The author explores common polygons---squares, triangles, trapezoids---and uses the free online grapher, Desmos, to derive single equations that generate the shapes. Throughout, the author discribes his inquiry-based approaches---the trials and errors he employs in his eventual discovery of various shape-generating equations

Iobst, Christopher Simon

OSU Libraries Digital Journal Publishing (Ohio State University)

Shapes and Their Equations:Experimentation with DesmosChristopher Simon Iobst, Columbus AcademyAbstract: The author explores common polygons—squares, triangles, trapezoids—and uses the freeonline grapher, Desmos, to derive single equations and inequalities that generate the shapes. Throughout,the author describes his inquiry-based approaches—the trials and errors that he employs in his eventualdiscovery of various shape-generating expressions. Keywords: polygons, graphing, inquiry, multiplerepresentations1 IntroductionMathematics has always interested me. As a student at Columbus Academy, I have the opportunityto study with incredible teachers who spark new interests and help me find answers to mathematicalquestions. Lately, I have been researching different shapes and their equations. For instance, theequation, x2+ y2 = 1, generates a circle centered at the origin with radius 1 unit. While my textbookprovides equations for conic sections—ellipses (including circles), hyperbolas, and parabolas—theauthors provide no single equation to generate other simple shapes (e.g., squares and triangles).Ever since I discovered this, I have been on a quest to find different ways to express common shapeswith single equations. Using Desmos, a free online graphing utility, I’ve worked to develop, test,and revise conjectures and share findings with teachers and classmates.2 The Search For A SquareMy first idea was the square. The more common way I was taught to make a square was by creatinglines at different x and y values. The results would look somewhat like Figure 1.Fig. 1: Graphing a square with four perpendicular lines (see https:// tinyurl.com/squarewithlines).Ohio Journal of School Mathematics 79 Page 27This was okay, but I just wanted the square. I didn’t want lines that went on forever in differentdirections. Of course we could also restrict the domain or range of these lines, such as typingy = 1{−1 ≤ x ≤ 1}, to restrict the graph to a square, but this was not satisfying. I started playingwith the circle equation. I found that if you were to make the powers of x and y larger while keepingthem equal to each other, a square-like shape would be created. I got all the way to x512 + y512 = 1before I realized that it would never create a perfect square, because at each angle of the square itwould still be a curve, as shown in Figures 2 and 3. To experiment further with this, I used a sliderin Desmos. To explore my work, navigate to the following link: https://tinyurl.com/almostsquare.Fig. 2: Graph of x110 + y110 = 1, a square-like shape (see https:// tinyurl.com/almostsquare)..Fig. 3: Local behavior at one “vertex” of the square-like shape.Then one of my teachers suggested that there is an equation for a square. This equation was:|x|+ |y| = 1. This created something that resembled a rotated square. It technically was a square;it had four sides that were equal in length. However, it was rotated about the origin as shown inFigure 4. The main goal was to find an equation that would create a non-rotated square. Soon, I wasable to find one using many different equations. I wanted to use infinite values for m and n, whichdidn’t work since Desmos did not allow such input. Then, I tried to find an equation for segmentsso I could construct a square, but then I realized that this wouldn’t be a single equation.Page 28 Ohio Journal of School Mathematics 79Fig. 4: Graph of a rotated square generated by equation |x|+ |y| = 1 (see https:// tinyurl.com/rotatedsquare).Finally it occurred to me that an equation that generates a square actually generates a set of points.This observation led me to the equation max(abs(x), abs(y)) = 1. It’s instructive to look at a table ofvalues for sample x and y to better understand the behavior of the expression.x y abs(x) abs(y) max(abs(x),abs(y)) max(abs(x),abs(y))=1?0 0 0 0 0 no0 0.5 0 0.5 0.5 no1 0 1 0 1 yes1 0.1 1 0 1 yes1 0.2 1 0 1 yes1 0.9 1 0 1 yes1 1 1 1 1 yes1 1.1 1 1.1 1.1 noAs the table suggests, all points (1, y) with 0 < y < 1 satisfy max(abs(x), abs(y)) = 1. I used themax function to ensure the use of every point in the coordinate