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1. Math 101 Core Competency in Mathematics Dept. of Mathematical Sciences Northern Illinois University http://www.math.niu.edu Professor Richard Blecksmith [email protected] http://www.math.niu.edu/∼richard/math101 2. 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 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 5. Instructor Prof. Richard Blecksmith • • • • Office: Watson Hall 344 Phone: (815)753-1835 Office Hours: M W 3–4, W 1–2 and by appointment 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 318 7. 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 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 230 + 220 = 450, the guaranteed cutoff for a C. 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 xy ) – 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 TI83 14. Some calculating problems • Compute 2 × 3 + 5 – Pressing 2 × 3 + 5 = on your calculator gives the answer: 11, – as expected. • Compute 2 + 3 × 5 – Pressing 2 + 3 × 5 = on your calculator gives the answer: 17, 6 – which may be a surprise if you expected your calculator to add 2 + 3 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 2 + 3 × 5 = as 2 + (3 × 5) = 2 + 15 = 17. 16. Another Example 2+3 • Suppose you want to calculate 5 • 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 + 0.6 = 2.6 • To make your calculator add the 2 and 3 first, • you must use parentheses: (2 + 3)/5 = • which gives the correct answer 1 17. Order of Operations • 1. Parenetheses • 2. Functions (such as square root) 7 • • • • 3. Exponentiation 4. Multiplication and Division 5. Addition and Subtraction .05 240 Example: To calculate 1000 1 + 12 • Enter: 1000 × (1 + .05/12)ˆ240 = to get the answer • 2712.64 18. Find the Maximum • Which is larger? 624 or 2510 ? • My calculator says – 624 = 4.73838133832E18 – 520 = 9.53674316406E13 • 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 624 = 4.73838133832 × 1018 520 = 9.53674316406 × 1013 624 is a 19 digit number 520 is only a 14 digit number So 624 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 – 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 = 1.414 22. The Quadratic Formula The equation ax2 + bx + c = 0 has the solution √ −b ± b2 − 4ac x= 2a • Find the roots of the equation 2x2 + 5x − 11 = 0 • Plug a = 2, b = 5, and c = −11 into the formula 9 • p 52 − 4 · 2 · (−11) √ 2·2 −5 ± 52 + 4 · 2 · 11 = 2·2 x= −5 ± 23. Quadratic Formula Cont’d • • • • • On your compute these two answers separately √ calculate, 2 (−5 + (5 + 4 × 2 × 11))/(2 × 2) = Answers: √ 1.4075 (−5 − (52 + 4 × 2 × 11))/(2 × 2) = Answers: −3.9075 24. Using the Memory There is a short cut in computing the previous problem. √ The expression 52 + 4 × 2 × 11 was computed twice. To save time, you could compute it once and save the result in memory. • • • • • On √ a 2graphing calculator, compute (5 + 4 × 5 × 11) → 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 • 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? 26. Conversions Metric to U.S. 1 millimeter [mm] 0.03937 in 1 centimeter [cm] 10 mm 0.3937 in 1 meter [m] 100 cm 1.0936 yd 1 kilometer [km] 1000 m 0.6214 mile U.S. to metric 1 inch [in] 1 foot [ft] 12 in 1 yard [yd] 3 ft 1 mile 1760 yd 1 int nautical mile 2025.4 yd 2.54 cm 0.3048 m 0.9144 m 1.6093 km 1.853 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 • 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 = 5.52 square yards = 30.25 square yards • 1 acre = 160 × 30.25 square yards = 4840 square yards • Now a collegiate football field measures 100 by 53 13 yards. 