Connecting Algebra to Geometry

It's interesting that for the most part, Algebra and Geometry are taught as two different classes so students often have difficulty seeing any connection, even when algebraic expressions are used instead of degrees.

One connection is that any two dimensional shape can be represented on a coordinate system via one or more equations and intersection points.  Actually any graph is a geometric representation of the algebraic statement.

Furthermore, the area and perimeter of any two dimensional shape can be expressed in an algebraic equations such as the area of a parallelogram is always A = length x width.  While three dimensional shapes such as Surface area, and Volume have their own equations.  These algebraic equations represent the physical found in geometry.

In addition, the Pythagorean Theorem provides an algebraic method of finding the length of missing sides for triangles and can be used to determine the type of triangle without needing to draw each triangle.  So if a student is given sides, they can use the theorem to determine if it is an acute, right, or obtuse triangle.  One activity I do in class is have students classify triangles using the Pythagorean Theorem and then have students draw the actual triangles using the given lengths to confirm their algebraic answers.

We also have algebraic equations based on the number of triangles in a polygon so we can find the number of sides, each interior angle, or the measurement of each exterior angle.  The formula can found simply by looking at the patterns of the number of triangles within a quadrilateral on up to a heptagon or further. 

We see transformations all the time in both geometry and algebra.  We see the transformations via certain additions to the parent equation such as y = x -3 indicates the line crosses the y axis through (0,-3) instead of  (0,0).  We can see transformations listed algebraically via h and k while they are quite visible if drawn.  There are other clues indicating a shape has been flipped or dilated.  All of these transformations can be represented by algebraic equations or visually through geometry.

I think its important to stress the relationship between the two topics so students learn there is a connection between the two and that they are closely related.  We can use technology in the form of GeoGebra or Desmos depending on what is needed.  Rather than teach them as we normally teach to subjects, we need to stress the connections, stress that algebra provides the mathematical representation of geometric figures.

Let me know what you think.  I'd love to hear.