Connections Between Geometry and Algebra

Updated

The Desmos Geometry Tool allows you to explore and move between concrete and abstract representations of geometric objects. The Desmos Graphing Calculator is built into the Geometry Tool, so you can measure, compute, work with compound transformations, and more. You can construct in a blank workspace, on a grid, in the coordinate plane, or even with an image as a background. By using sliders, lists, and algebraic functions in your constructions, you can notice, wonder, make generalizations, and build a deeper understanding of geometric and algebraic relationships.

 

Make Dynamic Transformations

Rotations

In this example, you’ll see three defined rotations: one with a constant value used for the angle of rotation, and two others where a list and a slider variable define the angle of rotation.

Questions to consider:

  • Why do you think a 60 degree rotation was used to define the first rotation in this example?
  • How are the slider bounds and the list values in the example similar? How are they different?

Animation of the Geometry tool showing a slider starting at 0 and triangle rotating around the origin. 

Composite Transformations

In the Geometry Tool, transformations are defined as functions. If you use the transformation tools from the toolbar, these functions are defined automatically. When you drag a transformation token from the navigator into the expression list, you can apply the function to other objects. Or, you can use the functions to build composite transformations.

Open the Composite Transformation example to explore the pattern created by a rotation plus dilation composition.

Questions to consider:

  • What do you notice about the rectangle's area as you change its width?
  • What is the greatest area you can make for a rectangle with that fixed perimeter?
  • How does this relate to the parabola formed by points plotted (width, area)?
  • What other connections between the rectangle and parabola can you make?

Rotation and Dilation composition of a polygon. Screenshot.

 

Explore Area and Perimeter

Triangle Perimeter and Area

In this example, you can move object tokens to the expression list to see how the object functions were defined. Then, you can make generalizations about relationships between edge length, angles, area, and perimeter of a triangle. Open the Triangle Perimeter and Area example to explore.

Questions to consider:

  • Which location of the glider point leads the largest perimeter of the triangle? How about the smallest perimeter?
  • What do you notice about the area of the triangle as the slider plays?

GIF a triangle with a vertex on a line gliding along the line.

 

Rectangle Area

You can use the expression list to explore geometric relationships algebraically.

Open the Maximize the Area of a Rectangle example and use the slider to choose a fixed perimeter for the rectangle. Then, drag point C to change the rectangle's width.

You can use this graph to explore the maximum area of a rectangle given a fixed perimeter. You can also learn more about the substitution function.

Questions to consider:

  • What do you notice about the rectangle's area as you change its width?
  • What is the greatest area you can make for a rectangle with that fixed perimeter?
  • What other connections between the rectangle and the parabola can you make?

Maximizing the area of a rectangle. Screenshot.

 

Learn More

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