Chapter 1
Notes for Instructors
Content
The first chapter of Long and DeTemple provides a foundation for the rest of the book.
It is important that you cover this chapter, especially if you plan to require your students
to read the textbook. I say this, in part, because P´olya’s Problem Solving Principles are
covered in Section 1.3. Every example problem in the textbook follows P´olya’s problems
solving methodology. Although it is important to cover Chapter 1, it should be noted that it
is not hard to imagine how one could spend months on the problems in Chapter 1. I devoted
3 weeks to chapter 1, but, in retrospect, I think it would have been better to spend only two
weeks on this chapter..
There are many problems in Chapter 1 which could be used in an elementary classroom.
This helps to provide some motivation for the course. There are also more sophisticated
problems. It should be noted that some students may find the more sophisticated problems
easier than those which appear to be more appropriate for an elementary classroom. College
students have been trained to use variables whenever possible. Many problems can be
solved without using variables. For example, elementary students will often attempt to solve
problems by working examples. Using a variable is one of the problem solving strategies
discussed in Chapter 1, but there are many others which college students will need to recall.
Notes and Suggestions
Since you will not have the luxury of spending months on Chapter 1, I want to highlight
some important ideas and topics in Chapter 1 that will be useful to the students later in the
course.
•P´olya’s Problem Solving Principles: This approach is used to solve problems
throughout the textbook. It is particularly important that you highlight the “Look
Back” step, a step which is often ignored.
•Pascal’s Triangle: This triangle is important because it can be used in the chapter
on probability to evaluate combinations.
•Gauss’s Trick: I like this trick. I suppose that one could teach this course without
discussing Gauss’s Trick, but I have found problems later in the text for which this
trick comes in handy. Moreover, I think there are some good ideas which come along
with problems that require Gauss’s trick. First, Gauss’s trick uses a variable in a
clever way. Second, students will be required to add equations. (This does appear to
be a novel idea to some of them.) Finally, I like to use Gauss’s trick to find the sum
of the first nterms in an arithmetic progression because these problems have several
layers. Students must begin by finding a formula for the nth term of the arithmetic
progression. They must also understand how to get the formula for the (n−1)st term.
That is, students can often understand why 1 + 2 + ···+ 100 = 100 + 99 + · · · + 1, but
it is more difficult to see that 1 + 3 + ···+ (2n−1) = (2n−1) + (2n−3) + ···+ 1.
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