Free Response Questions Included in Training Module

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1 Free Response Questions Included in Training Module Copyright 2011 Laying the Foundation, Inc. All right reserved. The materials included in these files are intended for noncommercial use by educators for course and test preparation; permission for any other use must be sought from Laying the Foundation, Inc. Teachers may reproduce them, in whole or in part, in limited quantities, for faceto-face teaching purposes but may not mass distribute the materials, electronically or otherwise. This permission does not apply to any third-party copyrights contained herein. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here.

2 YES Part B Pre-Calculus 2010 Danh rides his bicycle to school along a straight road starting from home at t = 0 minutes. For the first eight minutes, Danh s velocity, in miles per minute, is modeled by the piecewise linear function whose graph is provided. v Danh s Velocity velocity in miles per minute t v(t) time in minutes t (a) How fast is Danh riding when t = 2 minutes? Include units in the answer. (b) Write an equation for the portion of the velocity function between 2 and 4 minutes. Using your equation, calculate how fast Danh is riding when t = 3.2 minutes. (c) Danh decreases his velocity at a constant rate from t = 8 minutes to t = 12 minutes so that his velocity at t = 12 minutes will be 0 miles per minute. On the graph provided, extend the velocity function to include this new information. Mark each grid point that lies on the new portion of the velocity function. Since the area between the velocity function and the t-axis represents the distance that Danh travels, what is the total distance that Danh will travel from t = 0 to t = 12 minutes? Show the work that leads to your answer. Include units in the answer. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 1

3 YES Part B Pre-Calculus 2010 Sierra is also riding her bicycle to school along a straight road starting at her home. Her velocity for 0 t 12 is given by the function w( t) sin t where t is time in minutes and w(t) is the velocity in miles per minute. (d) How much faster is Danh traveling than Sierra at t = 2 minutes? Show the work that leads to your answer. After evaluating the trigonometric value, use 3 to approximate the final answer to the nearest tenth of a mile per minute. (e) The total distance that Sierra travels during t minutes of her ride can be determined using the 4 function, d( t) 1 cos t To the nearest tenth of a mile, how far will she have traveled when t = 12 minutes? (f) If both Sierra and Danh arrive at school 12 minutes after they leave home, who lives farthest from school? To the nearest tenth of a mile, what is the difference in their distances between home and school? Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 2

4 YES Part B Pre-Calculus 2010 Rubric (a) miles (a) 1 pt: 1 pt: correct answer with units 0.2 min (b) v( t) 0.1( t 4) (b) 2 pts: 1 pt: correct velocity equation miles v(3.2) 0.08 minute (c) total distance = 1 mi 1 mi (4 min) (1min 2 min) 2 min 2 min 1 mi mi 1 mi (1min) (4 min) min min 2 min 1 pt: correct answer with correct analysis or correct evaluation of 3.2 in student s velocity equation (c) 2 pts: 1 pt: area work shown for at least one portion of the graph (d) (e) = 1.5 miles miles w(2) sin 2 sin min Danh travels miles faster. 30 min d(12) 1 cos 12 (2) 1.6 miles pt: correct total distance with correct units with appropriate area work (d) 2 pt: 1 pt: correct answer for Sierra s velocity with appropriate work shown 1 pt: correct difference, to nearest tenth of a mile, based on student s answers in (a) and (d) (e) 1 pt: 1 pt: correct answer to nearest tenth of a mile with appropriate work shown (f) = 0.1 miles. Sierra lives 0.1 miles farther from school than Danh. (f) 1 pt: 1 pt: correct name with correct difference, to nearest tenth of a mile, based on student s answers in (c) and (e) On the 2010 exam, there was an error in the question in part (e) which affected the answers for both part (e) and part (f). The correction has been made in both the question and the rubric in this printed version. Student samples show the previous version where credit in part (e) for d(12) 1 cos 12 (2) 1.2 miles Danh lives 0.3 miles farther from school than Sierra. 3 d( t) 1 cos t Students received and in part (f) for miles; Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 3

5 YES Part B Pre-Calculus 2010 Notes: In part (e) of the question, students were given the equation for the distance Sierra traveled; 4 however, there was a copy error. The correct equation is dt () 1 cos t The rubric was written based on the equation stated in part (e); however, if a student had corrected the equation and then answered the questions based on that equation, the student would have received credit. (b) Accept any correct equation in any form. Student may earn the second point by substituting into an equation that is not stated explicitly. mi For example, the student could write: 0.1(3.2) and earn the second point min without earning the first point. Another example would be when the student says Danh is mi slowing at 0.1, so (3.2 2) (1.2) min Student may earn the second point using similar triangles: 0.1 x x x 0.08 (c) Area work can be shown on the graph by dividing the graph into geometric figures and determining the area for that geometric portion. (d) 1 st pt: Must have 1 2 substituted for sin. 6 Do not accept early substitution of 3 for where student must evaluate No bald answer. 3 sin 6. 2 nd pt: Accept twice as fast. Student can recoup point lost for units from part (a) if answer to part (a) did not include incorrect units. Read with points can be earned if completely accurate and stated to the tenths. Answer must be based on student s work and previous answers. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 4