plane. The absolute value functionensures that (x, y) are plotted for negative x and y. The resulting graph is shown in Figure 5.Fig. 5: Square generated bymax(abs(x), abs(y)) = 1 (see https:// tinyurl.com/maxfunctionsquare).Ohio Journal of School Mathematics 79 Page 293 Explorations With Other ShapesThe previous equation, max(abs(x), abs(y)) = 1, can be altered to produce different shapes. Specif-ically, we can produce rectangles, triangles, and trapezoids of all different sizes. We can use thesquare equation from Figure 5 as a starting point for other findings. Altering this equation leads tothe discovery of shapes such as triangles, trapezoids, and rectangles. The equations of these shapeshave a similar format to that of the square. Let’s start by considering a triangle.3.1 TriangleThe triangle-generating expression, max(abs(−2y), abs(4x) + 2y) ≤ 1, is shown in Figure 6. Notethat the constant term on the right appears to represent the height of the triangle—if we set theright side of the inequality to 1, the triangle’s distance from the origin is 1. Does that property holdfor any value of the parameter? Drag on the slider and explore! The factor multiplied by x controls thewidth of the triangle. Smaller values appear to make the triangle wider. Note, too, that the triangleappears isosceles—the vertices on either side appear to be equidistant from the y-axis.Fig. 6: Graph of triangle and its interior (see https:// tinyurl.com/triangleequation).3.2 RectangleThe generating expression for a rectangle is similar to that of the square. For example, consider theinequality max(abs(2y), abs(4x)− 22) ≤ 2. The values that multiply by x and y give the rectangleits height and width.Fig. 7: The graph of a rectangle and its interior (see https:// tinyurl.com/rectangleequation).Surprisingly, setting the x and y coefficients equal does not generate a square. Compare the results ofthe following expressions in Desmos. Which generates a square? (a) max(abs(2y), abs(2x)− 22) ≤ 2;(b) max(abs(2y), abs(6x)− 22) ≤ 2Page 30 Ohio Journal of School Mathematics 793.3 TrapezoidSince a trapezoid can be viewed as a truncated triangle, the generating expression for the two shapesare closely related. Consider, for instance, the expression max(abs(5y), abs(4x) + 2y) ≤ 2. Notethat the coefficient of y in the absolute value appears to be larger than the constant value on theother side. Does this property hold for any trapezoids? The link https://tinyurl.com/trapezoidgraphcontains a more generalized graph that can be manipulated to explore this conjecture (and manyothers) in greater depth.Fig. 8: The graph of an isosceles trapezoid and its interior (see https:// tinyurl.com/trapezoidequation).You can make many new shapes using the expressions I’ve provided by changing numerical valuesof parameters. The beauty of these expressions is their versatility—all are readily manipulatedusing parameters and sliders in Desmos.4 SummaryThis project has provided me with answers to questions that I didn’t even know I had. My hope isthat the expressions I’ve provided can be used to spark further mathematical explorations. The useof Desmos and other electronic tools that create a more dynamic approach to mathematics provedinvaluable to me in this project.I would like to thank Chris Bolognese, my math teacher, who helped me fine tune this project andgive me pointers and ideas throughout my research. Mathematics is not just lines on graphs, but itis the movement and manipulation of those lines on graphs that make it so interesting. I hope morestudents will ask and explore their own questions, developing their own methods for investigatingproblems of their own design. Such work benefits the mathematical knowledge of the researcher,but also impacts the knowledge of student peers and teachers. I firmly believe that teachers shouldencourage their students to pursue their own projects and ideas that will broaden their knowledgeof mathematics. I’m positive that other students like myself can research concepts such as these.I will certainly continue my exploration of mathematics and I hope to uncover more interestingmathematics with Desmos.Christopher Simon Iobst, iobstc20@columbusacademy.org, is currently a student at ColumbusAcademy in Gahanna, Ohio.Ohio Journal of School Mathematics 79 Page 31

Shapes and Their Equations: Experimentation with Desmos

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Shapes and Their Equations: Experimentation with Desmos

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