# Chapter 5 – Skills in Teaching Mathematics

How wonderful it would be if we knew everything about students, teaching, mathematics, and the teaching of mathematics. We could bottle it, sell it, become rich, and solve a lot of problems for everyone in the process. We all know there is no magic formula for teaching mathematics. In the process of attempting to find keys, we often stumble and miss objectives, but we also learn a variety of things that can be cataloged and used at a later date in other settings. This chapter deals with a broad collection of ideas, concepts, and strategies that should be useful in a variety of teaching environments. As you work through your career and attempt these suggestions, you will compile a list of methods and activities that will help your students learn mathematics.

## Exercises

### Exercise 5.1

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

### Exercise 5.2

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

### Exercise 5.3

- 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 5.4

- Do the tower puzzle for five disks and show the solution using base 2 numeration.
- 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.
- There is a similar rule for the base 2 numeration solution of the tower puzzle using 1s rather than 0s. Describe it.
- Locate two additional websites that provide interactive versions of the Tower of Hanoi.

### Exercise 5.5

- Define the generalized formula for the
*n*^{th}heptagonal and octagonal numbers. - 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. - Define a generalization for the common differences of each counting number column in Figure 5.5 of the text.

### Exercise 5.6

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

### Exercise 5.7

- 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.
- 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 5.8

- 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?
- Locate a website other than MacTutor that can be used as an excellent resource or reference for the history of mathematics.
- Create a word problem that would appeal to a student in a secondary mathematics setting.
- 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 5.9

- Describe some of the earliest uses of numbers and numerals.
- Document the beginning of irrational numbers and how they were used in early computations.
- 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.
- Describe how the concept of numbers has evolved through the ages.
- Trace the evolution of place value systems.
- Describe the development of 0 from its rudimentary conceptualizations through how we write and use it today.
- Document the beginning of fractions and how they were used in early computations. Be sure to investigate computation involving unit fractions.
- Document the beginning of decimals and how they were used in early computations.
- Describe the development of negative numbers and how they were used in early computations.
- Document the beginning of complex numbers and how they were used in early computations.
- Document the beginning of transfinite numbers and how they were used in early computations.
- Investigate regular polygons. How did the names originate? How were they constructed?

### Exercise 5.10

- 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.
- 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?

## Problem Solving Challenges

**Computational Madness**

- Simplify the following expression.

Hint: Try using x = 1234567891

## Answer

Answer/solution: 1234567890.

Let 1234567891 = N, then 1234567890 = N – 1 and 1234567892 = N + 1.

Therefore,

$$\begin{array}{l}\frac{1234567890}{{1234567891}^{2}-(1234567890)(1234567892)}=\\ \frac{N-1}{{N}^{2}-(N-1)(N+1)}=\frac{N-1}{{N}^{2}-({N}^{2}-1)}=\frac{N-1}{1}=1234567890\end{array}$$- 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

Answer: There are many possibilities. One such answer is

$$\frac{1}{5}=\frac{52769}{13845}$$Others include:

- 2769 / 13845
- 2973 / 14865
- 9237 / 46185
- 2697 / 13485
- 2937 / 14685
- 2967 / 14835
- 3297 / 16485
- 3729 / 18645
- 6297 / 31485
- 9723 / 48615
- 9627 / 48135
- 7629 / 38145

## Additional Learning Activities

**1. **Create a formula that shows how to determine the number of handshakes given to any number of people. Show that your formula works by adding a column to the table in the first exercise and completing it for each of the last three entries, showing your work.

You might notice that these are the triangular numbers.

The idea of generating formulas is common in the world of mathematics. We look at situations and try to come up with useful generalizations. Often these generalizations give us formulas that can be used in similar situations and that save time trying to find answers for each separate case. You have used many of these universally acceptable generalizations over your career. For example, formulas for area and perimeter, distances, rates, times, tax computations, and so on are among such generalizations.

**Your Turn**

- Develop a formula for finding the sum of any set of consecutive counting numbers, beginning with any number.

- Given that $$\frac{\mathrm{n}(\mathrm{n}+1)}{2}$$ gives the sum of the first n counting numbers, starting with 1, and that n
^{2}is the sum of the first n odd counting numbers, then it seems as if $$\frac{n(n+1)}{2}-{n}^{2}$$ should give the sum of the first n even counting numbers.

But $$\frac{n-{n}^{2}}{2}$$ is negative! What is wrong?

- What famous mathematician is associated with $$\frac{\mathrm{n}(\mathrm{n}+1)}{2}$$ which gives the sum of the first n counting numbers?