Mathematics Project. Class:10 Date of submission :

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1 Mathematics Project Class:10 Date of submission : General Instructions: - The project should be hand written in about 5-8 A4 size sheets - Credit will be given to original and creative use of material/ pictures/ drawings / methods of illustrating as well as creative use of Mathematics - The project must be presented in a neatly spiral bound A4 size folder. - Cover page must have Name, Class, and Title of Project, followed by a page with the table of contents. There must also be a page at the end as conclusion. - Your last page should contain references and bibliography Please cut out the table below and place on the last page of your project. Name: Class Roll No. Assessment of the project work: Identification and statement of project Design of project Procedure adopted Write up of project Interpretation of result Viva Total :

2 Any one of the following projects may be chosen A) Linear Equations 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:

3 Notice that the train and the train contain the same rods but are listed separately. If you use identical rods in a different 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 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

4 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 1. Do your background research, 2. Organize your known facts, and 3. Spend some time thinking about the problem and you should be able to come up with the proof. 4. Present your findings in a 3-4 pages handwritten report.

5 D) 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 : o pencil, o paper, o compass, and o 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

6 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 1. Do your background research, 2. Organize your known facts, 3. Perform the experiments for 3 different semi circles 4. Tabulate your findings 5. Mathematically prove the result 6. Present your work in 3-4 handwritten pages. E) Golden Ratio In Mathematics and the Arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately Eg 1, 1, 2, 3, 5, 8, 13, 21, 34,... Let us consider 5, = = 1.6 approx. 8 5 = 1.6 This sequence of numbers is known as the Fibonacci sequence Let's build an approximation of the golden rectangle using square grid paper.. At each step we will calculate the ratio of length to width. Remember the Golden Ratio is approximately Step 1: Start with a square 1 by 1. Ratio Step 2: Add another 1 by 1 square.

7 Ratio Step 3: Add a 2 X 2 square. Ratio Step 4: Add a 3 X 3 square. Ratio Step 5: Add a 5 X 5 square. Ratio Step 6: It's Your Turn! Build the next approximation on square grid paper.

8 Then observe the pattern and answer the following: Length n = Length n + width n = The lengths of the various rectangles are 1, 2, 3, 5,..., Notice that each term is found by adding the 2 previous ones, the sequence known as the Every step of the rectangle will produce a ratio closer and closer to the golden ratio of approximately Which things around you are made in the golden ratio? Measure the following with a centimeter measuring tape or stick. (Round measurements to the nearest.5 cm.) Then rank the items from closest (1) to the golden ratio to least close (4). Rank Item Measured Length(cm) Width(cm) Ratio(L:W) TV screen Calculator Math Textbook Notebook paper Rs 100 Bill Find an item that has a ratio even closer to the golden ratio. List its dimensions and ratio below. Item Measured Length(cm) Width(cm) Ratio(L:W)

9 Challenge Problem Take an 8 1/2 X 11 inch sheet of paper. Measure and make a golden rectangle by making one straight cut. Give the dimensions of all possible golden rectangles you could cut from this sheet of paper.