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Common Core State Standards AlignmentAdvanced Placement®
By Beth Hart, Elaine Carman, Danielle Luisier and Natasha Vasavada
REsEaRCH REpoRt 2011-8
ConneCFor further information, visit www.collegeboard.org/research
Beth Hart is the director of English language arts curriculum and standards at the College Board.
Elaine Carman is the director of mathematics curriculum and standards at the College Board.
Danielle Luisier is the director of standards and curriculum alignment services at the College Board.
Natasha Vasavada is the executive director of standards and curriculum alignment services at the College Board.
About the College Board
The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of more than 5,900 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success — including the SAT® and the Advanced Placement Program®. The organization also serves the education community through research and advocacy on behalf of students, educators and schools.
© 2011 The College Board. College Board, Advanced Placement Program, AP, SAT and the acorn logo are registered trademarks of the College Board. College Board Standards for College Success, inspiring minds and ReadiStep are trademarks owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Printed in the United States of America.
connection
ContentsIntroduction.............................................................................................................. 2
Content.for.Comparison........................................................................................... 3
Methodology............................................................................................................. 4
Summary.of.the.Alignment.Between.the.Common.Core.State.Standards..and.AP.English.and.Mathematics.Courses........................................................... 7
Demonstration.of.Alignment.Methodology.Table................................................. 9
References............................................................................................................... 13
Appendixes
Appendix.A:.AP.English.—.Common.Core.State.Standards.Side-by-Side.Alignment.Table............................................................................................. 15
Appendix.B:.AP.Calculus,.AP.Statistics,.and.AP.Computer.Science.A.—.Common.Core.State.Standards.Side-by-Side.Alignment.Table................ 41
2 College Board Research Reports
Common Core State Standards Alignment
How the Common Core State Standards Prepare Students to Engage in AP® CoursesIntroduction
The College Board has been a strong advocate for and played an active role in the development of the Common Core State Standards. As part of this collaboration, the College Board helped draft the standards and helped shape the initiative by providing executive guidance on the Common Core Advisory Committee. The goal of the Common Core State Standards — to establish a common set of rigorous expectations to prepare students for college and career readiness — strongly reflects the guiding mission and values of the College Board, as well as our programs and services.
Prior to the development of these Common Core State Standards, the College Board established blue ribbon panels in English language arts, mathematics and statistics, and science to define the skills and knowledge students must develop and master to succeed in college and the workforce in the 21st century. This effort resulted in the development of the College Board Standards for College Success™. These standards for English Language Arts, Mathematics and Statistics, and Science were based on empirical research conducted by the University of Oregon’s Center for Educational Policy Research in collaboration with the Association of American Universities. The standards are benchmarked against the Advanced Placement Program® (AP®) as well as national and international frameworks, including NAEP, TIMSS, and PISA (Wiley, Wyatt, & Camara, 2010). These standards were an important resource in the development of the Common Core State Standards because they were based and benchmarked on empirical evidence related to college success.
The College Board recognizes that defining clear, consistent, and rigorous standards is just the start of ensuring educational improvement. The College Board actively works with states and districts as they build their Common Core implementation plans, and we continue to provide guidance and technical assistance in the areas of:
• comprehensive alignment services;
• benchmarking standards, assessments, and performance level indicators against college outcomes;
• Common Core implementation plan development; and
• professional development.
In support of near-term goals, the College Board has developed this alignment study to document the existing correspondence between the Common Core State Standards and selected AP courses.
This report describes how the Common Core State Standards can prepare students to engage in the following courses: AP English Literature and Composition, AP English Language and Composition, AP Calculus AB, AP Calculus BC, AP Statistics, and AP Computer Science A. As new curricula and assessments emerge and existing curricula and assessments are enhanced, the College Board will conduct additional studies to understand the alignment of other forms of curricula and assessments that may be administered in support of the Common Core State Standards.
3College Board Research Reports
Common Core State Standards Alignment
Content for ComparisonCommon.Core.State.Standards
The Common Core State Standards articulate the knowledge and skills students need to be ready to succeed in college and careers. They were designed to be (1) anchored in research and evidence; (2) aligned to college and workplace expectations; (3) rigorous, clear, and consistent; and (4) reflective of best practices in international frameworks. The standards span kindergarten through 12th grade.
The.Advanced.Placement.Program®
AP is a rigorous academic program built on the commitment, passion, and hard work of students and educators from both secondary schools and higher education. With more than 30 courses in a wide variety of subject areas, AP provides willing and academically prepared high school students with the opportunity to study and learn at the college level.
Each AP course is modeled upon a comparable college course, and college and university faculty play a vital role in ensuring that AP courses align with college-level standards. For example, through the AP Course Audit, AP teachers submit their syllabi for review and approval by college faculty. Only courses using syllabi that meet or exceed the college-level curricular and resource requirements for each AP course are authorized to carry the “AP” label.
AP courses culminate in a suite of college-level assessments developed and scored by college and university faculty members as well as experienced AP teachers. AP Exams are an essential part of the AP experience, enabling students to demonstrate their mastery of college-level course work. Strong performance on AP Exams is rewarded by colleges and universities worldwide. More than 90 percent of four-year colleges and universities in the United States grant students credit, placement, or both on the basis of successful AP Exam scores.
Although the Common Core State Standards create a strong foundation for many AP courses, College Board content teams focused this study within the English language arts and mathematics disciplines. The following AP courses were considered within the scope of this study:
AP.English.Literature.and.CompositionAn AP English Literature and Composition course engages students in the careful reading and critical analysis of imaginative literature. As they read, students consider a work’s structure, style, and themes, as well as specific elements such as the use of figurative language, imagery, symbolism, and tone. Such reading should be accompanied by thoughtful discussion and writing about those books in the company of one’s fellow students.
AP.English.Language.and.CompositionAn AP course in English Language and Composition engages students in becoming
AP.provides.willing.
and.academically.
prepared.high.school.
students.with.the.
opportunity.to.study.
and.learn.at.the.
college.level.
4 College Board Research Reports
Common Core State Standards Alignment
skilled readers of prose written in a variety of rhetorical contexts, and in becoming skilled writers who compose for a variety of purposes. Both their writing and their reading should make students aware of the interactions among a writer’s purposes, audience expectations, and subjects, as well as the way genre conventions and the resources of language contribute to effectiveness in writing.
AP.Calculus.AB.and.AP.Calculus.BCAP Calculus AB and AP Calculus BC are primarily concerned with developing the students’ understanding of the concepts of calculus and providing experience with its methods and applications. The courses emphasize a multi-representational approach to calculus, with concepts, results, and problems expressed graphically, numerically, analytically, and verbally. Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics taught in Calculus AB plus additional topics, but both courses are intended to be challenging and demanding; they require a similar depth of understanding of common topics.
AP.StatisticsAn AP course in Statistics introduces students to the major tools and concepts for collecting, representing, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: exploring data, sampling and experimentation, anticipating patterns, and statistical inference.
AP.Computer.Science.AAn AP course in Computer Science A emphasizes object-oriented programming methodology with a concentration on problem solving and algorithm development. It includes the study of data structures, design, and abstraction. A large part of the course is built around the development of computer programs that correctly solve a given problem, while developing the use of logic and formal methods of programming.
MethodologyResearchers and organizations have developed different methodologies and approaches to align standards (Brown & Conley, 2007; Rothman, 2003; Webb, 1997, 1999; Porter, 2002; Achieve, 2007, 2010; College Board, 2008, 2010; Blank, 2007), and the alignment approaches vary in terms of rigor and accuracy. The College Board employs rigorous alignment methodologies as it compares standards, curricula, and assessment frameworks and consistently adheres to the following alignment principles, which are recognized as best practices by thought leaders in the field:
• Deep and careful consideration must be given to the full intent and scope of each standards statement, with attention to both the content and skill components of the standard.
• Alignments must be made to the most specific component or grain size of the standard.
• The course materials utilized in the alignment study must clearly and explicitly outline the course objectives and expected student performances.
• Consideration and attention must be given to both content and cognitive demand (rigor) when developing alignments between standards and course materials.
These alignment principles are apparent in the alignment methodologies developed by
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Common Core State Standards Alignment
Norman Webb and Andrew Porter, as well as in the nationally recognized work from organizations such as Achieve, WestEd, the Education Policy Improvement Center (EPIC), the Center for Assessment, and the Surveys of Enacted Curriculum (SEC) Project led by the Council of Chief State School Officers in partnership with the University of Wisconsin.
For this study, the College Board applied these alignment principles to compare the Common Core State Standards to selected AP courses. The comparison is nuanced in that the Common Core State Standards are designed to articulate the knowledge and skills students need to be ready to succeed in college and careers, and AP courses and exams are designed to represent the level of a first-year college course. In light of this distinction, alignments between the Common Core State Standards and AP courses should not be interpreted as linkages of content or skills at the same level of rigor or challenge, but as areas where there is an identifiable bridge from one framework to another — a link from the Common Core State Standards to a specific AP course. Similarly, “gaps” should not be interpreted as Common Core State Standards that do not prepare students for AP, but as valuable, general content that contributes to a student’s preparedness as a whole.
The purpose of this alignment study is to identify the Common Core State Standards that prepare students to engage in selected AP courses. Correlations between the Common Core State Standards and the AP courses are indicated in the alignment tables (appendixes A and B) by an “X” next to the Common Core State Standard under the appropriate course column.
In order for a Common Core State Standard to be considered aligned to an AP course, evidence of the extension of that Common Core State Standard must be cited from at least one document from the collection of AP course materials. Evidence of the extension of a Common Core State Standard for English Language Arts will appear in the AP English courses as content and skills similar to those of the Common Core State Standards but in the differing context of first-year college-level coursework. The relationship between a Common Core State Standard for Mathematics and the evidence of the extension in the AP Calculus courses, AP Statistics, and AP Computer Science A will appear as more of a traditional prerequisite relationship between the Common Core State Standards and the AP courses.
For further explanation of the relationship between an aligned Common Core State Standard and an AP course, see the Demonstration of Alignment Methodology table.
The.comparison.
is.nuanced.in.that.
the.Common.Core.
State.Standards.are.
designed.to.articulate.
the.knowledge.and.
skills.students.need.to.
be.ready.to.succeed.
in.college.and.careers,.
and.AP.courses.and.
exams.are.designed.
to.represent.the.level.
of.a.first-year.college.
course.
6 College Board Research Reports
Common Core State Standards Alignment
AP.Course.MaterialsThe following AP course materials informed this study:
• AP Course Descriptions–Detailed descriptions of the AP courses and exams.
• AP Curricular Requirements–The set of curricular and resource expectations that college faculty nationwide have established for college-level courses.
• AP Teacher’s Guides–Additional information on the AP course, exam, curricular requirements, available resources, and sample lessons and syllabi.
• AP Syllabus Development Guides–Subject-specific documents that contain multiple samples of evidence for each requirement, illustrating the variety of ways curricular requirements can be met in a syllabus.
• AP Released Exams–Documents that include an AP Exam that was once administered as a live exam and supporting information such as answer keys and statistical information about student performance when the exam was administered.
• AP Exam Specifications–Statistical and content-based blueprints for the design of AP Exams.
This collection of resources presents a detailed view of the content knowledge and skills students should master upon completion of an AP course.
The College Board aligned the AP English courses, the AP Calculus courses, AP Statistics, and AP Computer Science A to the following Common Core State Standards grade levels and strands:
AP.English.Literature.and.AP.English.Language:Both AP English Literature and AP English Language were aligned to the Common Core State Standards for English Language Arts grade bands 9–10 and 11–12, as well as to the College and Career Readiness Anchor Standards1 (Anchor Standards).
All strands of the Common Core State Standards for English Language Arts were considered for this alignment study (i.e., all Anchor Standards, Reading Standards for Literature, Reading Standards for Informational Text, Writing Standards, Speaking and Listening Standards, and Language Standards/Progressive Skills)2.
AP.Calculus,.AP.Statistics,.and.AP.Computer.Science.A:AP Calculus AB, AP Calculus BC, AP Statistics, and AP Computer Science A were aligned to the Common Core State Standards for Mathematics grade 6 through high school, as well as to the Standards for Mathematical Practice. Middle school mathematics standards were included in this alignment study because content within the Common Core State Standards at the middle school grade levels had been traditionally identified within high school-level coursework in most state standards. In addition, in order to accurately demonstrate the body of prerequisite knowledge and skills required to be prepared for an AP mathematics courses, it was necessary to include the full middle school and high school Common Core State Standards.
1. The College and Career Readiness Anchor Standards define general literacy expectations and are meant to be considered in tandem with the Common Core State Standards high school grade bands.
2. The Standards for Literacy in History/Social Studies, Science, and Technical Subjects were not included within the scope of this alignment study. These literacy standards are designed to inform literacy instruction across the disciplines and so would not be appropriate for an alignment to the AP English courses.
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Common Core State Standards Alignment
It is important to note that the intent of the Common Core State Standards is to provide focused standards that all students should meet in order to be college and career ready. Additional mathematics content that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by a (+) within the Common Core State Standards for Mathematics and the alignment table.
Summary of the Alignment Between the Common Core State Standards and AP English and Mathematics CoursesGiven the nature of the alignment between the Common Core State Standards and AP courses, it is more appropriate to discuss the degrees of alignment in terms of concentration instead of strength of coverage. As expected, the highest concentration of alignment between the Common Core State Standards and AP courses occurs within the strand most specific to the AP course (e.g., the highest concentration of alignment between the Common Core State Standards and AP English Language occurs within the Reading Standards for Informational Text). Overall, the alignment indicates that a student following a curriculum based on the Common Core State Standards would be well prepared for a number of AP courses.
AP.English.Literature.and.CompositionThere is a high concentration of alignment between the Common Core State Standards for English Language Arts and AP English Literature, specifically in the areas of the Reading Anchor Standards, the Reading Standards for Literature at grade bands 9–10 and 11–12, the Writing Anchor Standards, the Writing Standards at grade bands 9–10 and 11–12, the Language Anchor Standards, and the Language Standards and Progressive Skills at grade bands 9–10 and 11–12. The few Common Core State Standards in these strands that do not align to AP English Literature should not be considered superfluous. Some standards, such as those that reference comparing artistic media, will enhance a student’s preparation for AP but are not directly evidenced in the AP materials.
The scarcity of alignment between the Reading Standards for Informational Text and AP English Literature is to be expected. While informational text may be useful for students conducting a critical analysis of fiction, the focus of AP English Literature is decidedly imaginative literature. Similarly, there is less frequent alignment between AP English Literature and the groups of Common Core State Standards in Writing and Language that address the more specific elements of research and the use of reference materials. There is also a lower concentration of alignment between the Speaking and Listening Standards and AP English Literature. This limited alignment should not be interpreted to mean that these standards do not contribute to a student’s preparation for AP. Speaking and listening
Overall,.the.alignment.
indicates.that.a.
student.following.a.
curriculum.based.on.
the.Common.Core.
State.Standards.would.
be.well.prepared.for.a.
number.of.AP.courses.
8 College Board Research Reports
Common Core State Standards Alignment
skills, such as those addressed in the Common Core State Standards, are expected to be mastered by students prior to their entering an AP course and so are not emphasized in the AP English Course Description. In addition, these skills, typically evaluated through classroom observation, are unlikely to appear in materials related to assessment.
AP.English.Language.and.CompositionThe most concentrated alignment between the Common Core State Standards for English Language Arts and AP English Language occurs in the areas of the Reading Anchor Standards and the Reading Standards for Informational Text at grade bands 9–10 and 11–12. In addition, there is a high concentration of alignment in the areas of the Writing Anchor Standards, the Writing Standards at grade bands 9–10 and 11–12, the Language Anchor Standards, and the Language Standards and Progressive Skills at grade bands 9–10 and 11–12. Because the AP English Course Description notes that fiction and poetry can have a place in the AP English Language course, there is also significant alignment between the Reading Standards for Literature and AP English Language.
There is limited alignment between the Speaking and Listening Standards and AP English Language. Much like the relationship between the Speaking and Listening Standards and AP English Literature, speaking and listening skills are valuable to a student entering AP English Language; however, these skills are considered prerequisite and best measured through classroom observation. For these reasons, speaking and listening skills are not emphasized as part of the AP English Course Description or exam materials, but they are evident within AP classroom practice.
AP.Calculus.AB.and.AP.Calculus.BCThe Common Core State Standards for Mathematics contain additional standards, identified by a (+), to indicate concepts linked to more advanced work in mathematics, work that may lead to a course such as AP Calculus. A majority of these “+” standards were linked to AP Calculus AB and AP Calculus BC as prerequisite skills, which illustrates how AP Calculus is part of a logical progression in an advanced mathematics course plan.
There is a high concentration of alignment between the Common Core State Standards for Mathematics at the grade 6 through high school level within the Standards for Mathematical Practice as well as in the conceptual categories of Number and Quantity, Algebra, Functions, and Geometry. As anticipated, a less concentrated alignment to the Common Core State Standards at the grade 6 through high school level occurs in the conceptual category of Statistics and Probability. Standards that align to AP Calculus BC but not to AP Calculus AB are those standards that provide support for the study of content unique to AP Calculus BC such as polar and vector functions.
AP.StatisticsAP Statistics should also be viewed as part of a logical progression in an advanced mathematics course plan based upon the Common Core State Standards. There is a high concentration of alignment between the Common Core State Standards for Mathematics at the grade 6 through high school level and AP Statistics within the Standards for Mathematical Practice as well as in the conceptual categories of Statistics and Probability, Number and Quantity, Algebra, and Functions. While the alignment to the Common Core State Standards at the grade 6 through high school level is less concentrated in the conceptual category of Geometry, it is important to understand that the AP Statistics course is considered an option for any secondary school student who has successfully completed a second course in algebra. Teachers and curriculum developers should not interpret a less concentrated alignment in conceptual categories to mean that there is no benefit to mastering those
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Common Core State Standards Alignment
uncorrelated standards prior to taking an AP Statistics course. Those skills will enhance the student’s mathematical abilities and afford the student a more rounded opportunity for success in AP Statistics.
AP.Computer.Science.AAP Computer Science A aligns to the Common Core State Standards for Mathematics in a less concentrated manner. As stated within the AP Computer Science A documents, the prerequisite skills needed to be successful in an AP Computer Science A course include knowledge of basic algebra and functions as well as problem-solving skills. Concentration of alignment to the Common Core State Standards for Mathematics at the grade 6 through high school level occurs primarily within the Standards for Mathematical Practice and within the conceptual categories of Algebra and Number and Quantity. A less concentrated alignment to the Common Core State Standards is evident within the conceptual categories of Functions and Geometry.
Despite the lower concentration of alignment in some conceptual categories, it can be concluded that the Common Core State Standards for Mathematics effectively prepare students to participate in AP Computer Science A. Students learning the mathematics of the Common Core State Standards will have experienced mathematical content and skills that go beyond the minimum requirements of the mathematics skills needed to be prepared to take the AP Computer Science A course. Teachers and curriculum developers should not interpret a lack of correlation to mean that there is no benefit to mastering the uncorrelated standards prior to taking an AP Computer Science A course. Those skills will enhance the student’s mathematical abilities and afford the student a more rounded opportunity for success in AP Computer Science A.
Demonstration of Alignment Methodology TableThis table illustrates the methodology used in the alignment study by highlighting examples of the relationship between the Common Core State Standards and AP courses. In order for a Common Core State Standard to be considered aligned to an AP course, evidence of the extension of that Common Core State Standard must be cited from least one of the documents from the collection of AP course materials. The full alignment tables (appendixes A and B) demonstrate a macroscopic view of this methodology.
