhead 

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.

appicon Square Graphs Tool 

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 incorrect data.

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 questions:

  • 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.]

pdficon  Graphing What Activity Sheet 

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:

  1. Identify the independent and dependent variables;
  2. Determine the function family to which each graph belongs; and,
  3. 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 overhead projector.)

1110 overhead  Graph Chart 

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.

pdficon  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.]

References 

  • 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.

Assessments 

  1. Have students respond to the following journal prompt:

    What function families would you expect to find if you were to look for relationships among the measures of a non‑square rectangle? Why? 

    [The relationships among measures of a non‑square rectangle do not easily lead to functions of one variable unless a side length is specified. If a side length is indicated, relationships that match the functions found for the square may be identified. For example, when the length of a rectangle is 7 units, the width can take on infinitely many values, so there are infinitely many rectangles with length 7. The perimeter of one of these rectangles would be given by p(w) = 2w + 14, a linear relationship that is slightly different from the linear relationship we found for the square. Similarly, the diagonal length as a function of the width is d(w) = (w2 + 72)½. As with the diagonal of the square, the relationship here involves a square root function.

    If a side length is not specified, the relationships could be represented as functions of two variables, which might be an extension of this lesson.]

  2. Ask students to develop functions that describe the relationship among measures in a regular pentagon and a golden triangle. (See the Pentagon and Golden Triangle activity sheet.)
    pdficon  Pentagon and Golden Triangle Activity Sheet 

Extensions  

  1. As suggested in the answer to the first Assessment Option above, students could describe the area, perimeter, and diagonal length of a non-square rectangle as a function of two variables, the length and width of the rectangle.
  2. Launch the Square Perimeter Diagonal applet. Allow students to use this applet while answering the questions on the Square Perimeter Diagonal activity sheet.
    pdficon  Square Perimeter Diagonal Activity Sheet 
  3. Arcavi (2000) presents the following problem:
    Triangle ABC is an isosceles triangle with AB = AC = 5 units. When you drag point B or C, the lengths of the legs will remain 5 units, but the shape—and supposedly the area—of the triangle will change. Predict the graph of the area of ΔABC as a function of BC.
    Students may think that the area will involve a quadratic relationship, but that is not the case here; however, do not pursue this issue when first presenting the problem. Instead, open the Isosceles Triangle Investigation, and allow students to explore this question. After the investigation, ask students:
    • How does the graph you see match your prediction?
    • Determine the value of BC for which the maximum area occurs.
    • Derive an algebraic rule for the area of ΔABC as a function of BC.

      [To create an algebraic rule, students may have difficulty knowing how many variables to use and what relationships to establish. It helps to highlight that variables are quantities that change, implying that BC might be a good candidate for a variable. Students may use the Pythagorean Theorem to compute the height of the triangle, and the area function is given by

      A = \frac{{x\sqrt {100 - {x^2}} }}{4}

      where A is the area and x is the value of BC.

     
 

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 cmand the other in cm.] 

Teacher Reflection 

  • 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 effective?
 

Learning Objectives

Students will:

  • Identify the family of a function represented by relationships between measures of a square
  • Determine which measures are the independent and dependent variables in functional relationships
  • Explain how differences between members of the same function family (e.g., different slopes) relate to the measures of a square (e.g., comparing the graph of perimeter as a function of side length, to the graph of diagonal length as a function of side length)

NCTM Standards and Expectations

  • Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.
  • Analyze properties and determine attributes of two- and three-dimensional objects.