# Chapter 12 – Geometry

For many students, geometry is the outside mathematical subject that you either love or hate. For many students, geometry has no connection to algebra or other mathematical concepts. Connecting geometry to algebraic concepts is important as students continue their mathematical learning. Hands-on activities that include the use of technology can allow students to visualize geometric concepts and discover important mathematical properties. Allowing students to learn in groups from other students can be key to their development.

## Exercises

### Exercise 12.1

- Compare and contrast the concept of distance when considered in Euclidean and taxicab geometry.
- The van Hiele levels have been modified to reflect current research in mathematics education. How have the levels been modified? Can the five levels developed in 1984 still be used effectively?
- Examples of specific activities and situations involving student placement in the van Hiele model can be found in a variety of articles and books, including some listed with this question. Select one of the articles listed, or another that discusses the van Hiele model, and present a written response to it.

Burger, W. F. and Shaughnessy, J. M. (1986). ‘Characterizing the van Hiele levels of development in geometry,’ *Journal for Research in Mathematics Education*, 17, 31{en}48.

Geddes, D., Fuys, D., and Tischler, R. (1985). *An Investigation of the Van Hiele Model of Thinking in Geometry Among Adolescents (Final Report). *Research in Science Education (RISE) Program of the National Science Foundation, Grant No. SED 7920640. Washington, DC: National Science Foundation.

Grouws, D. A. (ed.) (1992). *Handbook of Research on Mathematics Teaching and Learning*. A project of the National Council of Teachers of Mathematics. New York: Macmillan.

Hoffer, A. (1983). ‘Van Hiele-based research,’ in R. Lesh and M. Landau (eds.), *Acquisition of Mathematical Concepts and Processes* (pp. 205{en}277). New York: Academic Press.

Lindquist, M. M. (ed.) (1987). ‘Learning and teaching geometry, K–12,’ in *1987 NCTM Yearbook*. Washington, DC: National Council of Teachers of Mathematics.

Usiskin, Z. (1982). *Van Hiele Levels and Achievement in Secondary School Geometry*. Final report, Cognitive Development and Achievement in Secondary School Geometry Project. Chicago: University of Chicago.

van Hiele, P. M. (1986). *Structure and Insight: A Theory of Mathematics Education*. Orlando, FL: Academic Press.

### Exercise 12.2

- Do you think a student who is at van Hiele Level 2 is capable of doing a problem more than one way as was discussed in problem solving? Explain your position.
- Do you see a need for concern that students entering a high school geometry course that appears to be van Hiele Level 3 are only at van Hiele Level 2? Why or why not?
- Compare and contrast the basic tenets of Piaget and the van Hieles. Conclude with a position statement dealing with which seems most logical to you, and support it with some explanation of how you arrived at your conclusion.

### Exercise 12.3

- Compare and contrast the geometry expectations for elementary students in Grades K–5 within the Common Core Standards and NCTM’s Standards 2000. Are topics missing in the Common Core Standards that you feel should be in the elementary curriculum? Are there topics in the Common Core Standards you feel should be omitted?
- Investigate a scope and sequence chart that accompanies an elementary mathematics series. List the major geometry topics covered, the grade level at which they are presented, and the number of times the students are exposed to each concept throughout the elementary curriculum.
- At what point in the book are the major geometry topics introduced? Are these topics repeated from the prior year?

### Exercise 12.4

- Determine whether or not “proofs” like that shown in Figure 12.10 of the text are limited to regular polygons. Show an example to support your position.
- One US president did a couple proofs of the Pythagorean theorem. Name the president and show examples of his proofs.
- Do the Pythagorean theorem “proof” shown in Figure 12.10 using semicircles.

### Exercise 12.5

- Describe the sequence of steps necessary to bisect an angle by folding it with paper. You should do this before you describe it. Give your instructions to a novice to determine if they are clear enough to produce the desired results.
- Describe the sequence of steps necessary to produce a parabola using paper folding. You should do this before you describe it. Give your instructions to a novice to determine if they are clear enough to produce the desired results.
- Do a Patty Paper construction. This should be different from the ones described or assigned in this text. Cite the reference used to obtain the problem. Describe the sequence of steps used to complete the construction.

### Exercise 12.6

- Is the figure created in Figure 12.12 of the text an exact duplicate or distorted? Give a geometric defense of your response.
- Is there a need for an informal geometry course in the curriculum that precedes the typical high school geometry course? Why or why not?
- Rather than a separate informal geometry course, should the topics be intertwined into the current preformal geometry curriculum as is now done in most districts? Why or why not?

### Exercise 12.7

- Describe how you would deal with the problem described earlier that uses Hero’s formula to find the area of a triangle in a class of appropriately ready students.

### Exercise 12.8

- Create a lesson plan having the students complete a formal proof for the theorem shown in Figure 11.27.

Select two theorems typically found in a formal geometry class and develop lessons designed to have students complete a formal proof of each theorem. In each lesson the selected theorem should be established intuitively with the students prior to the formal proof. The selected theorems should not be ones discussed in this text. The selected theorems do not have to be related.

## Problem Solving Challenges

- The sum of the interior angles of two regular convex polygons is 21. The sum of the number of diagonals of each polygon is 85. How many sides does each polygon have?

Hint: Might consider that Number of Diagonals of a Polygon = **n(n-3)/2**

## Answer

1. Answer/solution: 8 and 13.

If a and b are the number of sides of the two regular polygons, then a + b = 21. The number of diagonals of a polygon can be found by $$\frac{a(a-3)}{2}$$ where a is the number of sides of the polygon. Therefore, $$\frac{a(a-3)}{2}+\frac{b(b-3)}{2}=85$$

Since a + b = 21, then a = 21 – b.

Substituting

$$\begin{array}{l}\frac{(21-b)((21-b)-3)}{2}+\frac{b(b-3)}{2}=85\\ \frac{(21-b)(18-b)}{2}+\frac{b(b-3)}{2}=85\\ (21-b)(18-b)+b(b-3)=170\\ 378-18b-21b+{b}^{2}+{b}^{2}-3b=170\\ 378-42b+2{b}^{2}=170\\ {b}^{2}-21b+189=85\\ {b}^{2}-21b+104=0\\ (b-13)(b-8)=0\end{array}$$- A princess is in love with a dashing knight. Unfortunately, the King prefers another suitor for his daughter. The King has locked the princess in the castle tower. The castle is surrounded by a square moat that is 10 yards wide. The knight is attempting to cross the moat, but only has two 9.75 yard planks, and no way to fasten them together. How can the brave knight bridge the moat?

Hint: Might use a board on the corner of the moat.

## Answer

Answer/solution: 14.

Answer: Place one board diagonally across the corner of the moat. Place the second board from the midpoint of the first board to the corner of the castle to form a T.

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.