**General Instructions:**

- **The project should be hand written**

- **Credit will be given to original and creative use of material/pictures/drawings/methods of illustrating **

- **The project must be presented in a neatly bound simple folder.**

**Any one of the following projects may be chosen**

Project Assignment

Think of a question that asks about a cause and effect relationship between two measurable quantities. (eg.. does fingernail length affect typing speed?)

1. Write two different "how does _____ affect _____" questions.

2. Select the question that makes the most sense to you and explain why you have chosen it.

3. Write a hypothesis to answer your question.

4. Graph your data using appropriate choices of scales and axis.

5. In pencil, draw your "best" line.

6. Find the equation of your line.

Respond to the following questions

7. What do the variables in your equation represent? What does the equation represent?

8. Was your data positively correlated, negatively correlated or neither? Give possible explanations for the relationships or absence of relationships that you see in the data.

9. Use your equation to predict two data points not represented by the data. How good do you think these estimates are? why?

10. What information does the slope indicate?

Present your findings in a 3-4 pages handwritten report. Graph must be included.

**B)** __Integer trains__

You can use rods of integer sizes to build "trains" that all share a common length. A "train of length 5" is a row of rods whose combined length is 5. Here are some examples:

Notice that the

� How many trains of length 5 are there?

� Repeat for length 6

� Repeat for length 7

� Come up with a formula for the number of trains of length *n*. (Assume you have rods of every possible integer length available.) Prove that your formula is correct.

� Come up with an algorithm that will generate all the trains of length *n*.

� Create trains of lengths 6,7. Record any findings, conclusions in 3-4 pages of handwritten work.

**C)����� Area of an Arbelos**

Objective

**Objective: Prove that the area of the arbelos (white shaded region) is equal to the area of circle CD.**

What is an arbelos? The arbelos is the white region in the figure, bounded by three semicircles. The diameters of the three semicircles are all on the same line segment, *AB*, and each semicircle is tangent to the other two. The arbelos has been studied by mathematicians since ancient times, and was named, apparently, for its resemblance to the shape of a round knife (called an arbelos) used by leatherworkers in ancient times.

An interesting property of the arbelos is that its area is equal to the area of the circle with diameter *CD*. *CD* is along the line tangent to semicircles *AC* and *BC* (*CD* is thus perpendicular to *AB*). *C* is the point of tangency, and *D* is the point of intersection with semicircle *AB*. Can you prove that the area of circle *CD* equals the area of the arbelos?

To do this project, you should do research that enables you to use the following terms and concepts:

- right triangles,
- circumscribing a circle about a triangle,
- similar triangles,
- area of a circle,
- Tangents are perpendicular to radii at the point of contact.

Materials and Equipment

For the proof,� you'll need :

- pencil,
- paper,
- compass, and
- straight edge.

Experimental Procedure

- Do your background research,
- Organize your known facts, and
- Spend some time thinking about the problem and you should be able to come up with the proof.
- Present your findings in a 3-4 pages handwritten report.

**D)���� The Birthday paradox**

Objective

The objective of this project is to prove whether or not the birthday paradox holds true by looking at random groups of 23 or more people.

Introduction

The Birthday Paradox states that in a random gathering of 23 people, there is a 50% chance that two people will have the same birthday. Is this really true?

Experimental Procedure

1) First you will need to collect birth dates for random groups of 23 or more people. Ideally you would like to get 10-12 groups of 23 or more people so you have enough different groups to compare. Here are a couple of ways that you can find a number of randomly grouped people.

- You could use birthday lists from your own school for different classes.

Take the class lists of about 12 sections. Pass these around each of these classes and collect the birth date data

- Use the birth dates of players on major teams. (Note: this information can easily be found on the internet).

2) Next you will need to sort through all the birth dates you have collected and see if the Birthday Paradox holds true for the random groups of people you collected. How many of your groups have two or more people with the same birthday? Based on the birthday paradox, how many groups would you expect to find that have two people with the same birthday?

3) Tabulate/Organize your data and findings in a 3-4 pages� handwritten report

E) ������� **Perimeters of Semi Circles**

Objective

The objective of this project is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Introduction

The figure below shows a semicircle (AE) with a series of smaller semicircles (AB, BC, CD, DE,) constructed inside it. As you can see, the sum of the diameters of the four smaller semicircles is equal to the diameter of the large semicircle. The area of the larger semicircle is clearly greater than the sum of the four smaller semicircles. What about the perimeter?

Your goal is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Materials and Equipment

- For the proof, you'll need :
- pencil,
- paper,
- compass, and
- straight edge.
- Here's a suggestion for your display: in addition to your background research and your proof, you can make a model of� the Figure� with colored paper. Use a compass and straightedge to construct the semicircles. Cut pieces of string or yarn equal to the arc-lengths of the semicircles. You can use these to demonstrate that the perimeter lengths are indeed equal. Repeat for 3 different measurements of semi circles.

Experimental Procedure

- Do your background research,
- Organize your known facts,
- Perform the experiments for 3 different semi circles
- Tabulate your findings
- Mathematically prove the result
- Present your work in 3-4 handwritten pages.