AP Course Common Core State Standard Evidence of Extension into APAP English Literature
Demonstrate.knowledge.of.eighteenth-,.nineteenth-.and.early.twentieth-century.foundational.works.of.American.literature,.including.how.two.or.more.texts.from.the.same.period.treat.similar.themes.or.topics..(RL.11-12.9)
By.the.time.the.student.completes.AP.English.Literature.and.Composition,.she.or.he.will.have.studied.during.high.school.literature.from.both.British.and.American.writers,.as.well.as.works.written.in.several.genres.from.the.sixteenth.century.to.contemporary.times..The.works.selected.for.the.course.should.require.careful,.deliberate.reading.that.yields.multiple.meanings..(AP.English.Literature.and.Composition:.Curricular.Requirements)
Write.arguments.to.support.claims.in.an.analysis.of.substantive.topics.or.texts.using.valid.reasoning.and.relevant.and.sufficient.evidence..(W.CCR.1)
Writing.is.an.integral.part.of.the.AP.English.Literature.and.Composition.course.and.exam..Writing.assignments.focus.on.the.critical.analysis.of.literature.and.include.expository,.analytical.and.argumentative.essays..(AP.English.Course.Description)
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Common Core State Standards Alignment
AP Course Common Core State Standard Evidence of Extension into APAP English Language
Gather.relevant.information.from.multiple.print.and.digital.sources,.assess.the.credibility.and.accuracy.of.each.source,.and.integrate.the.information.while.avoiding.plagiarism..(W.CCR.8)
The.course.teaches.research.skills,.and.in.particular,.the.ability.to.evaluate,.use.and.cite.primary.and.secondary.sources..The.course.assigns.projects.such.as.the.researched.argument.paper,.which.goes.beyond.the.parameters.of.a.traditional.research.paper.by.asking.students.to.present.an.argument.of.their.own.that.includes.the.analysis.and.synthesis.of.ideas.from.an.array.of.sources..(AP.English.Language.and.Composition:.Curricular.Requirements)
Demonstrate.command.of.the.conventions.of.standard.English.grammar.and.usage.when.writing.or.speaking..(L.CCR.1)
The.intense.concentration.on.language.use.in.the.course.enhances.students’.ability.to.use.grammatical.conventions.appropriately.and.to.develop.stylistic.maturity.in.their.prose..(AP.English.Course.Description)
AP Calculus AB
Understand the relationship between zeros and factors of polynomials.Identify.zeros.of.polynomials.when.suitable.factorizations.are.available,.and.use.the.zeros.to.construct.a.rough.graph.of.the.function.defined.by.the.polynomial..(A-APR.3)
Students.enrolled.in.an.AP.Calculus.AB.course.will.study.limits.of.functions.and.will.develop.an.intuitive.understanding.of.the.limiting.process,.the.ability.to.calculate.limits.using.algebra,.and.estimate.limits.from.graphs.or.tables.of.data..(AP.Calculus.Course.Description)
AP Calculus BC
Represent and model with vector quantities.Solve.problems.involving.velocity.and.other.quantities.that.can.be.represented.by.vectors..(N-VM.3)
Students.enrolled.in.an.AP.Calculus.BC.course.will.study.the.applications.of.derivatives..Students.will.analyze.planar.curves.given.in.parametric.form,.polar.form.and.vector.form,.including.velocity.and.acceleration..(AP.Calculus.Course.Description)
AP Calculus AB & AP Calculus BC
Use appropriate tools strategically.Mathematically.proficient.students.consider.the.available.tools.when.solving.a.mathematical.problem..These.tools.might.include.….a.calculator,.a.spreadsheet,.a.computer.algebra.system,.a.statistical.package,.or.dynamic.geometry.software..Proficient.students.….make.sound.decisions.about.when.each.of.these.tools.might.be.helpful,.recognizing.both.the.insight.to.be.gained.and.their.limitations..….They.are.able.to.use.technological.tools.to.explore.and.deepen.their.understanding.of.concepts..(MP.5)
Students.enrolled.in.an.AP.Calculus.course.will.study.the.analysis.of.graphs..With.the.aid.of.technology,.graphs.of.functions.are.often.easy.to.produce..The.emphasis.is.on.the.interplay.between.the.geometric.and.analytic.information.and.on.the.use.of.calculus.both.to.predict.and.to.explain.the.observed.local.and.global.behavior.of.a.function..(AP.Calculus.Course.Description)
AP Statistics Make inferences and justify conclusions from sample surveys, experiments, and observational studies.Use.data.from.a.sample.survey.to.estimate.a.population.mean.or.proportion;.develop.a.margin.of.error.through.the.use.of.simulation.models.for.random.sampling..(S-IC.4)
Students.enrolled.in.an.AP.statistics.course.will.study.four.major.themes,.one.of.which.is.exploring.data..Exploratory.analysis.of.data.makes.use.of.graphical.and.numerical.techniques.to.study.patterns.and.departures.from.patterns..Emphasis.is.placed.on.interpreting.information.from.graphical.and.numerical.displays.and.summaries..This.includes.constructing.and.interpreting.graphical.displays.of.distributions.of.univariate.data.(dot.plot,.stem.plot,.histogram,.cumulative.frequency.plot)..Students.will.study.center.and.spread,.clusters.and.gaps,.outliers.and.other.unusual.features,.and.shape..(AP.Statistics.Course.Description)
11College Board Research Reports
Common Core State Standards Alignment
AP Course Common Core State Standard Evidence of Extension into APAP Statistics Construct viable arguments and critique the
reasoning of others.Mathematically.proficient.students.understand.and.use.stated.assumptions,.definitions,.and.previously.established.results.in.constructing.arguments..They.make.conjectures.and.build.a.logical.progression.of.statements.to.explore.the.truth.of.their.conjectures..….They.reason.inductively.about.data,.making.plausible.arguments.that.take.into.account.the.context.from.which.the.data.arose..Mathematically.proficient.students.are.also.able.to.compare.the.effectiveness.of.two.plausible.arguments,.distinguish.correct.logic.or.reasoning.from.that.which.is.flawed,.and.—.if.there.is.a.flaw.in.an.argument.—.explain.what.it.is..….Students.learn.to.determine.domains.to.which.an.argument.applies..Students.at.all.grades.can.listen.or.read.the.arguments.of.others,.decide.whether.they.make.sense,.and.ask.useful.questions.to.clarify.or.improve.the.arguments..(MP.3)
Students.enrolled.in.an.AP.statistics.course.will.study.four.major.themes,.one.of.which.is.statistical.inference..Statistical.inference.guides.the.selection.of.appropriate.models..Models.and.data.interact.in.statistical.work:.models.are.used.to.draw.conclusions.from.data,.while.the.data.are.allowed.to.criticize.and.even.falsify.the.model.through.inferential.and.diagnostic.methods..Inference.from.data.can.be.thought.of.as.the.process.of.selecting.a.reasonable.model,.including.a.statement.in.probability.language.of.how.confident.one.can.be.about.the.selection..(AP.Statistics.Course.Description)
AP Computer Science A
Understand the concept of a function and use function notation.Understand.that.a.function.from.one.set.(called.the.domain).to.another.set.(called.the.range).assigns.to.each.element.of.the.domain.exactly.one.element.of.the.range..If.f.is.a.function.and.x.is.an.element.of.its.domain,.then.f(x).denotes.the.output.of.f.corresponding.to.the.input.x..The.graph.of.f.is.the.graph.of.the.equation y = f(x)..(F-IF.1)
A.student.in.the.AP.Computer.Science.A.course.should.be.comfortable.with.functions.and.the.concepts.found.in.the.uses.of.functional.notation,.such.as.f(x) = x + 2.and.f(x) = g(h(x))..(AP.Computer.Science.A.Course.Description)
Reason abstractly and quantitatively.Mathematically.proficient.students.make.sense.of.quantities.and.their.relationships.in.problem.situations..They.bring.two.complementary.abilities.to.bear.on.problems.involving.quantitative.relationships:.the.ability.to.decontextualize.—.to.abstract.a.given.situation.and.represent.it.symbolically.….—.and.the.ability.to.contextualize,.to.pause.as.needed.during.the.manipulation.process.in.order.to.probe.into.the.referents.for.the.symbols.involved..Quantitative.reasoning.entails.habits.of.creating.a.coherent.representation.of.the.problem.at.hand;.considering.the.units.involved;.attending.to.the.meaning.of.quantities,.not.just.how.to.compute.them;.and.knowing.and.flexibly.using.different.properties.of.operations.and.objects..(MP.2)
By.the.time.students.complete.the.AP.Computer.Science.course,.they.should.be.able.to.design.and.implement.solutions.to.problems.by.writing,.running.and.debugging.computer.programs..To.do.this,.they.should.be.able.to.develop.and.select.appropriate.algorithms.and.data.structures.to.solve.problems..(AP.Computer.Science.A.Course.Description)
13College Board Research Reports
Common Core State Standards Alignment
ReferencesAchieve. (2007). Aligned expectations? A closer look at college admissions and placement
tests. Washington, DC: Author.
Achieve. (2010). A comparison of the Common Core State Standards for mathematics to the American Diploma Project Benchmarks for mathematics. Washington, DC: Author.
Blank, R. K. (2007). Alignment content analysis of NAEP 2007 Mathematics assessment using the Surveys of Enacted Curriculum methodology. National Center for Education Statistics.
Brown, R. S., & Conley, D. T. (2007). Comparing state high school assessments to standards for success in entry-level university courses. Educational Assessment, 12(2), 137–160.
Carter, D. C. (2007). AP Computer Science Teacher’s Guide. Retrieved from the College Board website: http://apcentral.collegeboard.com/apc/members/repository/ap07_compsci_teachersguide.pdf
College Board. (2007). Syllabus Development Guide: AP English Language and Composition. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/SDG-EnglishLanguage.pdf
College Board. (2007). Syllabus Development Guide: AP English Literature and Composition. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/ repository/SDGEnglishLiterature.pdf
College Board. (2008). AP® English Literature and Composition 2008 Free-Response Questions. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/ap08_eng_lit_frq.pdf
College Board. (2008). 2007 AP Statistics Released Exam. New York: The College Board.
College Board. (2008). Report on the rigor and alignment to college readiness of the Colorado Model Content Standards for Reading and Writing and Mathematics. New York: The College Board.
College Board. (2010). 2009 AP Computer Science A Released Exam. New York: The College Board.
College Board. (2010). AP Computer Science A Course Description. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf
College Board. (2010). AP Calculus Course Description. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf
College Board. (2010). AP English Course Description. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/ap-english-course-description.pdf
College Board. (2010). AP English Language and Composition: Curricular Requirements. Retrieved from College Board website: http://www.collegeboard.com/html/apcourseaudit/courses/english_curricular_requirements.html
College Board. (2010). AP English Literature and Composition: Curricular Requirements.Retrieved from College Board website: http://www.collegeboard.com/html/apcourseaudit/courses/english_lit_curricular_requirements.html
College Board. (2010). AP Statistics Course Description. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/ap-statistics-course-description.pdf
14 College Board Research Reports
Common Core State Standards Alignment
College Board. (2010). How the College Board Standards for College Success align to the Common Core State Standards. New York: The College Board.
College Board. (2011). Assembly specifications grid for Advanced Placement Calculus AB/Calculus BC multiple choice sections. Unpublished test specifications.
Howell, M. (2007). AP Calculus Teacher’s Guide. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/members/repository/ap07_calculus_teachersguide_2.pdf
Legacy, M. (2008). AP Statistics Teacher’s Guide. Retrieved from College Board website: http://apcentral.collegeboard.com/apc/public/repository/AP_Statistics_Teacher_Guide.pdf
Porter, A. C. (2002). Measuring the content of instruction: Uses in research and practice.Educational Researcher, 31(7), 3–14.
Rothman, R. (2003, March). Imperfect matches: The alignment of standards and tests. Paper commissioned by the Committee on Test Design for K–12 Science Achievement.
Vasavada, N., Carman, E., Hart, B., & Luisier, D. (2010). Common Core State Standards alignment: ReadiStep™, PSAT/NMSQT® and SAT®. (College Board Research Report 2010-5). Retrieved from College Board website: http://professionals.collegeboard.com/data-reports-research/cb/common_core_alignments
Webb, N. L. (1997). Criteria for alignment of expectations and assessments in mathematics and science education. (Research Monograph No. 6). Washington, DC: Council of Chief State School Officers.
Webb, N. L. (1999). Alignment of science and mathematics standards and assessments in four states. (Research Monograph No. 18). Madison, WI: University of Wisconsin-Wisconsin Center for Education Research.
Wiley, A., Wyatt, J., & Camara, W. J. (2010). The development of a multidimensional college readiness index. (College Board Research Report 2010-3). Retrieved from College Board website: http://professionals.collegeboard.com/profdownload/pdf/10b_3110_CollegeReadiness_RR_WEB_110315.pdf
15College Board Research Reports
Appendixes
Appendix.AAP English — Common Core State Standards Side-by-Side Alignment Table
17College Board Research Reports
Appendixes
AP.
Eng
lish.
—.C
omm
on.C
ore.
Stat
e.St
anda
rds.
Side
-by-
Side
.Alig
nm
ent.
Tab
le
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eC
CR
Rea
din
gK
ey.Id
eas.
and.
Det
ails
Rea
d.cl
osel
y.to
.det
erm
ine.
wha
t.th
e.te
xt.s
ays.
expl
icit
ly.a
nd.t
o.m
ake.
log
ical
.infe
renc
es.fr
om.it
;.ci
te.s
peci
fic.t
extu
al.e
vide
nce.
whe
n.w
riti
ng.o
r.sp
eaki
ng.t
o.su
ppor
t.co
nclu
sion
s.d
raw
n.fr
om.t
he.
text
.
R.C
CR
.1X
X
Det
erm
ine.
cent
ral.i
deas
.or.
them
es.o
f.a.t
ext.
and.
anal
yze.
thei
r.de
velo
pmen
t;.su
mm
ariz
e.th
e.ke
y.su
ppor
ting
.det
ails
.and
.idea
s.
R.C
CR
.2X
X
Ana
lyze
.how
.and
.why
.ind
ivid
uals
,.eve
nts,
.and
.id
eas.
deve
lop.
and.
inte
ract
.ove
r.th
e.co
urs
e.of
.a.
text
.
R.C
CR
.3X
X
Cra
ft.a
nd.
Stru
ctu
reIn
terp
ret.
wor
ds.a
nd.p
hra
ses.
as.t
hey.
are.
used
.in
.a.t
ext,
.incl
udin
g.de
term
inin
g.te
chn
ical
,.co
nno
tati
ve,.a
nd.fi
gu
rati
ve.m
ean
ings
,.and
.an
alyz
e.ho
w.s
peci
fic.w
ord.
choi
ces.
shap
e.m
ean
ing.
or.t
one.
R.C
CR
.4X
X
Ana
lyze
.the
.str
uctu
re.o
f.tex
ts,.i
nclu
din
g.ho
w.
spec
ific
.sen
tenc
es,.p
arag
raph
s,.a
nd.la
rger
.po
rtio
ns.
of.t
he.t
ext.
(e.g
.,.a.
sect
ion,
.cha
pter
,.sc
ene,
.or.
stan
za).r
elat
e.to
.eac
h.ot
her.
and.
the.
who
le.
R.C
CR
.5X
X
Ass
ess.
how
.poi
nt.o
f.vie
w.o
r.pu
rpos
e.sh
apes
.the
.co
nten
t.an
d.st
yle.
of.a
.tex
t.R
.CC
R.6
XX
Inte
gra
tion
.of.
Kno
wle
dge.
and.
Idea
s
Inte
gra
te.a
nd.e
valu
ate.
cont
ent.
pres
ente
d.in
.d
iver
se.fo
rmat
s.an
d.m
edia
,.inc
lud
ing.
visu
ally
.an
d.q
uant
itat
ivel
y,.a
s.w
ell.a
s.in
.wor
ds.*
R.C
CR
.7X
Del
inea
te.a
nd.e
valu
ate.
the.
arg
um
ent.
and.
spec
ific
.cla
ims.
in.a
.tex
t,.in
clud
ing.
the.
valid
ity.
of.t
he.r
easo
nin
g.as
.wel
l.as.
the.
rele
vanc
e.an
d.su
ffic
ienc
y.of
.the
.evi
denc
e.
R.C
CR
.8X
X
Ana
lyze
.how
.tw
o.or
.mor
e.te
xts.
add
ress
.sim
ilar.
them
es.o
r.to
pics
.in.o
rder
.to.
build
.kno
wle
dge.
or.
to.c
ompa
re.t
he.a
ppro
ache
s.th
e.au
thor
s.ta
ke.
R.C
CR
.9X
X
Ran
ge.o
f.R
ead
ing.
and.
Leve
l.of.T
ext.
Com
plex
ity
Rea
d.an
d.co
mpr
ehen
d.co
mpl
ex.li
tera
ry.
and.
info
rmat
iona
l.tex
ts.in
depe
nden
tly.
and.
prof
icie
ntly
.
R.C
CR
.10
XX
18 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
e9-
10R
ead
ing.
Stan
dard
s.fo
r.L
iter
atu
re
Key
.Idea
s.an
d.D
etai
lsC
ite.
stro
ng.a
nd.t
horo
ugh.
text
ual.e
vide
nce.
to.
supp
ort.
anal
ysis
.of.w
hat.
the.
text
.say
s.ex
plic
itly
.as
.wel
l.as.
infe
renc
es.d
raw
n.fr
om.t
he.t
ext.
RL
.9-1
0.1
XX
Det
erm
ine.
a.th
eme.
or.c
entr
al.id
ea.o
f.a.t
ext.
and.
anal
yze.
in.d
etai
l.its
.dev
elop
men
t.ov
er.t
he.c
ours
e.of
.the
.tex
t,.in
clud
ing.
how
.it.e
mer
ges.
and.
is.
shap
ed.a
nd.r
efin
ed.b
y.sp
ecif
ic.d
etai
ls;.p
rovi
de.
an.o
bjec
tive
.su
mm
ary.
of.t
he.t
ext.
RL
.9-1
0.2
X
Ana
lyze
.how
.com
plex
.cha
ract
ers.
(e.g
.,.th
ose.
wit
h.m
ulti
ple.
or.c
onfli
ctin
g.m
otiv
atio
ns)
.dev
elop
.ov
er.t
he.c
ours
e.of
.a.t
ext,
.inte
ract
.wit
h.ot
her.
char
acte
rs,.a
nd.a
dvan
ce.t
he.p
lot.
or.d
evel
op.t
he.
them
e.
RL
.9-1
0.3
X
Cra
ft.a
nd.
Stru
ctu
reD
eter
min
e.th
e.m
ean
ing.
of.w
ords
.and
.ph
rase
s.as
.the
y.ar
e.us
ed.in
.the
.tex
t,.in
clud
ing.
figu
rati
ve.
and.
con
nota
tive
.mea
nin
gs;.a
naly
ze.t
he.
cum
ula
tive
.impa
ct.o
f.spe
cific
.wor
d.ch
oice
s.on
.m
ean
ing.
and.
tone
.(e.g
.,.ho
w.t
he.la
ngua
ge.e
voke
s.a.
sen
se.o
f.tim
e.an
d.pl
ace;
.how
.it.s
ets.
a.fo
rmal
.or.
info
rmal
.ton
e).
