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![Page 1: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/1.jpg)
Reasoning in the Presence of the
Calculator on the SAT
Good ideas in teaching precalculus and…
Rutgers University
March 16, 2012
![Page 2: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/2.jpg)
Presenters
• Brian O’Reilly• The College Board
• Robin K. O’Callaghan• The College Board
![Page 3: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/3.jpg)
SAT UPDATE
Pre-calculus Conference
Rutgers University
March 16, 2012
![Page 4: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/4.jpg)
SAT takers in class of 2011:– 44% were minority
students– 27% reported that English
was not exclusively spoken at home
– 45% intend to be first-generation, college-going students
• The SAT is administered by over 7,000 test centers in more than 170 countries!
Who Took the SAT® This Year?
4
The 2011 cohort was the largest, most diverse group ever!
The 2011 cohort was the largest, most diverse group ever!
![Page 5: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/5.jpg)
SAT® Participation: 20 Year Trend + 59%
Note: SAT participation data for 1990-2006 includes students in each year’s graduating class who took the SAT through March of their senior year. SAT participation data for 2011 includes all students in each year’s graduating class who took the SAT through June of their senior year.
Cohort
![Page 6: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/6.jpg)
SAT® Participation Growing; High School Graduates Declining for Six More Years
0%
1%
2%
3%
4%
5%
6%
2007 2008 2009 2010 2011
% C
hang
e vs
. 200
7 Co
hort
US SAT TakersUS High School
Graduates
CohortSource: 2011 College-Bound Seniors Total Group Report; WICHE (2008)
![Page 7: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/7.jpg)
Average SAT® Math Scores by High School GPA
8
Students earning higher grades in high school earned higher math scores on the SAT.
Students earning higher grades in high school earned higher math scores on the SAT.
Average SAT Math Scores
Aver
age
SAT
Mat
h Sc
ores
High School GPA
![Page 8: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/8.jpg)
Average SAT® Math Scores byAverage High School Math Grades
9
Aver
age
SAT
Mat
h Sc
ores
Average High School Math Grade
Students earning higher math grades in high school earned higher math scores on the SAT.Students earning higher math grades in high school earned higher math scores on the SAT.
Average SAT Math Scores
![Page 9: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/9.jpg)
Average SAT® Math Scores by Years of Math Study in High School
10
Average SAT Math Scores
Students taking more years of math in high school earned higher math scores on the SAT.Students taking more years of math in high
school earned higher math scores on the SAT.
Aver
age
SAT
Mat
h Sc
ores
Years of Math Study in High School
![Page 10: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/10.jpg)
Average SAT® Math Scores by Highest Level Math Course Taken
Aver
age
SAT
Mat
h Sc
ores
Highest Level Math Course Taken
Average SAT Math Scores
Students taking higher level math courses earned higher math scores on the SAT.
Students taking higher level math courses earned higher math scores on the SAT.
![Page 11: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/11.jpg)
Math Subject Tests vs. the SAT® Math Section
12
Math 1 Subject Test
Math 1 Subject Test
Math 2 Subject Test
Math 2 Subject Test
SAT Math Section
SAT Math Section
60 minutes(1 section)
60 minutes(1 section)
70 minutes(3 sections: 20,25,25)
70 minutes(3 sections: 20,25,25)
3 years3 years More than 3 yearsMore than 3 years 3 years3 years
44 multiple choice10 student produced44 multiple choice
10 student produced50 multiple choice50 multiple choice
LengthLength
Years of StudyYears of Study
FormatFormat
Test AttributeTest Attribute
Recommended Course Work
Recommended Course Work
• Two years of algebra • One year of geometry • Two years of algebra • One year of geometry
• Two years of algebra • One year of geometry• Some precalculus /
trigonometry
• Two years of algebra • One year of geometry• Some precalculus /
trigonometry
• Two years of algebra • One year of geometry • Two years of algebra • One year of geometry
What’s AssessedWhat’s
AssessedMath content/subject-area knowledge and
the ability to apply that knowledgeMath content/subject-area knowledge and
the ability to apply that knowledge
Mathematical reasoning,problem solving &
critical thinking
Mathematical reasoning,problem solving &
critical thinking
![Page 12: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/12.jpg)
Mathematics Subject Test Selection
13
• Students who have taken trigonometry and/or precalculus and received grades of B (or better) should probably select Math Level 2.
Consider taking the test that covers topics learned most recently (when content is fresh in the student’s mind).
