MATH 2030 – Spring 2010
Project 1 – Due: 4 March 2010 (tentative)

You must complete the project in a group of two to three students. Normally, all members of the group will receive the same grade; however, the instructor reserves the right to conduct individual interviews over the content of the project and to assign different grades to different members of the group.

Introduction

We have discussed several aspects of solving mathematical problems in class. For practice in mathematical problem solving, you are to completely solve five (10 points each) of the problems listed below (choosing at least one problem from each section). A grading rubric will be posted on the course website.

Final Report

You are to present a written report describing your findings. Give a well-organized explanation and details about how the problem was approached and explored so that the reader can follow/construct/understand your work with minimal effort. You will be graded on the quality and clarity of your written presentation as well as the mathematical accuracy of your paper.

Specifically,

• Each problem must be on separate pages.
• Restate the problem in your own words.
• Clearly explain your reasoning.
• Include numerical, graphical data (to the extent that it is applicable) to support your arguments and conjectures. Label diagrams, tables, graphs, or other visual representations you used.
• Include multiple approaches/representations (numerical, graphical and symbolic) to the problem and the solution.
• Provide an algebraic proof/solution for your conjectures/observations where it is applicable.
• Discuss possible extensions for one of the problems and explore/investigate the extensions.
• You may submit the project once prior to 25 February 2010 to receive feedback.
• INDIVIDUALLY, and on a separate piece of paper (NOT submitted with the solutions to the problems), discuss your participation in the group AND discuss the participation of your partner(s) (2 points deducted if not submitted).

Please follow these guidelines when preparing your report:

• Use complete, grammatically-correct sentences. For each problem include a title and the problem statement.
• Projects should include a mixture of tables, graphs, equations, and written explanations. Your textbook can serve as a model. For example, look at the equations, tables, and graphs in your textbook and notice the way the text inserts these into the flow of its written explanations. In particular, note that an equation is usually easier to read if centered on a line by itself.
• Projects must be typed. You may write equations and formulas in by hand or you can use the built-in equation editor in Microsoft Word. Click on Insert on the command bar, select Object, and select Microsoft Equation.
• Graphs from your calculator may be printed or copied into a word processor using TI-Graph Link software.
• Tables and graphs can be made and printed or copied into a word processor using Excel software. I am willing to help you build an Excel spreadsheet but you will need to tell me what you want the spreadsheet to compute.
• You are not writing a user's manual on how to use the calculator. Do not write about which keys you pressed on your calculator.
• Turn in one project per group. All work must be your own. The instructor will keep all projects. Make a copy for your files before turning in your paper.

Section 1

1. How many squares, of any size, are on a 3×3 checkerboard? 4×4? 8×8? n×n?

2. What is (x–a)(x–b)(x–c)···(x–y)(x–z)?

3. Find the units digit 7¹⁸⁹. Determine a general rule for finding the units digit of 7ⁿ, where n is any whole number.

4. In the range of numbers between 1 and 209, how many times will the digits of each number add to 8? In the range of numbers between 1 and x, how many times will the digits of each number add to 8? Give your answer in terms of x.

Section 2

5. Find an equation that relates the area of a kite to its diagonals.

6. Find a number that is the sum of two squares and whose square is also the sum of two squares.

7. A group of positive integers are divisible by 3 and by 5. When divided by 7, each integer in the group has a remainder of 4. How do each of the terms in the group relate? Write a formula that will predict the n^th term in the group.

8. The values of a, b, c, and d are 1, 2, 3, and 4 but not in that order. Find the largest possible value of ab + bc + cd + da.

Section 3

9. Consider the unit fraction 1/n, for n ≥ 50. What patterns do you notice in the decimal form of these unit fractions? What do you notice about the terminating decimals? The repeating decimals? The decimals that repeat after a delay? Which unit fraction (for n ≥ 50) has the decimal expansion of longest period?

10. Consider the Fibonacci's sequence: 1, 1, 2, 3, 5, 8, 13, ... where each term is generated by the sum of the previous two terms. Now divide successive terms, such as 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ... Explain what eventually happens to this ratio. What happens to the ratio if the first two terms of the Fibonacci's sequence were numbers other than 1?

11. How many zeros are at the end of the whole number 1000! (i.e., 1000·999·998···3·2·1)?

12. A fraction containing the digits 1 through 9 is called a "pandigital fraction". Find a pandigital fraction equal to 1/2. How many of such fractions equal to 1/2 exist?

Section 4

13. Consider two-digit numbers that end with 5. Find a method to square them in your head.

14. How many triangles can be constructed with integral sides and a perimeter of 15?

15. Pick any two positive integers (they can be the same number); add the two numbers and multiply the two numbers. Add the sum and product. What is the largest number less than 50 that you cannot get with this process?

16. The number of days in a year, 365, is a special number because it is the sum of three squared numbers. Find these three numbers. Find all possible sets of numbers. Is 366, the number of days in a leap year, just as special?