about advanced placement mathematics

Advanced Placement Mathematics Overview

Advanced Placement is not just a test, not simply a course, it is a program.

The Advanced Placement Program • “is a cooperative educational endeavor between secondary schools and

colleges and universities.” • is committed to educational excellence and educational equity. • is composed of 37 courses.

Benefits of the AP Program for Students • Students can earn college credit and/or advanced standing at over 90

percent of colleges and universities in the United States. • Students gain college-level analytical, communication, and study skills as

well as academic confidence. • Students can gain state or national recognition.

The AP Effect on Students Students who complete AP Courses are: • Better prepared academically • More likely to choose challenging majors • Likely to complete more college-level work • Likely to perform significantly better than students who did not take AP

courses • More likely to exercise leadership • More likely to graduate with a double major • Twice as likely to go into advanced study

Benefits of the AP Program for Schools • Provides challenging academic program for motivated students • Provides a rewarding professional development opportunity for teachers • Demonstrates a school’s commitment to high academic standards • Enhances a school’s overall academic reputation • Provides useful data to guide school reform

AP Course Audit The AP Course Audit was created as a means to provide teachers and administrators with clear guidelines on the curricular and resource requirements for AP courses. The AP Course Audit also helps colleges and universities better interpret secondary school courses marked “AP” on students’ transcripts. To receive authorization from the College Board to label a course “AP,” schools must demonstrate how their courses meet or exceed these requirements, which colleges and universities expect to see within a college-level curriculum.

There are three Advanced Placement Examinations for mathematics offered by the College Board: AP Calculus AB The exam covers differential and integral calculus topics that are typically included in introductory calculus courses at the college level. Because graphing calculator use is an integral part of the course, the exam contains questions that require students to use a graphing calculator. AP Calculus BC The exam covers the same differential and integral calculus topics that are included in the Calculus AB exam, plus additional topics in differential and integral calculus, and polynomial approximations and series. This is material that would be included in a two-semester calculus sequence at the college level. Because graphing calculator use is an integral part of the course, the exam contains questions that require students to use a graphing calculator. Both courses emphasize a multirepresentational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections among these representations also are important. Both require a similar depth of understanding, and both courses are intended to be challenging and demanding. AP Statistics The following major topics are covered: exploring data (describing patterns and departures from patterns); sampling and experimentation (planning and conducting a study); anticipating patterns (producing models using probability and simulation); and statistical inference (estimating population parameters and testing hypotheses). Advanced Placement Examinations • AP Examinations are composed of multiple-choice sections and free-

response questions with the exception of Studio Art • Free-response questions are scored by college professors and experienced

AP teachers at June readings • In scoring multiple choice questions, one-fourth of the number of questions

answered incorrectly will be subtracted from the number of questions answered correctly, to discourage random guessing.

• Examination scores range from 1 to 5 5 – Extremely Well Qualified 4 – Well Qualified 3 – Qualified 2 – Possibly Qualified 1 – No Recommendation

AP Calculus AB and AP Calculus BC Exam Format

% of Grade Number of Questions

Minutes Allotted Calculator Use

Section I Multiple Choice 50

Part A 28 55 No calculator

Part B

17 50 Graphing calculator required

Section II Free Response 50

Part A 3 problems 45 Graphing calculator

required Part B

3 problems 45 No calculator

AP Statistics Exam Format

% of Grade Number of Questions

Minutes Allotted Calculator Use

Section I Multiple Choice 50 40 90

Section II Free Response 50

Part A

75% of Section II 5 problems Approx 65

Part B 25% of Section II 1 problem Approx 25

Graphing calculator expected

What Is Pre-AP? Pre-AP is a set of content-specific strategies to: • Build rigorous curricula • Promote access to AP for all students • Introduce skills, concepts, and assessment methods to prepare students for

success when they take AP and other challenging courses Pre-AP prepares growing numbers of students, especially those traditionally underrepresented, for the challenges offered by the Advanced Placement Program. Pre-AP is a concerted effort to fulfill the College Board’s mission to champion educational excellence for all students. What is a Vertical Team? A vertical team is a group of educators from different grade levels in a given discipline who work cooperatively to develop and implement a vertically aligned program. AP Vertical Teams are groups of educators from different grade levels in a given discipline who work cooperatively to develop and implement a vertically aligned program aimed at helping students acquire the skills necessary for success in the Advanced Placement Program. Why Create AP Vertical Teams?

