As any effective teacher can testify, finding meaningful, applicable math problems is only half of the challenge. The other, arguably more demanding, half is discovering how these problems can be successfully integrated into classroom instruction. In order to do this, a teacher must first determine for which grades and courses the problem is appropriate. In the example problem regarding Farmer Fran’s fence from the previous pages, the key tasks of this problem can be broken up into two parts.
The first part completed with a spreadsheet requires a student to understand how length and width are involved in calculating the perimeter and area of a rectangle. As outlined by the Alabama State Department of Education (ASDOE) (2015), this first part of the problem addresses topics covered in 7th grade geometry and applies specifically to the following standard:
16. Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms [7-G6] (ASDOE, 2015, p. 58).
The second part completed with Desmos requires a student to recognize the relationship between length and area of the corral in order to plot these coordinates on a graph. A student must then recognize that these points lie on a parabolic curve and use his or her knowledge of the quadratic equation and factoring to determine the equation of the curve that passes through the given points. The ASDOE (2015) assigns these topics to the Algebra I curriculum, and this second part of the problem addresses the following standards:
28. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* [F-IF4]
31. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]
a. Graph linear and quadratic functions, and show intercepts, maxima, and minima. [F-IF7a]
32. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context [F-IF8a] (ASDOE, 2015, p. 85).
While the above standards specifically refer to graphing a given function, it is important to recognize that finding a quadratic equation given its graph (as done in the example problem) requires students to exercise the same knowledge and understanding demanded by these standards.
Recognizing that various parts of this problem are appropriate for different levels of curricula can help a teacher determine which parts of this problem are appropriate for his or her students, and consequently, which technology should be involved. A seventh grade math teacher might incorporate only the first part of this problem, using it as a scaffolding activity to help students see how spreadsheets can be a useful tool for analyzing area. As stressed by Niess (2005), problems like this “help your students gain skills with the spreadsheet in a piecemeal fashion, in a way that keeps the activity’s focus on the mathematics.” Thus, students not only strengthen their understanding of perimeter and area, they acquire skills with spreadsheets that expand their technology toolbox with which they can solve math problems in the future.
An Algebra I teacher might extend this problem to include the graphing application after analyzing the data with spreadsheets. Doing this enables students to see how the familiar concepts of length and area can be modeled with a parabolic equation. Using both spreadsheets and Desmos helps students to make connections as they see how the same conclusion can be drawn from both the spreadsheet and graphical representations. This connection across representations further validates the use of this technology in the classroom (Dick and Hollebrands, 2011).
While the problem regarding Farmer Fran’s corral may have seemed overtly simplistic at first, this analysis demonstrates the potential of this problem to encompass topics from 7th grade geometry to algebra and to incorporate the use of both spreadsheet and graphical software. Furthermore, this problem serves as a means by which students can increase their relational understanding of mathematical concepts, which Skemp (1978) explains as “knowing both what to do and why.” So as students build technological problem-solving skills with this problem, they simultaneously grasp a more complete comprehension of how perimeter and area are related and why given elements of a quadratic function are significant.
Having determined that this problem deserves classroom attention, students should start working as soon as possible- Farmer Fran will be wanting her new corral!