It is essential to have a pre calculus course that gathers all the topics covered in pre calculus just in one subject. This subject is overwhelming for many students. This is because they have to have thorough knowledge of several other subjects. The student should have solid base in algebra, geometry, and trigonometry. Special attention should be given to binomial theorem, conic sections, complex numbers and real numbers.
Calculus is the branch of science that deals with rates of change. In algebra the student is taught about a function that represents a physical quantity, for example the amount of water in a tank relative to time. However; in calculus the interest is in the rate at which water is changing in a tank relative to time. One can see clearly how calculus completes the picture after algebra. Algebra answers one question and calculus answers a different question and they together give the whole picture.
Calculus requires that the student have a good knowledge of algebra. A good tactic in teaching algebra in a pre calculus course is to point out the points that most students have problems with. Students studying algebra have particular problems with exponents and logarithms.
It is good to use calculators to do laboratory experiments to get the students to understand exponents. For example the teacher can ask the students to use their calculators to find the exponents of 0, 2, 4, 100, and 200. The students will see how the change of the output gets very steep with small changes in the input. Then the teacher should ask the students to use their calculator to plot the graph of an exponent function. The students should examine the curve and see how little change in input cause a dramatic change in output. The same experiments should be done with logarithms. The teacher should choose a base and chooses values to take the logarithms of. The teacher should let students try forbidden values like negative values and see how the calculator responds. Also the teacher should let the students plot a logarithmic function and spot the forbidden values on the graph.
Geometry is very important for calculus. The most important concept in geometry related to calculus is the area. The student should see how he can find the area of a shape by slicing it into simpler shapes and then adding the individual areas of these shapes. The student should see that we can simplify the circumference calculation by simplifying the circle by considering it a polygon. This simplification causes some error margin. However; the error decreases as we increase the number of sides of the polygon. The error becomes null when the number of sides become infinite.
Although the need of trigonometry does not come at the beginning of a calculus class students should know the basic trigonometric functions and their relations. The students should appreciate that the cosine is a sine with a shift and the sine is a cosine with a shift. They should see their graphs simultaneously on a calculator.
After knowing what should be taught in a pre calculus course one should choose a suitable textbook to be assigned to the class. Choosing a textbook is not an easy task. Some students would not do well in a course only because the textbook was not friendly or easy to understand. The textbook should be easy to understand with lots of examples and starts right from the beginning. The teacher explanation for concepts should be in close resemblance to the textbook so the students do not get confused.
For every new concept or idea the teacher should solve several demonstration problems. This really allows the student to follow up with the ideas discussed in class.
The teacher should allow the students to ask as many questions as needed to understand. The students should know that their questions are welcome and that it is important for the teacher that they understand the concepts.
Throughout the course the students should do laboratory experiments using calculators to dissolve the concepts. They would need an algebra calculator, a basic scientific calculator and a graphing calculator. There are many new innovative software calculators like kinematics calculators and many others. These calculators allow the teacher to be creative in how he creates mathematics laboratory experiments.