1 a football field = 5333 square yards 3 • So a football field is about 10% bigger than an acre. 29. A Football Field 1 acre 0 10 20 30 40 50 40 Go Huskies! 30 20 10 0 12 30. Area Conversions Metric to U.S. 1 sq cm [cm2 ] 100 mm2 0.1550 in2 1 sq m [m2 ] 10,000 cm2 1.1960 yd2 1 hectare [ha] 10,000 m2 2.4711 acres 1 sq km [km2 ] 100 ha 0.3861 mile2 U.S. to metric 1 sq inch [in2 ] 1 sq foot [ft2 ] 144 in2 1 sq yd [yd2 ] 9 ft2 1 acre 4840 yd2 1 sq mile [mile2 ] 640 acres 6.4516 cm2 0.0929 m2 0.8361 m2 4046.9 m2 2.59 km2 31. Fluids and Weights Metric to U.S. 1 gram [g] 1,000 mg 1 kilogram [kg] 1,000 g 1 tonne [t] 1,000 kg 1 liter [`] 0.0353 oz 2.2046 lb 1.1023 ton 1.0569 qt U.S. to metric 1 fluid ounce 29.574 ml 1 pint 16 fl oz 0.4731 ` 1 gallon 4 qt 3.7854 ` 1 pound [lb] 16 oz 0.4536 kg 13 32. 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 1.2374 Canadian dollars = 1 U. S. dollar. The metric conversion is 1 liter = .26417 gallons. What is the price in U.S. currency for one gallon of gasoline in Canada? Criticize the following actual student solutions: $16,540 $0.023 33. What’s the correct answer? dollar S. dollar liter 0.749 Canadian × 1.23741 U. liter Canadian dollar × .26417 gallons = 0.749 U. S. dollar 1.2374×.26417 gallon S. dollar = 2.29 U. gallon Two years ago this seemed expensive Now it seems cheap! 14 34. Killer Summation • The computation S = 1 + 2 + 3 + 4 + 5 + · · · + 999 + 1000 could be done on a hand calculator by entering each number, starting with 1 and ending with 1000. • 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 =1 + 2 + 3 + 4 + 5 + · · · + 999 + 1000 • S =1000 + 999 + · · · + 5 + 4 + 3 + 2 + 1. • Now add each entry in the top row with the entry in the row directly below it: 1 + 1000 = 1001, 2 + 999 = 1001, 3 + 998 = 1001, etc. • You get 1000 sums of 1001. • So 2S = 1000 × 1001 = 1001000. • Dividing by two gives us the value S = 1001000/2 = 500500. 36. Exercises: Try the “backwards” method on: • • • • • 1. 1 + 3 + 5 + 7 + 9 + · · · + 95 + 97 + 99. 2. 11 + 12 + 13 + · · · + 98 + 99 + 100. 3. 5 + 10 + 15 + 20 + · · · + 190 + 195 + 200. 4. 2 + 5 + 8 + 11 + 14 + 17 + · · · + 992 + 995 + 998. Sequences like these are called arithmetic progressions. 15 • Note that we obtain each term by adding the same number to the previous term. 37. 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 + x2 + x3 + · · · + xn−1 + xn . • 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 + x3 + · · · + xn−1 + xn + xn+1 S =1 + x + x2 + x3 + · · · + xn−1 + xn . • 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 xn+1 in line 1 and the 1 in line 2. • Hence xS − S = xn+1 − 1 39. Geom. Sequences Completed • • • • Start with: xS − S = xn+1 − 1 Pull out S: (x − 1)S = xn+1 − 1 Divide by x − 1 to get a formula for S: n+1 S = x x−1−1 16 xn+1 − 1 • 1 + x + x + ···x = x−1 2 n 40. Example Compute the sum 1 + 2 + 4 + 8 + 16 + 32 • 1 + 2 + 4 + 8 + 16 + 32 = 1 + 21 + 22 + 23 + 24 + 25 • In the formula xn+1 − 1 2 n • 1 + x + x + ···x = x−1 • x = 2 and n = 5 • Plugging these values into the formula gives 6 −1 • 1 + 2 + 4 + 8 + 16 + 32 = 22−1 • Since 26 = 64 63 • 1 + 2 + 4 + 8 + 16 + 32 = 64−1 2−1 = 1 = 63