6 YES Part B Pre-Calculus 2010 (e) Answer must be based on student s work and previous answers. (f) Answer must be based on student s work and previous answers. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 5

7 Sample A

8 Sample A

9 Sample A 9 points Part Points Earned 2010 Free Response Part A Pre-Calculus Question 2 Calculator Scoring Commentary a 1 Student states correct answer with correct units. b 1 Student states correct velocity equation. b 1 Student correctly evaluates 3.2 in velocity equation. c 1 c 1 d 1 Student shows appropriate area work for at least one section of the graph. Student shows appropriate area work, computes correct total distance, and labels answer with correct units. Student correctly calculates Sierra s velocity with appropriate work shown. d 1 Student computes correct difference. e 1 Student computes correct distance. f 1 Student states who lives farther and computes correct difference. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

10 Sample B

11 Sample B

12 Sample B 7 points 2010 Free Response Part A Pre-Calculus Question 2 Calculator Part Points Earned Scoring Commentary a 1 Student states correct answer with correct units. b 1 Student states correct velocity equation. b 1 Student correctly evaluates 3.2 in velocity equation. c 1 c 1 d 1 Student shows appropriate area work for at least one section of the graph. Student shows appropriate area work, computes correct total distance, and labels answer with correct units. Student correctly calculates Sierra s velocity with appropriate work shown. d 1 Student computes correct difference. e 0 f 0 Student does not compute correct distance. Student correctly evaluates cosπ but multiplied the numerator and denominator of the fraction by 2. Student states who lives farther but does not compute correct difference. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

13 Sample C

14 Sample C

15 Sample C 5 points 2010 Free Response Part A Pre-Calculus Question 2 Calculator Part Points Earned Scoring Commentary a 1 Student states correct answer with correct units. b 0 Student does not state correct velocity equation. b 0 Student does not correctly evaluate 3.2 in velocity equation. c 1 c 1 Student shows appropriate area work for at least one section of the graph. Student shows appropriate area work, computes correct total distance, and labels answer with correct units. d 0 Student substitutes t = 2 appropriately but does not simplify. d 0 Student does not attempt to compute the difference. e 1 Student computes correct distance. f 1 Student states who lives farther and computes correct difference. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

16 EOC Section II Pre-Calculus 2009 The function v(t) represents the velocity of a particle moving on a horizontal line at any time t 0. If the velocity is positive, the particle is moving to the right. If the velocity is negative, the particle is moving to the left. (a) A particle has a velocity function, () sin ( ) vt a bt c d, with the minimum velocity at the point (1, 4) and the maximum velocity at the point (4, 2). (i) Evaluate the constants a, b, c, and d and record the function, indicating the correct numerical value for each constant. (ii) What is the value of v(0)? (iii) On the grid provided below, sketch the velocity function for 0 t 7. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

17 EOC Section II Pre-Calculus 2009 (b) A second particle has a velocity function defined by vt () asin( bt) where the velocity is measured in centimeters per second as the particle moves on a horizontal line at any time t 0, where a and b are constants such that a 0andb 0. (i) On the grid provided below, scale the axes in terms of a and b and then sketch one period of the function. v t (ii) In terms of b, what are the first three times the particle is at rest? (iii) In terms of b, what is the first interval where the particle is moving to the left? (iv) The area between the velocity function and the t-axis represents the distance traveled by the particle. Approximate how far the particle travel to the right between the first two times the particle is at rest by inscribing an isosceles triangle under the positive portion of the function so that the vertex is at the maximum point of the function and the base is on the t-axis. The answer will be in terms of a and b. v t Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

18 EOC Section II Pre-Calculus 2009 RUBRIC () 3sin vt () 3sin t (a) (i) vt t (a) 4 pts: 1 pt: two correct constants 1 pt: two additional correct constants (ii) (iii) 1 v(0) pt: correct value for v(0) 1 pt: correct graph with points at (0, 4), (4, 2), and (7, 4)and either at 2.5, 1 or 5.5, 1 (b) (i) (ii) (iii) a a v 2 t 0,, b b 2 t b b t 2 b (iv) The distance the particle travels is a units. 2b a v (b) 5 pt: 1 pt: correct sine function shape 1 pt: correct labels on t-and v-axes 1 pt: at least two correct zeroes based on student s graph if in terms of b 1 pt: correct interval based on student s graph if in terms of b 1 pt: correct answer based on student s graph if in terms of b t 2 b Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