⇒ Standards ⇒ Inclusion ⇒ Innovation ⇒ Coordination ⇒ Empowerment ⇒

⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

Enthusiasm AP Vertical Teams – Factors for Success:

Committed Members Administrative Support Vision and Rationale Team-Building Experiences Leadership Knowledge of curricula at all grade levels Understanding of AP standards Awareness of skills required for success in AP courses

How Are AP Vertical Teams Developed? • by raising awareness of the AP Program • by generating administrative support • by organizing a team and choosing a facilitator • by creating and implementing curricular reform • by cultivating team building skills • by developing community support • by assessing effectiveness AP Vertical Teams – Major Activities • Vertically align curriculum • Assess student progress toward Advanced Placement standard • Develop Pre-AP Strategies AP Vertical Teams Resources: The Advanced Placement Mathematics Vertical Teams Toolkit Setting the Cornerstones: Building the Foundation of AP Vertical Teams Setting the Cornerstones is a two-day interactive workshop on forming discipline-based teams of teachers and administrators to improve academic performance and participation in the College Board’s Advanced Placement Program. PREREQUISITES for AP Calculus (from page 7 of the Course Description)

Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students. Courses in which they study: Algebra, Geometry, Trigonometry, Analytic geometry, and Elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions.

Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the

values of the trigonometric functions of numbers such as 0, , , ,6 4 3π π π and .2

π

PREREQUISITES for AP Statistics Second-year algebra.

Resources AP Central apcentral.collegeboard.com Pre-AP http://apcentral.collegeboard.com/apc/public/preap/index.html AP Calculus AB and BC Course Descriptions (“the Acorn book”) http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf AP Calculus AB and BC Teachers’ Guide http://apcentral.collegeboard.com/apc/members/repository/ap07_calculus_teachersguide_2.pdf AP Statistics Course Description http://apcentral.collegeboard.com/apc/public/repository/ap08_statistics_coursedesc.pdf AP Statistics Teachers’ Guide http://apcentral.collegeboard.com/apc/public/repository/AP_Statistics_Teacher_Guide.pdf

The AP Mathematics exams in 2009 are administered on:

Tuesday, May 4 (Statistics)

Wednesday, May 5 (Calculus AB, Calculus BC)

Adapting Advanced Placement Free Response Questions for the Vertical

Team

AP® CALCULUS AB

2003 SCORING GUIDELINES (Form B)

Copyright © 2003 by College Entrance Examination Board. All rights reserved.

Available at apcentral.collegeboard.com.

6

Question 5

Let f be a function defined on the closed interval [0,7]. The graph of

f, consisting of four line segments, is shown above. Let g be the

function given by 2

( ) ( ) .x

g x f t dt=

(a) Find ( )3 ,g ( )3 ,g and ( )3 .g

(b) Find the average rate of change of g on the interval 0 3.x

(c) For how many values c, where 0 3,c< < is ( )g c equal to the

average rate found in part (b)? Explain your reasoning.

(d) Find the x-coordinate of each point of inflection of the graph of

g on the interval 0 7.x< < Justify your answer.

(a) ( )3

2

1(3) ( ) 4 2 3

2g f t dt= = + =

(3) (3) 2g f= =

0 4

(3) (3) 24 2

g f= = =

3 :

1 : (3)

1 : (3)

1 : (3)

g

g

g

(b) (3) (0)

3

g g =

3

0

1( )

3f t dt

= ( )1 1 1 7(2)(4) (4 2)

3 2 2 3+ + =

2 :

3

01 : (3) (0) ( )

1 : answer

g g f t dt=

(c) There are two values of c.

We need 7

( ) ( )3

g c f c= =

The graph of f intersects the line 7

3y = at two

places between 0 and 3.

