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critique of several limitations that should be noted by potential users.

tion of a square inscribed in a circle, as illustrated in Figure 5. To recreate this yourself, first, click on the circle tool to create a circle. Second, click on the line tool and create a line the goes through the center point. Third, click on more tools and choose perpendicular line ; then select the line and center point of the circle. Fourth, click on the point tool and create points at each intersection of the lines and circle and click on the select tool and label each point. Fifth, click on the segment tool and cre ate line segments , , , . Sixth, click on the angle tool to find the measure for each angle and click each side of the square and click on more tools to label the length for each side of the square to verify the properties of a square hold. The cor rectness of the construction can be checked by per forming the “ drag test. ” To do this, click on the se lect tool and choose a point to drag. Arbitrarily

Investigating Concepts in High School Geome try

Desmos Geometry tools, like GeoGebra, allows for students to develop conceptual understanding of definitions, postulates, and theorems by providing a way for students to visualize them. For example, students can create a triangle using the polygon tool , label vertices by clicking on each vertex using more tools , find the angle measures using the angle tool , and find side lengths of the triangle by click ing on the side and more tools . Then, students can relate the sides and angles of a triangle by discov ering that the longest side and largest angle are op posite of each other (see Figure 2).

Another example has students prove what happens when a composition of two reflections occur. If the reflections occurred with parallel lines (reflection in parallel lines theorem) then the composition re sults in a translation (see Figure 3). If the reflec tions occurred with intersecting lines (reflection in intersecting lines theorem) then the composition results in a rotation (see Figure 3). Desmos Geome try allows for multiple compositions of transfor mations. Figure 2: Ordering the sides by length, given angle measures, ordering the angles by degree given side lengths and discover ing the relationship between angles and sides

Figure 5: Constructions of a square inscribed in a circle

move that point around on the screen. The con struction should hold; it should move rigidly and retain its shape, demonstrating that the construction is rigid and that it is a square by preserving the properties of a square.

Table 3: Composition in parallel lines and composition in intersecting lines

Investigating Concepts in Middle School Mathe matics

Using Desmos to Construct and Justify

Desmos Geometry can also be used in middle school mathematics classes to teach geometry con cepts, such as to verify the Pythagorean Theorem

Desmos Geometry can be used to investigate con structions and to justify the reasoning used in the process. An example can be found in the construc

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