19 Sample A

20 Sample A

21 2009 Free Response Part B Pre-Calculus Question 2 Sample A 9 points Part Points Earned a 1 a 1 a 1 a 1 b 1 b 1 b 1 b 1 Scoring Commentary Student correctly determines the amplitude to be half the distance between the maximum velocity and the minimum velocity, a = 3 Since half the period of the function is 3, 2π π 6 = and b =. b 3 Student determines d = 1 as the minimum velocity + the amplitude. Student also determine the first time where v = 1 to be at 2.5 so c = 2.5. Student correctly calculates v (0) and earns the point. π vt () = 3sin ( t 2.5) 3 π 5π 3 v(0) = 3sin (0 2.5) 1= 3sin 1= 1= Student earns the point for the graph by showing the points (1, 4), (4, 2), a point near (7, 4),and a point near (2.5, 1) or near the point (5.5, 1). Student draws the basic shape of a sine function passing through the origin. Student shows a and a on the vertical axis to coincide with the maximum of the function. Student also shows 2 π on the b horizontal axis to coincide with the end of the first cycle of the sine function. Point is earned. Student correctly determines that the first three zeros occur π 2π when t = 0,,. b b Since the particle is moving left when vt () 0, the correct π 2π interval is, b b. The endpoints are ignored. Student earns the point. 1 π π a = 2 b 2b b 1 The triangle s area is ( a) Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

22 Sample B

23 Sample B

24 2009 Free Response Part B Pre-Calculus Question 2 Sample B 7 points Part Points Earned Scoring Commentary a 1 Student determine a = 3, and d = 1 to earn one point. a 0 a 0 a 1 Student lets b = 6 which is the period of the function instead of 2π solving b =. The student select c = 2 instead of c = 2.5. period This point is not earned. Student begins this calculation correctly using the equation determined in part (a), but declares sin(12) = 1 and point is not awarded. Student draws the basic shape of a sine function passing through the origin. b 1 b 1 b 1 Student shows a and a on the vertical axis to coincide with the maximum of the function. Student also shows 2 π on the b horizontal axis to coincide with the end of the first cycle of the sine function. Point is earned. Student correctly determines that the first three zeros occur π 2π when t = 0,,. b b Since the particle is moving left when vt () 0, the correct π 2π interval is, b b. Student earns the point. 1 π π a = 2 b 2b b 1 The triangle s area is ( a) b 1 Student draws the basic shape of a sine function passing through the origin. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

25 Sample C

26 Sample C

27 2009 Free Response Part B Pre-Calculus Question 2 Sample C 5 points Part Points Earned Scoring Commentary a 1 a 1 a 1 Student correctly shifts the function to the left 1 2 and determines a = 3 to correspond with the minimum point on the sine function. Student determines d = 1 as the minimum velocity + the π amplitude and determines that b = 3 Student correctly calculates v (0) and earns the point. π 1 5π vt () = 3sin t+ 1= 3sin 1= a 0 Student does not draw a graph and does not earn the point. b 1 Student draws the basic shape of a sine function passing through the origin. b 1 Student shows a and a on the vertical axis to coincide with the maximum of the function. Student also shows 2 π on the b horizontal axis to coincide with the end of the first cycle of the sine function. Point is earned. b 0 Student does not answer and does not earn the point. b 0 Student does not answer and does not earn the point. b 0 Student does not answer and does not earn the point. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

28 Posttest Part B Pre-Calculus Let vt () represent the velocity in inches per second of a particle moving along a horizontal line for t 0. Use the graph and the corresponding table of values to answer the following questions v(t) (in/ sec) t (seconds) t v(t) (sec) (in/sec) (a) For what value(s) of t, if any, is the particle not moving? Justify your answer and include units. (b) For what value(s) of t, if any, is the particle moving to the right? Justify your answer. (c) For what approximate t value is the speed of the particle the greatest over the interval [0, 5]? Justify your answer and include units.

29 Posttest Part B Pre-Calculus 2005 (d) Find the average rate of change in velocity, called the average acceleration, over the interval, [2, 4]. Show your work and units of measurement. (e) Divide the interval [0, 3] into three equal partitions and use triangles and trapezoids to estimate the total distance traveled by the particle from 0 to 3 seconds. Show your work and units of measurement.

