Which Mathematics Skills are Necessary for Freshman-level Coursework in Quantitative Fields?

Transcription

1 Page 1 of 22 Which Mathematics Skills are Necessary for Freshman-level Coursework in Quantitative Fields? A method for answering this question: Faculty in quantitative fields frequently point out that it is the lack of math skills, not the lack of prior study in the relevant field, which holds students back in their Freshman courses. A 2006 College Board study (see reached this conclusion and suggested that the math in question is not advanced. It is high school-level or below. The set of math problems that is attached represents an effort to delineate more precisely the math skills necessary for success in beginning quantitative courses. The problems will not be used to test students. Instead, they will be given to OUS faculty as a basis for defining the specific mathematical skills they expect in the repertoire of entering students. We want to understand the particular mathematical reasoning, manipulation and problem-solving skills that are assumed by OUS faculty who teach beginning courses in a range of fields that depend on mathematics. Our primary interest is not in math courses, but rather in the courses designed to introduce freshmen to fields like the basic sciences, engineering, psychology, and economics. These courses are typically populated by prospective majors, but courses for non-majors that require significant application of mathematics will also be included. How this approach differs from others: The attempt to understand faculty expectations in mathematics is not a new endeavor, but previous queries asked about the relevance of broad mathematics subject areas, such as algebra, geometry or trigonometry, rather than specific kinds of problems or concepts within those areas. Information from those earlier studies has informed the current K-12 Mathematics Standards, but there are still persistent calls from the Oregon Department of Education and K-12 teachers for a more precise description of OUS expectations. To respond, we have assembled this group of math problems from a variety of sources: the high school exit exams used by several states, an online source of illustrative math problems for high school teachers, and our own problem sets. The problems are intended to encompass the range of math skills that introductory courses in a variety of quantitative fields assume, although the subsets required for individual fields will likely vary considerably. Moreover, some problems may not correspond to skills that are truly essential for beginning students in any quantitative field. Identifying those will be useful in defining a reasonable high school/university boundary in mathematics. It s important to appreciate that we are not asking about all of the math students will need to see them through the full course of study in a quantitative field physics or

2 Page 2 of 22 engineering, for instance. We assume that such students will take the university math (e.g. calculus, differential equations) that equips them for the advanced work in their fields. We are focusing exclusively on the entry point: the math that students need to begin their studies in physics, chemistry, economics and so forth. Procedure: We will identify OUS faculty teaching entry-level courses in the following fields: o Biology o Business (esp. Accounting and Finance) o Chemistry o Computer Science o Economics o Engineering (esp. Mechanical & Electrical) o Environmental Science o Geology o Physics o Psychology We will distribute the problems to these faculty in the form of a survey that asks faculty to rate the relevance of the skill represented by each problem for the courses they teach. We will collect the responses, sort them by discipline, and look for common patterns Potential Results and Possible Use: We anticipate that the importance of certain specific skills will vary among fields, but that a common core of key skills will emerge. We may be wrong, of course, but that s why we re conducting a survey, rather than operating on our own hunches. We think that information about both the core and the field-specific skills will be useful to K- 12 teachers preparing students for university work. In addition, we suspect that some commonly-taught aspects of mathematics may turn out to be less important to OUS faculty than presently assumed. That shouldn t be viewed as an automatic signal to jettison them, but it could be useful in efforts to simplify K-12 Math Standards.

3 Page 3 of 22 Before evaluating the math problems, please give us the following information about your course. If you teach courses that require different math preparation, please describe only the course to which your responses on this survey apply. If you wish to fill out additional surveys for your other courses, you are welcome to do so. 1. Disciplinary field (Please check one.) Biology Business (sub-discipline: ) Chemistry Computer Science Economics Engineering (sub-discipline: ) Environmental Science Geology Physics Psychology Other (please specify: ) 2. Level and type of course (Please check one.) Introductory: Intended for non-majors Introductory: Intended for majors Intermediate: Intended for majors who are unusually strong students or have had prior exposure to the field. Other (please specify: ) 3. Institution where course is offered (Please check one.) Oregon Community College Oregon University System Other 4-year college or university in Oregon

4 Page 4 of 22 Using Specific Problems to Define Math Skills Necessary for Quantitative Courses What kinds of math problems correspond to skills that you assume students will bring to your course? On the scale below each question, indicate the importance for success in your course that students be able to solve this type of problem, where 1 = extremely important and 5 = totally unimportant. If important skills have been omitted, please list them in the space at the end. 1. One hundred is multiplied by a number between 0 and 1. The answer has to be A. less than 0. B. between 0 and 50 but not 25. C. between 0 and 100 but not 50. D. between 0 and The square root of 150 is between A. 10 and 11. B. 11 and 12. C. 12 and 13. D. 13 and Which of the following is an example of a graph showing exponential growth? A. B. C. D

