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.
YES
Part B
Pre-Calculus 2010
velocity in miles per minute
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
t
t
v(t)
0
0
2
0.2
4
0
5
0
6
0.3
7
0.3
8
0.2
time in minutes
(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: www.ltftraining.org
1
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
15  12 
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
5
 12  
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: www.ltftraining.org
2
YES
Part B
Pre-Calculus 2010
Rubric
(a) 1 pt:
miles
min
(a)
0.2
(b)
v(t )  0.1(t  4)
v(3.2)  0.08
(c)
1 pt:
correct answer with units
(b) 2 pts: 1 pt:
correct velocity equation
1 pt:
miles
minute
total distance =
(c) 2 pts: 1 pt:
correct answer with correct
analysis or correct evaluation
of 3.2 in student’s velocity
equation
area work shown for at least
one portion of the graph
1
mi  1 
mi 

(4 min)  0.2
   0.3
 (1min  2 min) 
2
min  2 
min 

1
mi
mi  1
mi 


(1min)  0.3
 0.2
  (4 min)  0.2

2
min
min
2
min




1 pt:
= 1.5 miles
(d)
w(2) 

  
    miles
sin   2   sin   
15  12  15  6  30 min
(d) 2 pt:
1 pt:
1 pt:
Danh travels

miles
faster.
0.2 
 0.2  0.1  0.1
30
min
(e)
4
8

 4
d (12)  1  cos  12    (2)   1.6 miles
5
12
5
5


(e) 1 pt:
1 pt:
(f)
1.6 – 1.5 = 0.1 miles.
(f) 1 pt:
1 pt:
correct total distance with
correct units with appropriate
area work
correct answer for Sierra’s
velocity with appropriate
work shown
correct difference, to nearest
tenth of a mile, based on
student’s answers in (a)
and (d)
correct answer to nearest
tenth of a mile with
appropriate work shown
correct name with correct
difference, to nearest tenth of
Sierra lives 0.1 miles farther from school
a mile, based on student’s
than Danh.
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
3


3
  
d (t )  1  cos  t   .
5
 12  
Students received
credit in part (e) for d (12)  1  cos  12    (2)   1.2 miles and in part (f) for 1.5 1.2  0.3 miles;
5
12
5
5


3
6

Danh lives 0.3 miles farther from school than Sierra.
Copyright © 2010 Laying the Foundation®, Inc. Dallas, TX. All rights reserved. Visit: www.ltftraining.org
3
Part B
YES
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 d (t )  1  cos  t   . The rubric was
5
 12  
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
and earn the second point
For example, the student could write: 0.1(3.2)  0.4  0.08
min
without earning the first point. Another example would be when the student says Danh is
mi
slowing at 0.1
, so 0.2  0.1(3.2  2)  0.2  0.2(1.2)  0.2  0.12  0.08 .
min
Student may earn the second point using similar triangles:
0.1
0.1 x

1 0.8
x
x  0.08
0.8
1
(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) 1st pt: Must have
1

substituted for sin .
2
6
3
Do not accept early substitution of 3 for  where student must evaluate sin   .
6
No bald answer.
2nd 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: www.ltftraining.org
4
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: www.ltftraining.org
5
Sample A
Sample A
2010 Free Response Part A
Pre-Calculus Question 2
Calculator
Sample A – 9 points
Part
Points
Earned
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
Student shows appropriate area work for at least one section of
the graph.
c
1
Student shows appropriate area work, computes correct total
distance, and labels answer with correct units.
d
1
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.
Scoring Commentary
Copyright © 2010 Laying the Foundation®, Inc. Dallas, TX. All rights reserved. Visit: www.LTFtraining.org
Sample B
Sample B
2010 Free Response Part A
Pre-Calculus Question 2
Calculator
Sample B – 7 points
Part
Points
Earned
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
Student shows appropriate area work for at least one section of
the graph.
c
1
Student shows appropriate area work, computes correct total
distance, and labels answer with correct units.
d
1
Student correctly calculates Sierra’s velocity with appropriate
work shown.
d
1
Student computes correct difference.
e
0
Student does not compute correct distance. Student correctly
evaluates cos π but multiplied the numerator and denominator
of the fraction by 2.
f
0
Student states who lives farther but does not compute correct
difference.
Scoring Commentary
Copyright © 2010 Laying the Foundation®, Inc. Dallas, TX. All rights reserved. Visit: www.LTFtraining.org
Sample C
Sample C
2010 Free Response Part A
Pre-Calculus Question 2
Calculator
Sample C – 5 points
Part
Points
Earned
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
Student shows appropriate area work for at least one section of
the graph.
c
1
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.
Scoring Commentary
Copyright © 2010 Laying the Foundation®, Inc. Dallas, TX. All rights reserved. Visit: www.LTFtraining.org
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, v(t )  a sin  b(t  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: www.layingthefoundation.org
EOC
Section II
Pre-Calculus 2009
(b)
A second particle has a velocity function defined by v(t )  a sin(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  0 and b  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: www.layingthefoundation.org
EOC
Section II
Pre-Calculus 2009
RUBRIC


