2. 1 2008 USM Summer Math Institute Co-Sponsored by Institutions of Higher Learning (IHL) U.S. Department of Education (No Child Left Behind Funding) The Center for Science and Mathematics Education The Department of Mathematics College of Science and Technology College of Education and Psychology Day Sixteen Repeating Decimals, Irrational Numbers, Number Density, Compound and Simple Interest (CellSheet and Spreadsheet), and Expanded Form and Patterns

3. 2 2006 USM Summer Math Institute Your Mathematics Instructors, Staff and Partners Ms. Michelle Green, Co-Director and Co- Instructor for (SM) 2 I – Stringer Attendance Center – National Board Certified/Early Adolescence – Email mgreen@westjasper.k12.ms.us, Phone 601- 428-5508mgreen@westjasper.k12.ms.us Dr. Myron Henry, Director and Co-Instructor for (SM) 2 I – Department of Mathematics and the Center for Science and Mathematics Education – Johnson Science Tower 314 – Email myron.henry@usm.edu, Phone 601-266-4739 or 266-6516myron.henry@usm.edu All Participants (that’s you)

4. 3 Day Sixteen of (SM) 2 I 1.Repeating decimals 2.Virginia and the irrational numbers 3.Compound and simple interest calculated by you on the calculator (i. e., more on Cellsheets)

5. 4 Virginia, terminating decimals are fractions! Are numbers with repeating decimals fractions ( i.e., rational numbers) ?

6. 5 But what about a number like

7. Convert the Repeating Decimals to Rational Numbers Algebraic Means 6

8. Work Space

9. 8 Observations: All exponents of the square of the rational number in primes to powers are even. If 2 is part of the prime factorization, then after simplification, it at most appears in either the numerator or denominator but not both. This pattern pertains for any rational number. Virginia, your friends have told you all numbers are rational numbers? Is that true? A rational number out of the air! It’s a square out of the air! Simplify the square of the rational.

10. 9 Find a number that is not rational For any positive whole numbers a and b (we may assume they have no common factors), suppose Case I. If in the prime factorization to powers of squared does not contain the prime number 2 in either the numerator or denominator, then both a and b are odd numbers. But then which means an odd number equals an even number, a contradiction.

11. 10 Case II. If in the prime factorization to powers of squared contains the prime number 2 (to an even power) in denominator, then the numerator a 2 is an odd number and which again means an odd number equals an even number, a contradiction. Case III. If in the prime factorization to powers of squared contains the prime number 2 (must be to an even power) in numerator, then the denominator b 2 is an odd number and Then which again means an odd number b 2 equals an even number, a contradiction. Virginia, this means, is not a rational number!

12. 11 Virginia your friends have told you this Thanksgiving cartoon is accurate?

14. So, Virginia, what about these numbers? Rational numbers or irrational numbers?

16. Yes, Virginia, there are numbers that are not rational numbers! By Myron Henry, former editor of the Peru (IN) Sun. Acknowledgement: Thanks to Ms. Melanie Fuller, eighth grade mathematics teacher, Rod Paige Middle School, Lawrence County School District, for her important role in the conversation leading to Yes, Virginia, there are numbers that are not rational numbers. exists!

17. Yes Virginia, π ≠ 22/7

18. 17

24. 23 Simple Interest Eric deposited $10,000 in a savings account for his daughter Karen’s freshman year of college. The bank pays 8% tax-deferred simple interest. His daughter is currently beginning her senior year in high school. How much will Eric’s account be worth when his daughter begins college next year? Set the problem up Look for a pattern!

25. 24 Simple Interest Eric deposited $10,000 in a savings account for his daughter Karen’s freshman year of college. The bank pays 8% tax-deferred simple interest. His daughter is currently beginning her junior year in high school. How much will Eric’s account be worth when his daughter begins college in two years? What if she starts in three years? Look for a pattern!

26. 25 Compound Interest Eric deposited $10,000 in a savings account for his daughter Karen’s freshman year of college. The bank pays 8% tax-deferred interest annually compounded quarterly. His daughter is currently beginning her senior year in high school. How much will Eric’s account be worth when his daughter begins college next year? Set the problem up

27. 26 Look for a pattern!

28. 27 If Karen is just beginning her junior year of high school, how much will Eric’s account be worth when she starts college in two years? What if Karen waits for three years before beginning college? Look for a pattern!

29. 28 Jim wants to deposit money in a account to buy a good quality used car in three years. The following interest rates are available to him. 1.6.1% simple interest. 2.6.0% compounded annually 3.5.96% compounded semiannually 4.5.92% compounded quarterly. Find how much Jim has after three years if he invests $10,000. Which account should he choose to maximize his return? More Examples Simple Interest: A = P·( 1 + n · r)

30. 29 Simple Interest: A = P·( 1 + n · r)

31. 30 To have $10,000 available in five years under scenarios 2 and 4, how much must Jim invest now ?

32. 31 A Check

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35. 34

37. Compute the value of the expressions: 1 H 9 + 2, 12 H 9 + 3, and 123 H 9 + 4. Use patterns and number sense to determine what you would expect are the values of 1234 H 9 + 5 and 12345 H 9 + 6. Use patterns and number sense to write down the pattern for the numbers that end with + 7, + 8, +9, and +10. Use expanded form (powers of ten) to demonstrate and explain why the pattern observed above works for 123 H 9 + 4 and 1234 H 9 + 5 (hint: think 9 = 10 – 1 and the distributive law).

38. Compute the value of the expressions:1 H 9 + 1, 22 H 9 + 2, and 333 H 9 + 3 (Now remember the distributive law at that 9 = 10 - 1.) By observing patterns and employing number sense, describe in words what you would expect are the values of : 4444 H 9 + 4 and 55555 H 9 + 5. Use expanded form (powers of ten) to demonstrate and explain why the pattern observed above works for 333 H 9 + 3 and 4444 H 9 + 4.

39. Use expanded form (powers of ten) to demonstrate and explain a pattern 1  9 +2, 12  9 +3, 123 H 9 + 4 and 1234 H 9 + 5

41. Use expanded form (powers of ten) to demonstrate and explain a pattern 1  9 +1, 22  9 + 2, 333 H 9 + 3 and 4444 H 9 + 4