Consider the requirements of the colleges and/or programs in which students are interested.
• Students who are sufficiently prepared to take Math Level 2 but take Level 1 in hopes of receiving a higher score may not do as well as they expect to.
• Students who have taken trigonometry and/or precalculus and received grades of B (or better) should probably select Math Level 2.
Consider taking the test that covers topics learned most recently (when content is fresh in the student’s mind).
Consider the requirements of the colleges and/or programs in which students are interested.
• Students who are sufficiently prepared to take Math Level 2 but take Level 1 in hopes of receiving a higher score may not do as well as they expect to.
Choosing Between the Mathematics Levels 1 & 2 Subject TestsChoosing Between the Mathematics Levels 1 & 2 Subject Tests
![Page 13: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/13.jpg)
SAT Subject Tests™ & AP® – How Do They Compare?
14
• Cover more advanced topics or in greater depth, which is more reflective of a college-level course
• Primarily used for college course credit & placement
• Indication of rigor of courses taken in high school
• Cover more advanced topics or in greater depth, which is more reflective of a college-level course
• Primarily used for college course credit & placement
• Indication of rigor of courses taken in high school
SAT Subject TestsSAT Subject Tests AP ExamsAP Exams
• Assess knowledge of fundamental concepts and the ability to apply that knowledge
• Primarily used for college admissions
• Indication of interest in specific subjects
• Assess knowledge of fundamental concepts and the ability to apply that knowledge
• Primarily used for college admissions
• Indication of interest in specific subjects
High-school-level tests that indicate a student’s readiness to
take college-level courses in specific subject areas
College-level tests that assess a student’s knowledge, skills and
abilities learned in the corresponding AP courses
SAT Subject Tests can also provide students without access to certain AP classes the
opportunity to demonstrate subject-matter achievement.
![Page 14: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/14.jpg)
The Timing of AP® Exams and SAT Subject Tests™ Makes a Difference
15
• Timing – SAT Subject Tests provide AP students who have not yet taken the AP Exam an opportunity to demonstrate knowledge and achievement during the admission process.
• Most students do not take any AP math course until their senior year; AP Exams are in May.• Subject Tests in Mathematics are offered six times a year and can be taken by students in
any grade.
• Timing – SAT Subject Tests provide AP students who have not yet taken the AP Exam an opportunity to demonstrate knowledge and achievement during the admission process.
• Most students do not take any AP math course until their senior year; AP Exams are in May.• Subject Tests in Mathematics are offered six times a year and can be taken by students in
any grade.
![Page 15: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/15.jpg)
SAT Now on a School Day
• Two SAT School Day test dates in 2011-12 in selected districts and states.
• Administering the test on a school day eliminates barriers of
transportationfamily work obligations
that may hinder weekend testing, helping to get even more students on the road to college.
![Page 16: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/16.jpg)
Increasing Access: State Initiatives, Fee Waivers
States providing/paying for SAT• Maine administers May SAT to all public school juniors• Texas pays SAT fees for any junior for any test date• Delaware administers school-day SAT to all juniors, as will Idaho
Fee Waivers• More than just the waiver of test fees — low-income students get
access to a set of tools that help them realize their college dreams.• Over 350,000 low-income students (21% of all SAT-takers) in the
class of 2011 benefitted from SAT Fee Waiver Services, a 77% increase since 2007
• Over $37 million in SAT fees and services provided to students in grades 9 – 12 during 2010-11 school year.
![Page 17: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/17.jpg)
The SAT Mathematics Test
![Page 18: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/18.jpg)
SAT Mathematics Content Areas
• Number and Operations (20–25%)• Algebra and Functions (35–40%)• Geometry and Measurement (25–30%)• Data Analysis, Statistics, and Probability
(10–15%)
![Page 19: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/19.jpg)
Multiple Choice
Key = E
![Page 20: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/20.jpg)
Student-Produced Response
Key = 9.8, 49/5
![Page 21: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/21.jpg)
SPR Answer Grid
![Page 22: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/22.jpg)
The SAT Mathematics Development Committee
![Page 23: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/23.jpg)
College Board Committees
• A Committee in the early 1990s advised the College Board on 1994 changes to the SAT.
• A Committee formed in 2002 to advise the College Board on introducing changes to the SAT.
• The present Committee continues to advise the College Board on the SAT.