2 : 1 : answer of 2

1 : reason

Note: 1/2 if answer is 1 by MVT

(d) x = 2 and x = 5

because g f= changes from increasing to

decreasing at x = 2, and from decreasing to

increasing at x = 5.

2 :

1 : 2 and 5 only

1 : justification

(ignore discussion at 4)

x x

x

= =

=

AP® CALCULUS AB 2003 SCORING COMMENTARY (Form B)

Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com.

6

Question 5

This problem presented the graph of a function f and asked questions about the function g defined as the definite integral of f over the interval from 2 to x. While it was possible to find an explicit algebraic representation of f as a piecewise-linear function, the questions were designed to be more easily answered using the Fundamental Theorem of Calculus and the geometric meaning of the integral. Part (a) required the use of the Fundamental Theorem of Calculus, the recognition of integral as area, and the ability to read the derivative of a linear function from its graph. Part (b) asked for the average rate of change of the function g, which required translating the difference in the values of g into a definite integral of f, and evaluating that definite integral from the graph of the

function. Students could use ( )3g from part (a) and compute ( )0g separately. Part (c) required solving ( ) 73

f c =

and recognizing that there are exactly two solutions. Although this looks like a problem involving the Mean Value Theorem, it actually is not, though it could be used to initiate classroom discussion of the Mean Value Theorem. Part (d) required students to find a point of inflection by reading the graph of the derivative. Sample A (Score 9) The student earned all 9 points. Sample B (Score 7) The student earned 7 points: 2 points in part (a), 2 points in part (b), 1 point in part (c), and 2 points in part (d). In part (a), the student did not find ( )3 ,g′′ losing that point. In part (c), the student gave a correct justification, but used an incorrect equation, and therefore lost the answer point.

David

Rectangle

David

Rectangle

David

Rectangle

David

Rectangle

David

Underline

David

Underline

Adaptation of 2003 Calculus AB Form B question 5 for Middle Grades

1. For what values of x is y = 0? 2. Find all values of x for which y = 2. 3. For what values of x is y positive? 4. For what values of x is y negative? 5. For what values of x is the graph increasing? 6. For what values of x is the graph decreasing? 7. What is the maximum value of y? 8. For what value of x does the graph reach its maximum value?


9. What is the minimum value of y? 10. For what values of x does the graph reach its minimum value? 11. Find the area of the region enclosed by the graph and the x-axis for 0 ? Show 4x≤ ≤ your work.


Adaptation of 2003 Calculus AB Form B question 5 for Algebra 1

D

B

A 1. Find the y-intercept of the graph of f. 2. What is the slope of segment OA ? 3. Write the equation of the line passing through p 4. Verify that (1, 2) is a point on the line OA . 5. Find f (2). 6. For which values of x is ( ) 2f x = ? 7. For which values of x is ( )f x positive? 8. For which values of x does ( )f x have positive s

Copyright © 2003 by College Entrance Examination Boarapcentral.collegeboard.com.

E

C

oints O and A.

lope?

d. All rights reserved. Available at

9. Find the slope of segment AB . 10. Write the equation of the line passing through point A and B.

11. Find the length of segment AB . 12. Graph . On the graph shade the area enclosed by the graphs of( ) 2g x = ( )f x and

. ( )g x 13. Find the area of the region enclosed by the graphs of ( )f x and . Show the

work that leads to your conclusion. ( )g x


Adaptation of 2003 Calculus AB Form B question 5 for Geometry (1)

1. Find the slope of ( )f x if 0 2x≤ ≤ . 2. Segments OA and AB are not perpendicular; however, segments BC and CE

are perpendicular. Justify that this is a true statement. 3. Find the length of the segment OA . 4. Graph . Label the points on the new graph as

. Title the graph as “Graph of h”. ( ) ( ) 1h x f x= +

', ', ', ', ', 'O A B C D E

y

4

1

8

5

6

2

73-2-2

99

7

5

3

1

-2-2

8642-1-1

99x


5. What is the slope of if 0( )h x 2x≤ ≤ ? Explain your reasoning. 6. What statement can you make about the slopes and the lengths of the segments

and ' ', and ' ', and ' ', and ' ',as well as and ' '?OA O A AB A B BC B C CE C E ED E D

