Designing a 5-hour Precalculus Course to be a Direct Pathway to Calculus

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1 Designing a 5-hour Precalculus Course to be a Direct Pathway to Calculus Lesa L. Beverly beverlyll@sfasu.edu Stephen F. Austin State University Higher Education Conference October 2006 p. 1/?

2 Common Calculus Mistakes Higher Education Conference October 2006 p. 2/?

3 Common Calculus Mistakes ln (x + y) = lnx + lny Higher Education Conference October 2006 p. 2/?

4 Common Calculus Mistakes ln (x + y) = lnx + lny x2 + y 2 = x + y Higher Education Conference October 2006 p. 2/?

5 Common Calculus Mistakes ln (x + y) = lnx + lny x2 + y 2 = x + y x2 = x Higher Education Conference October 2006 p. 2/?

6 Common Calculus Mistakes ln (x + y) = lnx + lny x2 + y 2 = x + y x2 = x 3 log 3 ( 5) = 5 Higher Education Conference October 2006 p. 2/?

7 Common Calculus Mistakes ln (x + y) = lnx + lny x2 + y 2 = x + y x2 = x 3 log 3 ( 5) = 5 5 log 2 4 = log Higher Education Conference October 2006 p. 2/?

8 Calculus Students Trouble Spots Higher Education Conference October 2006 p. 3/?

9 Calculus Students Trouble Spots Algebra Higher Education Conference October 2006 p. 3/?

10 Calculus Students Trouble Spots Algebra Trigonometry Higher Education Conference October 2006 p. 3/?

11 Calculus Students Trouble Spots Algebra Trigonometry Analytic Geometry Higher Education Conference October 2006 p. 3/?

12 SFASU s Precalculus Higher Education Conference October 2006 p. 4/?

13 SFASU s Precalculus Designed specifically to prepare students for the calculus sequence Higher Education Conference October 2006 p. 4/?

14 SFASU s Precalculus Designed specifically to prepare students for the calculus sequence 5-hour one semester course that can replace up to 9 hours of prerequisites Higher Education Conference October 2006 p. 4/?

15 SFASU s Precalculus Designed specifically to prepare students for the calculus sequence 5-hour one semester course that can replace up to 9 hours of prerequisites Refresher course Higher Education Conference October 2006 p. 4/?

16 Algebra Higher Education Conference October 2006 p. 5/?

17 Algebra Functions (including exponentials and logarithms) Higher Education Conference October 2006 p. 5/?

18 Algebra Functions (including exponentials and logarithms) definition Higher Education Conference October 2006 p. 5/?

19 Algebra Functions (including exponentials and logarithms) definition domain and range Higher Education Conference October 2006 p. 5/?

20 Algebra Functions (including exponentials and logarithms) definition domain and range graphs Higher Education Conference October 2006 p. 5/?

21 Algebra Functions (including exponentials and logarithms) definition domain and range graphs one to one functions Higher Education Conference October 2006 p. 5/?

22 Algebra Functions (including exponentials and logarithms) definition domain and range graphs one to one functions inverse functions Higher Education Conference October 2006 p. 5/?

23 Algebra Functions (including exponentials and logarithms) definition domain and range graphs one to one functions inverse functions piecewise-defined functions Higher Education Conference October 2006 p. 5/?

24 Sample Problems Higher Education Conference October 2006 p. 6/?

25 Sample Problems Define "function" and give an example of a function from real life (not a mathematics classroom example). Higher Education Conference October 2006 p. 6/?

26 Sample Problems Define "function" and give an example of a function from real life (not a mathematics classroom example). A student attempted to solve the inequality x by multiplying both sides of the x 3 inequality by x 3 to get x This led to a solution of {x x 4}. Is the student correct? Explain. Higher Education Conference October 2006 p. 6/?

27 Sample Problems (cont.) Suppose the graph of the rational function R(x) = p(x) has a vertical asymptote at q(x) x = 5. What information does this provide about the graph of the function? What does the vertical asymptote tell you about p(x) and q(x)? Higher Education Conference October 2006 p. 7/?