RL
.9-1
0.4
XX
Ana
lyze
.how
.an.
auth
or’s
.cho
ices
.con
cern
ing.
how
.to
.str
uctu
re.a
.tex
t,.o
rder
.eve
nts.
wit
hin
.it.(e
.g.,.
para
llel.p
lots
),.an
d.m
anip
ula
te.t
ime.
(e.g
.,.pa
cing
,.fla
shba
cks)
.cre
ate.
such
.effe
cts.
as.m
yste
ry,.
ten
sion
,.or.
surp
rise
.
RL
.9-1
0.5
XX
Ana
lyze
.a.p
arti
cula
r.po
int.
of.v
iew
.or.
cultu
ral.
expe
rien
ce.r
efle
cted
.in.a
.wor
k.of
.lite
ratu
re.fr
om.
outs
ide.
the.
Un
ited
.Sta
tes,
.dra
win
g.on
.a.w
ide.
read
ing.
of.w
orld
.lite
ratu
re.
RL
.9-1
0.6
X
Inte
gra
tion
.of.
Kno
wle
dge.
and.
Idea
s
Ana
lyze
.the
.rep
rese
ntat
ion.
of.a
.su
bjec
t.or
.a.
key.
scen
e.in
.tw
o.d
iffe
rent
.art
isti
c.m
ediu
ms,
.in
clud
ing.
wha
t.is
.em
phas
ized
.or.
abse
nt.in
.eac
h.tr
eatm
ent.
(e.g
.,.A
uden
’s.“
Mus
ée.d
es.B
eau
x.A
rts”
.an
d.B
reug
hel’s
.Lan
dsca
pe w
ith
the
Fall
of Ic
arus
).
RL
.9-1
0.7
(Not
.app
licab
le.t
o.lit
erat
ure
)R
L.9
-10.
8
Ana
lyze
.how
.an.
auth
or.d
raw
s.on
.and
.tra
nsf
orm
s.so
urc
e.m
ater
ial.i
n.a.
spec
ific
.wor
k.(e
.g.,.
how
.Sh
akes
pear
e.tr
eats
.a.t
hem
e.or
.top
ic.fr
om.O
vid.
or.
the.
Bib
le.o
r.ho
w.a
.late
r.au
thor
.dra
ws.
on.a
.pla
y.by
.Sh
akes
pear
e).
RL
.9-1
0.9
X
19College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eR
ange
.of.
Rea
din
g.an
d.Le
vel.o
f.Tex
t.C
ompl
exit
y
By.
the.
end.
of.g
rade
.9,.r
ead.
and.
com
preh
end.
liter
atu
re,.i
nclu
din
g.st
orie
s,.d
ram
as,.a
nd.
poem
s,.in
.the
.gra
des.
9–10
.tex
t.co
mpl
exit
y.ba
nd.p
rofic
ient
ly,.w
ith.
scaf
fold
ing.
as.n
eede
d.at
.th
e.h
igh.
end.
of.t
he.r
ange
..By.
the.
end.
of.g
rade
.10
,.rea
d.an
d.co
mpr
ehen
d.lit
erat
ure
,.inc
lud
ing.
stor
ies,
.dra
mas
,.and
.poe
ms,
.at.
the.
hig
h.en
d.of
.the
.gra
des.
9–10
.tex
t.co
mpl
exit
y.ba
nd.
inde
pend
entl
y.an
d.pr
ofic
ient
ly.
RL
.9-1
0.10
XX
11-1
2R
ead
ing.
Stan
dard
s.fo
r.L
iter
atu
re
Key
.Idea
s.an
d.D
etai
lsC
ite.
stro
ng.a
nd.t
horo
ugh.
text
ual.e
vide
nce.
to.
supp
ort.
anal
ysis
.of.w
hat.
the.
text
.say
s.ex
plic
itly
.as
.wel
l.as.
infe
renc
es.d
raw
n.fr
om.t
he.t
ext,
.in
clud
ing.
dete
rmin
ing.
whe
re.t
he.t
ext.
leav
es.
mat
ters
.unc
erta
in.
RL
.11-
12.1
XX
Det
erm
ine.
two.
or.m
ore.
them
es.o
r.ce
ntra
l.ide
as.
of.a
.tex
t.an
d.an
alyz
e.th
eir.
deve
lopm
ent.
over
.the
.co
urse
.of.t
he.t
ext,
.incl
udin
g.ho
w.t
hey.
inte
ract
.an
d.bu
ild.o
n.on
e.an
othe
r.to
.pro
duce
.a.c
ompl
ex.
acco
unt;.
prov
ide.
an.o
bjec
tive
.sum
mar
y.of
.the
.te
xt.
RL
.11-
12.2
X
Ana
lyze
.the
.impa
ct.o
f.the
.aut
hor’s
.cho
ices
.re
gard
ing.
how
.to.
deve
lop.
and.
rela
te.e
lem
ents
.of.
a.st
ory.
or.d
ram
a.(e
.g.,.
whe
re.a
.sto
ry.is
.set
,.how
.th
e.ac
tion
.is.o
rder
ed,.h
ow.t
he.c
hara
cter
s.ar
e.in
trod
uced
.and
.dev
elop
ed).
RL
.11-
12.3
X
Cra
ft.a
nd.
Stru
ctu
reD
eter
min
e.th
e.m
ean
ing.
of.w
ords
.and
.ph
rase
s.as
.the
y.ar
e.us
ed.in
.the
.tex
t,.in
clud
ing.
figu
rati
ve.
and.
con
nota
tive
.mea
nin
gs;.a
naly
ze.t
he.im
pact
.of
.spe
cific
.wor
d.ch
oice
s.on
.mea
nin
g.an
d.to
ne,.
incl
udin
g.w
ords
.wit
h.m
ulti
ple.
mea
nin
gs.o
r.la
ngua
ge.t
hat.
is.p
arti
cula
rly.fr
esh,
.eng
agin
g,.o
r.be
auti
ful..
(Inc
lude
.Sha
kesp
eare
.as.
wel
l.as.
othe
r.au
thor
s.)
RL
.11-
12.4
XX
Ana
lyze
.how
.an.
auth
or’s
.cho
ices
.con
cern
ing.
how
.to.
stru
ctu
re.s
peci
fic.p
arts
.of.a
.tex
t.(e
.g.,.
the.
choi
ce.o
f.whe
re.t
o.be
gin
.or.
end.
a.st
ory,
.the
.ch
oice
.to.
prov
ide.
a.co
med
ic.o
r.tr
agic
.res
olut
ion)
.co
ntri
bute
.to.
its.
over
all.s
truc
ture
.and
.mea
nin
g.as
.wel
l.as.
its.
aest
heti
c.im
pact
.
RL
.11-
12.5
XX
20 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eA
naly
ze.a
.cas
e.in
.wh
ich.
gra
spin
g.po
int.
of.v
iew
.re
qu
ires
.dis
ting
uis
hin
g.w
hat.
is.d
irec
tly.
stat
ed.
in.a
.tex
t.fr
om.w
hat.
is.r
eally
.mea
nt.(e
.g.,.
sati
re,.
sarc
asm
,.iro
ny,.o
r.u
nder
stat
emen
t).
RL
.11-
12.6
XX
Inte
gra
tion
.of.
Kno
wle
dge.
and.
Idea
s
Ana
lyze
.mu
ltipl
e.in
terp
reta
tion
s.of
.a.s
tory
,.d
ram
a,.o
r.po
em.(e
.g.,.
reco
rded
.or.
live.
prod
ucti
on.
of.a
.pla
y.or
.rec
orde
d.no
vel.o
r.po
etry
),.ev
alua
ting
.ho
w.e
ach.
vers
ion.
inte
rpre
ts.t
he.s
ourc
e.te
xt..
(Inc
lude
.at.
leas
t.on
e.pl
ay.b
y.Sh
akes
pear
e.an
d.on
e.pl
ay.b
y.an
.Am
eric
an.d
ram
atis
t.)
RL
.11-
12.7
(Not
.app
licab
le.t
o.lit
erat
ure
)R
L.1
1-12
.8
Dem
onst
rate
.kno
wle
dge.
of.e
ight
eent
h-,.
nin
etee
nth-
.and
.ear
ly-t
wen
tiet
h-ce
ntu
ry.
fou
ndat
iona
l.wor
ks.o
f.Am
eric
an.li
tera
ture
,.in
clud
ing.
how
.tw
o.or
.mor
e.te
xts.
from
.the
.sam
e.pe
riod
.tre
at.s
imila
r.th
emes
.or.
topi
cs.
RL
.11-
12.9
X
Ran
ge.o
f.R
ead
ing.
and.
Leve
l.of.T
ext.
Com
plex
ity
By.
the.
end.
of.g
rade
.11,
.rea
d.an
d.co
mpr
ehen
d.lit
erat
ure
,.inc
lud
ing.
stor
ies,
.dra
mas
,.and
.poe
ms,
.in
.the
.gra
des.
11–C
CR
.tex
t.co
mpl
exit
y.ba
nd.
prof
icie
ntly
,.wit
h.sc
affo
ldin
g.as
.nee
ded.
at.t
he.
hig
h.en
d.of
.the
.ran
ge..B
y.th
e.en
d.of
.gra
de.1
2,.
read
.and
.com
preh
end.
liter
atu
re,.i
nclu
din
g.st
orie
s,.d
ram
as,.a
nd.p
oem
s,.a
t.th
e.h
igh.
end.
of.t
he.g
rade
s.11
–CC
R.t
ext.
com
plex
ity.
band
.in
depe
nden
tly.
and.
prof
icie
ntly
.
RL
.11-
12.1
0X
X
9-10
Read
ing.
Stan
dard
s.fo
r.In
form
atio
nal.
Text
Key
.Idea
s.an
d.D
etai
lsC
ite.
stro
ng.a
nd.t
horo
ugh.
text
ual.e
vide
nce.
to.
supp
ort.
anal
ysis
.of.w
hat.
the.
text
.say
s.ex
plic
itly
.as
.wel
l.as.
infe
renc
es.d
raw
n.fr
om.t
he.t
ext.
RI.9
-10.
1X
Det
erm
ine.
a.ce
ntra
l.ide
a.of
.a.t
ext.
and.
anal
yze.
its.
deve
lopm
ent.
over
.the
.cou
rse.
of.t
he.t
ext,
.in
clud
ing.
how
.it.e
mer
ges.
and.
is.s
hape
d.an
d.re
fine
d.by
.spe
cific
.det
ails
;.pro
vide
.an.
obje
ctiv
e.su
mm
ary.
of.t
he.t
ext.
RI.9
-10.
2X
Ana
lyze
.how
.the
.aut
hor.
un
fold
s.an
.ana
lysi
s.or
.se
ries
.of.i
deas
.or.
even
ts,.i
nclu
din
g.th
e.or
der.
in.w
hic
h.th
e.po
ints
.are
.mad
e,.h
ow.t
hey.
are.
intr
oduc
ed.a
nd.d
evel
oped
,.and
.the
.con
nect
ion
s.th
at.a
re.d
raw
n.be
twee
n.th
em.
RI.9
-10.
3X
21College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eC
raft
.and
.St
ruct
ure
Det
erm
ine.
the.
mea
nin
g.of
.wor
ds.a
nd.p
hra
ses.
as.t
hey.
are.
used
.in.a
.tex
t,.in
clud
ing.
figu
rati
ve,.
con
nota
tive
,.and
.tec
hn
ical
.mea
nin
gs;.a
naly
ze.
the.
cum
ula
tive
.impa
ct.o
f.spe
cific
.wor
d.ch
oice
s.on
.mea
nin
g.an
d.to
ne.(e
.g.,.
how
.the
.lang
uage
.of.a
.co
urt
.opi
nio
n.d
iffe
rs.fr
om.t
hat.
of.a
.new
spap
er).
RI.9
-10.
4X
Ana
lyze
.in.d
etai
l.how
.an.
auth
or’s
.idea
s.or
.cl
aim
s.ar
e.de
velo
ped.
and.
refi
ned.
by.p
arti
cula
r.se
nten
ces,
.par
agra
phs,
.or.
larg
er.p
orti
ons.
of.a
.tex
t.(e
.g.,.
a.se
ctio
n.or
.cha
pter
).
RI.9
-10.
5X
Det
erm
ine.
an.a
utho
r’s.p
oint
.of.v
iew
.or.
purp
ose.
in.a
.tex
t.an
d.an
alyz
e.ho
w.a
n.au
thor
.use
s.rh
etor
ic.
to.a
dvan
ce.t
hat.
poin
t.of
.vie
w.o
r.pu
rpos
e.
RI.9
-10.
6X
Inte
gra
tion
.of.
Kno
wle
dge.
and.
Idea
s
Ana
lyze
.var
ious
.acc
ount
s.of
.a.s
ubj
ect.
told
.in.
dif
fere
nt.m
ediu
ms.
(e.g
.,.a.
pers
on’s
.life
.sto
ry.in
.bo
th.p
rint
.and
.mu
ltim
edia
),.de
term
inin
g.w
hic
h.de
tails
.are
.em
phas
ized
.in.e
ach.
acco
unt
.
RI.9
-10.
7X
Del
inea
te.a
nd.e
valu
ate.
the.
arg
um
ent.
and.
spec
ific
.cla
ims.
in.a
.tex
t,.a
sses
sing
.whe
ther
.the
.re
ason
ing.
is.v
alid
.and
.the
.evi
denc
e.is
.rel
evan
t.an
d.su
ffic
ient
;.ide
ntif
y.fa
lse.
stat
emen
ts.a
nd.
falla
ciou
s.re
ason
ing.
RI.9
-10.
8X
Ana
lyze
.sem
inal
.U.S
..doc
um
ents
.of.h
isto
rica
l.an
d.lit
erar
y.si
gn
ific
ance
.(e.g
.,.W
ash
ing
ton’
s.Fa
rew
ell.A
ddre
ss,.t
he.G
etty
sbu
rg.A
ddre
ss,.
Roo
seve
lt’s.
Fou
r.F
reed
oms.
spee
ch,.K
ing’
s.“L
ette
r.fr
om.B
irm
ingh
am.J
ail”
),.in
clud
ing.
how
.the
y.ad
dre
ss.r
elat
ed.t
hem
es.a
nd.c
once
pts.
RI.9
-10.
9X
Ran
ge.o
f.R
ead
ing.
and.
Leve
l.of.T
ext.
Com
plex
ity
By.
the.
end.
of.g
rade
.9,.r
ead.
and.
com
preh
end.
liter
ary.
non
ficti
on.in
.the
.gra
des.
9–10
.tex
t.co
mpl
exit
y.ba
nd.p
rofic
ient
ly,.w
ith.
scaf
fold
ing.
as.n
eede
d.at
.the
.hig
h.en
d.of
.the
.ran
ge..B
y.th
e.en
d.of
.gra
de.1
0,.r
ead.
and.
com
preh
end.
liter
ary.
non
ficti
on.a
t.th
e.h
igh.
end.
of.t
he.g
rade
s.9–
10.t
ext.
com
plex
ity.
band
.inde
pend
entl
y.an
d.pr
ofic
ient
ly.
RI.9
-10.
10X
11-1
2R
ead
ing.
Stan
dard
s.fo
r.In
form
atio
nal.
Tex
t
Key
.Idea
s.an
d.D
etai
lsC
ite.
stro
ng.a
nd.t
horo
ugh.
text
ual.e
vide
nce.
to.
supp
ort.
anal
ysis
.of.w
hat.
the.
text
.say
s.ex
plic
itly
.as
.wel
l.as.
infe
renc
es.d
raw
n.fr
om.t
he.t
ext,
.in
clud
ing.
dete
rmin
ing.
whe
re.t
he.t
ext.
leav
es.
mat
ters
.unc
erta
in.
RI.1
1-12
.1X
22 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eD
eter
min
e.tw
o.or
.mor
e.ce
ntra
l.ide
as.o
f.a.t
ext.
and.
anal
yze.
thei
r.de
velo
pmen
t.ov
er.t
he.c
ours
e.of
.th
e.te
xt,.i
nclu
din
g.ho
w.t
hey.
inte
ract
.and
.bu
ild.
on.o
ne.a
noth
er.t
o.pr
ovid
e.a.
com
plex
.ana
lysi
s;.
prov
ide.
an.o
bjec
tive
.su
mm
ary.
of.t
he.t
ext.
RI.1
1-12
.2X
Ana
lyze
.a.c
ompl
ex.s
et.o
f.ide
as.o
r.se
que
nce.
of.
even
ts.a
nd.e
xpla
in.h
ow.s
peci
fic.in
div
idua
ls,.
idea
s,.o
r.ev
ents
.inte
ract
.and
.dev
elop
.ove
r.th
e.co
urs
e.of
.the
.tex
t.
RI.1
1-12
.3X
Cra
ft.a
nd.
Stru
ctu
reD
eter
min
e.th
e.m
ean
ing.
of.w
ords
.and
.ph
rase
s.as
.the
y.ar
e.us
ed.in
.a.t
ext,
.incl
udin
g.fig
ura
tive
,.co
nno
tati
ve,.a
nd.t
ech
nic
al.m
ean
ings
;.ana
lyze
.ho
w.a
n.au
thor
.use
s.an
d.re
fine
s.th
e.m
ean
ing.
of.a
.ke
y.te
rm.o
r.te
rms.
over
.the
.cou
rse.
of.a
.tex
t.(e
.g.,.
how
.Mad
ison
.def
ines
.fact
ion
in.F
eder
alis
t N
o.
10).
RI.1
1-12
.4X
Ana
lyze
.and
.eva
luat
e.th
e.ef
fect
iven
ess.
of.t
he.
stru
ctu
re.a
n.au
thor
.use
s.in
.his
.or.
her.
expo
siti
on.
or.a
rgu
men
t,.in
clud
ing.
whe
ther
.the
.str
uctu
re.
mak
es.p
oint
s.cl
ear,.
conv
inci
ng,.a
nd.e
ngag
ing.
RI.1
1-12
.5X
Det
erm
ine.
an.a
utho
r’s.p
oint
.of.v
iew
.or.
purp
ose.
in.a
.tex
t.in
.wh
ich.
the.
rhet
oric
.is.p
arti
cula
rly.
effe
ctiv
e,.a
naly
zing
.how
.sty
le.a
nd.c
onte
nt.
cont
ribu
te.t
o.th
e.po
wer
,.per
suas
iven
ess,
.or.
beau
ty.o
f.the
.tex
t.
RI.1
1-12
.6X
Inte
gra
tion
.of.
Kno
wle
dge.
and.
Idea
s
Inte
gra
te.a
nd.e
valu
ate.
mu
ltipl
e.so
urc
es.o
f.in
form
atio
n.pr
esen
ted.
in.d
iffe
rent
.med
ia.o
r.fo
rmat
s.(e
.g.,.
visu
ally
,.qua
ntit
ativ
ely)
.as.
wel
l.as.
in.w
ords
.in.o
rder
.to.
add
ress
.a.q
uest
ion.
or.s
olve
.a.
prob
lem
.
RI.1
1-12
.7X
Del
inea
te.a
nd.e
valu
ate.
the.
reas
onin
g.in
.se
min
al.U
.S..t
exts
,.inc
lud
ing.
the.
appl
icat
ion.
of.c
onst
itut
iona
l.pri
ncip
les.
and.
use.
of.le
gal.
reas
onin
g.(e
.g.,.
in.U
.S..S
upre
me.
