Sketching a graph by hand or tweaking a triangle construction can be a tedious task for your students and can use up valuable class time that could be better spent learning math concepts. Using graphing software not only helps simplify this process, it offers visual representations that are impossible using pencils and graph paper.
Geogebra is a free mathematical software tool teachers can use at various levels from secondary to graduate math. Students can create difficult geometric figures quickly and manipulate the consequences by making slight to drastic changes. They can test conjectures, accurately represent proofs and solve algebraic problems with visual support.
The inclusion of problem solving and reasoning procedures in the Common Core State Standards strongly supports the need for visual software to enable students to expand beyond creating sketches on grid paper. Using graphing software can also help students address the Research and Information Fluency indicator within the ISTE Standards for Students and help teachers address the ISTE Standards for Teachers indicator on designing and developing digital age learning experiences.
Plus, students can save their work as GGB files so they can revisit it later to repeat problems or modify as problems become more in depth.
Geogebra also allows descriptive labeling for better comprehension. Students can label coordinates, lines, intersections and angles with words, such as “inner circle” and “outer circle” or “interior angle” and “exterior angle,” rather than using simple variables that require a key.
Teachers can use this software to easily create examples without dimensional limitations and then insert them into instructional materials as figures.
Get started with early math.
Use this software in early math classes to enhance student knowledge of concepts that will be developed later on. For example, introduce variables in linear equations by moving the cursor along a line to illustrate the infinite coordinate pairs that share the same relationship. As the students develop and create conjectures naturally, the figure modifications can occur with an engineered method by the adjustment of specific input values. The representation from both perspectives allows for a differentiated classroom to be developed with assessment at many levels.
You can use this software to different areas of mathematics.
Geometry. One Common Core high school geometry standard is to make formal geometric constructions. To address this, many geometry teachers have students complete these constructions using a compass and straightedge.
Geogebra allows you to take this a step further. Using a toolbar, students create any diagram or shape. Once students have finished their construction, they have the option of clicking and dragging on any point in their construction.
We taught our geometry students how to locate the four different triangle centers, I had them drag a vertex of their triangle around the screen and observe where each center was located in relation to the type of triangle they were working with. This visual illustrated that two of the four triangle centers will always be located inside of their triangle, while the other two triangle centers would move onto or outside of the triangle if it was a right or obtuse triangle.
You can also use this software when teaching similarity. With Geogebra, students can create variables that show a ratio. This can be handy for students because they could intentionally create similar polygons using this feature and then manipulate their construction to observe different properties of similar polygons.
For example, you can teach your students that all corresponding sides of a polygon need to be proportional by asking them to create a variable that displays the scale factor of each set of corresponding sides. Then students could move a vertex of one of their polygons and determine if the two polygons are still similar.
Algebra. Connect algebra to geometry by using Geogebra to display the prelude to geometric connections, and show various algebraic functions. Students can easily solve linear equations and linear inequalities without the common limitations of visualizing parameters before graphing.
Making multiple graphs can quickly represent the relationship between linear functions, and when adjustments are made, students can visualize algebraic changes. Topics can progress seamlessly with the ease of making simple one-step changes from linear to quadratic equations.
You can design lessons to show the impact when the exponent increases from one to two, or line to parabola. While extensive constructions are created in Geogebra, the algebraic contributions can also be completed within the application.
Kathleen Rulkowski has a masters in education and a bachelor’s degree in industrial and operations engineering. Rachel Kreuzer has a master's in education and a bachelor’s degree in material science engineering. She spent seven years as a metallurgist for Worthington Industries.