Teaching Secondary Mathematics

Chapter 4: Exercises

Exercise 4.1

1. Describe how statistics and algebra could be related in a middle school setting.
2. Relate number theory and geometry in an environment appropriate for a geometry student.

Exercise 4.2

1. Find a number trick that could be used to motivate students to learn the standard addition algorithm.
2. Find a number trick that could be used to motivate students to learn the standard subtraction algorithm.

Exercise 4.3

1. List five weird questions that could be used to stimulate students to learn mathematics. At least two should be appropriate for middle school and at least two for high school.

Exercise 4.4

1. Do the tower puzzle for five disks and show the solution using base 2 numeration.
2. Can you define a process that will predict where to move a disk? That is, if the three pegs the disks are placed on are named A, B, and C, devise a system that would say where a specific disk in the sequence of moves would be placed.
3. There is a similar rule for the base 2 numeration solution of the tower puzzle using 1s rather than 0s. Describe it.
4. Locate two additional websites that provide interactive versions of the Tower of Hanoi. 

Exercise 4.5

1. Define the generalized formula for the nth heptagonal and octagonal numbers.
2. Describe connections for pentagonal or hexagonal number sets as was done for triangular numbers, the handshake problem, and joining vertices of a convex n-gon with line segments.
3. Define a generalization for the common differences of each counting number column in Figure 4.6.

Exercise 4.6

1. Why do we teach the sequence Algebra I, Geometry, Algebra II?
2. What are the names of the women who invented the bulletproof vest, fire escapes, windshield wipers, and laser printers?
3. Identify another set of numbers that will react like 13 × 7 = 28 in a similar setting.

Exercise 4.7

1. Do the problem 16,873 ÷ 47 using long division. List all the potential problem areas or places where a student might be expected to make errors.
2. Ask a group of students to write a personal answer to the following question, “What do you want to learn?”  Allow 5{en}10 minutes to write an answer.  Do not put restrictions on the question such as “…in math class, what do you want to learn?” Review the student responses. What did you learn?

Exercise 4.8

1. Provide a paragraph-style summary of the last meeting of the class you are using this text with. Swap papers with another student from the class. How similar are the papers? What, if any, are the differences?
2. Locate a website other than MacTutor that can be used as an excellent resource or reference for the history of mathematics.
3. Create a word problem that would appeal to a student in a secondary mathematics setting.
4. Pick a secondary textbook explanation you feel is unclear. Rewrite it, eliminating all areas of concern. Compare your explanation with that of the author’s and describe the strengths and weaknesses of each.

Exercise 4.9

1. Describe some of the earliest uses of numbers and numerals.
2. Document the beginning of irrational numbers and how they were used in early computations.
3. Research at least one of the following topics and prepare a written summary of its development: Egyptian pyramids, golden section, golden ratio, Fibonacci numbers, networks, twisted surfaces, computational short cuts, percent, or measurement.
4. Describe how the concept of numbers has evolved through the ages.
5. Trace the evolution of place value systems.
6. Describe the development of 0 from its rudimentary conceptualizations through how we write and use it today.
7. Document the beginning of fractions and how they were used in early computations. Be sure to investigate computation involving unit fractions.
8. Document the beginning of decimals and how they were used in early computations.
9. Describe the development of negative numbers and how they were used in early computations.
10. Document the beginning of complex numbers and how they were used in early computations.
11. Document the beginning of transfinite numbers and how they were used in early computations.
12. Investigate regular polygons. How did the names originate? How were they constructed?

Exercise 4.10

1. Look at a recent series of mathematics texts designed for Grades 6, 7, and 8. Determine how much material is repeated from year to year. Defend why you feel this is or is not appropriate.
2. If you could communicate with the lead editor for a middle school mathematics text publisher, what would you suggest to make the texts more appealing and useful for you, as a teacher, and for your students?

Chapter 4: Computational Madness

 

Question 1

Simplify the following expression.

Answer    [Click to reveal...]

 

 

Question 2

By arranging the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 it is possible to come up with a fraction equivalent to one-eighth. For example:

Your task is to arrange the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 to come up with an equivalent fraction to one-fifth.

Answer    [Click to reveal...]

Chaper 4: Videos

 

Introduction

Sticky Question

Problem Solving