7. represents a translation of f up one unit. Write the equation of a new

function g(x) that will translate f down 2 units. ( )h x

8. Explain why the statements that were made in question 6 will also be true for the

corresponding segments of the new function g. Generalize your statement for any function . ( ) ( )m x f x k= +

9. Write an equation for a new function p(x) in terms of f(x) that will translate f two

units to the right. 10. Write a conjecture about the slopes and the lengths of segments of corresponding

segments that have been translated vertically or horizontally. 11. Segment AB is a reflection of segment OA about which vertical line? 12. Find the slope of AB and compare it to the slope of OA that you calculated in

question 1.


13. Compare the slopes of segment BC and CE . These two segments are also

reflections of each other about the vertical line 5x = . 14. Use your results from questions 12 and 13 to make a conjecture about the slopes

of segments that are reflections about a vertical line. 15. Draw a new function k(x) that is reflection of f(x) about the y-axis. Write an

equation in terms of f(x) for k(x). y

10

6

-6

-4

4

2

-8

8 -2-10 -10

8

4

-2

-6

-10-10

6 2-4-8 10x

16. Using the grid provided in question 15, draw a new function m(x) that is reflected

about the x-axis. Write an equation in terms of f(x) for m(x).


17. If a bug crawls along the function f beginning at point O and stopping at point D,

how far does it travel? 18. If the same bug crawls along any one of the functions, h, g, p, k, or m what

statement can you make about the distance that the bug will travel? Explain your reasoning.

19. Generalize a statement about the slopes and lengths of line segments under a

reflection about a vertical line. 20. Generalize a statement about the slopes and lengths of line segments under a

reflection about a horizontal line.


Adaptation of 2003 Calculus AB Form B question 5 for Geometry (2)

D

C

B

A

1. Graph . ( ) ( ) 1h x f x= +

y

4

1

8

5

6

2

73-2-2

99

7

5

3

1

-2-2

8642-1-1

99x

2. Devise a plan for finding the area bounded by the x-axis, the y-axis, the curve

h(x), and the line x = 7. Explain your plan. 3. Calculate the area described in question 2. Show the calculations that support

your answer.

4.Graph . ( ) ( ) 3g x f x= +

y

4

1

8

5

6

2

73-2-2

99

7

5

3

1

-2-2

8642-1-1

99x

5. Devise a plan for finding the area bounded by the x-axis, the y-axis, the curve

g(x), and the line x = 7 then explain your plan. 6. Calculate the area described in question 5. Show the calculations that support

your answer. 7. Compare your answers from question 3 and 6. By looking at the geometric

differences in the two figures, give a geometric reason why the difference between the two answers is 14.

8. Without drawing the figure, if ( ) ( ) 5k x f x= + , what is the area bounded by the x-

axis, the y-axis, the curve g(x), and the line x = 7? Explain your reasoning.

Adaptation of 2003 Calculus AB Form B question 5 for Algebra 2/Precalculus Let f be a function defined on the closed interval [0, 7]. The graph of f, consisting of four line segments, is shown above. 1. Write the piecewise function for f(x). 2. Find the values of x for which f(x) = 1. 3. For what interval(s) of x does f have a rate of change of 2? Explain your answer. 4. On which interval of x is the rate of change of f the greatest? Justify your answer.


For questions 5 – 8, express, in terms of the function f, a function g that will have the following characteristics: 5. The maximum value of g is located at the point (2,5). 6. The maximum value of g is located at the point (2,2). 7. The minimum value of g is located at the point (2, -4). 8. The range of g is [0,4]. Each of the following transformations on the function f will change the point at which the maximum value of the function f will occur. Given that k > 1, explain the effect of each transformation. 9. ( ) ( )g x f x k= + 10. ( ) ( )g x f x k= − 11. ( ) ( )g x k f x= ⋅ 12. ( ) ( )g x f kx=


2003 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)

Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available to AP professionals at apcentral.collegeboard.com and to

students and parents at www.collegeboard.com/apstudents.