(a) (i) v(t )  3sin   t  2.5    1
3



v(t )  3sin   t  0.5    1
3

(a) 4 pts: 1 pt: two correct constants
1 pt: two additional correct constants
1
2
1 pt: correct value for v(0)
(ii) v(0)  2
1 pt: correct graph with points at
(0,  4), (4, 2), and (7,  4) and
either at  2.5,  1 or  5.5,  1
(iii)
(b) (i) a
(b) 5 pt:
v
t
(ii) t  0,

b
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
2
b
–a
(iii)
1 pt:
 2
b
t 
,
b
2
b
(iv) The distance the particle travels is
a
units.
2b
a
v
t
2
b
®
Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Sample A
Sample A
2009 Free Response Part B
Pre-Calculus Question 2
Sample A – 9 points
Part
a
a
Points
Earned
Scoring Commentary
1
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
and b = .
6=
3
b
1
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.
a
⎛π
⎞
v(t ) = 3sin ⎜ (t − 2.5) ⎟
⎝3
⎠
1
⎛π
⎞
⎛ −5π
v(0) = 3sin ⎜ (0 − 2.5) ⎟ − 1 = 3sin ⎜
⎝3
⎠
⎝ 6
3
⎞
⎟ − 1 = − − 1 = −2.5
2
⎠
a
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).
b
1
Student draws the basic shape of a sine function passing
through the origin.
1
Student shows a and –a on the vertical axis to coincide with
2π
on the
the maximum of the function. Student also shows
b
horizontal axis to coincide with the end of the first cycle of the
sine function. Point is earned.
1
Student correctly determines that the first three zeros occur
π 2π
when t = 0, ,
.
b b
b
1
Since the particle is moving left when v(t ) ≤ 0 , the correct
⎡ π 2π ⎤
interval is ⎢ , ⎥ . The endpoints are ignored. Student earns
⎣b b ⎦
the point.
b
1
The triangle’s area is
b
b
®
1⎛π ⎞
πa
⎜ ⎟(a) =
2⎝ b ⎠
2b
Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Sample B
Sample B
2009 Free Response Part B
Pre-Calculus Question 2
Sample B – 7 points
Part
Points
Earned
a
1
Student determine a = 3, and d = –1 to earn one point.
0
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.
a
0
Student begins this calculation correctly using the equation
determined in part (a), but declares sin(12) = 1 and point is not
awarded.
a
1
Student draws the basic shape of a sine function passing
through the origin.
1
Student shows a and –a on the vertical axis to coincide with
2π
the maximum of the function. Student also shows
on the
b
horizontal axis to coincide with the end of the first cycle of the
sine function. Point is earned.
1
Student correctly determines that the first three zeros occur
π 2π
when t = 0, ,
.
b b
b
1
Since the particle is moving left when v(t ) ≤ 0 , the correct
⎡ π 2π ⎤
interval is ⎢ , ⎥ . Student earns the point.
⎣b b ⎦
b
1
The triangle’s area is
b
1
Student draws the basic shape of a sine function passing
through the origin.
a
b
b
Scoring Commentary
®
1⎛π ⎞
πa
⎜ ⎟(a) =
2⎝ b ⎠
2b
Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Sample C
Sample C
2009 Free Response Part B
Pre-Calculus Question 2
Sample C – 5 points
Part
Points
Earned
Scoring Commentary
1
and
2
determines a = −3 to correspond with the minimum point on
the sine function.
Student correctly shifts the function to the left
a
1
Student determines d = –1 as the minimum velocity + the
a
1
amplitude and determines that b =
π
3
Student correctly calculates v (0) and earns the point.
a
1
⎡π ⎛ 1 ⎞⎤
⎛ 5π
v(t ) = −3sin ⎢ ⎜ t + ⎟ ⎥ − 1 = −3sin ⎜
⎝ 6
⎣ 3 ⎝ 2 ⎠⎦
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
2π
the maximum of the function. Student also shows
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.
®
⎞
⎟ − 1 = −2.5
⎠
Copyright © 2009 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Posttest
Part B
Pre-Calculus 2005
3. Let v(t ) 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.
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
t
(sec)
v(t) (in/ sec)
t (seconds)
1
2
3
4
5
6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
v( t )
(in/sec)
0
-5.625
-8
-7.875
-6
-3.125
0
2.625
4
3.375
0
-6.875
-18
(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.
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.
Posttest
Part B
Pre-Calculus 2005
3. Let v(t ) 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)
5
answer the following questions.
4
3
