Teaching Secondary Mathematics

Chapter 13: Exercises

Exercise 13.1

1. It is said that you learn something best when you teach it. That is true{em}but does that give you license to use a class as guinea pigs? How much should you know about a topic before embarking on the study of it with a class?
2. You undoubtedly have had some logic as a part of your undergraduate program. Out of that information, what could be inserted into a precalculus course and why? If you have not had logic beyond basic truth tables, research the subject to determine what should be included in the precalculus class. As a part of your research, you should include a description of how much time it will take you to learn the material well enough to teach it.
3. Part 2 of this exercise mentions learning material prior to teaching it. Does this imply you will be lecturing? Is lecturing more acceptable in an advanced course, because these are more capable students and there is so much information to cover? Why or why not?

Exercise 13.2

Using a graphing calculator or software, do the following:

 

1. Graph one of each type of the functions:

   f(x) = Constant

   f(x) = Linear

   f(x) = Quadratic

   f(x) = Polynomial

   f(x) = Rational

   f(x) = Exponential

   f(x) = Logarithmic

   Select any three of these functions and describe their similarities and differences. List the main points you would bring out to students if you were comparing and contrasting the selected three in a precalculus class.

2. Graph at least four trigonometric functions:

   f(x) = sin(x)

   f(x) = cos(x)

   f(x) = tan(x)

   f(x) = csc(x)

   f(x) = sec(x)

   f(x) = cot(x)

   It is rather common to have an elaborate explanation of the development of sine using a unit circle. Use the unit circle to explain why one of the other trigonometric functions behaves as it does. Build your discussion in the form of a lesson plan. You should incorporate technology.

3. Graph one of each type of the functions:

   f(x) = Power

   f(x) = Reciprocal

   f(x) = Absolute value

   f(x) = Trigonometric inverse

   f(x) = Greatest-integer

   f(x) = Piecewise

   Which of these functions were you exposed to prior to a precalculus or calculus course? Is it reasonable to expect similar backgrounds from students taking the course at the time you are reading this question? Why or why not?

4. Which of the graphs in parts 1–3 of this exercise accepts a vertical or horizontal shift?
5. Which of the graphs in parts 1–3 of this exercise can be compressed or stretched? Describe an example of one of the compressions or stretches in the format that would be appropriate for students in a precalculus class.
6. What change or rotation would make one graph from each of parts 1–3 of this exercise no longer fit the definition of a function? Develop a lesson plan for each of the three examples you select.
7. Do you think a beginning teacher should be assigned to teach a precalculus class? Why or why not?

Exercise 13.3

1. The text discusses dealing with f(g(x)) when f(x) = 2x and g(x) = 3x - 4. Describe the advantages and disadvantages of approaching this by using a graphing calculator or function plotting software.
2. When f(x) = x2 + 5x - 6 and g(x) = x2 + 7, what is f(g(x))? What is g(f(x))? Does f(g(x)) = g(f(x)) in general? When does g(f(x)) = f(g(x)), if ever? Develop a lesson plan for this problem. It should include technology, and you should assume the students have the appropriate skills and background with the calculator or software selected.
3. Describe how you would determine a class background on compound functions.
4. How extensive should the treatment of compound functions be in a precalculus class? Defend your position.

Exercise 13.4

1. Find a different derivation of the quadratic formula. Compare and contrast it with the one presented here. Are the differences significant or mostly cosmetic and author preference? How should secondary students react to these different avenues to arrive at the same destination? Why?
2. Can the quadratic formula be introduced to students prior to the traditional algebra class? Defend your position.
3. Are there other derivations like that of the quadratic formula given here that students should have seen in prerequisite courses for college algebra? If you say yes, list at least three and discuss their value to the precalculus course. If you say no, defend your position, part of which should include a rationalization for why the particular derivation should or should not be included in the precalculus course.

Exercise 13.5

1. Create a lesson designed to teach a class about a quadratic function (parabola) that is symmetric and opens downward.
2. Using part 1 of this exercise, modify the lesson so the function shows the other two possible cases of roots. Do you think this is too much to cover in one day? Why or why not?

Chapter 13: Problem Solving Challenges

 

Question 1

What would be the units digit of ?

Answer    [Click to reveal...]

 

 

Question 2

There is a hill that is two miles from the base to the top on the north side and one mile from the top to the bottom on the south side. Jack has an old car that can only go up the hill at an average speed of 40 miles per hour, but he can race down the hill as fast as he desires. What will Jack's average speed have to be going down the south side of the hill to average 60 miles per hour over the entire hill?

Answer    [Click to reveal...]

Chaper 13: Videos

 

Introduction

Sticky Question

Problem Solving