GO ON TO THE NEXT PAGE. 10

5. Contestants on a game show spin a wheel like the one shown in the figure above. Each of the four outcomes on this wheel is equally likely and outcomes are independent from one spin to the next.

� � The contestant spins the wheel.

� If the result is a skunk, no money is won and the contestant’s turn is finished.

� If the result is a number, the corresponding amount in dollars is won. The contestant can then stop with those winnings or can choose to spin again, and his or her turn continues.

� If the contestant spins again and the result is a skunk, all of the money earned on that turn is lost and the turn ends.

� The contestant may continue adding to his or her winnings until he or she chooses to stop or until a spin results in a skunk.

(a) What is the probability that the result will be a number on all of the first three spins of the wheel?

(b) Suppose a contestant has earned $800 on his or her first three spins and chooses to spin the wheel again. What is the expected value of his or her total winnings for the four spins?

(c) A contestant who lost at this game alleges that the wheel is not fair. In order to check on the fairness of the wheel, the data in the table below were collected for 100 spins of this wheel.

Result Skunk $100 $200 $500

Frequency 33 21 20 26

Based on these data, can you conclude that the four outcomes on this wheel are not equally likely? Give appropriate statistical evidence to support your answer.

AP® STATISTICS 2003 SCORING GUIDELINES (Form B)


13

Question 5

Solution Part (a):

P(a number on all 3 spins) = [P(number)]3 since the outcomes are independent from spin to spin

= 33 0.4219

4� � =� �

Part (b):

Winnings 0 900 1000 1300 Probability 0.25 0.25 0.25 0.25 E(winnings) = = 0(0.25)+900(0.25)+1000(0.25)+1300(0.25) = 800i ix p�

Or E(winnings on 4th spin) = -800(0.25) + 100(0.25) + 200(0.25) + 500(0.25) = 0 So E(winnings) = initial amount + E(winnings on 4th spin) = 800 + 0 = 800

Part (c): Element 1: States a correct pair of hypotheses

H0: The four outcomes are equally likely 1 2 3 41or 4

p p p p� �= = = =� �

Ha: The four outcomes are not equally likely (or at least one pi differs from 14

)

Element 2: Identifies a correct test (by name or by formula) and checks appropriate conditions.

Chi-square test (for goodness of fit) 2

2 ( )Obs ExpExp

�−

=�

Conditions: Outcomes of spins of the wheel are independent and large sample size.

The problem states that successive spins of the wheel are independent.

The expected counts are all equal to 25, which is greater than 5 (or 10), so the sample size is large enough to proceed.



14

Question 5 (cont’d)

Element 3: Correct mechanics, including the value of the test statistic, df, and p-value (or rejection region)

Expected counts: 25 for each of the 4 cells

2 2 2

2 ( ) (33 25) (26 25) 4.2425 25

Obs ExpExp

�− − −= = + + =� �

df = 4-1 = 3 p-value = .2367 (from tables p-value > 0.10, from Graphing Calculator: p-value = 0.23669, from table rejection region for 0.05� = is 7.81, 0.01� = is 11.34)

Element 4: Using the results of the statistical test, states a correct conclusion in the context of the problem.

Because the p-value is greater than the stated � (or because the p-value is large, or because the test statistic does not fall in the rejection region), fail to reject H0. There is not convincing evidence that the four outcomes on the wheel are not equally likely. That is, we don’t have convincing evidence against the conjecture that the four outcomes on the wheel are equally likely.