30 3. Let vt () represent the velocity in inches per second of a particle moving along a horizontal line for t 0. Use the graph and the corresponding table of values to v(t) (in/ sec) answer the following questions. 5 4 (a) For what value(s) of t, if any, is the particle not 3 2 moving? Justify your answer and include units. 1 (b) For what value(s) of t, if any, is the particle moving to the right? Justify your answer. (c) For what approximate t value is the speed of the particle the greatest over the interval [0, 5]? Justify your answer and include units. (d) Find the average rate of change in velocity, called the average acceleration, over the interval, [2, 4]. Show your work and units of measurement. (e) Divide the interval [0, 3] into three equal partitions and use triangles and trapezoids to estimate the total distance traveled by the particle from 0 to 3 seconds. Show your work and units of measurement. (a) t 0, 3, 5 because vt () 0 seconds t (seconds) Posttest Part B Pre-Calculus 2005 t v(t) (sec) (in/sec) (a) 2 pts: 1 pt: three correct answers 1 pt: explanation (b) (3, 5) seconds because vt () 0 (b) 2 pts: 1 pt: correct interval note: ignore [, ] and (, ) 1 pt: explanation (c) t 1.25 seconds because speed = vt ( ) or because the sign shows direction and does not affect speed (c) 2 pts: 1 pt: answer in [1, 1.5) 1 pt: explanation v(4) v(2) in (d) sec (d) 1 pt: 1 pt: answer (e) 14 inches (e) 1 pt: 1 pt: answer Units pt: correct units in (a), (b), or (c) and in (d) and (e)

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37 SCORING GUIDELINE FOR SAMPLE PAPERS 2005 LTF PRE-CALCULUS QUESTION 3: Sample A: (9 points earned) (a) 1-1 (b) 1-1 (c) 1-1 (d) 1 (e) 1 Units: 1 Sample B: (7 points earned) (a) 1-1 (b) 1-1 (c) 1-1 (d) 1 (e) 0 Units: 0 Sample C: (5 points earned) (a) 1-1 (b) 0-1 (c) 1-1 (d) 0 (e) 0 Units: 0 1

38 Free Response Questions Included in Other Training Modules Copyright 2011 Laying the Foundation, Inc. All right reserved. The materials included in these files are intended for noncommercial use by educators for course and test preparation; permission for any other use must be sought from Laying the Foundation, Inc. Teachers may reproduce them, in whole or in part, in limited quantities, for faceto-face teaching purposes but may not mass distribute the materials, electronically or otherwise. This permission does not apply to any third-party copyrights contained herein. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here.

39 EOC Section II Pre-Calculus t 2 t The function, vt () sin, graphed below, represents the velocity, in millimeters per 4 6 second, of a particle moving on a horizontal line for the time interval [0, 12] seconds. 100 v 50 0 t (a) When is the particle at rest? Include units in your answer. (b) At what times do the two relative maximums occur? What are the two relative maximum velocities? Include the units in your answers. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

40 EOC Section II Pre-Calculus 2008 (c) Using two triangles with bases of equal length, approximate the area between the curve and the horizontal axis. Show the computations that lead to your answer. Write a sentence explaining what the area represents in terms of the problem situation. Include the units as a part of your explanation. (d) Based on the answer calculated in part (c), what is the average velocity for the given interval? Show the computations that lead to your answer and include the units for the answer. (e) What is the first time that the particle is traveling at the average velocity? Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

41 2008 Pre-Calculus Rubric Question 1 (Calculator Allowed) 45t 2 t The function, vt () sin, graphed below, represents the velocity, in 4 6 millimeters per second, of a particle moving on a horizontal line for the time interval [0, 12] seconds v (a) When is the particle at rest? Include units in your answer. (b) At what times do the two relative maximums occur? What are the two relative maximum velocities? Include the units in your answers. (c) Using two triangles with bases of equal length, approximate the area between the curve and the horizontal axis. Show the computations that lead to your answer. Write a sentence explaining what the area represents in terms of the problem situation. Include the units as a part of your explanation. (d) Based on the answer calculated in part (c), what is the average velocity for the given interval? Show the computations that lead to your answer and include the units for the answer. (e) What is the first time that the particle is traveling at the average velocity? 0 t (a) (b) (a) t 0, 6, 12 seconds 1 pt: 1 pt: must have t = 6 and no incorrect times (b) mm mm 2 pt: 1 pt: correct first velocity max vel and sec sec 1 pt: correct second velocity (c) 1 1 (6)(36.738) (6)( ) mm 2 2 The particle traveled mm in 12 sec. (c) 4 pt: 1 pt: heights based on student s answers in part (b) 1 pt: width of 6 1 pt: correct answer 1 pt: explanation (must include mm as units) (d) (d) mm mm or pt: 1 pt: student s answer in part(c) 12 sec sec divided by 12 (e) (e) t = or seconds 1 pt: 1 pt: solves v(t) = student s answer in part (d) (0 /1 if student s answer in part (d) is zero) Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

