1. Math 101. Core Competency in Mathematics. richard/math101

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1 1. Math 101 Core Competency in Mathematics Dept. of Mathematical Sciences Northern Illinois University Professor Richard Blecksmith richard/math Core Competency The mathematics core competency requirement can be satisfied by passing Math 101 (with a D or higher) obtaining a grade of C or better in Math 155 (trig) Math 201 (math for elementary education) Math 206 (discrete mathematics) Math 210 (finite mathematics) Math 211 (business calculus) Math 229 (calculus) obtaining credit for one of the courses listed above, except Math 101, through credit by examination (Advanced Placement) 1

2 2 3. Core Competency Cont d obtaining a grade of C or better in Stat 208, Stat 301, Stat 350, Ieng 335, or Ubus 223; and obtaining a grade of C or better in Math 110 an ACT mathematics score of at least 24 an SAT mathematics score of at least 560 an A- or B-level placement on the mathematics placement exam obtaining equivalent transfer credit passing the Mathematics Core Competency Exam 4. Text Text: Mathematical Thinking and Quantitative Reasoning By: Linda Sons, Peter Nicholls, and Joseph Stephen third edition Kendall/Hunt Publishers Plus Additional Handouts Available on the Math 101 Webpage Prof. Richard Blecksmith 5. Instructor Office: Watson Hall 344 Phone: (815) Office Hours: M W 3 4, W 1 2 and by appointment

3 3 6. Recitation Frank Gambino B1 Tues 2:30 3:20 DU 340 B3 Tues 3:30 4:20 DU 418 B5 Thurs 2:30 3:20 FW 201 B7 Thurs 3:30 4:20 DU 348 Kenny Albright B2 Tues 2:30 3:20 WZ 103B B4 Tues 3:30 4:20 DU 268 B6 Thurs 2:30 3:20 DU 340 B8 Thurs 3:30 4:20 DU Important Dates Test 1: Wed Sept 27, 2006 Last Day to Withdraw: Fri Oct 20, 2006 Test 2: Fri Oct 27, 2006 Test 3: Mon Nov 20, 2006 Thanksgiving week All makeups will resolved at the end of the semester Final Exam: Wed Dec 13, 8-9:50 P.M. 8. Tests and Quizzes 3 tests worth 100 points each Final exam worth 200 points Homework (9/11) for 90 points Recitation Quizzes (9/11) for 90 points In-class MiniQuizzes (10/11) for 30 points Projects (2) for 20 points each Total: 750 points

4 4 9. Grading Scale The scale will be at least as generous as A: 637 points (85%) B: 562 points (75%) C: 450 points (60%) D: 375 points (50%) Note: passing the final exam (with a score of at least 100 points out of 200) automatically guarantees that you will pass the course. 10. How to Pass the Course To get a C in this course, turn in the all the homework and study for the quizzes and miniquizzes Homework: 85 pts Recitation Quizzes: 80 pts Miniquizzes: 25 pts Projects: 40 pts Total: 230 points 11. How to Pass the Course Now suppose you know how to solve 30% of the test questions: 30% of 500 = 150 points Guess on the remaining 350 points The probability of guessing correctly is 1/5 1/5 of 350 = 70 points Total: 220 points = 450, the guaranteed cutoff for a C.

5 5 12. Observations This strategy requires that you: work very hard on the non test part of the course know something (30 percent of the material) are an average guesser The secret is to get as many easy points as you can come to class and recitation regularly work consistently 13. Calculators You will need a calculator for this course. It needs to have an exponentiation button (ˆ or x y ) parentheses buttons ( and ) at least one memory There are two types of calculators which you may use: scientific or business graphing calculator such as the TI Some calculating problems Compute Pressing = on your calculator gives the answer: 11, as expected. Compute Pressing = on your calculator gives the answer: 17,

6 6 which may be a surprise if you expected your calculator to add to get 5 and then multiply by a seond five to get 25 for an answer. 15. What happened? When evaluating an expression involving +,,, and, without parentheses, your calculator will compute the multiplication and divisions first Your calculator computes = as 2 + (3 5) = = Another Example Suppose you want to calculate If you press the buttons 2 + 3/5 = on your calculator, the answer will be 2.6 Without parentheses, your calculator is thinking 2 + 3/5 = = 2.6 To make your calculator add the 2 and 3 first, you must use parentheses: (2 + 3)/5 = which gives the correct answer Order of Operations 1. Parenetheses 2. Functions (such as square root)

7 7 3. Exponentiation 4. Multiplication and Division 5. Addition and Subtraction Example: To calculate 1000 ( ) Enter: 1000 (1 +.05/12)ˆ240 = to get the answer Find the Maximum Which is larger? 6 24 or 25 10? My calculator says 6 24 = E = E13 The second number is NOT larger than the first even though 9.5 > 4.7. Why not? 19. Scientific Notation These two numbers are written in scientific notation 6 24 = = is a 19 digit number 5 20 is only a 14 digit number So 6 24 is the larger number Can you think of a way to tell which number is bigger without using a calculator? 20. Precision How many digits should you write down?

8 8 As many as on the calculator display? Just 2 or 3? The correct answer is: as many as you need to solve the problem. Common sense should be your guide: For example, when working with money An exta value meals costs $3.55 My new car lists for 16, 549 Bill Gates is worth 55 billion dollars 21. Precision Cont d As a rule of thumb, your answer should be to an many significant digits as the data you used in the calculations. When giving a purely mathematical solution, however, you should avoid saying something like: The square root of 2 is 1 arguing that your answer is accurate to one digit, because 2 is just a one digit number. Instead, use at least four digits: 2 = The Quadratic Formula The equation ax 2 + bx + c = 0 has the solution x = b ± b 2 4ac 2a Find the roots of the equation 2x 2 + 5x 11 = 0 Plug a = 2, b = 5, and c = 11 into the formula

9 9 x = 5 ± ( 11) 2 2 = 5 ± Quadratic Formula Cont d On your calculate, compute these two answers separately ( 5 + ( ))/(2 2) = Answers: ( 5 ( ))/(2 2) = Answers: Using the Memory There is a short cut in computing the previous problem. The expression was computed twice. To save time, you could compute it once and save the result in memory. On a graphing calculator, compute ( ) X Then the two roots are calculated ( 5 + X)/4 = and ( 5 X)/4 = On a scientific calculator, use the Store Memory and Recall Memory keys. 25. Size comparisons Which is larger: a liter or pepsi or a quart of coca-cola? What is the difference in ounces?

10 10 In American track, the quarter mile race has been replaced by the 400 meter race. Which is larger: a quarter of a mile or 400 meters? What is the difference in feet? Which is larger: a football field (between the sidelines and from endzone to endzone) or one acre? What is the difference in square feet? Are you over one billion seconds old? Are your parents? Exactly how many years old is this? Metric to U.S. 26. Conversions 1 millimeter [mm] in 1 centimeter [cm] 10 mm in 1 meter [m] 100 cm yd 1 kilometer [km] 1000 m mile U.S. to metric 1 inch [in] 2.54 cm 1 foot [ft] 12 in m 1 yard [yd] 3 ft m 1 mile 1760 yd km 1 int nautical mile yd km 27. How Big is an Acre? Which is bigger: an acre or a football field? Webster Unabridged Dictionary: An acre is a U. S. and English measurement, meant to represent the amount of land a yoke of oxen could plow in one day.

11 11 One acre equals 160 square rods What s a rod? By Webster again, one rod equals 5.5 yards 28. An Acre Cont d 1 square rod = square yards = square yards 1 acre = square yards = 4840 square yards Now a collegiate football field measures 100 by yards. a football field = square yards So a football field is about 10% bigger than an acre. 29. A Football Field 1 acre Go Huskies!

12 12 Metric to U.S. 30. Area Conversions 1 sq cm [cm 2 ] 100 mm in 2 1 sq m [m 2 ] 10,000 cm yd 2 1 hectare [ha] 10,000 m acres 1 sq km [km 2 ] 100 ha mile 2 U.S. to metric 1 sq inch [in 2 ] cm 2 1 sq foot [ft 2 ] 144 in m 2 1 sq yd [yd 2 ] 9 ft m 2 1 acre 4840 yd m 2 1 sq mile [mile 2 ] 640 acres 2.59 km 2 Metric to U.S. 31. Fluids and Weights 1 gram [g] 1,000 mg oz 1 kilogram [kg] 1,000 g lb 1 tonne [t] 1,000 kg ton 1 liter [l] qt U.S. to metric 1 fluid ounce ml 1 pint 16 fl oz l 1 gallon 4 qt l 1 pound [lb] 16 oz kg

13 The Price of Gasoline A few years ago, I put the following problem on an exam: In Canada gasoline costs 74.9 Canadian cents per liter. The money exchange rate is Canadian dollars = 1 U. S. dollar. The metric conversion is 1 liter = gallons. What is the price in U.S. currency for one gallon of gasoline in Canada? Criticize the following actual student solutions: $16,540 $ What s the correct answer? Canadian dollar 1 U. S. dollar liter liter Canadian dollar gallons U. S. dollar = gallon U. S. dollar = 2.29 gallon Two years ago this seemed expensive Now it seems cheap!

14 Killer Summation The computation S = could be done on a hand calculator by entering each number, starting with 1 and ending with The chances for making an error, by pressing a wrong key while entering a digit, is very great. Besides, it takes a long time to enter a thousand numbers! 35. A better way: Write the list once, and directly below it, write the same list with the numbers in reverse order: S = S = Now add each entry in the top row with the entry in the row directly below it: = 1001, = 1001, = 1001, etc. You get 1000 sums of So 2S = = Dividing by two gives us the value S = /2 = Exercises: Try the backwards method on: Sequences like these are called arithmetic progressions.

15 Note that we obtain each term by adding the same number to the previous term Another Sequence A second type of sum we will meet in this class is obtained by multiplying the previous term by a common factor. These are called geometric progressions. If the first term is 1, then the sum of a geometric progression is S = 1 + x + x 2 + x x n 1 + x n. The trick for adding a geometric progression is to multiply S by x on a line just above S: 38. Geom. Sequences Continued xs =x + x2 + x x n 1 + x n + x n+1 S =1 + x + x 2 + x x n 1 + x n. Now subtact the second line from the first and notice that almost every term in the first row cancels a term in the seond row. The only uncancelled terms are the x n+1 in line 1 and the 1 in line 2. Hence xs S = x n Geom. Sequences Completed Start with: xs S = x n+1 1 Pull out S: (x 1)S = x n+1 1 Divide by x 1 to get a formula for S: S = xn+1 1 x 1

16 x + x 2 + x n = xn+1 1 x Example Compute the sum = In the formula 1 + x + x 2 + x n = xn+1 1 x 1 x = 2 and n = 5 Plugging these values into the formula gives = Since 2 6 = = = 63 1 = 63