5 Page 5 of If a is a positive number and b is a negative number, which expression is always positive? A. a b B. a+b C. a b D. a b 5. a. Len runs a mile in 8 minutes. At this rate how long will it take him to run a 26-mile marathon? A minutes B hours C minutes D hours b. Which of the following problems can be solved using the same arithmetic operations that are used to solve the problem above? A. Len runs 26 miles in 220 minutes. How long does it take him to run each mile? B. A librarian has 356 books to place on 18 shelves. Each shelf will contain the same number of books. How many books can the librarian place on each shelf? C. A cracker box weighs 200 grams. What is the weight of 100 boxes? D. Each basket of strawberries weighs 60 grams. How many baskets can be filled from 500 grams of strawberries? 6. Which equation does NOT represent the line graphed? A. 3y = 2x 6 B. 2y 3x = 6 C. y = (3/2)x-3 D. 2y = 3x 6

6 Page 6 of The radius of the earth s orbit is 150,000,000,000 meters. What is this number in scientific notation? A B C D The cost of an afternoon movie ticket last year was $4.00. This year an afternoon movie ticket costs $5.00. What is the percent increase of the ticket from last year to this year? A. 10% B. 20% C. 25% D. 40% 9. What is the solution of the system of equations shown on the right? y = 3x 5 y = 2x A. (1,-2) B. (1,2) C. (5,10) D. (-5, -10) 10. Which of the following is equivalent to 7-6 x 7 4? A B C. 7 2 D What is the value of (2 6 x 2 4 )/2 5? A. 4 B. 10 C. 16 D What is the absolute value of 4? A. 4 B. 1/4 C. 1/4 D. 4

7 Page 7 of In each of the pairs below, circle the larger value or circle both if they re the same. Log Log Log ln100 Log ln1 10 x log Log log x log Factor. One factor is given. f(x) = 4x 3-4x x Factor: x Find the measure of angle a, given that l m, m EFG = 30 and FH FI. A. 30 B. 60 C. 90 D CAT is similar to DOG. What is the measure of DG? A. 16 B. 18 C. 20 D. 24

8 Page 8 of A bucket contains 3 bottles of apple juice, 2 bottles of orange juice, 6 bottles of tomato juice, and 8 bottles of water. If Kira randomly selects a bottle, what is the probability that she will select a drink other than water? A. 3/4 B. 11/19 C. 8/19 D. 1/4 18. The table below shows the number of real estate transactions by type for a city. Real Estate Transactions Type of Property Sold Number of Sales Single-Family Residence 157 Condo/Townhouse 17 Mobile Home 6 Multi-Family 2 Commercial 15 Land 255 Total 452 Based on the information in the table, which statement is true? A. More than half of the sales were single-family residences. B. More sales occurred for land than in all other areas combined. C. The number of condo/townhouse sales was more than 10% of the total sales. D. The number of mobile home and multifamily sales combined was twice the number of commercial sales. 19. Which of the following is equivalent to (6x-2)(6x-2)(6x+2)? A. (6x-2) 3 B. (6x+2) 3 C. 2(6x-2)(6x+2) D. (6x-2) 2 (6x+2) 20. Solve for x. 2 x 3 = 7 A. 5 B. 2 C. 2 D One millimeter is A. 1/1000 of a meter B. 1/100 of a meter. C. 100 meters. D meters.

9 Page 9 of Determine the midpoint of a segment whose endpoints are (-3, 4) and (3, 10). A. (0, 7) B. (3, 7) C. (0, -3) D. (-3, 3) 23. The coordinates of point A are (-5, 3). If A is reflected over the y-axis, then translated 3 units right and 4 units down, the new coordinates of A are: A. (5, 3) B. (8, -1) C. (-2, -1) D. (-2, -7) 24. The volume of the cone is? cm3. A. 2592π B. 3240π C. 7776π D. 9720π 25. What is the volume left in the cylinder after the shaded cone region is removed? A. 1,024π in3 B. 1,536π in3 C. 2,048π in3 D. 4,096π in3

10 Page 10 of The points (1, 1), (2, 3), (4, 3), and (5, 1) are the vertices of a polygon. What type of polygon is formed by these points? A. Triangle B. Trapezoid C. Parallelogram D. Pentagon 27. Joe has two fair dice (faces numbered 1-6) and a fair coin. He rolls the dice and tosses the coin. What is the probability Joe rolls one three (3), one four (4), and tosses a head? A. 1/216 B. 1/36 C. 1/24 D. 5/6 28. Use the quadratic formula to solve the equation: -5x x - 7 = Which of the following is true of e, the base of the natural log system? A. Its value is close to 3 B. Its value is 10 C. Its value is 0 D. It has no numerical value

11 Page 11 of The perimeter, P, of a square may be found by using the formula (1/4) P =sq rt A, where A is the area of the square. What is the perimeter of the square with an area of 36 square inches? A. 9 inches B. 12 inches C. 24 inches D. 72 inches 31. Colleen solved the equation 2 (2x +5) = 8 using the following steps. Given: 2 (2x +5) = 8 Step 1: 4 x + 10 = 8 Step 2: 4 x = 2 Step 3: x = ½ To get from Step 2 to Step 3, Colleen A. divided both sides by 4. B. subtracted 4 from both sides. C. added 4 to both sides. D. multiplied both sides by What are the coordinates of the x-intercept of the line 3x + 4 y = 12? A. (x=0, y=3) B. (x=3, y=0) C. (x=0, y=4) D. (x=4, y=0) 33. Which of the following statements describes parallel lines? A. Same y-intercept but different slopes B. Same slope but different y-intercepts C. Opposite slopes but same x-intercepts D. Opposite x-intercepts but same y-intercept

