# Connecting Mathematical Concepts & Procedures

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.

Mathematical concepts can be linked to mathematical procedures. Many real-world problems can be solved by first applying a mathematical concept to said problem and then carrying out a corresponding procedure (or mathematical operation) to derive an actual solution . There is a plethora of examples to illustrate the importance of connecting mathematical concepts to mathematical procedures.

## Connecting Mathematical Concepts and Procedures

Often, math can be applied to real-world problems. In doing so, one might have to first conceptualize the mathematical nature of a problem and then carry out a corresponding mathematical procedure. We seek to define and link a couple of mathematical concepts with their respective procedures e.g. involving circumference of a circle and a length along a right triangle via the Pythagorean Theorem.

For instance, say Sally is walking the path of a perfect circle around a lake. Perhaps, she would like to quantify the distance of one revolution or trip around the circle. How might she approach this problem? This is where the value in connecting mathematical concepts to procedures can be illustrated. The concept serves as the context for the procedure e.g. Sally is walking in a circular path and wishes to quantify the distance around one full circle. The procedure serves as the means of finding a finite solution to said problem. For instance, what is the mathematical operation for finding the actual distance around a circle?

### Concept of Circumference

To recap we can view the concept as follows: Sally is walking a circle and wishes to find how far she travels in one trip around the circle. Note that the distance around one full circle is given as the circumference. Circumference is represented by 2πr, where π is a constant -just given as 3.14- and r is the radius of the circle, the distance to the edge of the circle from the center.

### Procedure for Circumference

The corresponding procedure for finding this distance is given by a specific mathematical operation. Let's say that the radius of the circle is given as 100 yards. What is the circumference? The circumference can be calculated by substituting r = 100 into 2πr and simplifying the expression: 2(3.14)(100) = 628 yards.

In summary, as Sally wanted to find how far she was walking around a circular path, we developed a concept of what constitutes a complete circle i.e. circumference or 2πr. We then carried out the corresponding procedure for finding the exact value of the circumference (noting the circle's radius r = 100 yards) via 2πr = 2(3.14)(100) = 628 yards.

#### The Theorem of Pythagoras -Connecting a Concept and a Procedure

Let's say that Bob is 5 feet from the base of a building with a height of 15 feet. Suppose that Bob wishes to extend a ladder to the roof of this building. We can use the Pythagorean Theorem to determine how long the ladder must extend to reach the roof.

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