Cou
rt.m
ajor
ity.
opin
ion
s.an
d.d
isse
nts)
.and
.the
.pre
mis
es,.
purp
oses
,.and
.arg
um
ents
.in.w
orks
.of.p
ubl
ic.
advo
cacy
.(e.g
.,.T
he F
eder
alis
t,.pr
esid
enti
al.
add
ress
es).
RI.1
1-12
.8X
23College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eA
naly
ze.s
even
teen
th-,.
eigh
teen
th-,.
and.
nin
etee
nth-
cent
ury
.fou
ndat
iona
l.U.S
..doc
um
ents
.of
.his
tori
cal.a
nd.li
tera
ry.s
ign
ific
ance
.(inc
lud
ing.
The
.Dec
lara
tion
.of.I
ndep
ende
nce,
.the
.Pre
ambl
e.to
.the
.Con
stit
utio
n,.t
he.B
ill.o
f.Rig
hts,
.and
.L
inco
ln’s
.Sec
ond.
Inau
gu
ral.A
ddre
ss).f
or.t
heir
.th
emes
,.pu
rpos
es,.a
nd.r
heto
rica
l.fea
ture
s.
RI.1
1-12
.9X
Ran
ge.o
f.R
ead
ing.
and.
Leve
l.of.T
ext.
Com
plex
ity
By.
the.
end.
of.g
rade
.11,
.rea
d.an
d.co
mpr
ehen
d.lit
erar
y.no
nfic
tion
.in.t
he.g
rade
s.11
–CC
R.t
ext.
com
plex
ity.
band
.pro
ficie
ntly
,.wit
h.sc
affo
ldin
g.as
.nee
ded.
at.t
he.h
igh.
end.
of.t
he.r
ange
..By.
the.
end.
of.g
rade
.12,
.rea
d.an
d.co
mpr
ehen
d.lit
erar
y.no
nfic
tion
.at.
the.
hig
h.en
d.of
.the
.gra
des.
11–
CC
R.t
ext.
com
plex
ity.
band
.inde
pend
entl
y.an
d.pr
ofic
ient
ly.
RI.1
1-12
.10
X
CC
RW
riti
ngT
ext.
Type
s.an
d.P
urp
oses
*
Wri
te.a
rgu
men
ts.t
o.su
ppor
t.cl
aim
s.in
.an.
anal
ysis
.of
.su
bsta
ntiv
e.to
pics
.or.
text
s,.u
sing
.val
id.
reas
onin
g.an
d.re
leva
nt.a
nd.s
uff
icie
nt.e
vide
nce.
W.C
CR
.1X
X
Wri
te.in
form
ativ
e/ex
plan
ator
y.te
xts.
to.e
xam
ine.
and.
conv
ey.c
ompl
ex.id
eas.
and.
info
rmat
ion.
clea
rly.a
nd.a
ccu
rate
ly.t
hro
ugh.
the.
effe
ctiv
e.se
lect
ion,
.org
aniz
atio
n,.a
nd.a
naly
sis.
of.c
onte
nt.
W.C
CR
.2X
X
Wri
te.n
arra
tive
s.to
.dev
elop
.rea
l.or.
imag
ined
.ex
peri
ence
s.or
.eve
nts.
usin
g.ef
fect
ive.
tech
niq
ue,.
wel
l-cho
sen.
deta
ils,.a
nd.w
ell-s
truc
ture
d.ev
ent.
seq
uenc
es.
W.C
CR
.3X
X
Pro
duct
ion.
and.
Dis
trib
utio
n.of
.Wri
ting
Pro
duce
.cle
ar.a
nd.c
oher
ent.
wri
ting
.in.w
hic
h.th
e.de
velo
pmen
t,.o
rgan
izat
ion,
.and
.sty
le.a
re.
appr
opri
ate.
to.t
ask,
.pu
rpos
e,.a
nd.a
udie
nce.
W.C
CR
.4X
X
Dev
elop
.and
.str
eng
then
.wri
ting
.as.
need
ed.b
y.pl
ann
ing,
.rev
isin
g,.e
dit
ing,
.rew
riti
ng,.o
r.tr
ying
.a.
new
.app
roac
h.
W.C
CR
.5X
X
Use
.tec
hno
log
y,.in
clud
ing.
the.
Inte
rnet
,.to.
prod
uce.
and.
publ
ish.
wri
ting
.and
.to.
inte
ract
.and
.co
llabo
rate
.wit
h.ot
hers
.
W.C
CR
.6
24 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eR
esea
rch.
to.B
uild
.an
d.P
rese
nt.
Kno
wle
dge
Con
duct
.sho
rt.a
s.w
ell.a
s.m
ore.
sust
aine
d.re
sear
ch.p
roje
cts.
base
d.on
.focu
sed.
que
stio
ns,
.de
mon
stra
ting
.und
erst
and
ing.
of.t
he.s
ubj
ect.
und
er.in
vest
igat
ion.
W.C
CR
.7X
X
Gat
her.
rele
vant
.info
rmat
ion.
from
.mu
ltipl
e.pr
int.
and.
dig
ital
.sou
rces
,.ass
ess.
the.
cred
ibili
ty.a
nd.
accu
racy
.of.e
ach.
sou
rce,
.and
.inte
gra
te.t
he.
info
rmat
ion.
wh
ile.a
void
ing.
plag
iari
sm.
W.C
CR
.8X
Dra
w.e
vide
nce.
from
.lite
rary
.or.
info
rmat
iona
l.te
xts.
to.s
uppo
rt.a
naly
sis,
.ref
lect
ion,
.and
.re
sear
ch.
W.C
CR
.9X
X
Ran
ge.o
f.W
riti
ngW
rite
.rou
tine
ly.o
ver.
exte
nded
.tim
e.fr
ames
.(tim
e.fo
r.re
sear
ch,.r
efle
ctio
n,.a
nd.r
evis
ion)
.and
.sho
rter
.ti
me.
fram
es.(a
.sin
gle.
sitt
ing.
or.a
.day
.or.
two)
.for.
a.ra
nge.
of.t
asks
,.pu
rpos
es,.a
nd.a
udie
nces
.
W.C
CR
.10
XX
9-10
Wri
ting
Tex
t.Ty
pes.
and.
Pu
rpos
esW
rite
.arg
um
ents
.to.
supp
ort.
clai
ms.
in.a
n.an
alys
is.
of.s
ubs
tant
ive.
topi
cs.o
r.te
xts,
.usi
ng.v
alid
.re
ason
ing.
and.
rele
vant
.and
.su
ffic
ient
.evi
denc
e.
Intr
oduc
e.pr
ecis
e.cl
aim
(s),.
dis
ting
uis
h.th
e.cl
aim
(s).f
rom
.alte
rnat
e.or
.opp
osin
g.cl
aim
s,.a
nd.
crea
te.a
n.or
gan
izat
ion.
that
.est
ablis
hes.
clea
r.re
lati
onsh
ips.
amon
g.cl
aim
(s),.
cou
nter
clai
ms,
.re
ason
s,.a
nd.e
vide
nce.
W.9
-10.
1aX
X
Dev
elop
.cla
im(s
).and
.cou
nter
clai
ms.
fair
ly,.
supp
lyin
g.ev
iden
ce.fo
r.ea
ch.w
hile
.poi
ntin
g.ou
t.th
e.st
reng
ths.
and.
limit
atio
ns.
of.b
oth.
in.a
.man
ner.
that
.ant
icip
ates
.the
.aud
ienc
e’s.
know
ledg
e.le
vel.
and.
conc
ern
s.
W.9
-10.
1bX
X
Use
.wor
ds,.p
hra
ses,
.and
.cla
uses
.to.
link.
the.
maj
or.
sect
ion
s.of
.the
.tex
t,.c
reat
e.co
hesi
on,.a
nd.c
lari
fy.
the.
rela
tion
ship
s.be
twee
n.cl
aim
(s).a
nd.r
easo
ns,
.be
twee
n.re
ason
s.an
d.ev
iden
ce,.a
nd.b
etw
een.
clai
m(s
).and
.cou
nter
clai
ms.
W.9
-10.
1cX
X
Est
ablis
h.an
d.m
aint
ain.
a.fo
rmal
.sty
le.a
nd.
obje
ctiv
e.to
ne.w
hile
.att
end
ing.
to.t
he.n
orm
s.an
d.co
nven
tion
s.of
.the
.dis
cipl
ine.
in.w
hic
h.th
ey.a
re.
wri
ting
.
W.9
-10.
1dX
X
Pro
vide
.a.c
oncl
udin
g.st
atem
ent.
or.s
ecti
on.
that
.follo
ws.
from
.and
.sup
port
s.th
e.ar
gu
men
t.pr
esen
ted.
W.9
-10.
1eX
25College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eW
rite
.info
rmat
ive/
expl
anat
ory.
text
s.to
.exa
min
e.an
d.co
nvey
.com
plex
.idea
s,.c
once
pts,
.and
.in
form
atio
n.cl
early
.and
.acc
ura
tely
.th
roug
h.th
e.ef
fect
ive.
sele
ctio
n,.o
rgan
izat
ion,
.and
.ana
lysi
s.of
.co
nten
t.
Intr
oduc
e.a.
topi
c;.o
rgan
ize.
com
plex
.idea
s,.
conc
epts
,.and
.info
rmat
ion.
to.m
ake.
impo
rtan
t.co
nne
ctio
ns.
and.
dis
tinc
tion
s;.in
clud
e.fo
rmat
ting
.(e
.g.,.
head
ings
),.g
raph
ics.
(e.g
.,.fig
ure
s,.t
able
s),.
and.
mu
ltim
edia
.whe
n.us
efu
l.to.
aid
ing.
com
preh
ensi
on.
W.9
-10.
2aX
X
Dev
elop
.the
.top
ic.w
ith.
wel
l-cho
sen,
.rel
evan
t,.
and.
suff
icie
nt.fa
cts,
.ext
ende
d.de
fin
itio
ns,
.co
ncre
te.d
etai
ls,.q
uota
tion
s,.o
r.ot
her.
info
rmat
ion.
and.
exam
ples
.app
ropr
iate
.to.
the.
aud
ienc
e’s.
know
ledg
e.of
.the
.top
ic.
W.9
-10.
2bX
X
Use
.app
ropr
iate
.and
.var
ied.
tran
siti
ons.
to.li
nk.
the.
maj
or.s
ecti
ons.
of.t
he.t
ext,
.cre
ate.
cohe
sion
,.and
.cl
arif
y.th
e.re
lati
onsh
ips.
amon
g.co
mpl
ex.id
eas.
and.
conc
epts
.
W.9
-10.
2cX
X
Use
.pre
cise
.lang
uage
.and
.dom
ain-
spec
ific
.vo
cabu
lary
.to.
man
age.
the.
com
plex
ity.
of.t
he.
topi
c.
W.9
-10.
2dX
X
Est
ablis
h.an
d.m
aint
ain.
a.fo
rmal
.sty
le.a
nd.
obje
ctiv
e.to
ne.w
hile
.att
end
ing.
to.t
he.n
orm
s.an
d.co
nven
tion
s.of
.the
.dis
cipl
ine.
in.w
hic
h.th
ey.a
re.
wri
ting
.
W.9
-10.
2eX
X
Pro
vide
.a.c
oncl
udin
g.st
atem
ent.
or.s
ecti
on.
that
.follo
ws.
from
.and
.sup
port
s.th
e.in
form
atio
n.or
.exp
lana
tion
.pre
sent
ed.(e
.g.,.
arti
cula
ting
.im
plic
atio
ns.
or.t
he.s
ign
ific
ance
.of.t
he.t
opic
).
W.9
-10.
2fX
Wri
te.n
arra
tive
s.to
.dev
elop
.rea
l.or.
imag
ined
.ex
peri
ence
s.or
.eve
nts.
usin
g.ef
fect
ive.
tech
niq
ue,.
wel
l-cho
sen.
deta
ils,.a
nd.w
ell-s
truc
ture
d.ev
ent.
seq
uenc
es.
Eng
age.
and.
orie
nt.t
he.r
eade
r.by
.set
ting
.out
.a.
prob
lem
,.sit
uati
on,.o
r.ob
serv
atio
n,.e
stab
lish
ing.
one.
or.m
ulti
ple.
poin
t(s)
.of.v
iew
,.and
.intr
oduc
ing.
a.na
rrat
or.a
nd/o
r.ch
arac
ters
;.cre
ate.
a.sm
ooth
.pr
ogre
ssio
n.of
.exp
erie
nces
.or.
even
ts.
W.9
-10.
3aX
X
Use
.nar
rati
ve.t
ech
niq
ues,
.suc
h.as
.dia
log
ue,.
paci
ng,.d
escr
ipti
on,.r
efle
ctio
n,.a
nd.m
ulti
ple.
plot
.lin
es,.t
o.de
velo
p.ex
peri
ence
s,.e
vent
s,.a
nd/o
r.ch
arac
ters
.
W.9
-10.
3bX
X
Use
.a.v
arie
ty.o
f.tec
hn
ique
s.to
.seq
uenc
e.ev
ents
.so
.tha
t.th
ey.b
uild
.on.
one.
anot
her.
to.c
reat
e.a.
cohe
rent
.who
le.
W.9
-10.
3cX
X
26 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eU
se.p
reci
se.w
ords
.and
.phr
ases
,.tel
ling.
deta
ils,.a
nd.
sens
ory.
lang
uage
.to.
conv
ey.a
.viv
id.p
ictu
re.o
f.the
.ex
peri
ence
s,.e
vent
s,.s
etti
ng,.a
nd/o
r.cha
ract
ers.
W.9
-10.
3dX
X
Pro
vide
.a.c
oncl
usio
n.th
at.fo
llow
s.fr
om.a
nd.
refle
cts.
on.w
hat.
is.e
xper
ienc
ed,.o
bser
ved,
.or.
reso
lved
.ove
r.th
e.co
urs
e.of
.the
.nar
rati
ve.
W.9
-10.
3e
Pro
duct
ion.
and.
Dis
trib
utio
n.of
.Wri
ting
Pro
duce
.cle
ar.a
nd.c
oher
ent.
wri
ting
.in.w
hic
h.th
e.de
velo
pmen
t,.o
rgan
izat
ion,
.and
.sty
le.a
re.
appr
opri
ate.
to.t
ask,
.pu
rpos
e,.a
nd.a
udie
nce.
.(G
rade
-spe
cific
.exp
ecta
tion
s.fo
r.w
riti
ng.t
ypes
.ar
e.de
fine
d.in
.sta
ndar
ds.1
–3.a
bove
.)
W.9
-10.
4X
X
Dev
elop
.and
.str
eng
then
.wri
ting
.as.
need
ed.
by.p
lan
nin
g,.r
evis
ing,
.ed
itin
g,.r
ewri
ting
,.or.
tryi
ng.a
.new
.app
roac
h,.fo
cusi
ng.o
n.ad
dre
ssin
g.w
hat.
is.m
ost.
sig
nif
ican
t.fo
r.a.
spec
ific
.pu
rpos
e.an
d.au
die
nce.
.(Ed
itin
g.fo
r.co
nven
tion
s.sh
ould
.de
mon
stra
te.c
omm
and.
of.L
ang
uage
.sta
ndar
ds.
1–3.
up.t
o.an
d.in
clud
ing.
gra
des.
9–10
.on.
page
.54.
)
W.9
-10.
5X
X
Use
.tec
hno
log
y,.in
clud
ing.
the.
Inte
rnet
,.to.
prod
uce,
.pu
blis
h,.a
nd.u
pdat
e.in
div
idua
l.or.
shar
ed.w
riti
ng.p
rodu
cts,
.tak
ing.
adva
ntag
e.of
.te
chno
log
y’s.
capa
city
.to.
link.
to.o
ther
.info
rmat
ion.
and.
to.d
ispl
ay.in
form
atio
n.fle
xibl
y.an
d.dy
nam
ical
ly.
W.9
-10.
6
Res
earc
h.to
.Bu
ild.
and.
Pre
sent
.K
now
ledg
e
Con
duct
.sho
rt.a
s.w
ell.a
s.m
ore.
sust
aine
d.re
sear
ch.p
roje
cts.
to.a
nsw
er.a
.que
stio
n.(in
clud
ing.
a.se
lf-ge
nera
ted.
que
stio
n).o
r.so
lve.
a.pr
oble
m;.
narr
ow.o
r.br
oade
n.th
e.in
qu
iry.
whe
n.ap
prop
riat
e;.
synt
hesi
ze.m
ulti
ple.
sou
rces
.on.
the.
subj
ect,
.de
mon
stra
ting
.und
erst
and
ing.
of.t
he.s
ubj
ect.
und
er.in
vest
igat
ion.
W.9
-10.
7X
X
Gat
her.
rele
vant
.info
rmat
ion.
from
.mu
ltipl
e.au
thor
itat
ive.
prin
t.an
d.d
igit
al.s
ourc
es,.u
sing
.ad
vanc
ed.s
earc
hes.
effe
ctiv
ely;
.ass
ess.
the.
usef
uln
ess.
of.e
ach.
sou
rce.
in.a
nsw
erin
g.th
e.re
sear
ch.q
uest
ion;
.inte
gra
te.in
form
atio
n.in
to.t
he.
text
.sel
ecti
vely
.to.
mai
ntai
n.th
e.flo
w.o
f.ide
as,.
avoi
din
g.pl
agia
rism
.and
.follo
win
g.a.
stan
dard
.fo
rmat
.for.
cita
tion
.
W.9
-10.
8X
27College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eD
raw
.evi
denc
e.fr
om.li
tera
ry.o
r.in
form
atio
nal.
text
s.to
.sup
port
.ana
lysi
s,.r
efle
ctio
n,.a
nd.
rese
arch
.
App
ly.g
rade
s 9–
10 R
ead
ing
stan
dard
s.to
.lit
erat
ure
.(e.g
.,.“A
naly
ze.h
ow.a
n.au
thor
.dra
ws.
on.a
nd.t
ran
sfor
ms.
sou
rce.
mat
eria
l.in.
a.sp
ecif
ic.
wor
k.[e
.g.,.
how
.Sha
kesp
eare
.tre
ats.
a.th
eme.
or.
topi
c.fr
om.O
vid.
or.t
he.B
ible
.or.
how
.a.la
ter.
auth
or.
dra
ws.
on.a
.pla
y.by
.Sha
kesp
eare
]”).
W.9
-10.
9aX
X
App
ly.g
rade
s 9–
10 R
ead
ing
stan
dard
s.to
.lit
erar
y.no
nfic
tion
.(e.g
.,.“D
elin
eate
.and
.eva
luat
e.th
e.ar
gu
men
t.an
d.sp
ecif
ic.c
laim
s.in
.a.t
ext,
.as
sess
ing.
whe
ther
.the
.rea
son
ing.
is.v
alid
.and
.the
.ev
iden
ce.is
.rel
evan
t.an
d.su
ffic
ient
;.ide
ntif
y.fa
lse.
stat
emen
ts.a
nd.fa
llaci
ous.
reas
onin
g”).
W.9
-10.
9bX
Ran
ge.o
f.W
riti
ngW
rite
.rou
tine
ly.o
ver.
exte
nded
.tim
e.fr
ames
.(tim
e.fo
r.re
sear
ch,.r
efle
ctio
n,.a
nd.r
evis
ion)
.and
.sho
rter
.ti
me.
fram
es.(a
.sin
gle.
sitt
ing.
or.a
.day
.or.
two)
.for.
a.ra
nge.
of.t
asks
,.pu
rpos
es,.a
nd.a
udie
nces
.
