Introduction to Detailed Notes
This is a set of notes that have been made on reading the textbook. There
is no real attempt to have comments on absolutely everything in the
book noted here. At the same time, there is supplementary material here that
is not in the book.
After writing out the notes for the first few sections, it has become clear that there is a tendency to make this a "teaching" document. As much as possible, efforts will be made to not do this. Rather, if there is teaching material to be presented then that will be done in separate pages, with pointers inserted here.
The important word here is "Review". The material is presented in short order, without a significant amount of explanation and example. If this is a review for the student, then there should not be much problem. However, for some students this will be material that they have either not seen before, or at least have not mastered before.
The discussion of sets in the text is quite short. A longer treatment can be found in the Detail Notes on Sets page.
In mathematics we classify and group numbers into different sets. In addition to the presentation in the text, you may want to review the Sets of Numbers page.
The text jumps into using the calculators with only minor explanation. You may want to see the Show Calculator Keys page to see a representation of the calculator.
The calculator treatment of rational numbers could use some more explanation. See the
The reference in Q7 on page 5 to using the MODE menu could use a further explanation. See
The diagram in CB7 is an example of a Venn Diagram. It is a picture of the relationship between sets. There is no importance to the size of the circles and areas. Rather, the important concept in the Venn diagram is whether or not circles and areas intersect (overlap) or are separate. For example, the region marked as N for the Natural numbers is completely inside the region marked as W (the Whole numbers) because every natural number is also a Whole number. The Whole number region, W, is bigger than is the Natural number region, N, because there is at least one Whole number that is not a Natural number, namely, zero.
We have a small problem with CB8 on page 6 since it states that "All the sets of numbers consist of signed numbers." The idea of "sign" is really introduced with the integers, the set Z. There are no negative numbers in the set of Natural numbers, N, nor are there any in the Whole numbers, W. Still, it is important to recognize numbers as representing both magnitude and direction.
The rules for performing the basic operations on real numbers are presented in CB9 through CB11. Once you understand the rules, they almost make sense. However, they can be confusing, especially since the rule for addition involves subtraction, and the rule for subtraction changes problems into addition. It is nice to know that the calculator will be able to do all of this correctly, assuming that we can enter the problem correctly, but students should be both comfortable and proficient in doing these operations mentally with reasonably small numbers. To that end, there are four calculator programs that are available to provide practice in doing the basic operations with positive and negative integers. Those programs are called DRLADD, DRLSUB, DRLMUL, and DRLDIV. They will be available to transfer to your calculator in class. They will be on the calculators that I bring to class. They will be on the PC in the classroom for use with the TI-GraphLink software. And, for those with access to a copy of the TI-GraphLink program on another machine, the practice programs are available for downloading here (PC-users: you may need to right-click on the link to save the file onto your machine; Mac-user: you may need to hold down the mouse button on the link):
| TI-83 | TI-85 | TI-86 | TI-89 | |
| DRLADD | Click Here | Click Here | Click Here | not available |
| DRLSUB | Click Here | Click Here | Click Here | not available |
| DRLMUL | Click Here | Click Here | Click Here | not available |
| DRLDIV | Click Here | Click Here | Click Here | not available |
This section launches into a quick and dirty introduction to relations, and to the terms associated with relations, namely, domain, range, ordered pairs, and coordinates. That is a lot to cover in a few pages. There was a time when these topics would be given an entire chapter. There was a time when these topics were not even considered. How does one choose just how much to do? For us, at this time and in this course, we seem to have reached an understanding that we need to at least talk about the the terms and concepts, but we do not want to spend too much time them. Perhaps we believe that students already know the material, or perhaps we just find it too boring to really cover. In any case, hopefully, the short presentation in the textbook makes sense to you. If it does not, you might consider checking out the relation page that has been put out on the web.
I like this section,and have little to add to it. There is a discussion of graphing relations in the relation page that might be of interest. However, the material in the text does a good job of graphing relations.
One small issue that might arise in this section is the use of the word "continuous". We bring up the idea of "continuous" so that we can fill in the points between integers and move from a graph of "dots" to a graph where the dots merge to form a line or a curve. The section uses "continuous" in the intuitive sense of being "connected" and the examples, where the graph becomes a line, are certainly "continuous". What we are really talking about, however, is relations that have a continuous (connected) domain. Later in the chapter, we look at "step functions" which are a kind of relation. Those step functions have a continuous domain, but they are not continous because they jump from level to level. A short presentation of step relations is given in the relation web page.
I agree that understanding functions is extremely important in terms of success in mathematics courses. There are two things that I would add here. First, even though functions may seem difficult and strange, we introduce them in the most simple form that we can. We begin our work with functions where there is one input (the x-coordinate, or element of the domain) and one output (the y-coordinate, or element of the range). In real life, it is rarely the case that we have this simple situation. For example, in many courses, final grades are determined by exam scores, test scores, quiz scores, homework grades, attendance, and class participation. These are all recognized and accepted inputs to determine the final grade. This is a function of many variables. We start by an in-depth study of functions with one input and one output, knowing that at some point we will have to carry our ideas of functions to more complex situations.
The second point that I think we need to make is that we seem to do everything we can to confuse students with respect to functions and function notation. Thus, we will see
In all of the cases above, we understand that the x value is an element of the domain. We take that value, multiply by 3 and subtract 4. That gives us the corresponding element of the range. The "function" is at once the rule that we follow and the set of ordered pairs that follow the rule.
I am working on pages that gives directions and screens for graphing functions on the TI calculators. At this time, web pages exist for:
Again, the text has a good presentation of the concepts and use of mathematical models. I would stress that models are just that, models. They are not the real thing. When we choose a model we immediately begin to neglect certain portions of the real world because we we simply can not take all of the variables into account.
©Roger M. Palay
Saline, MI 48176
September, 1999