Scoring

Part (a): 1 for correct answer (including Binomial calculation) = 0.4219

12

if answer is 34

or 31 0.0156

4� � =� �

or 31 (3) 0.047

4� � =� �

or 0.4219 with no work

Part (b): 1 if the expected value, 800, is correct (except for minor computational errors)

12

if expected value is computed as

�� 800 + expected winnings on one spin, or 800 + 200 = 1000 ��E(outcome on one spin)=200 but then solution breaks down ��E(winnings on 4th spin)=0 but then solution breaks down

��With fairly major computational errors 3200e.g., 3

� ��



15

Question 5 (cont’d)

0 if

�� answer of 800 is given but no work is shown or bad logic, e.g., 4(200) �� expected value formula is given but no calculations are done �� outcomes are set up correctly but no expected value is calculated

Part (c): 12

for each element of the test that is correct

1. statement of hypotheses 2. identification of test and check of sample size condition 3. correct mechanics 2 4.24� = , df=3, p-value=0.2367, and/or 2

3,0.05 7.81� = 4. statement of conclusion (fail to reject)

If both an α and a p-value are given, the linkage is implied. If no α is given, the solution must be explicit about the linkage by giving a correct interpretation of the p-value or explaining how the conclusion follows from the p-value.

NOTE: If the p-value in element 3 is incorrect but the conclusion is consistent with the computed p-value, element 4 can be considered as correct.

4 Complete Response

Score of 4 from parts (a) through (c)

3 Substantial Response Score of 3 from parts (a) through (c)

2 Developing Response

Score of 2 from parts (a) through (c)

1 Minimal Response Score of 1 from parts (a) through (c)

IF A PAPER IS BETWEEN TWO SCORES (FOR EXAMPLE, 2 ½ PARTS) USE A HOLISTIC APPROACH TO DETERMINE WHETHER TO SCORE UP OR DOWN DEPENDING ON THE STRENGTH OF THE RESPONSE AND COMMUNICATION.

Adaptation of 2003 Statistics Form B question 5 for Middle Grades

skunk 100 200 500 Contestants on a game show spin a wheel like the one shown in the figure above. Each of the four outcomes on this wheel is equally likely and outcomes are independent from one spin to the next. The contestant spins the wheel. If the result is a skunk, no money is won. If the result is a number, the corresponding amount in dollars is won.

Note: P(a) means the probability that a occurs. 1. P(100) = 2. P(not skunk) = 3. P(200 or 500) = 4. P(300) = 5. P(100 or 200 or 500 or skunk) = 6. P(contestant earns < $500) = 7. P(contestant earns > $200) = 8. P(contestant spins two skunks in a row) =


9. Do a simulation of 20 spins and record your results in the table below.

Individual spins 200 100 500 Skunk Number of spins Percentage

10. Collect the data from the entire class and record the results in the table below.

Pooled data 200 100 500 skunk Number of spins Percentage

11. Theoretically, what percentage of the spins should land on: a. 200? b. 100? c. 500? d. skunk? 12. Compare your experimental results to the theoretical results. How closely do they match? 13. On average, what is the expected payout? Why? 14. Based on the pooled experimental results, what is the average payout?


Adaptations of AP problems: Answer Key

2003 Calculus AB Form B question 5 for Middle Grades:

1. 0, 4, and 6 2. 1 and 3 3. Between 0 and 4, and between 6 and 7 4. Between 4 and 6 5. Between 0 and 2, and between 5 and 7 6. Between 2 and 5 7. 4 8. 2 9. –1 10. 5 11. 8 square units. Check students’ work.

2003 Calculus AB Form B question 5 for Algebra 1:

1. 0 2. 2 3. 2y x=4. Check students’ work. 5. (2) 4f =6. 1and 3x x= =7. Between 0 and 4, and between 6 and 7 8. Between 0 and 2, and between 5 and 7 9. –2 10. 2 8y x= − +

11. 20 2 5 4.47= ≈ units 12. See graph. 13. 10.5 square units

2003 Calculus AB Form B question 5 for Geometry (1):

1. 2 2. Check students’ work. 3. 20 2 5 4.47= ≈ 4. See graph. 5. 2 6. The slopes and the lengths have not changed. 7. ( ) ( ) 2g x f x= −8. In any , the new segments are parallel to the old ones, so they have the same

slope. Their lengths have not been altered. ( ) ( )m x f x k= +

9. ( ) ( 2)p x f x= −


10. When a segment is translated vertically or horizontally, their slopes and lengths will not change.

11. 2x =12. Slope of 2;slope of 2OA AB= = − 13. Slope of 1;slope of 1BC CE= − = 14. The slopes will be opposites of each other. 15. See graph: ( ) ( )k x f x= −16. See graph: ( ) ( )m x f x= −

17. 4 5 3 2 13.2+ ≈ units 18. When a segment is reflected about a vertical line, the length does not change, but the slope

becomes the opposite. 19. When a segment is reflected about a horizontal line, the length does not change, but the slope

becomes the opposite. 2003 Calculus AB Form B question 5 for Algebra 2/Precalculus:

1.