![Page 24: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/24.jpg)
SAT Mathematics TD Committee
Deborah Hughes Hallett, co-chair University of Arizona, Tucson, AZJ. T. Sutcliffe, co-chair Saint Mark’s School, Dallas, TX
Ann Davidian General Douglas MacArthur HS, Levittown, NY
Katherine Halvorsen Smith College, Northhampton, MA
Paul J. Karafiol Walter Payton College Prep HS, Chicago, IL
Kenneth Millett University of California, Santa Barbara, Santa Barbara, CA
Jose Molina Bellarmine College Preparatory, San Jose, CA
Thomas Morley Georgia Tech, Atlanta, GA
Dwight Pierre Borough of Manhattan Community College, New York, NY
![Page 25: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/25.jpg)
Committee Decisions
• The SAT should remain a reasoning test. • There will not be a no-calculator section of
the SAT.• Students should be able to bring the
calculator they use in their classroom.• Questions admitting multiple solution
strategies (including those involving the calculator) are encouraged.
• Questions involving multiple representations of material are encouraged.
![Page 26: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/26.jpg)
What is the Calculator Policy for the SAT?
![Page 27: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/27.jpg)
The Calculator Policy
• Every question on these tests can be solved without a calculator; however, using a calculator on some questions may be helpful to students. A scientific or graphing calculator is recommended.
• Calculator use is permitted on all sections of the SAT Mathematics test.
• Most calculators are permitted, including those with (CAS) computer algebra systems.
![Page 28: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/28.jpg)
The Calculator/Equity Issue
• The SAT Mathematics test contains questions requiring students to simplify algebraic expressions or to solve equations.
• Students can do this algebra “by hand.”• Students who have and can use a calculator
equipped with CAS (computer algebra system) may be able do this algebra by “pushing buttons.”
![Page 29: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/29.jpg)
What is the Calculator/Equity
Study ?
![Page 30: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/30.jpg)
Questions to be Answered
• What effect, if any, does CAS have on mathematics performance on the SAT?
• What models of calculators are students using on the SAT?
• How are students using their calculators on the SAT?
• Do students who have calculators with CAS have an advantage?
![Page 31: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/31.jpg)
The Study Design
• A short survey at the end of one SAT administration to be answered by all students at that administration
• Two specially designed variable sections, each given to a subset of students at the administration; one pretest contains 6 CAS-active questions; the other contains 6 algebra questions of comparable difficulty
![Page 32: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/32.jpg)
The Study Design
Survey questions:• What calculator did you use on the
Mathematics sections of the test?• Is this the calculator that you use in your
classroom?• How did you use your calculator?
![Page 33: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/33.jpg)
Types of Calculators Brought
![Page 34: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/34.jpg)
Examples of Possible CAS-active Questions
![Page 35: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/35.jpg)
![Page 36: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/36.jpg)
![Page 37: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/37.jpg)
![Page 38: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/38.jpg)
![Page 39: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/39.jpg)
![Page 40: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/40.jpg)
![Page 41: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/41.jpg)
Examples of Algebraic
Translation Problems
![Page 42: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/42.jpg)
P+ = 10%
![Page 43: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/43.jpg)
P+ = 57%
![Page 44: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/44.jpg)
P+ = 15%
![Page 45: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/45.jpg)
P+ = 11%
![Page 46: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/46.jpg)
P+ = 27%
![Page 47: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/47.jpg)
P+ = 34%
![Page 48: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/48.jpg)
Possibly CAS-active but Easy
![Page 49: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/49.jpg)
P+ = 71%
![Page 50: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/50.jpg)
P+ = 71%
![Page 51: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/51.jpg)
P+ = 72%
![Page 52: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/52.jpg)
Calculator Not an Advantage
![Page 53: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/53.jpg)
P+ = 22%
![Page 54: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/54.jpg)
P+ = 10%
![Page 55: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/55.jpg)
P+ = 11%
![Page 56: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/56.jpg)
P+ = 20%
![Page 57: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/57.jpg)
P+ = 11%
![Page 58: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/58.jpg)
P+ = 23%
![Page 59: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/59.jpg)
Calculator Used with Reasoning
![Page 60: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/60.jpg)
P+ = 31%
![Page 61: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/61.jpg)
P+ = 40%
![Page 62: Reasoning in the Presence of the Calculator on the SAT Good ideas in teaching precalculus and… Rutgers University March 16, 2012.](https://reader038.fdocuments.in/reader038/viewer/2022110210/56649eb25503460f94bb9ac5/html5/thumbnails/62.jpg)
P+ = 23%
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