W.9
-10.
10X
X
11-1
2W
riti
ngT
ext.
Type
s.an
d.P
urp
oses
Wri
te.a
rgu
men
ts.t
o.su
ppor
t.cl
aim
s.in
.an.
anal
ysis
.of
.su
bsta
ntiv
e.to
pics
.or.
text
s,.u
sing
.val
id.
reas
onin
g.an
d.re
leva
nt.a
nd.s
uff
icie
nt.e
vide
nce.
Intr
oduc
e.pr
ecis
e,.k
now
ledg
eabl
e.cl
aim
(s),.
esta
blis
h.th
e.si
gn
ific
ance
.of.t
he.c
laim
(s),.
dis
ting
uis
h.th
e.cl
aim
(s).f
rom
.alte
rnat
e.or
.op
posi
ng.c
laim
s,.a
nd.c
reat
e.an
.org
aniz
atio
n.th
at.lo
gic
ally
.seq
uenc
es.c
laim
(s),.
cou
nter
clai
ms,
.re
ason
s,.a
nd.e
vide
nce.
W.1
1-12
.1a
XX
"Dev
elop
.cla
im(s
).and
.cou
nter
clai
ms.
fair
ly.a
nd.
thor
ough
ly,.s
uppl
ying
.the
.mos
t.re
leva
nt.e
vide
nce.
for.
each
.wh
ile.p
oint
ing.
out.
the.
stre
ngth
s.an
d.lim
itat
ion
s.of
.bot
h.in
.a.m
anne
r.th
at.a
ntic
ipat
es.
the.
aud
ienc
e’s.
know
ledg
e.le
vel,.
conc
ern
s,.v
alue
s,.
and.
poss
ible
.bia
ses.
W.1
1-12
.1b
XX
Use
.wor
ds,.p
hra
ses,
.and
.cla
uses
.as.
wel
l.as.
vari
ed.s
ynta
x.to
.lin
k.th
e.m
ajor
.sec
tion
s.of
.th
e.te
xt,.c
reat
e.co
hesi
on,.a
nd.c
lari
fy.t
he.
rela
tion
ship
s.be
twee
n.cl
aim
(s).a
nd.r
easo
ns,
.be
twee
n.re
ason
s.an
d.ev
iden
ce,.a
nd.b
etw
een.
clai
m(s
).and
.cou
nter
clai
ms.
W.1
1-12
.1c
XX
Est
ablis
h.an
d.m
aint
ain.
a.fo
rmal
.sty
le.a
nd.
obje
ctiv
e.to
ne.w
hile
.att
end
ing.
to.t
he.n
orm
s.an
d.co
nven
tion
s.of
.the
.dis
cipl
ine.
in.w
hic
h.th
ey.a
re.
wri
ting
.
W.1
1-12
.1d
XX
28 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eP
rovi
de.a
.con
clud
ing.
stat
emen
t.or
.sec
tion
.th
at.fo
llow
s.fr
om.a
nd.s
uppo
rts.
the.
arg
um
ent.
pres
ente
d.
W.1
1-12
.1e
Wri
te.in
form
ativ
e/ex
plan
ator
y.te
xts.
to.e
xam
ine.
and.
conv
ey.c
ompl
ex.id
eas,
.con
cept
s,.a
nd.
info
rmat
ion.
clea
rly.a
nd.a
ccu
rate
ly.t
hro
ugh.
the.
effe
ctiv
e.se
lect
ion,
.org
aniz
atio
n,.a
nd.a
naly
sis.
of.
cont
ent.
Intr
oduc
e.a.
topi
c;.o
rgan
ize.
com
plex
.idea
s,.
conc
epts
,.and
.info
rmat
ion.
so.t
hat.
each
.new
.el
emen
t.bu
ilds.
on.t
hat.
wh
ich.
prec
edes
.it.t
o.cr
eate
.a.u
nif
ied.
who
le;.i
nclu
de.fo
rmat
ting
.(e
.g.,.
head
ings
),.g
raph
ics.
(e.g
.,.fig
ure
s,.t
able
s),.
and.
mu
ltim
edia
.whe
n.us
efu
l.to.
aid
ing.
com
preh
ensi
on.
W.1
1-12
.2a
XX
Dev
elop
.the
.top
ic.t
horo
ugh
ly.b
y.se
lect
ing.
the.
mos
t.si
gn
ific
ant.
and.
rele
vant
.fact
s,.e
xten
ded.
defi
nit
ion
s,.c
oncr
ete.
deta
ils,.q
uota
tion
s,.o
r.ot
her.
info
rmat
ion.
and.
exam
ples
.app
ropr
iate
.to.
the.
aud
ienc
e’s.
know
ledg
e.of
.the
.top
ic.
W.1
1-12
.2b
XX
Use
.app
ropr
iate
.and
.var
ied.
tran
siti
ons.
and.
synt
ax.t
o.lin
k.th
e.m
ajor
.sec
tion
s.of
.the
.tex
t,.
crea
te.c
ohes
ion,
.and
.cla
rify
.the
.rel
atio
nsh
ips.
amon
g.co
mpl
ex.id
eas.
and.
conc
epts
.
W.1
1-12
.2c
XX
Use
.pre
cise
.lang
uage
,.dom
ain-
spec
ific
.vo
cabu
lary
,.and
.tec
hn
ique
s.su
ch.a
s.m
etap
hor,.
sim
ile,.a
nd.a
nalo
gy.
to.m
anag
e.th
e.co
mpl
exit
y.of
.th
e.to
pic.
W.1
1-12
.2d
XX
Est
ablis
h.an
d.m
aint
ain.
a.fo
rmal
.sty
le.a
nd.
obje
ctiv
e.to
ne.w
hile
.att
end
ing.
to.t
he.n
orm
s.an
d.co
nven
tion
s.of
.the
.dis
cipl
ine.
in.w
hic
h.th
ey.a
re.
wri
ting
.
W.1
1-12
.2e
XX
Pro
vide
.a.c
oncl
udin
g.st
atem
ent.
or.s
ecti
on.
that
.follo
ws.
from
.and
.sup
port
s.th
e.in
form
atio
n.or
.exp
lana
tion
.pre
sent
ed.(e
.g.,.
arti
cula
ting
.im
plic
atio
ns.
or.t
he.s
ign
ific
ance
.of.t
he.t
opic
).
W.1
1-12
.2f
Wri
te.n
arra
tive
s.to
.dev
elop
.rea
l.or.
imag
ined
.ex
peri
ence
s.or
.eve
nts.
usin
g.ef
fect
ive.
tech
niq
ue,.
wel
l-cho
sen.
deta
ils,.a
nd.w
ell-s
truc
ture
d.ev
ent.
seq
uenc
es.
Eng
age.
and.
orie
nt.t
he.r
eade
r.by
.set
ting
.out
.a.
prob
lem
,.sit
uati
on,.o
r.ob
serv
atio
n.an
d.it
s.si
gn
ific
ance
,.est
ablis
hin
g.on
e.or
.mu
ltipl
e.po
int(
s).o
f.vie
w,.a
nd.in
trod
ucin
g.a.
narr
ator
.and
/or
.cha
ract
ers;
.cre
ate.
a.sm
ooth
.pro
gre
ssio
n.of
.ex
peri
ence
s.or
.eve
nts.
W.1
1-12
.3a
XX
29College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eU
se.n
arra
tive
.tec
hn
ique
s,.s
uch.
as.d
ialo
gue
,.pa
cing
,.des
crip
tion
,.ref
lect
ion,
.and
.mu
ltipl
e.pl
ot.
lines
,.to.
deve
lop.
expe
rien
ces,
.eve
nts,
.and
/or.
char
acte
rs.
W.1
1-12
.3b
XX
Use
.a.v
arie
ty.o
f.tec
hn
ique
s.to
.seq
uenc
e.ev
ents
.so
.tha
t.th
ey.b
uild
.on.
one.
anot
her.
to.c
reat
e.a.
cohe
rent
.who
le.a
nd.b
uild
.tow
ard.
a.pa
rtic
ula
r.to
ne.a
nd.o
utco
me.
(e.g
.,.a.
sen
se.o
f.mys
tery
,.su
spen
se,.g
row
th,.o
r.re
solu
tion
).
W.1
1-12
.3c
XX
Use
.pre
cise
.wor
ds.a
nd.p
hra
ses,
.tel
ling.
deta
ils,.
and.
sen
sory
.lang
uage
.to.
conv
ey.a
.viv
id.p
ictu
re.
of.t
he.e
xper
ienc
es,.e
vent
s,.s
etti
ng,.a
nd/o
r.ch
arac
ters
.
W.1
1-12
.3d
XX
Pro
vide
.a.c
oncl
usio
n.th
at.fo
llow
s.fr
om.a
nd.
refle
cts.
on.w
hat.
is.e
xper
ienc
ed,.o
bser
ved,
.or.
reso
lved
.ove
r.th
e.co
urs
e.of
.the
.nar
rati
ve.
W.1
1-12
.3e
Pro
duct
ion.
and.
Dis
trib
utio
n.of
.Wri
ting
Pro
duce
.cle
ar.a
nd.c
oher
ent.
wri
ting
.in.w
hic
h.th
e.de
velo
pmen
t,.o
rgan
izat
ion,
.and
.sty
le.a
re.
appr
opri
ate.
to.t
ask,
.pu
rpos
e,.a
nd.a
udie
nce.
.(G
rade
-spe
cific
.exp
ecta
tion
s.fo
r.w
riti
ng.t
ypes
.ar
e.de
fine
d.in
.sta
ndar
ds.1
–3.a
bove
.)
W.1
1-12
.4X
X
Dev
elop
.and
.str
eng
then
.wri
ting
.as.
need
ed.
by.p
lan
nin
g,.r
evis
ing,
.ed
itin
g,.r
ewri
ting
,.or.
tryi
ng.a
.new
.app
roac
h,.fo
cusi
ng.o
n.ad
dre
ssin
g.w
hat.
is.m
ost.
sig
nif
ican
t.fo
r.a.
spec
ific
.pu
rpos
e.an
d.au
die
nce.
.(Ed
itin
g.fo
r.co
nven
tion
s.sh
ould
.de
mon
stra
te.c
omm
and.
of.L
ang
uage
.sta
ndar
ds.
1–3.
up.t
o.an
d.in
clud
ing.
gra
des.
11–1
2.on
.pag
e.54
.)
W.1
1-12
.5X
X
Use
.tec
hno
log
y,.in
clud
ing.
the.
Inte
rnet
,.to.
prod
uce,
.pu
blis
h,.a
nd.u
pdat
e.in
div
idua
l.or
.sha
red.
wri
ting
.pro
duct
s.in
.res
pon
se.t
o.on
goin
g.fe
edba
ck,.i
nclu
din
g.ne
w.a
rgu
men
ts.o
r.in
form
atio
n.
W.1
1-12
.6
30 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eR
esea
rch.
to.B
uild
.an
d.P
rese
nt.
Kno
wle
dge
Con
duct
.sho
rt.a
s.w
ell.a
s.m
ore.
sust
aine
d.re
sear
ch.p
roje
cts.
to.a
nsw
er.a
.que
stio
n.(in
clud
ing.
a.se
lf-ge
nera
ted.
que
stio
n).o
r.so
lve.
a.pr
oble
m;.
narr
ow.o
r.br
oade
n.th
e.in
qu
iry.
whe
n.ap
prop
riat
e;.
synt
hesi
ze.m
ulti
ple.
sou
rces
.on.
the.
subj
ect,
.de
mon
stra
ting
.und
erst
and
ing.
of.t
he.s
ubj
ect.
und
er.in
vest
igat
ion.
W.1
1-12
.7X
X
Gat
her.
rele
vant
.info
rmat
ion.
from
.mu
ltipl
e.au
thor
itat
ive.
prin
t.an
d.d
igit
al.s
ourc
es,.u
sing
.ad
vanc
ed.s
earc
hes.
effe
ctiv
ely;
.ass
ess.
the.
stre
ngth
s.an
d.lim
itat
ion
s.of
.eac
h.so
urc
e.in
.ter
ms.
of.t
he.t
ask,
.pu
rpos
e,.a
nd.a
udie
nce;
.inte
gra
te.
info
rmat
ion.
into
.the
.tex
t.se
lect
ivel
y.to
.mai
ntai
n.th
e.flo
w.o
f.ide
as,.a
void
ing.
plag
iari
sm.a
nd.
over
relia
nce.
on.a
ny.o
ne.s
ourc
e.an
d.fo
llow
ing.
a.st
anda
rd.fo
rmat
.for.
cita
tion
.
W.1
1-12
.8X
Dra
w.e
vide
nce.
from
.lite
rary
.or.
info
rmat
iona
l.te
xts.
to.s
uppo
rt.a
naly
sis,
.ref
lect
ion,
.and
.re
sear
ch.
App
ly.g
rade
s 11
–12
Rea
din
g st
anda
rds.
to.
liter
atu
re.(e
.g.,.
“Dem
onst
rate
.kno
wle
dge.
of.
eigh
teen
th-,.
nin
etee
nth-
.and
.ear
ly-t
wen
tiet
h-ce
ntu
ry.fo
und
atio
nal.w
orks
.of.A
mer
ican
.lit
erat
ure
,.inc
lud
ing.
how
.tw
o.or
.mor
e.te
xts.
from
.th
e.sa
me.
peri
od.t
reat
.sim
ilar.
them
es.o
r.to
pics
”).
W.1
1-12
.9a
XX
App
ly.g
rade
s 11
–12
Rea
din
g st
anda
rds.
to.li
tera
ry.
non
ficti
on.(e
.g.,.
“Del
inea
te.a
nd.e
valu
ate.
the.
reas
onin
g.in
.sem
inal
.U.S
..tex
ts,.i
nclu
din
g.th
e.ap
plic
atio
n.of
.con
stit
utio
nal.p
rinc
iple
s.an
d.us
e.of
.lega
l.rea
son
ing.
[e.g
.,.in
.U.S
..Sup
rem
e.C
ourt
.C
ase.
maj
orit
y.op
inio
ns.
and.
dis
sent
s].a
nd.t
he.
prem
ises
,.pu
rpos
es,.a
nd.a
rgu
men
ts.in
.wor
ks.o
f.pu
blic
.adv
ocac
y.[e
.g.,
The
Fed
eral
ist,.
pres
iden
tial
.ad
dre
sses
]”).
W.1
1-12
.9b
X
Ran
ge.o
f.W
riti
ngW
rite
.rou
tine
ly.o
ver.
exte
nded
.tim
e.fr
ames
.(tim
e.fo
r.re
sear
ch,.r
efle
ctio
n,.a
nd.r
evis
ion)
.and
.sho
rter
.ti
me.
fram
es.(a
.sin
gle.
sitt
ing.
or.a
.day
.or.
two)
.for.
a.ra
nge.
of.t
asks
,.pu
rpos
es,.a
nd.a
udie
nces
.
W.1
1-12
.10
XX
CC
RSp
eaki
ng.a
nd.
Lis
ten
ing
Com
preh
ensi
on.
and.
.C
olla
bora
tion
Pre
pare
.for.
and.
part
icip
ate.
effe
ctiv
ely.
in.a
.ra
nge.
of.c
onve
rsat
ion
s.an
d.co
llabo
rati
ons.
wit
h.d
iver
se.p
artn
ers,
.bu
ildin
g.on
.oth
ers’
.idea
s.an
d.ex
pres
sing
.the
ir.o
wn.
clea
rly.a
nd.p
ersu
asiv
ely.
SL.C
CR
.1X
31College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eIn
teg
rate
.and
.eva
luat
e.in
form
atio
n.pr
esen
ted.
in.
div
erse
.med
ia.a
nd.fo
rmat
s,.in
clud
ing.
visu
ally
,.q
uant
itat
ivel
y,.a
nd.o
rally
.
SL.C
CR
.2
Eva
luat
e.a.
spea
ker’s
.poi
nt.o
f.vie
w,.r
easo
nin
g,.
and.
use.
of.e
vide
nce.
and.
rhet
oric
.SL
.CC
R.3
Pre
sent
atio
n.of
.Kno
wle
dge.
and.
Idea
s
Pre
sent
.info
rmat
ion,
.find
ings
,.and
.sup
port
ing.
evid
ence
.suc
h.th
at.li
sten
ers.
can.
follo
w.t
he.li
ne.
of.r
easo
nin
g.an
d.th
e.or
gan
izat
ion,
.dev
elop
men
t,.
and.
styl
e.ar
e.ap
prop
riat
e.to
.tas
k,.p
urp
ose,
.and
.au
die
nce.
SL.C
CR
.4
Mak
e.st
rate
gic
.use
.of.d
igit
al.m
edia
.and
.vis
ual.
dis
play
s.of
.dat
a.to
.exp
ress
.info
rmat
ion.
and.
enha
nce.
und
erst
and
ing.
of.p
rese
ntat
ion
s.
SL.C
CR
.5
Ada
pt.s
peec
h.to
.a.v
arie
ty.o
f.con
text
s.an
d.co
mm
un
icat
ive.
task
s,.d
emon
stra
ting
.com
man
d.of
.form
al.E
nglis
h.w
hen.
ind
icat
ed.o
r.ap
prop
riat
e.
SL.C
CR
.6
9-10
Spea
king
.and
.L
iste
nin
g.St
anda
rds
Com
preh
ensi
on.
and.
.C
olla
bora
tion
Init
iate
.and
.par
tici
pate
.effe
ctiv
ely.
in.a
.ran
ge.o
f.co
llabo
rati
ve.d
iscu
ssio
ns.
(one
-on-
one,
.in.g
roup
s,.
and.
teac
her-
led)
.wit
h.d
iver
se.p
artn
ers.
on.g
rade
s 9–
10 t
opic
s, t
exts
, and
issu
es,.b
uild
ing.
on.o
ther
s’.
idea
s.an
d.ex
pres
sing
.the
ir.o
wn.
clea
rly.a
nd.
pers
uasi
vely
.
Com
e.to
.dis
cuss
ion
s.pr
epar
ed,.h
avin
g.re
ad.a
nd.
rese
arch
ed.m
ater
ial.u
nder
.stu
dy;.e
xplic
itly
.dra
w.
on.t
hat.
prep
arat
ion.
by.r
efer
ring
.to.
evid
ence
.from
.te
xts.
and.
othe
r.re
sear
ch.o
n.th
e.to
pic.
or.is
sue.
to.
stim
ula
te.a
.tho
ught
ful,.
wel
l-rea
sone
d.ex
chan
ge.
of.id
eas.
SL.9
-10.
1aX
Wor
k.w
ith.
peer
s.to
.set
.ru
les.
for.
colle
gia
l.d
iscu
ssio
ns.
and.
deci
sion
-mak
ing.
(e.g
.,.in
form
al.
con
sen
sus,
.tak
ing.
vote
s.on
.key
.issu
es,.
pres
enta
tion
.of.a
ltern
ate.
view
s),.c
lear
.goa
ls.a
nd.
dead
lines
,.and
.ind
ivid
ual.r
oles
.as.
need
ed.
SL.9
-10.