12 Page 12 of a. What is the solution of the system of equations shown below? y = 3x - 5 y = 2 x A. (x= 1, y=2) B. (x=1, y=2) C. (x=5, y=10) D. (x= 5, y= 10 ) b. To what does this solution correspond? A. A single point B. Two points C. A straight line D. A plane 35. Which of the following is smallest? A B C D Ricardo runs 10 miles each Saturday. If he doubles his usual speed, he can run the 10 miles in one hour less than his usual time. What is his usual speed? A. 2 miles/hour B. 3 miles/hour C. 4 miles/hour D. 5 miles/hour

13 Page 13 of If you spin the spinner two times, what is the probability it will land on A both times? A. 1/4 B. 3/8 C. 6/8 D. 9/ The graph on the right represents the closing price of a share of a certain stock for each day of a week. Which day had the greatest increase in the value of this stock over that of the previous day? A. Tuesday B. Wednesday C. Thursday D. Friday 39. The scatter plot on the right shows the ages of some children and the distance each child lives from school. Which statement BEST describes the relationship between age and distance from school? A. As age increases, the distance from school increases B. As age increases, the distance from school decreases. C. As age increases, the distance from school remains constant. D. There is no relationship between age and distance from school.

14 Page 14 of Express in logarithmic form (For example: log = 3): 10 2 = = = You have 1 milliliter (ml) of sugar syrup that contains 10 gram of sugar per liter. How much water would you need to add to the 1 ml to produce syrup with the following concentrations of sugar? a. 1 gram sugar per liter: A. None B. 2 ml C. C. 9 ml D. 10 ml b gram sugar per liter: B. 3 ml B. 4 ml C. 25 ml D. 39 ml 42. A car has an original value of $20,000. The value decreases at a rate of 18 % each year. Part A: Write a function where f(x) represents the value of the car in dollars and x represents years. Part B: After how many years will the car be worth less than 1/2 of the original value? Show or explain your work. 43. Find the center, foci, vertices and co-vertices of this ellipse: (X - 2) 2 + (Y + 6) 2 =

15 Page 15 of The graph of the equation y = 3x is reflected over the y axis. What is the equation of the image? A. y = - 3x B. y = - (1/3) x C. y = (1/3) x D. y = 3 x 45. Calculate: log log 2 32 log log Joe wants to know how much Calcium is provided in the food and vitamin supplements he consumes. He finds Calcium content listed in the following ways on various packages. Which is a well-defined number that corresponds to the highest Calcium content? A. 2% B. 10 milligrams per milliliter (mg/ml) C. 10 grams per liter (g/l) D. 50% of daily requirement/serving

16 Page 16 of Consider the diagram: The function h(x)=(1/98)x 2 describes h(x), the height of part of a rollercoaster track, where x is the horizontal distance in feet from the center of this section of the track. The towers that support this part of the track are the same height and are 150 feet apart. Which is the best estimate of the height of the towers? A feet B feet C feet D feet 48. Use an inverse matrix to solve the linear system: -9X + 7Y = X -1Y = What is the slope of the line shown in the graph on the right? A. -2 B. -1/2 C. 1/2 D Solve this equation by factoring: 7½ X X -157½ = - 67½

17 Page 17 of 22 What s missing? Please use the space provided as question 51 on the electronic survey to list any mathematics skills that are important for your course, but are not represented among these 50 problems.

18 Page 18 of 22 Answers 1. D 26. B 2. C 27. B 3. D 28. x = 1, 1.4 (see solution at end) 4. A 29. A 5.a. D 30. C b. C 6. A 31. A 7. B 32. D 8. C 33. B 9. C 34.a. C b. A 10. C 35. D 11. D 36. D 12. D 13. log D ln B Same 39. D 10 x log log = log log = 5 log = x(x+5)(x-6) 41.a. C 15. D b. D 16. B 42.a. f(x) = $20,000 X (0.18 x $20,000) 17. B b. 3 years (see solution below) 18. B 43. Center: (2, -6) 19. D Vertices: (-3, -6) & (7, -6) 20. D Co-vertices: (2, -9) & (2, -3) 21. A Foci: (-2, -6) & (6, -6) 22. A 44. A 23. B B 46. A 25. A 47. A 48. x = 10, y = -6 (see solution below) 49. C 50. factors: (x-2)(x+6); x = 2, -6

19 Page 19 of 22 Solutions b± - 4ac ± ± ± ± , = b. Yearly decrease in value = 0.18 ($20,000) = $3600/yr Car will be worth less than ½ original value when $3600/yr x Xyrs >$10,000. At this point, X > $10,000/$3600/yr = 2.77 yrs x -132 = -3-1 y -24 Inverse of the A = -1/30-7/30 1/10-3/10 X = 10, 7 = 6