2 , 0 22 8, 2

( )4, 4 5

6, 5 7

x xx x

f xx x

x x

≤ ≤⎧⎪− + ≤ ≤⎪= ⎨− + ≤ ≤⎪⎪ − ≤⎩

4

≤

2. 1 1, 3 , and 72 2

3. - check students’ work. 0 x< < 224. - check students’ work. 0 x< <

5. ( ) ( ) 1g x f x= +6. ( ) ( ) 2g x f x= −7. ( ) ( )g x f x= −

8. [ ]4( ) ( ) 15

g x f x= +

9. The graph will be shifted up by k units. 10. The graph will be shifted to the right by k units. 11. The graph will be stretched vertically by a factor of k. 12. The graph will be shrunk horizontally by a factor of k.

2003 Calculus AB/BC question 1 for Middle Grades:

1. See graph. 2. See graph. 3. 7 units. Check explanation. 4. See graph. 5. (8, 1) 6. No, it divides AB into 2 unequal segments.


7. 4 units. Check explanation. 8. Scalene – no two sides have the same length. 9. 14 square units. Check students’ work.

2003 Calculus AB/BC question 1 for Algebra 1:

1. See graph. 2. (8,5), (2,1) and (9,1). Check students’ work. 3. See graph. 4. 17( 4.12), 52( 7.21),and 7.≈ ≈ 5. 14 square units 6. Answers will vary: one example is (0,0), (0,2), (7,0), and (7,2). 7. Check students’ work. 8. Answers will vary: previous example yields 0, 0, 7,and 2.y x x y= = = =

2003 Statistics Form B question 5 for Middle Grades:

1. 14

2. 34

3. 12

4. 0 5. 1

6. 34

7. 14

8. 116

9. Answers will vary. 10. Answers will vary. 11. 25% each. 12. Answers will vary. 13. $200 14. Answers will vary.

2003 Calculus AB Form B question 5 for Geometry (2):

1. See graph. 2. Check students’ work. Should describe dividing into simpler figures and adding areas. 3. 14.5 square units. 4. See graph.


5. Check students’ work. Should describe dividing into simpler figures and adding areas. 6. 28.5 square units 7. The 7 by 2 rectangle at the bottom adds 14 square units. 8. 42.5 square units – check explanation.


WHAT MAKES A GOOD PRE-AP MATHEMATICS PROBLEM? (from Advanced Placement And TEKS: A Lighthouse Initiative For Texas Mathematics Classrooms) A good pre-AP mathematics problem or activity: • Has a clear connection to the vocabulary, skills, concepts or habits of mind necessary for success in Advanced Placement mathematics courses. • Goes beyond a minimalist approach to addressing the TEKS. • Can serve multiple purposes, such as addressing an Algebra 1 TEK, reviewing a middle school geometry skill and introducing an AP Calculus concept. • Should go beyond simple drill and recall. There should be a greater emphasis on analysis, application and synthesis of material. • Requires students to engage in an extended chain of reasoning. Problems should require more than one step and might cover more than one topic. • Might be completely different from problems that the teacher has demonstrated in class, though based on the same concept. Students are expected to apply their knowledge in novel situations with very little teacher direction. • Requires students to develop their reading and interpretation skills using verbal, graphical, analytical and numerical prompts. • Asks students to communicate their thoughts orally and/or in writing. Students must be able to justify their work in clear, concise, and well-written sentences. • Stretches the students in ways that might make them uncomfortable. The solving of problems might take several attempts. They might have to hear someone else’s explanation (preferably one of their peers) before they begin to develop understanding. • Should be graded based on the process and methods as well as the final answers. • Might require the thoughtful use of technology.

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