1b
Pro
pel.c
onve
rsat
ion
s.by
.pos
ing.
and.
resp
ond
ing.
to.q
uest
ion
s.th
at.r
elat
e.th
e.cu
rren
t.d
iscu
ssio
n.to
.bro
ader
.the
mes
.or.
larg
er.id
eas;
.act
ivel
y.in
corp
orat
e.ot
hers
.into
.the
.dis
cuss
ion;
.and
.cl
arif
y,.v
erif
y,.o
r.ch
alle
nge.
idea
s.an
d.co
nclu
sion
s.
SL.9
-10.
1c
Res
pond
.tho
ught
fully
.to.
div
erse
.per
spec
tive
s,.
sum
mar
ize.
poin
ts.o
f.ag
reem
ent.
and.
dis
agre
emen
t,.a
nd,.w
hen.
war
rant
ed,.q
ualif
y.or
.ju
stif
y.th
eir.
own.
view
s.an
d.u
nder
stan
din
g.an
d.m
ake.
new
.con
nect
ion
s.in
.ligh
t.of
.the
.evi
denc
e.an
d.re
ason
ing.
pres
ente
d.
SL.9
-10.
1d
32 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eIn
teg
rate
.mu
ltipl
e.so
urc
es.o
f.in
form
atio
n.pr
esen
ted.
in.d
iver
se.m
edia
.or.
form
ats.
(e.g
.,.vi
sual
ly,.q
uant
itat
ivel
y,.o
rally
).eva
luat
ing.
the.
cred
ibili
ty.a
nd.a
ccu
racy
.of.e
ach.
sou
rce.
SL.9
-10.
2
Eva
luat
e.a.
spea
ker’s
.poi
nt.o
f.vie
w,.r
easo
nin
g,.
and.
use.
of.e
vide
nce.
and.
rhet
oric
,.ide
ntif
ying
.any
.fa
llaci
ous.
reas
onin
g.or
.exa
gger
ated
.or.
dis
tort
ed.
evid
ence
.
SL.9
-10.
3
Pre
sent
atio
n.of
.Kno
wle
dge.
and.
Idea
s
Pre
sent
.info
rmat
ion,
.find
ings
,.and
.sup
port
ing.
evid
ence
.cle
arly
,.con
cise
ly,.a
nd.lo
gic
ally
.suc
h.th
at.li
sten
ers.
can.
follo
w.t
he.li
ne.o
f.rea
son
ing.
and.
the.
orga
niz
atio
n,.d
evel
opm
ent,
.su
bsta
nce,
.and
.st
yle.
are.
appr
opri
ate.
to.p
urp
ose,
.aud
ienc
e,.a
nd.
task
.
SL.9
-10.
4
Mak
e.st
rate
gic
.use
.of.d
igit
al.m
edia
.(e.g
.,.te
xtua
l,.g
raph
ical
,.aud
io,.v
isua
l,.an
d.in
tera
ctiv
e.el
emen
ts).i
n.pr
esen
tati
ons.
to.e
nha
nce.
und
erst
and
ing.
of.fi
ndin
gs,.r
easo
nin
g,.a
nd.
evid
ence
.and
.to.
add.
inte
rest
.
SL.9
-10.
5
Ada
pt.s
peec
h.to
.a.v
arie
ty.o
f.con
text
s.an
d.ta
sks,
.de
mon
stra
ting
.com
man
d.of
.form
al.E
nglis
h.w
hen.
ind
icat
ed.o
r.ap
prop
riat
e..(S
ee.g
rade
s.9–
10.
Lan
gua
ge.s
tand
ards
.1.a
nd.3
.on.
page
s.54
.for.
spec
ific
.exp
ecta
tion
s.)
SL.9
-10.
6
11-1
2Sp
eaki
ng.a
nd.
Lis
ten
ing.
Stan
dard
s
Com
preh
ensi
on.
and.
.C
olla
bora
tion
Init
iate
.and
.par
tici
pate
.effe
ctiv
ely.
in.a
.ran
ge.o
f.co
llabo
rati
ve.d
iscu
ssio
ns.
(one
-on-
one,
.in.g
roup
s,.
and.
teac
her-
led)
.wit
h.d
iver
se.p
artn
ers.
on.g
rade
s 11
–12
topi
cs, t
exts
, and
issu
es,.b
uild
ing.
on.
othe
rs’.i
deas
.and
.exp
ress
ing.
thei
r.ow
n.cl
early
.an
d.pe
rsua
sive
ly.
Com
e.to
.dis
cuss
ion
s.pr
epar
ed,.h
avin
g.re
ad.a
nd.
rese
arch
ed.m
ater
ial.u
nder
.stu
dy;.e
xplic
itly
.dra
w.
on.t
hat.
prep
arat
ion.
by.r
efer
ring
.to.
evid
ence
.from
.te
xts.
and.
othe
r.re
sear
ch.o
n.th
e.to
pic.
or.is
sue.
to.
stim
ula
te.a
.tho
ught
ful,.
wel
l-rea
sone
d.ex
chan
ge.
of.id
eas.
SL.1
1-12
.1a
X
Wor
k.w
ith.
peer
s.to
.pro
mot
e.ci
vil,.
dem
ocra
tic.
dis
cuss
ion
s.an
d.de
cisi
on-m
akin
g,.s
et.c
lear
.goa
ls.
and.
dead
lines
,.and
.est
ablis
h.in
div
idua
l.rol
es.a
s.ne
eded
.
SL.1
1-12
.1b
Pro
pel.c
onve
rsat
ion
s.by
.pos
ing.
and.
resp
ond
ing.
to.q
uest
ion
s.th
at.p
robe
.rea
son
ing.
and.
evid
ence
;.en
sure
.a.h
eari
ng.fo
r.a.
full.
rang
e.of
.pos
itio
ns.
on.
a.to
pic.
or.is
sue;
.cla
rify
,.ver
ify,
.or.
chal
leng
e.id
eas.
and.
conc
lusi
ons;
.and
.pro
mot
e.d
iver
gent
.and
.cr
eati
ve.p
ersp
ecti
ves.
SL.1
1-12
.1c
33College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eR
espo
nd.t
houg
htfu
lly.t
o.di
vers
e.pe
rspe
ctiv
es;.
synt
hesi
ze.c
omm
ents
,.cla
ims,
.and
.evi
denc
e.m
ade.
on.a
ll.si
des.
of.a
n.is
sue;
.reso
lve.
cont
radi
ctio
ns.
whe
n.po
ssib
le;.a
nd.d
eter
min
e.w
hat.
addi
tion
al.
info
rmat
ion.
or.re
sear
ch.is
.requ
ired
.to.
deep
en.t
he.
inve
stig
atio
n.or
.com
plet
e.th
e.ta
sk.
SL.1
1-12
.1d
Inte
grat
e.m
ulti
ple.
sour
ces.
of.in
form
atio
n.pr
esen
ted.
in.d
iver
se.fo
rmat
s.an
d.m
edia
.(e.g
.,.vi
sual
ly,.q
uant
itat
ivel
y,.o
rally
).in.
orde
r.to
.mak
e.in
form
ed.d
ecis
ions
.and
.sol
ve.p
robl
ems,
.eva
luat
ing.
the.
cred
ibili
ty.a
nd.a
ccur
acy.
of.e
ach.
sour
ce.a
nd.
noti
ng.a
ny.d
iscr
epan
cies
.am
ong.
the.
data
.
SL.1
1-12
.2
Eva
luat
e.a.
spea
ker’s
.poi
nt.o
f.vie
w,.r
easo
nin
g,.
and.
use.
of.e
vide
nce.
and.
rhet
oric
,.ass
essi
ng.
the.
stan
ce,.p
rem
ises
,.lin
ks.a
mon
g.id
eas,
.wor
d.ch
oice
,.poi
nts.
of.e
mph
asis
,.and
.ton
e.us
ed.
SL.1
1-12
.3
Pre
sent
atio
n.of
.Kno
wle
dge.
and.
Idea
s
Pre
sent
.info
rmat
ion,
.find
ings
,.and
.sup
port
ing.
evid
ence
,.con
veyi
ng.a
.cle
ar.a
nd.d
isti
nct.
pers
pect
ive,
.suc
h.th
at.li
sten
ers.
can.
follo
w.
the.
line.
of.r
easo
nin
g,.a
ltern
ativ
e.or
.opp
osin
g.pe
rspe
ctiv
es.a
re.a
ddre
ssed
,.and
.the
.or
gan
izat
ion,
.dev
elop
men
t,.s
ubs
tanc
e,.a
nd.s
tyle
.ar
e.ap
prop
riat
e.to
.pu
rpos
e,.a
udie
nce,
.and
.a.
rang
e.of
.form
al.a
nd.in
form
al.t
asks
.
SL.1
1-12
.4
Mak
e.st
rate
gic
.use
.of.d
igit
al.m
edia
.(e.g
.,.te
xtua
l,.g
raph
ical
,.aud
io,.v
isua
l,.an
d.in
tera
ctiv
e.el
emen
ts).i
n.pr
esen
tati
ons.
to.e
nha
nce.
und
erst
and
ing.
of.fi
ndin
gs,.r
easo
nin
g,.a
nd.
evid
ence
.and
.to.
add.
inte
rest
.
SL.1
1-12
.5
Ada
pt.s
peec
h.to
.a.v
arie
ty.o
f.con
text
s.an
d.ta
sks,
.de
mon
stra
ting
.a.c
omm
and.
of.fo
rmal
.Eng
lish.
whe
n.in
dic
ated
.or.
appr
opri
ate.
.(See
.gra
des.
11–1
2.L
ang
uage
.sta
ndar
ds.1
.and
.3.o
n.pa
ge.5
4.fo
r.sp
ecif
ic.e
xpec
tati
ons.
)
SL.1
1-12
.6
CC
RL
ang
uage
Con
vent
ion
s.of
.Sta
ndar
d.E
nglis
h
Dem
onst
rate
.com
man
d.of
.the
.con
vent
ion
s.of
.st
anda
rd.E
nglis
h.g
ram
mar
.and
.usa
ge.w
hen.
wri
ting
.or.
spea
king
.
L.C
CR
.1X
X
Dem
onst
rate
.com
man
d.of
.the
.con
vent
ion
s.of
.st
anda
rd.E
nglis
h.ca
pita
lizat
ion,
.pu
nctu
atio
n,.a
nd.
spel
ling.
whe
n.w
riti
ng.
L.C
CR
.2X
X
34 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eK
now
ledg
e.of
.L
ang
uage
App
ly.k
now
ledg
e.of
.lang
uage
.to.
und
erst
and.
how
.lang
uage
.func
tion
s.in
.dif
fere
nt.c
onte
xts,
.to
.mak
e.ef
fect
ive.
choi
ces.
for.
mea
nin
g.or
.sty
le,.
and.
to.c
ompr
ehen
d.m
ore.
fully
.whe
n.re
adin
g.or
.lis
ten
ing.
L.C
CR
.3X
X
Voc
abu
lary
.A
cqu
isit
ion.
and.
Use
Det
erm
ine.
or.c
lari
fy.t
he.m
ean
ing.
of.u
nkn
own.
and.
mu
ltipl
e-m
ean
ing.
wor
ds.a
nd.p
hra
ses.
by.
usin
g.co
ntex
t.cl
ues,
.ana
lyzi
ng.m
ean
ing
ful.w
ord.
part
s,.a
nd.c
onsu
lting
.gen
eral
.and
.spe
cial
ized
.re
fere
nce.
mat
eria
ls,.a
s.ap
prop
riat
e.
L.C
CR
.4X
X
Dem
onst
rate
.und
erst
and
ing.
of.fi
gu
rati
ve.
lang
uage
,.wor
d.re
lati
onsh
ips,
.and
.nua
nces
.in.
wor
d.m
ean
ings
.
L.C
CR
.5X
X
Acq
uir
e.an
d.us
e.ac
cura
tely
.a.r
ange
.of.g
ener
al.
acad
emic
.and
.dom
ain-
spec
ific
.wor
ds.a
nd.
phra
ses.
suff
icie
nt.fo
r.re
adin
g,.w
riti
ng,.s
peak
ing,
.an
d.lis
ten
ing.
at.t
he.c
olle
ge.a
nd.c
aree
r.re
adin
ess.
leve
l;.de
mon
stra
te.in
depe
nden
ce.in
.gat
heri
ng.
voca
bula
ry.k
now
ledg
e.w
hen.
con
side
ring
.a.
wor
d.or
.ph
rase
.impo
rtan
t.to
.com
preh
ensi
on.o
r.ex
pres
sion
.
L.C
CR
.6X
X
9-10
Lan
gua
geC
onve
ntio
ns.
of.S
tand
ard.
Eng
lish
Dem
onst
rate
.com
man
d.of
.the
.con
vent
ion
s.of
.st
anda
rd.E
nglis
h.g
ram
mar
.and
.usa
ge.w
hen.
wri
ting
.or.
spea
king
.
Use
.par
alle
l.str
uctu
re.*
L.9
-10.
1aX
X
Use
.var
ious
.typ
es.o
f.ph
rase
s.(n
oun,
.ver
b,.
adje
ctiv
al,.a
dver
bial
,.par
tici
pial
,.pre
posi
tion
al,.
abso
lute
).and
.cla
uses
.(ind
epen
dent
,.dep
ende
nt;.
nou
n,.r
elat
ive,
.adv
erbi
al).t
o.co
nvey
.spe
cific
.m
ean
ings
.and
.add
.var
iety
.and
.inte
rest
.to.
wri
ting
.or
.pre
sent
atio
ns.
L.9
-10.
1bX
X
En
sure
.su
bjec
t-ve
rb.a
nd.p
rono
un-
ante
cede
nt.
agre
emen
t.L
.3.1
fX
X
Pro
duce
.com
plet
e.se
nten
ces,
.rec
ogn
izin
g.an
d.co
rrec
ting
.inap
prop
riat
e.fr
agm
ents
.and
.ru
n-on
s.L
.4.1
fX
X
Cor
rect
ly.u
se.fr
eque
ntly
.con
fuse
d.w
ord
s.(e
.g.,.
to/
too/
two;
the
re/t
heir
).L
.4.1
gX
X
Rec
ogn
ize.
and.
corr
ect.
inap
prop
riat
e.sh
ifts
.in.
verb
.ten
se.
L.5
.1d
XX
35College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eR
ecog
niz
e.an
d.co
rrec
t.in
appr
opri
ate.
shif
ts.in
.pr
onou
n.nu
mbe
r.an
d.pe
rson
.L
.6.1
cX
X
Rec
ogn
ize.
and.
corr
ect.
vag
ue.p
rono
un
s.(i.
e.,.o
nes.
wit
h.u
ncle
ar.o
r.am
big
uous
.ant
eced
ents
).L
.6.1
dX
X
"Rec
ogn
ize.
vari
atio
ns.
from
.sta
ndar
d.E
nglis
h.in
.th
eir.
own.
and.
othe
rs’.w
riti
ng.a
nd.s
peak
ing,
.and
.id
enti
fy.a
nd.u
se.s
trat
egie
s.to
.impr
ove.
expr
essi
on.
in.c
onve
ntio
nal.l
ang
uage
.
L.6
.1e
XX
Plac
e.ph
rase
s.an
d.cl
ause
s.w
ith
in.a
.sen
tenc
e,.
reco
gn
izin
g.an
d.co
rrec
ting
.mis
plac
ed.a
nd.
dang
ling.
mod
ifie
rs.
L.7
.1c
XX
Rec
ogn
ize.
and.
corr
ect.
inap
prop
riat
e.sh
ifts
.in.
verb
.voi
ce.a
nd.m
ood.
L.8
.1d
XX
Dem
onst
rate
.com
man
d.of
.the
.con
vent
ion
s.of
.st
anda
rd.E
nglis
h.ca
pita
lizat
ion,
.pu
nctu
atio
n,.a
nd.
spel
ling.
whe
n.w
riti
ng.
Use
.a.s
emic
olon
.(and
.per
haps
.a.c
onju
ncti
ve.
adve
rb).t
o.lin
k.tw
o.or
.mor
e.cl
osel
y.re
late
d.in
depe
nden
t.cl
ause
s.
L.9
-10.
2aX
X
Use
.a.c
olon
.to.
intr
oduc
e.a.
list.
or.q
uota
tion
.L
.9-1
0.2b
XX
Spel
l.cor
rect
ly.
L.9
-10.
2cX
X
Use
.pu
nctu
atio
n.(c
omm
as,.p
aren
thes
es,.d
ashe
s).
to.s
et.o
ff.no
nre
stri
ctiv
e/pa
rent
heti
cal.e
lem
ents
.L
.6.2
aX
X
Kno
wle
dge.
of.
Lan
gua
geA
pply
.kno
wle
dge.
of.la
ngua
ge.t
o.u
nder
stan
d.ho
w.la
ngua
ge.fu
ncti
ons.
in.d
iffe
rent
.con
text
s,.
to.m
ake.
effe
ctiv
e.ch
oice
s.fo
r.m
ean
ing.
or.s
tyle
,.an
d.to
.com
preh
end.
mor
e.fu
lly.w
hen.
read
ing.
or.
liste
nin
g.
Wri
te.a
nd.e
dit
.wor
k.so
.tha
t.it
.con
form
s.to
.th
e.g
uid
elin
es.in
.a.s
tyle
.man
ual.(
e.g.
,.ML
A
Han
dbo
ok,.T
ura
bian
’s.M
anua
l for
Wri
ters
).ap
prop
riat
e.fo
r.th
e.d
isci
plin
e.an
d.w
riti
ng.t
ype.
L.9
-10.
3aX
Cho
ose.
wor
ds.a
nd.p
hra
ses.
for.
effe
ct.
L.3
.3a
XX
Cho
ose.
punc
tuat
ion.
for.
effe
ct.
L.4
.3b
XX
Var
y.se
nten
ce.p
atte
rns.
for.
mea
nin
g,.r
eade
r/lis
tene
r.in
tere
st,.a
nd.s
tyle
.‡L
.6.3
aX
X
Mai
ntai
n.co
nsi
sten
cy.in
.sty
le.a
nd.t
one.
L.6
.3b
XX
Cho
ose.
lang
uage
.tha
t.ex
pres
ses.
idea
s.pr
ecis
ely.
and.
conc
isel
y,.r
ecog
niz
ing.
and.
elim
inat
ing.
wor
din
ess.
and.
redu
ndan
cy.
L.7
.3a
XX
Voc
abu
lary
.A
cqu
isit
ion.
and.
Use
Det
erm
ine.
or.c
lari
fy.t
he.m
ean
ing.
of.u
nkn
own.
and.
mu
ltipl
e-m
ean
ing.
wor
ds.a
nd.p
hra
ses.
base
d.on
.gra
des.
9–10
.rea
din
g.an
d.co
nten
t,.c
hoos
ing.
flexi
bly.
from
.a.r
ange
.of.s
trat
egie
s.
Use
.con
text
.(e.g
.,.th
e.ov
eral
l.mea
nin
g.of
.a.
sent
ence
,.par
agra
ph,.o
r.te
xt;.a
.wor
d’s.
posi
tion
.or.
func
tion
.in.a
.sen
tenc
e).a
s.a.
clue
.to.
the.
mea
nin
g.of
.a.w
ord.
or.p
hra
se.
L.9
-10.
4aX
X
36 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eId
enti
fy.a
nd.c
orre
ctly
.use
.pat
tern
s.of
.wor
d.ch
ange
s.th
at.in
dic
ate.
dif
fere
nt.m
ean
ings
.or.
part
s.of
.spe
ech.
(e.g
.,.an
alyz
e,.a
naly
sis,
.ana
lyti
cal;.
advo
cate
,.adv
ocac
y).
L.9
-10.
4b
Con
sult.
gene
ral.a
nd.s
peci
aliz
ed.r
efer
ence
.m
ater
ials
.(e.g
.,.d
icti
onar
ies,
.glo
ssar
ies,
.th
esau
ruse
s),.b
oth.
prin
t.an
d.d
igit
al,.t
o.fi
nd.t
he.
pron
unc
iati
on.o
f.a.w
ord.
or.d
eter
min
e.or
.cla
rify
.it
s.pr
ecis
e.m
ean
ing,
.its.
part
.of.s
peec
h,.o
r.it
s.et
ymol
ogy.
L.9
-10.
4c
Ver
ify.
the.
prel
imin
ary.
dete
rmin
atio
n.of
.the
.m
ean
ing.
of.a
.wor
d.or
.ph
rase
.(e.g
.,.by
.che
ckin
g.th
e.in
ferr
ed.m
ean
ing.
in.c
onte
xt.o
r.in
.a.
dic
tion
ary)
.
L.9
-10.
4dX
X
Dem
onst
rate
.und
erst
and
ing.
of.fi
gu
rati
ve.
lang
uage
,.wor
d.re
lati
onsh
ips,
.and
.nua
nces
.in.
wor
d.m
ean
ings
.
Inte
rpre
t.fig
ure
s.of
.spe
ech.
(e.g
.,.eu
phem
ism
,.ox
ymor
on).i
n.co
ntex
t.an
d.an
alyz
e.th
eir.
role
.in.
the.
text
.
L.9
-10.
5aX
X
Ana
lyze
.nua
nces
.in.t
he.m
ean
ing.
of.w
ord
s.w
ith.
sim
ilar.
deno
tati
ons.
L.9
-10.
5bX
X
Acq
uir
e.an
d.us
e.ac
cura
tely
.gen
eral
.aca
dem
ic.
and.
dom
ain-
spec
ific
.wor
ds.a
nd.p
hra
ses,
.su
ffic
ient
.for.
read
ing,
.wri
ting
,.spe
akin
g,.a
nd.
liste
nin
g.at
.the
.col
lege
.and
.car
eer.
read
ines
s.le
vel;.
dem
onst
rate
.inde
pend
ence
.in.g
athe
ring
.vo
cabu
lary
.kno
wle
dge.
whe
n.co
nsi
deri
ng.a
.w
ord.
or.p
hra
se.im
port
ant.
to.c
ompr
ehen
sion
.or.
expr
essi
on.
L.9
-10.
6X
X
11-1
2L
ang
uage
Con
vent
ion
s.of
.Sta
ndar
d.E
nglis
h
Dem
onst
rate
.com
man
d.of
.the
.con
vent
ion
s.of
.st
anda
rd.E
nglis
h.g
ram
mar
.and
.usa
ge.w
hen.
wri
ting
.or.
spea
king
.
App
ly.t
he.u
nder
stan
din
g.th
at.u
sage
.is.a
.mat
ter.
of.c
onve
ntio
n,.c
an.c
hang
e.ov
er.t
ime,
.and
.is.
som
etim
es.c
onte
sted
.
L.1
1-12
.1a
X
Res
olve
.issu
es.o
f.com
plex
.or.
cont
este
d.us
age,
.co
nsu
lting
.ref
eren
ces.
(e.g
.,.M
erri
am-W
ebst
er’s
D
icti
onar
y of
Eng
lish
Usa
ge, G
arne
r’s M
oder
n A
mer
ican
Usa
ge).a
s.ne
eded
.
L.1
1-12
.1b
En
sure
.su
bjec
t-ve
rb.a
nd.p
rono
un-
ante
cede
nt.
agre
emen
t.L
.3.1
fX
X
Pro
duce
.com
plet
e.se
nten
ces,
.rec
ogn
izin
g.an
d.co
rrec
ting
.inap
prop
riat
e.fr
agm
ents
.and
.ru
n-on
s.L
.4.1
fX
X
37College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eC
orre
ctly
.use
.freq
uent
ly.c
onfu
sed.
wor
ds.
(e.g
.,.to
/to
o/tw
o; t
here
/the
ir).
L.4
.1g
XX
Rec
ogn
ize.
and.
corr
ect.
inap
prop
riat
e.sh
ifts
.in.
verb
.ten
se.
L.5
.1d
XX
Rec
ogn
ize.
and.
corr
ect.
inap
prop
riat
e.sh
ifts
.in.
pron
oun.
num
ber.
and.
pers
on.
L.6
.1c
XX
Rec
ogn
ize.
and.
corr
ect.
vag
ue.p
rono
un
s.(i.
e.,.o
nes.
wit
h.u
ncle
ar.o
r.am
big
uous
.ant
eced
ents
).L
.6.1
dX
X
Rec
ogn
ize.
vari
atio
ns.
from
.sta
ndar
d.E
nglis
h.in
.th
eir.
own.
and.
othe
rs’.w
riti
ng.a
nd.s
peak
ing,
.and
.id
enti
fy.a
nd.u
se.s
trat
egie
s.to
.impr
ove.
expr
essi
on.
in.c
onve
ntio
nal.l
ang
uage
.
L.6
.1e
XX
Plac
e.ph
rase
s.an
d.cl
ause
s.w
ith
in.a
.sen
tenc
e,.
reco
gn
izin
g.an
d.co
rrec
ting
.mis
plac
ed.a
nd.
dang
ling.
mod
ifie
rs.
L.7
.1c
XX
Rec
ogn
ize.
and.
corr
ect.
inap
prop
riat
e.sh
ifts
.in.
verb
.voi
ce.a
nd.m
ood.
L.8
.1d
XX
Use
.par
alle
l.str
uctu
re.
L.9
-10.
1aX
X
Dem
onst
rate
.com
man
d.of
.the
.con
vent
ion
s.of
.st
anda
rd.E
nglis
h.ca
pita
lizat
ion,
.pu
nctu
atio
n,.a
nd.
spel
ling.
whe
n.w
riti
ng.
Obs
erve
.hyp
hena
tion
.con
vent
ion
s.L
.11-
12.2
aX
X
Spel
l.cor
rect
ly.
L.1
1-12
.2b
XX
Use
.pu
nctu
atio
n.(c
omm
as,.p
aren
thes
es,.d
ashe
s).
to.s
et.o
ff.no
nre
stri
ctiv
e/pa
rent
heti
cal.e
lem
ents
.L
.6.2
aX
X
Kno
wle
dge.
of.
Lan
gua
geA
pply
.kno
wle
dge.
of.la
ngua
ge.t
o.u
nder
stan
d.ho
w.la
ngua
ge.fu
ncti
ons.
in.d
iffe
rent
.con
text
s,.
to.m
ake.
effe
ctiv
e.ch
oice
s.fo
r.m
ean
ing.
or.s
tyle
,.an
d.to
.com
preh
end.
mor
e.fu
lly.w
hen.
read
ing.
or.
liste
nin
g.
Var
y.sy
ntax
.for.
effe
ct,.c
onsu
lting
.ref
eren
ces.
(e.g
.,.Tu
fte’
s.A
rtfu
l Sen
tenc
es).f
or.g
uid
ance
.as.
need
ed;.
appl
y.an
.und
erst
and
ing.
of.s
ynta
x.to
.the
.stu
dy.o
f.co
mpl
ex.t
exts
.whe
n.re
adin
g.
L.1
1-12
.3a
XX
Cho
ose.
wor
ds.a
nd.p
hra
ses.
for.
effe
ct.
L.3
.3a
XX
Cho
ose.
punc
tuat
ion.
for.
effe
ct.
L.4
.3b
XX
Mai
ntai
n.co
nsi
sten
cy.in
.sty
le.a
nd.t
one.
L.6
.3b
XX
Cho
ose.
lang
uage
.tha
t.ex
pres
ses.
idea
s.pr
ecis
ely.
and.
conc
isel
y,.r
ecog
niz
ing.
and.
elim
inat
ing.
wor
din
ess.
and.
redu
ndan
cy.
L.7
.3a
XX
38 College Board Research Reports
Appendixes
Eng
lish.
Lang
uage
.Art
s.Si
de-b
y-Si
de.A
lign
men
t.T
able
:.AP®
—C
omm
on.C
ore.
Stat
e.St
anda
rds
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Str
and
Ban
dS
tan
dar
dS
kil
lC
CS
S
IDA
P
Lit
erat
ure
AP
L
ang
uag
eV
ocab
ula
ry.
Acq
uis
itio
n.an
d.U
se
Det
erm
ine.
or.c
lari
fy.t
he.m
ean
ing.
of.u
nkn
own.
and.
mu
ltipl
e-m
ean
ing.
wor
ds.a
nd.p
hra
ses.
base
d.on
.gra
des
11–1
2 re
adin
g an
d co
nten
t,.ch
oosi
ng.
flexi
bly.
from
.a.r
ange
.of.s
trat
egie
s.
Use
.con
text
.(e.g
.,.th
e.ov
eral
l.mea
nin
g.of
.a.
sent
ence
,.par
agra
ph,.o
r.te
xt;.a
.wor
d’s.
posi
tion
.or.
func
tion
.in.a
.sen
tenc
e).a
s.a.
clue
.to.
the.
mea
nin
g.of
.a.w
ord.
or.p
hra
se.
L.1
1-12
.4a
XX
Iden
tify
.and
.cor
rect
ly.u
se.p
atte
rns.
of.w
ord.
chan
ges.
that
.ind
icat
e.d
iffe
rent
.mea
nin
gs.o
r.pa
rts.
of.s
peec
h.(e
.g.,.
conc
eive
, con
cept
ion,
co
ncei
vabl
e).
L.1
1-12
.4b
Con
sult.
gene
ral.a
nd.s
peci
aliz
ed.r
efer
ence
.m
ater
ials
.(e.g
.,.d
icti
onar
ies,
.glo
ssar
ies,
.th
esau
ruse
s),.b
oth.
prin
t.an
d.d
igit
al,.t
o.fi
nd.t
he.
pron
unc
iati
on.o
f.a.w
ord.
or.d
eter
min
e.or
.cla
rify
.it
s.pr
ecis
e.m
ean
ing,
.its.
part
.of.s
peec
h,.it
s.et
ymol
ogy,
.or.
its.
stan
dard
.usa
ge.
L.1
1-12
.4c
Ver
ify.
the.
prel
imin
ary.
dete
rmin
atio
n.of
.the
.m
ean
ing.
of.a
.wor
d.or
.ph
rase
.(e.g
.,.by
.che
ckin
g.th
e.in
ferr
ed.m
ean
ing.
in.c
onte
xt.o
r.in
.a.
dic
tion
ary)
.
L.1
1-12
.4d
XX
Dem
onst
rate
.und
erst
and
ing.
of.fi
gu
rati
ve.
lang
uage
,.wor
d.re
lati
onsh
ips,
.and
.nua
nces
.in.
wor
d.m
ean
ings
.
Inte
rpre
t.fig
ure
s.of
.spe
ech.
(e.g
.,.hy
perb
ole,
.pa
rado
x).in
.con
text
.and
.ana
lyze
.the
ir.r
ole.
in.t
he.
text
.
L.1
1-12
.5a
XX
Ana
lyze
.nua
nces
.in.t
he.m
ean
ing.
of.w
ord
s.w
ith.
sim
ilar.
deno
tati
ons.
L.1
1-12
.5b
XX
Acq
uir
e.an
d.us
e.ac
cura
tely
.gen
eral
.aca
dem
ic.
and.
dom
ain-
spec
ific
.wor
ds.a
nd.p
hra
ses,
.su
ffic
ient
.for.
read
ing,
.wri
ting
,.spe
akin
g,.a
nd.
liste
nin
g.at
.the
.col
lege
.and
.car
eer.
read
ines
s.le
vel;.
dem
onst
rate
.inde
pend
ence
.in.g
athe
ring
.vo
cabu
lary
.kno
wle
dge.
whe
n.co
nsi
deri
ng.a
.w
ord.
or.p
hra
se.im
port
ant.
to.c
ompr
ehen
sion
.or.
expr
essi
on.
L.1
1-12
.6X
X
41College Board Research Reports
Appendixes
Appendix.BAP Calculus, AP Statistics, and AP Computer Science A — Common Core State Standards Side-by-Side Alignment Table
43College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Und
erst
and.
rati
o.co
ncep
ts.a
nd.u
se.
rati
o.re
ason
ing.
to.
solv
e.pr
oble
ms.
Und
erst
and.
the.
conc
ept.
of.a
.rat
io.a
nd.
use.
rati
o.la
ngua
ge.t
o.de
scri
be.a
.rat
io.
rela
tion
ship
.bet
wee
n.tw
o.q
uant
itie
s.
6.R
P.1
XX
XX
6R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Und
erst
and.
rati
o.co
ncep
ts.a
nd.u
se.
rati
o.re
ason
ing.
to.
solv
e.pr
oble
ms.
Und
erst
and.
the.
conc
ept.
of.a
.un
it.r
ate.
a/b.
asso
ciat
ed.w
ith.
a.ra
tio.
a:b.
wit
h b
≠.0,
.and
.use
.rat
e.la
ngua
ge.in
.the
.co
ntex
t.of
.a.r
atio
.rel
atio
nsh
ip.
6.R
P.2
XX
XX
6R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Und
erst
and.
rati
o.co
ncep
ts.a
nd.u
se.
rati
o.re
ason
ing.
to.
solv
e.pr
oble
ms.
Use
.rat
io.a
nd.r
ate.
reas
onin
g.to
.sol
ve.
real
-wor
ld.a
nd.m
athe
mat
ical
.pro
blem
s,.
e.g.
,.by.
reas
onin
g.ab
out.
tabl
es.o
f.eq
uiv
alen
t.ra
tios
,.tap
e.d
iag
ram
s,.
dou
ble.
num
ber.
line.
dia
gra
ms,
.or.
equa
tion
s.
Mak
e.ta
bles
.of.e
qu
ival
ent.
rati
os.r
elat
ing.
qua
ntit
ies.
wit
h.w
hole
-nu
mbe
r.m
easu
rem
ents
,.fin
d.m
issi
ng.v
alue
s.in
.the
.ta
bles
,.and
.plo
t.th
e.pa
irs.
of.v
alue
s.on
.the
.co
ord
inat
e.pl
ane.
.Use
.ta
bles
.to.
com
pare
.rat
ios.
6.R
P.3a
XX
XX
6R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Und
erst
and.
rati
o.co
ncep
ts.a
nd.u
se.
rati
o.re
ason
ing.
to.
solv
e.pr
oble
ms.
Use
.rat
io.a
nd.r
ate.
reas
onin
g.to
.sol
ve.
real
-wor
ld.a
nd.m
athe
mat
ical
.pro
blem
s,.
e.g.
,.by.
reas
onin
g.ab
out.
tabl
es.o
f.eq
uiv
alen
t.ra
tios
,.tap
e.d
iag
ram
s,.
dou
ble.
num
ber.
line.
dia
gra
ms,
.or.
equa
tion
s.
Solv
e.u
nit
.rat
e.pr
oble
ms.
incl
udin
g.th
ose.
invo
lvin
g.u
nit
.pri
cing
.and
.con
stan
t.sp
eed.
6.R
P.3b
XX
XX
6R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Und
erst
and.
rati
o.co
ncep
ts.a
nd.u
se.
rati
o.re
ason
ing.
to.
solv
e.pr
oble
ms.
Use
.rat
io.a
nd.r
ate.
reas
onin
g.to
.sol
ve.
real
-wor
ld.a
nd.m
athe
mat
ical
.pro
blem
s,.
e.g.
,.by.
reas
onin
g.ab
out.
tabl
es.o
f.eq
uiv
alen
t.ra
tios
,.tap
e.d
iag
ram
s,.
dou
ble.
num
ber.
line.
dia
gra
ms,
.or.
equa
tion
s.
Fin
d.a.
perc
ent.
of.a
.q
uant
ity.
as.a
.rat
e.pe
r.10
0.(e
.g.,.
30%
.of.a
.qua
ntit
y.m
ean
s.30
/100
.tim
es.t
he.
qua
ntit
y);.s
olve
.pro
blem
s.in
volv
ing.
find
ing.
the.
who
le,.g
iven
.a.p
art.
and.
the.
perc
ent.
6.R
P.3c
XX
XX
6R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Und
erst
and.
rati
o.co
ncep
ts.a
nd.u
se.
rati
o.re
ason
ing.
to.
solv
e.pr
oble
ms.
Use
.rat
io.a
nd.r
ate.
reas
onin
g.to
.sol
ve.
real
-wor
ld.a
nd.m
athe
mat
ical
.pro
blem
s,.
e.g.
,.by.
reas
onin
g.ab
out.
tabl
es.o
f.eq
uiv
alen
t.ra
tios
,.tap
e.d
iag
ram
s,.
dou
ble.
num
ber.
line.
dia
gra
ms,
.or.
equa
tion
s.
Use
.rat
io.r
easo
nin
g.to
.co
nver
t.m
easu
rem
ent.
un
its;
.man
ipu
late
..an
d.tr
ansf
orm
.un
its.
appr
opri
atel
y.w
hen.
mu
ltipl
ying
.or.
div
idin
g.q
uant
itie
s.
6.R
P.3d
XX
XX
44 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
to.
div
ide.
frac
tion
s.by
.frac
tion
s.
Inte
rpre
t.an
d.co
mpu
te.q
uoti
ents
.of.
frac
tion
s,.a
nd.s
olve
.wor
d.pr
oble
ms.
invo
lvin
g.d
ivis
ion.
of.fr
acti
ons.
by.
frac
tion
s,.e
.g.,.
by.u
sing
.vis
ual.f
ract
ion.
mod
els.
and.
equa
tion
s.to
.rep
rese
nt.t
he.
prob
lem
.
6.N
S.1
XX
XX
6T
he.N
um
ber.
Syst
emC
ompu
te.f
luen
tly.
wit
h.m
ulti
-dig
it.
num
bers
.and
.find
.co
mm
on.fa
ctor
s.an
d.m
ulti
ples
.
Flu
entl
y.d
ivid
e.m
ulti
-dig
it.n
um
bers
.us
ing.
the.
stan
dard
.alg
orit
hm
.6.
NS.
2X
XX
X
6T
he.N
um
ber.
Syst
emC
ompu
te.f
luen
tly.
wit
h.m
ulti
-dig
it.
num
bers
.and
.find
.co
mm
on.fa
ctor
s.an
d.m
ulti
ples
.
Flu
entl
y.ad
d,.s
ubt
ract
,.mu
ltipl
y,.a
nd.
div
ide.
mu
lti-d
igit
.dec
imal
s.us
ing.
the.
stan
dard
.alg
orit
hm
.for.
each
.ope
rati
on.
6.N
S.3
XX
XX
6T
he.N
um
ber.
Syst
emC
ompu
te.f
luen
tly.
wit
h.m
ulti
-dig
it.
num
bers
.and
.find
.co
mm
on.fa
ctor
s.an
d.m
ulti
ples
.
Fin
d.th
e.g
reat
est.
com
mon
.fact
or.o
f.tw
o.w
hole
.nu
mbe
rs.le
ss.t
han.
or.e
qua
l.to.