28 Sample Problems (cont.) Analyze the graph of the function (x + 3)(2x 5) R(x) =. (You do not need to (x + 3)(x 4) graph it! Just tell me about it!) Higher Education Conference October 2006 p. 8/?

29 Sample Problems (cont.) Analyze the graph of the function (x + 3)(2x 5) R(x) =. (You do not need to (x + 3)(x 4) graph it! Just tell me about it!) Suppose you have a friend in college algebra who is confused about function inverses. What information would you share with your friend to help him better understand? Higher Education Conference October 2006 p. 8/?

30 True or False: Higher Education Conference October 2006 p. 9/?

31 True or False: ln (x + y) = lnx + lny Higher Education Conference October 2006 p. 9/?

32 True or False: ln (x + y) = lnx + lny Expected solution: False because the rule is lnx + lny = lnxy. Higher Education Conference October 2006 p. 9/?

33 True or False: ln (x + y) = lnx + lny Expected solution: False because the rule is lnx + lny = lnxy. Unexpected student solution: False because "you can t distribute a function over an argument". Higher Education Conference October 2006 p. 9/?

34 Trigonometry Higher Education Conference October 2006 p. 10/?

35 Trigonometry Definition of trigonometric functions from right triangles Higher Education Conference October 2006 p. 10/?

36 Trigonometry Definition of trigonometric functions from right triangles Special triangles Higher Education Conference October 2006 p. 10/?

37 Trigonometry Definition of trigonometric functions from right triangles Special triangles Circular functions Higher Education Conference October 2006 p. 10/?

38 Trigonometry Definition of trigonometric functions from right triangles Special triangles Circular functions Trigonometric identities Higher Education Conference October 2006 p. 10/?

39 Trigonometry Definition of trigonometric functions from right triangles Special triangles Circular functions Trigonometric identities Inverse trigonometric functions Higher Education Conference October 2006 p. 10/?

40 Trigonometry Definition of trigonometric functions from right triangles Special triangles Circular functions Trigonometric identities Inverse trigonometric functions Solving trigonometric equations Higher Education Conference October 2006 p. 10/?

41 Trigonometry Definition of trigonometric functions from right triangles Special triangles Circular functions Trigonometric identities Inverse trigonometric functions Solving trigonometric equations Solving triangles using Laws of Sines/Cosines Higher Education Conference October 2006 p. 10/?

42 Sample problems Higher Education Conference October 2006 p. 11/?

43 Sample problems Is sin (2 radians) = sin (2 )? Explain you answer. Higher Education Conference October 2006 p. 11/?

44 Sample problems Is sin (2 radians) = sin (2 )? Explain you answer. A trigonometry student thinks that cos 1 x and 1 are the same function. What information cosx can you give to the student that will help her understand the difference between the two functions? Higher Education Conference October 2006 p. 11/?

45 Sample problems Is sin (2 radians) = sin (2 )? Explain you answer. A trigonometry student thinks that cos 1 x and 1 are the same function. What information cosx can you give to the student that will help her understand the difference between the two functions? Is the equation sin 1 (sin x) = x always true? Explain. Higher Education Conference October 2006 p. 11/?

46 Analytic Geometry Higher Education Conference October 2006 p. 12/?

47 Analytic Geometry Cartesian coordinate system/distance formula Higher Education Conference October 2006 p. 12/?

48 Analytic Geometry Cartesian coordinate system/distance formula Conic sections Higher Education Conference October 2006 p. 12/?

49 Analytic Geometry Cartesian coordinate system/distance formula Conic sections Polar coordinates Higher Education Conference October 2006 p. 12/?

50 Analytic Geometry Cartesian coordinate system/distance formula Conic sections Polar coordinates Systems of equations Higher Education Conference October 2006 p. 12/?

51 Success??? Higher Education Conference October 2006 p. 13/?

52 Success??? To be determined. Higher Education Conference October 2006 p. 13/?

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only.

AP Calculus AB. Nevada Academic Standards that are assessable at the local level only. Calculus AB Priority Keys Aligned with Nevada Standards MA I MI L S MA represents a Major content area. Any concept labeled MA is something of central importance to the entire class/curriculum; it is a