42 2008 Pre-Calculus Rubric Question 1 (Calculator Allowed) Part (c): A student who uses the rounded velocity values rather than storing these values will have an answer of mm which will not earn the correct answer point. Units must be included in either the explanation or on the answer. If the answers from part (b) are f(3) = and f(9) = , total distance is 405 mm and the values in part (d), the average value is and the answer in part (e) is 3 seconds. To earn the third point of part (c), the student must show a sum, either as a correct equation or as a correct number value. The student who does writes a correct equation but continues to record an incorrect number value does not earn the point. Part (d): Student s answer will be the same as shown in the rubric if or is used. Part (e): Student s answer will be the same as shown in the rubric if or is used. Student s answer must be in the interval [0, 12]. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

43 Sample A

44 Sample A

45 2008 Free Response Part A Pre-Calculus Question 1 Sample A 9 points Part Points Earned Scoring Commentary a 1 Student records all three answers in the domain. b 1 Student correctly calculates the first relative maximum velocity. b 1 Student correctly calculates the second relative maximum velocity. c 1 Student uses the relative maximum values in mm/sec from part (b) as the heights for the triangles. c 1 Student uses 6 seconds as the base for each triangle. c 1 Student stores answers from (b) and uses the stored values to calculate the correct total of mm. c 1 Student correctly describes the area as the distance in millimeters that the particle traveled along a straight line during the 12 seconds. d 1 Student correctly divides the total distance by 12 seconds. e 1 Student correctly calculates the first time at which the particle travels at the average velocity. Additional Comments Excellent paper. The explanation in part (c) is clear and to the point. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

46 Sample B

47 Sample B

48 2008 Free Response Part A Pre-Calculus Question 1 Sample B 7 points Part Points Earned Scoring Commentary a 1 Student only calculates the answer of 6; however, this is the only value required to earn the point. b 1 Student correctly calculates the first relative maximum velocity. b 1 Student correctly calculates the second relative maximum velocity. c 1 Student uses the relative maximum values in mm/sec from part (b) as the heights for the triangles. c 1 Student uses 6 seconds as the base for each triangle. c 1 Student stores answers from (b) and uses the stored values to calculate the correct total of mm. c 0 The student does not describe the answer as distance and does not use the correct units. d 0 Student correctly sets up the division of the total distance by 12 seconds, but records an incorrect answer. e 1 The time recorded by the student is correct based on the answer given in part (d). Additional Comments Grading part (e) requires graphing the original velocity function and the line y = student s answer from part (d) to find the point of intersection or using the solver to calculate the intersection. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit:

49 Sample C

50 Sample C

51 2008 Free Response Part A Pre-Calculus Question 1 Sample C 5 points Part Points Earned Scoring Commentary a 1 Student records all three answers in the domain. b 1 Student correctly calculates the first relative maximum velocity. b 1 Student correctly calculates the second relative maximum velocity. c 1 Student uses the relative maximum values in mm/sec from part (b) as the heights for the triangles. c 1 Student uses 6 seconds as the base for each triangle. c 0 Student does not add the distances. Point not earned c 0 Student does not give an explanation. d 0 The student averages the two distance calculations instead of dividing the total distance by the total time. Point is not earned. e 0 The time recorded by the student is not correct based on the answer given in part (d). The student s answer should have been seconds. Additional Comments In part (c) the student must show a sum. Any of the following will earn the point. 1 1 (6)(36.738) + (6)( ), , , or If the student s answer is the student has not carried the decimal places from the calculator. Since the student is allowed to show the values rounded or truncated with 3 decimal places, the error is not evident until the actual sum is calculated. In part (e) the student must calculate the point of intersection of vt () and y = answer from part (d). To grade part (e) 1. Set up the window x [0, 12] and y [0, 120] using the scales shown of the graph. 2. Enter the velocity function in Y1 and 3. Enter A in Y2. 4. Store the student s answer from (d) in A and then draw the graph. 5. Use the intersect calculation in the calculator. Since the two graphs intersect several times, be sure to guess a time value close to the appropriate intersection point. Copyright 2008 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: