Unit 2 Cross Curricular - Internet Project

Who is behind this geometry idea anyway?

Introduction | Task | Process | Guidance | Conclusion | Questions

Introduction
Have you ever wondered who first developed some of the ideas you are learning in your geometry class? Many ideas we study were developed many years ago, but people today are also discovering new mathematics. Many mathematicians of the past were men, but in recent years women mathematicians have been making their mark. Two women born in Texas have made contributions to new mathematical ideas. Mary Ellen Rudin’s specialties include finding counterexamples in the field of topology. Mary Wheeler made and proved conjectures relating to applied mathematics. To find out more about famous mathematicians, the Internet is a great resource. The table shows how people use the Internet.

     In this project, you will be using the Internet to research a topic in geometry. You will then prepare a portfolio or poster to display your findings.

The Task
Your school is having a Mathematics Appreciation Day. One event for the day is a competition for the best research project on a mathematical topic. Since you are currently studying geometry, your teacher wants each student in your class to research a topic in geometry. She wants the topic to be related to Unit 2 in your textbook. You will need to prepare a portfolio or poster to display the results of your research. If you prefer, you can prepare a Web page with this information that others can view. Be sure that your portfolio or Web page contains the following information:

• a description of the geometry topic you are researching, for example, the Pythagorean Theorem or fractals;
• the names of at least two people from the past or the present that contributed to this topic;
• a written description of the contributions that each of the two people made to your chosen topic;
• a problem relating to your topic that was proposed or solved by one of the people you have chosen.

• Select a geometry topic to research from Unit 2 in your Geometry textbook. For help, try these Web sites.
  www-groups.dcs.st-and.ac.uk/~history/
  www2.bw.edu/~dcalvis/history.html
  aleph0.clarku.edu/~djoyce/mathhist/mathhist.html
  mathforum.org/isaac/mathhist.html
  www.tmeg.com/bab_mat/bab_mat.htm
  metalab.unc.edu/expo/vatican.exhibit/exhibit/d-mathematics/Greek_math.html
  www.amug.org/~jpaul/math.html
  archives.math.utk.edu/topics/history.html
• Find two people who contributed to or studied this topic. The people can be from many years ago, such as a famous Greek mathematician, or they can be researching the topic today.
• Write at least a one-page summary of the contributions that the two people you selected made to the topic you have chosen.
• Display one problem that one of your chosen people proposed or solved.
• Be creative. Add some additional data, information, or even pictures to your portfolio or Web page.

1. Make a timeline for the geometric topic that you have chosen to research. Mark the dates of important contributions to the topic.
2. Write a complete history of the topic you have chosen.
3. Research how the study of the geometric topic you have chosen has helped in other fields. For example, how has the study of fractals contributed to science?
4. Research chaos theory. How does it relate to fractals?
5. Research several careers that depend heavily upon mathematics. What mathematics courses do you need to take in high school to prepare for these careers?

• Present your project to your class or at a family night. Have parents and students work on a historic problem relating to the topic you chose to research.
• Present the information on a Web page. Have other students critique your project and help you to make improvements to your project.
• Write a one-page summary of your project, including what you have learned from researching the history of a geometric topic.

Lesson 4–6
Hans is looking on the Internet for geometry topics in Chapter 4. In his research he finds that Thales of Miletus, a Greek mathematician and philosopher, is credited with three important geometric ideas from Chapter 4 of his textbook.

1) The base angles of an isosceles triangle are equal.
2) Two triangles are congruent if they have two angles and the included side equal.
3) An angle in a semicircle is a right angle.
Hans plans to include a proof of 3, as shown below, in his geometry project.

Complete the proof by filling in the reason for each statement.

[Insert the diagram of the circle and inscribed triangle from the project book p. 7.]
Given: ΔABC in circle O with diameter
Prove: ABC is a right angle.

Lesson 5–1
In his Internet research, Paulo found a history of the Nine-Point Circle. This geometric figure has sometimes been attributed to mathematician Leonhard Euler in the 1700s. However, others attribute it to Karl Feuerbach, who published a proof about the circle in 1822. Paulo plans to include the construction of a Nine-Point Circle in his geometry project.

1. Carefully follow the steps below to construct your own Nine-Point Circle. After you have located and labeled the points in each step, erase any extra lines to avoid confusion.
   1. Draw a large, acute, scalene triangle and label the vertices X, Y, and Z.
   2. Construct the perpendicular bisectors of the sides of the triangle. Label the midpoints of the sides A, B, and D. Label the point where the perpendicular bisectors intersect as C, the circumcenter.
   3. Using the midpoints A, B, and D, draw the medians of the triangle. Label the point where the medians intersect as E, the centroid.
   4. Construct the altitude from each vertex to the opposite side. Label the points where the altitudes intersect the sides as H, J, and K. Label the point where the altitudes meet as O, the orthocenter.
   5. Draw the segments and Find the midpoint of each segment and label the points L, M, and N.
   6. Construct the angle bisector of each angle of ΔXYZ. Label the point of intersection of the angle bisectors as I, the incenter.
   7. Draw Bisect the segment and label the midpoint T.
   8. Draw a circle with center T and radius AT.
   9. If you have done the constructions very carefully, the points A, B, D, H, J, K, L, M, and N should lie on (or very close to the circle).
2. Construct two more Nine-Point Circles, one for a right triangle and one for an obtuse triangle.
3. Write several sentences comparing the Nine-Point Circle for an acute, a right, and an obtuse triangle.

Lesson 6–4
Mercedes is researching fractals for her project. She has learned that Warclaw Sierpinski, a Polish mathematician, is credited with introducing the Sierpinski Triangle, or gasket, in 1916. Later, Benoit Mandelbrot worked further with this well-known fractal. Mercedes plans to include a problem about the Sierpinski Triangle in her project. To make a Sierpinski Triangle, start with an equilateral triangle (Figure 1). Connect the midpoints of the sides of the triangle. Remove the middle triangle (white), leaving three colored triangles (Figure 2). Repeat this process. The four figures below show how the process continues.

1. Suppose the area of the first triangle is 16 square units. Find the total area of the colored triangles for the next three figures shown above. (Hint: Notice that in the second figure, one-fourth of the large triangle is to be cut out.)
2. How does the area of the colored triangles change as more and more triangles are cut out of the original triangle?
3. Suppose the side length of the first triangle is 8 units. Find the total perimeter of the colored triangles for each of the four figures shown above.
4. How does the perimeter of the colored triangles change as more and more triangles are cut out of the original triangle?

Lesson 6–7
Shonny has decided to research geometric mean for her project. She discovers that the ancient Greeks, possibly even Pythagoras, used a mean called the golden mean (also called the golden ratio or golden section.) A rectangle formed using the golden mean was often used in art and architecture as it was thought to be pleasing to the eye. The diagram shows a line segment that has been divided such that the ratio of the shorter portion to the longer portion is the golden ratio.

1. For the above segment, let x represent the longer segment. To find the golden ratio, write a proportion such that the longer of the two segments is the geometric mean between the shorter segment and the entire segment.
2. Use the Quadratic Formula to solve the proportion for x. Find the value in both radical and decimal form.