(a) For what value(s) of t, if any, is the particle not
2
1
moving? Justify your answer and include units.
t (seconds)
0
1
2
3
4
5
6
-1
(b) For what value(s) of t, if any, is the particle moving
-2
-3
to the right? Justify your answer.
-4
-5
(c) For what approximate t value is the speed of the
-6
-7
particle the greatest over the interval [0, 5]?
-8
-9
Justify your answer and include units.
-10
(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 seconds
t
(sec)
0
v(t)
(in/sec)
0
0.5
-5.625
1
-8
1.5
-7.875
2
-6
2.5
-3.125
3
0
3.5
2.625
4
4
4.5
3.375
5
0
5.5
-6.875
6
-18
(a) 2 pts: 1 pt: three correct answers
1 pt: explanation
because v(t )  0
(b) (3, 5) seconds
because v(t )  0
(c) t  1.25 seconds
(b) 2 pts: 1 pt: correct interval
note: ignore [ , ] and ( , )
1 pt: explanation
(c) 2 pts: 1 pt: answer in [1, 1.5)
1 pt: explanation
because speed = | v(t ) |
or
because the sign shows direction and
does not affect speed
(d)
v(4)  v(2)
in
5 2
42
sec
(e) 14 inches
(d) 1 pt: 1 pt: answer
(e) 1 pt: 1 pt: answer
Units pt: correct units in (a), (b), or (c)
and in (d) and (e)
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
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.
EOC
Section II
Pre-Calculus 2008
45t 2   t 
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.
The function, v(t ) 
v
100
50
t
0
(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: www.layingthefoundation.org
EOC
Section II
Pre-Calculus 2008
(c)
(d)
(e)
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.
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.
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: www.layingthefoundation.org
2008 Pre-Calculus Rubric
®
Question 1 (Calculator Allowed)
45t 2   t 
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.
The function, v(t ) 
v
100
50
(a) When is the particle at rest? Include units in your answer.
t
0
(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?
(a)
(a)
t  0, 6, 12 seconds
(b)
1 pt:
1 pt:
must have t = 6 and no
incorrect times
2 pt:
1 pt:
correct first velocity
1 pt:
correct second velocity
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)
1 pt:
1 pt:
student’s answer in part(c)
divided by 12
1 pt:
1 pt:
solves v(t) = student’s
answer in part (d)
(0 /1 if student’s answer in
part (d) is zero)
(b)
mm
mm
max vel  36.738
and 102.369
sec
sec
(c)
1
1
(6)(36.738)  (6)(102.369)  417.322 mm
2
2
The particle traveled 417.322 mm in 12 sec.
(d)
(c)
4 pt:
(d)
417.322...
mm
mm
 34.776
or 34.777
12
sec
sec
(e)
(e)
t = 3.099 or 3.100 seconds
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
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 417.321 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) = 33.75 and f(9) = 101.25, total distance is 405 mm and the
values in part (d), the average value is 33.75 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 417.322 or 417.321 is used.
Part (e):
 Student’s answer will be the same as shown in the rubric if 417.322 or 417.321 is used.
 Student’s answer must be in the interval [0, 12].
®
Copyright © 2008 Laying the Foundation , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Sample A
Sample A
2008 Free Response Part A
Pre-Calculus Question 1
Sample A – 9 points
Part
Points
Earned
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 417.322 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.
Scoring Commentary
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: www.layingthefoundation.org
Sample B
Sample B
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 417.322 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: www.layingthefoundation.org
Sample C
Sample C
2008 Free Response Part A
Pre-Calculus Question 1
Sample C – 5 points
Part
Points
Earned
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 8.464 seconds.
Scoring Commentary
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)(102.369) , 110.214 + 307.107, 110.214 + 307.108, or 417.322
2
2
If the student’s answer is 417.321 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 v(t ) 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: www.layingthefoundation.org