Mathematics Project

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Math= ematics Project

Class:10 &n= bsp; = ; &n= bsp; = Date of submission:07/07/08

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General Instructions: =

-= = The project should be hand written

-= = Credit wil= l be given to original and creative use of material/pictures/drawings/methods of illustrating

-= = The projec= t must be presented in a neatly bound simple folder.

-= = Your last = page should be

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= Name: &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; Class

= Roll No.

Assessment of the project work:<= /o:p>

Identificat= ion and statement of project	=
Design of p= roject	=
Procedure a= dopted	=
Write up of= project	=
Interpretat= ion of result	=

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Any one of the following projects may be chosen

A) Linear Equations=

Pr= oject 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 do= es _____ affect _____" questions.

2. Select the question that makes t= he 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&= quot; line.

6. Find the equation of your line.<= /span>

Re= spond to the following questions

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

8. Was your data positively correla= ted, negatively correlated or neither? Give possible explanations for the relationships or absence of relationships that you see in the data.<= /p>

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

10. What information does the slope indicate?

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

B) &n= bsp; Integer trains<= /span>

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= :

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Notice that the 1-2-2 train and the 2-1-2 train conta= in the same rods but are listed separately. If you use identical rods in a differe= nt order, this is a separate train.

· 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 tra= ins of length n. (Assume you have rods of every possible integer le= ngth 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

3D"Objective:

Objectiv= e: 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. T= he arbelos has been studied by mathematicians since ancient times, and was nam= ed, 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 a= bout a triangle,

similar triangles,

area of a circle,

Tangents are perpendicula= r 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 fac= ts, and

Spend some time thinking about the problem and you should be able to come up with the proof.
Present your findings i= n a 3-4 pages handwritten report.

D) The Birthday para= dox

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, the= re is a 50% chance that two people will have the same birthday.

Is this really true?

Experimental Procedure

First you will need to col= lect 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 h= ave 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 l= ists from your own school for different classes.

Take the class lists of about 12 sections. Pass these aro= und 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 colle= cted 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 bi= rthday? 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) &= nbsp; Peri= meters 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 semicirc= le.

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 a= rea 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 ne= ed :

pencil,

paper,

compass, and

straight edge.

Here's a suggestion for = your display: in addition to your background research and your proof, you c= an make a model of the Figure with colored pape= r. 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. Repe= at for 3 different measurements of semi circles.

Experimental Procedure

Do your background resea= rch,

Organize your known fact= s,

Perform the experiments = for 3 different semi circles

Tabulate your findings
Mathematically prove the result

Present your work in 3-4 handwritten pages.