# Using Multiple Representations of a Mathematical Concept

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Many mathematical concepts can be represented in multiple ways. Concepts include a point and its coordinates, area of a circle as a quadratic function of the radius, probability as the ratio of two areas and area of a plane as a definite integral.

## Representing Math Concepts in Multiple Ways

Imagine that a person is walking the path of a perfect circle and wishes to see how much distance he or she covers in one trip around that circle. Without having the actual circumference (or distance around the circle), it might seem hard to express initially. If given the radius, however, understanding that the circumference is a function of radius, the distance covered in one revolution about the circle can be easily found. Similarly, we might obtain the circumference of a circle with its diameter.

This is a very simple and straightforward example of how we may view a math concept, like circumference, in multiple ways. In this case, we viewed circumference as a function of both radius and diameter. Likewise, other mathematical concepts can be viewed in more than one way.

We can broaden our understanding of several math concepts by representing them in multiple ways. Let's illustrate this by looking at:

1. A point and its coordinates
2. The area of a circle as a quadratic function of the radius
3. Probability as the ratio of two areas
4. The area of a plane as a definite integral

## Points and its Coordinates

### 2-D Cartesian Coordinates vs. Polar Coordinates

The concept of a point in 2-D space can be represented in more than one way. Two of the most common representations are: points, as represented in a Cartesian (x, y) coordinate system:

and points, as represented in a polar coordinate system, where our polar coordinate below will be (r, θ), merely a different way of representing (x, y):

### 3-D Coordinates Cartesian Coordinates vs. Cylindrical Coordinates

Note that cylindrical coordinates can simply be seen as polar coordinates e.g. (r, θ) with a 3rd dimension: (r, θ φ). Phi (φ) and Theta (θ) are just angular determinates -as illustrated below. Again, (x, y, z) is analogous to the cylindrical coordinate (r, θ, φ),.

## The Area of a Circle as a Quadratic Function of the Radius

The radius r of a circle, as seen in the general equations for circumference C of a circle C = 2πr or area A = πr2 of a circle, can be represented in more than one way. For example, it can be represented as a standard form equation of a circle (i.e. as a square root of the sum of two binomials):

It can also be represented as a quadratic function. For instance, if we are given the polynomial x2 + y2 - 4x + 10y + 13 = 0, we can write its equivalent standard form by completing two squares:

1. x2 - 4x + _____ + y2 + 10y + _____ = -13
2. x2 - 4x + 4 + y2 + 10y + 25 = -13 + 4 + 25
3. (x - 2)(x - 2) + (y + 5)(y + 5) = 16 or
4. (x - 2)2 + (y + 5)2 = 42

Where h = 2, k = -5 and r = 4.

## Probability as the Ratio of Two Areas

Probability, or a fraction or ratio defining chance, can be illustrated as the ratio of two areas. Noting that all outcomes in nature take place along a bell curve:

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