Illuminations: All in the Family

All in the Family

 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.

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)

Materials

 Computer(s) with Internet connection Think of a Graph Overhead Graphing What Activity Sheet Graph Chart Overhead Square Perimeter Diagonal Activity Sheet (as an extension) Pentagon and Golden Triangle Activity Sheet (as an extension)

Questions for Students

 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.]

Assessment Options

 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.] 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.)

Extensions

 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. Launch the Square Perimeter Diagonal applet. Allow students to use this applet while answering the questions on the Square Perimeter Diagonal activity sheet. 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 where A is the area and x is the value of BC.]

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?

NCTM Standards and Expectations

 Algebra 9-12Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships. Geometry 9-12Analyze properties and determine attributes of two- and three-dimensional objects.

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.
 This lesson prepared by Gina Foletta and Rose Mary Zbiek.

1 period

Activities

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