100.
and.
the.
leas
t.co
mm
on.m
ulti
ple.
of.
two.
who
le.n
um
bers
.less
.tha
n.or
.eq
ual.
to.1
2..U
se.t
he.d
istr
ibut
ive.
prop
erty
.to.
expr
ess.
a.su
m.o
f.tw
o.w
hole
.nu
mbe
rs.
1–10
0.w
ith.
a.co
mm
on.fa
ctor
.as.
a.m
ulti
ple.
of.a
.su
m.o
f.tw
o.w
hole
.nu
mbe
rs.
wit
h.no
.com
mon
.fact
or.
6.N
S.4
XX
X
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
that
.pos
itiv
e.an
d.ne
gati
ve.
num
bers
.are
.use
d.to
geth
er.t
o.de
scri
be.
qua
ntit
ies.
havi
ng.o
ppos
ite.
dir
ecti
ons.
or.
valu
es.(e
.g.,.
tem
pera
ture
.abo
ve/b
elow
.ze
ro,.e
leva
tion
.abo
ve/b
elow
.sea
.leve
l,.cr
edit
s/de
bits
,.pos
itiv
e/ne
gati
ve.e
lect
ric.
char
ge);.
use.
posi
tive
.and
.neg
ativ
e.nu
mbe
rs.t
o.re
pres
ent.
qua
ntit
ies.
in.r
eal-
wor
ld.c
onte
xts,
.exp
lain
ing.
the.
mea
nin
g.of
.0.in
.eac
h.si
tuat
ion.
6.N
S.5
XX
XX
45College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
a.ra
tion
al.n
um
ber.
as.a
.po
int.
on.t
he.n
um
ber.
line.
.Ext
end.
num
ber.
line.
dia
gra
ms.
and.
coor
din
ate.
axes
.fam
iliar
.from
.pre
viou
s.g
rade
s.to
.rep
rese
nt.p
oint
s.on
.the
.line
.and
.in
.the
.pla
ne.w
ith.
nega
tive
.nu
mbe
r.co
ord
inat
es.
Rec
ogn
ize.
oppo
site
.sig
ns.
of.n
um
bers
.as.
ind
icat
ing.
loca
tion
s.on
.opp
osit
e.si
des.
of.0
.on.
the.
num
ber.
line;
.rec
ogn
ize.
that
.the
.op
posi
te.o
f.the
.opp
osit
e.of
.a.
num
ber.
is.t
he.n
um
ber.
itse
lf,.e
.g.,.
–(–3
).=.3
,.and
.th
at.0
.is.it
s.ow
n.op
posi
te.
6.N
S.6a
XX
XX
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
a.ra
tion
al.n
um
ber.
as.a
.po
int.
on.t
he.n
um
ber.
line.
.Ext
end.
num
ber.
line.
dia
gra
ms.
and.
coor
din
ate.
axes
.fam
iliar
.from
.pre
viou
s.g
rade
s.to
.rep
rese
nt.p
oint
s.on
.the
.line
.and
.in
.the
.pla
ne.w
ith.
nega
tive
.nu
mbe
r.co
ord
inat
es.
Und
erst
and.
sig
ns.
of.
num
bers
.in.o
rder
ed.
pair
s.as
.ind
icat
ing.
loca
tion
s.in
.qua
dra
nts.
of.t
he.c
oord
inat
e.pl
ane;
.re
cog
niz
e.th
at.w
hen.
two.
orde
red.
pair
s.d
iffe
r.on
ly.
by.s
ign
s,.t
he.lo
cati
ons.
of.
the.
poin
ts.a
re.r
elat
ed.b
y.re
flect
ion
s.ac
ross
.one
.or.
both
.axe
s.
6.N
S.6b
XX
XX
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
a.ra
tion
al.n
um
ber.
as.a
.po
int.
on.t
he.n
um
ber.
line.
.Ext
end.
num
ber.
line.
dia
gra
ms.
and.
coor
din
ate.
axes
.fam
iliar
.from
.pre
viou
s.g
rade
s.to
.rep
rese
nt.p
oint
s.on
.the
.line
.and
.in
.the
.pla
ne.w
ith.
nega
tive
.nu
mbe
r.co
ord
inat
es.
Fin
d.an
d.po
siti
on.in
tege
rs.
and.
othe
r.ra
tion
al.
num
bers
.on.
a.ho
rizo
ntal
.or
.ver
tica
l.nu
mbe
r.lin
e.d
iag
ram
;.fin
d.an
d.po
siti
on.
pair
s.of
.inte
gers
.and
.oth
er.
rati
onal
.nu
mbe
rs.o
n.a.
coor
din
ate.
plan
e.
6.N
S.6c
XX
XX
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
orde
ring
.and
.abs
olut
e.va
lue.
of.r
atio
nal.n
um
bers
.In
terp
ret.
stat
emen
ts.o
f.in
equa
lity.
as.s
tate
men
ts.
abou
t.th
e.re
lati
ve.p
osit
ion.
of.t
wo.
num
bers
.on.
a.nu
mbe
r.lin
e.d
iag
ram
.
6.N
S.7a
XX
.X
46 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
orde
ring
.and
.abs
olut
e.va
lue.
of.r
atio
nal.n
um
bers
.W
rite
,.int
erpr
et,.a
nd.
expl
ain.
stat
emen
ts.o
f.or
der.
for.
rati
onal
.nu
mbe
rs.
in.r
eal-w
orld
.con
text
s.
6.N
S.7b
XX
XX
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
orde
ring
.and
.abs
olut
e.va
lue.
of.r
atio
nal.n
um
bers
.U
nder
stan
d.th
e.ab
solu
te.
valu
e.of
.a.r
atio
nal.n
um
ber.
as.it
s.d
ista
nce.
from
.0.
on.t
he.n
um
ber.
line;
.in
terp
ret.
abso
lute
.val
ue.a
s.m
agn
itud
e.fo
r.a.
posi
tive
.or
.neg
ativ
e.q
uant
ity.
in.a
.re
al-w
orld
.sit
uati
on.
6.N
S.7c
XX
XX
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Und
erst
and.
orde
ring
.and
.abs
olut
e.va
lue.
of.r
atio
nal.n
um
bers
.D
isti
ngu
ish.
com
pari
son
s.of
.abs
olut
e.va
lue.
from
.st
atem
ents
.abo
ut.o
rder
.
6.N
S.7d
XX
.
6T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.n
um
bers
.to.
the.
syst
em.o
f.rat
iona
l.nu
mbe
rs.
Solv
e.re
al-w
orld
.and
.mat
hem
atic
al.
prob
lem
s.by
.gra
phin
g.po
ints
.in.a
ll.fo
ur.
qua
dra
nts.
of.t
he.c
oord
inat
e.pl
ane.
.In
clud
e.us
e.of
.coo
rdin
ates
.and
.abs
olut
e.va
lue.
to.fi
nd.d
ista
nces
.bet
wee
n.po
ints
.w
ith.
the.
sam
e.fi
rst.
coor
din
ate.
or.t
he.
sam
e.se
cond
.coo
rdin
ate.
6.N
S.8
XX
XX
6E
xpre
ssio
ns.
and.
Eq
uati
ons
App
ly.a
nd.
exte
nd.p
revi
ous.
und
erst
and
ings
.of
.ari
thm
etic
.to
.alg
ebra
ic.
expr
essi
ons.
Wri
te.a
nd.e
valu
ate.
num
eric
al.
expr
essi
ons.
invo
lvin
g.w
hole
-nu
mbe
r.ex
pone
nts.
6.E
E.1
XX
XX
6E
xpre
ssio
ns.
and.
Eq
uati
ons
App
ly.a
nd.
exte
nd.p
revi
ous.
und
erst
and
ings
.of
.ari
thm
etic
.to
.alg
ebra
ic.
expr
essi
ons.
Wri
te,.r
ead,
.and
.eva
luat
e.ex
pres
sion
s.in
.w
hic
h.le
tter
s.st
and.
for.
num
bers
.W
rite
.exp
ress
ion
s.th
at.
reco
rd.o
pera
tion
s.w
ith.
num
bers
.and
.wit
h.le
tter
s.st
and
ing.
for.
num
bers
.
6.E
E.2
aX
XX
X
47College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6E
xpre
ssio
ns.
and.
Eq
uati
ons
App
ly.a
nd.
exte
nd.p
revi
ous.
und
erst
and
ings
.of
.ari
thm
etic
.to
.alg
ebra
ic.
expr
essi
ons.
Wri
te,.r
ead,
.and
.eva
luat
e.ex
pres
sion
s.in
.w
hic
h.le
tter
s.st
and.
for.
num
bers
.Id
enti
fy.p
arts
.of.a
n.ex
pres
sion
.usi
ng.
mat
hem
atic
al.t
erm
s.(s
um
,.te
rm,.p
rodu
ct,.f
acto
r,.q
uoti
ent,
.coe
ffic
ient
);.vi
ew.o
ne.o
r.m
ore.
part
s.of
.an
.exp
ress
ion.
as.a
.sin
gle.
enti
ty.
6.E
E.2
bX
XX
X
6E
xpre
ssio
ns.
and.
Eq
uati
ons
App
ly.a
nd.
exte
nd.p
revi
ous.
und
erst
and
ings
.of
.ari
thm
etic
.to
.alg
ebra
ic.
expr
essi
ons.
Wri
te,.r
ead,
.and
.eva
luat
e.ex
pres
sion
s.in
.w
hic
h.le
tter
s.st
and.
for.
num
bers
.E
valu
ate.
expr
essi
ons.
at.s
peci
fic.v
alue
s.of
.th
eir.
vari
able
s..In
clud
e.ex
pres
sion
s.th
at.a
rise
.fr
om.fo
rmu
las.
used
.in.re
al-
wor
ld.p
robl
ems.
.Per
form
.ar
ith
met
ic.o
pera
tion
s,.
incl
udin
g.th
ose.
invo
lvin
g.w
hole
-nu
mbe
r.ex
pone
nts,
.in
.the
.con
vent
iona
l.or
der.
whe
n.th
ere.
are.
no.
pare
nthe
ses.
to.s
peci
fy.a
.pa
rtic
ula
r.or
der.
(Ord
er.o
f.O
pera
tion
s).
6.E
E.2
cX
XX
X
6E
xpre
ssio
ns.
and.
Eq
uati
ons
App
ly.a
nd.
exte
nd.p
revi
ous.
und
erst
and
ings
.of
.ari
thm
etic
.to
.alg
ebra
ic.
expr
essi
ons.
App
ly.t
he.p
rope
rtie
s.of
.ope
rati
ons.
to.
gene
rate
.eq
uiv
alen
t.ex
pres
sion
s.6.
EE
.3X
XX
X
6E
xpre
ssio
ns.
and.
Eq
uati
ons
App
ly.a
nd.
exte
nd.p
revi
ous.
und
erst
and
ings
.of
.ari
thm
etic
.to
.alg
ebra
ic.
expr
essi
ons.
Iden
tify
.whe
n.tw
o.ex
pres
sion
s.ar
e.eq
uiv
alen
t.(i.
e.,.w
hen.
the.
two.
expr
essi
ons.
nam
e.th
e.sa
me.
num
ber.
rega
rdle
ss.o
f.wh
ich.
valu
e.is
.su
bsti
tute
d.in
to.t
hem
).
6.E
E.4
XX
XX
48 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6E
xpre
ssio
ns.
and.
Eq
uati
ons
Rea
son.
abou
t.an
d.so
lve.
one-
vari
able
.eq
uati
ons.
and.
ineq
ualit
ies.
Und
erst
and.
solv
ing.
an.e
qua
tion
.or.
ineq
ualit
y.as
.a.p
roce
ss.o
f.an
swer
ing.
a.q
uest
ion:
.wh
ich.
valu
es.fr
om.a
.sp
ecif
ied.
set,
.if.a
ny,.m
ake.
the.
equa
tion
.or
.ineq
ualit
y.tr
ue?.
Use
.su
bsti
tuti
on.t
o.de
term
ine.
whe
ther
.a.g
iven
.nu
mbe
r.in
.a.
spec
ifie
d.se
t.m
akes
.an.
equa
tion
.or.
ineq
ualit
y.tr
ue.
6.E
E.5
XX
XX
6E
xpre
ssio
ns.
and.
Eq
uati
ons
Rea
son.
abou
t.an
d.so
lve.
one-
vari
able
.eq
uati
ons.
and.
ineq
ualit
ies.
Use
.var
iabl
es.t
o.re
pres
ent.
num
bers
.an
d.w
rite
.exp
ress
ion
s.w
hen.
solv
ing.
a.re
al-w
orld
.or.
mat
hem
atic
al.p
robl
em;.
und
erst
and.
that
.a.v
aria
ble.
can.
repr
esen
t.an
.un
know
n.nu
mbe
r,.or
,.de
pend
ing.
on.t
he.p
urp
ose.
at.h
and,
.any
.nu
mbe
r.in
.a.s
peci
fied.
set.
6.E
E.6
XX
XX
6E
xpre
ssio
ns.
and.
Eq
uati
ons
Rea
son.
abou
t.an
d.so
lve.
one-
vari
able
.eq
uati
ons.
and.
ineq
ualit
ies.
Solv
e.re
al-w
orld
.and
.mat
hem
atic
al.
prob
lem
s.by
.wri
ting
.and
.sol
ving
.eq
uati
ons.
of.t
he.fo
rm.x
.+.p
.=.q
.and
.px
= q
.for.
case
s.in
.wh
ich.
p,.q
.and
.x.a
re.
all.n
onne
gati
ve.r
atio
nal.n
um
bers
.
6.E
E.7
XX
XX
6E
xpre
ssio
ns.
and.
Eq
uati
ons
Rea
son.
abou
t.an
d.so
lve.
one-
vari
able
.eq
uati
ons.
and.
ineq
ualit
ies.
Wri
te.a
n.in
equa
lity.
of.t
he.fo
rm.
x.>
.c.o
r.x.
<.c
.to.
repr
esen
t.a.
con
stra
int.
or.c
ond
itio
n.in
.a.r
eal-w
orld
.or.
mat
hem
atic
al.p
robl
em..R
ecog
niz
e.th
at.in
equa
litie
s.of
.the
.form
.x.>
.c.o
r.x.
<.c
.hav
e.in
fin
itel
y.m
any.
solu
tion
s;.
repr
esen
t.so
luti
ons.
of.s
uch.
ineq
ualit
ies.
on.n
um
ber.
line.
dia
gra
ms.
6.E
E.8
XX
XX
6E
xpre
ssio
ns.
and.
Eq
uati
ons
Rep
rese
nt.
and.
anal
yze.
qua
ntit
ativ
e.re
lati
onsh
ips.
betw
een.
depe
nden
t.an
d.in
depe
nden
t.va
riab
les.
Use
.var
iabl
es.t
o.re
pres
ent.
two.
qua
ntit
ies.
in.a
.rea
l-wor
ld.p
robl
em.t
hat.
chan
ge.in
.rel
atio
nsh
ip.t
o.on
e.an
othe
r;.
wri
te.a
n.eq
uati
on.t
o.ex
pres
s.on
e.q
uant
ity,
.tho
ught
.of.a
s.th
e.de
pend
ent.
vari
able
,.in.
term
s.of
.the
.oth
er.q
uant
ity,
.th
ough
t.of
.as.
the.
inde
pend
ent.
vari
able
..A
naly
ze.t
he.r
elat
ion
ship
.bet
wee
n.th
e.de
pend
ent.
and.
inde
pend
ent.
vari
able
s.us
ing.
gra
phs.
and.
tabl
es,.a
nd.r
elat
e.th
ese.
to.t
he.e
qua
tion
.
6.E
E.9
XX
XX
49College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6G
eom
etry
Solv
e.re
al-w
orld
.an
d.m
athe
mat
ical
.pr
oble
ms.
invo
lvin
g.ar
ea,.
surf
ace.
area
,.and
.vo
lum
e.
Fin
d.th
e.ar
ea.o
f.rig
ht.t
rian
gles
,.oth
er.
tria
ngle
s,.s
peci
al.q
uad
rila
tera
ls,.a
nd.
poly
gon
s.by
.com
posi
ng.in
to.r
ecta
ngle
s.or
.dec
ompo
sing
.into
.tri
angl
es.a
nd.
othe
r.sh
apes
;.app
ly.t
hese
.tec
hn
ique
s.in
.the
.con
text
.of.s
olvi
ng.r
eal-w
orld
.and
.m
athe
mat
ical
.pro
blem
s.
6.G
.1X
XX
6G
eom
etry
Solv
e.re
al-w
orld
.an
d.m
athe
mat
ical
.pr
oble
ms.
invo
lvin
g.ar
ea,.
surf
ace.
area
,.and
.vo
lum
e.
Fin
d.th
e.vo
lum
e.of
.a.r
ight
.rec
tang
ula
r.pr
ism
.wit
h.fr
acti
onal
.edg
e.le
ngth
s.by
.pac
king
.it.w
ith.
un
it.c
ube
s.of
.the
.ap
prop
riat
e.u
nit
.frac
tion
.edg
e.le
ngth
s,.
and.
show
.tha
t.th
e.vo
lum
e.is
.the
.sam
e.as
.wou
ld.b
e.fo
und
.by.
mu
ltipl
ying
.th
e.ed
ge.le
ngth
s.of
.the
.pri
sm..A
pply
.th
e.fo
rmu
las.
V.=
.l.w
h.a
nd.V
.= b
h.
to.fi
nd.v
olu
mes
.of.r
ight
.rec
tang
ula
r.pr
ism
s.w
ith.
frac
tion
al.e
dge.
leng
ths.
in.
the.
cont
ext.
of.s
olvi
ng.r
eal-w
orld
.and
.m
athe
mat
ical
.pro
blem
s.
6.G
.2X
X
6G
eom
etry
Solv
e.re
al-w
orld
.an
d.m
athe
mat
ical
.pr
oble
ms.
invo
lvin
g.ar
ea,.
surf
ace.
area
,.and
.vo
lum
e.
Dra
w.p
olyg
ons.
in.t
he.c
oord
inat
e.pl
ane.
giv
en.c
oord
inat
es.fo
r.th
e.ve
rtic
es;.
use.
coor
din
ates
.to.
find
.the
.leng
th.o
f.a.
side
.join
ing.
poin
ts.w
ith.
the.
sam
e.fi
rst.
coor
din
ate.
or.t
he.s
ame.
seco
nd.
coor
din
ate.
.App
ly.t
hese
.tec
hn
ique
s.in
.th
e.co
ntex
t.of
.sol
ving
.rea
l-wor
ld.a
nd.
mat
hem
atic
al.p
robl
ems.
6.G
.3X
X
6G
eom
etry
Solv
e.re
al-
wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.in
volv
ing.
area
,.su
rfac
e.ar
ea,.a
nd.
volu
me.
Rep
rese
nt.t
hre
e-d
imen
sion
al.fi
gu
res.
usin
g.ne
ts.m
ade.
up.o
f.rec
tang
les.
and.
tria
ngle
s,.a
nd.u
se.t
he.n
ets.
to.fi
nd.t
he.
surf
ace.
area
.of.t
hese
.fig
ure
s..A
pply
.th
ese.
tech
niq
ues.
in.t
he.c
onte
xt.o
f.so
lvin
g.re
al-w
orld
.and
.mat
hem
atic
al.
prob
lem
s.
6.G
.4
6St
atis
tics
.and
.P
roba
bilit
yD
evel
op.
und
erst
and
ing.
of.s
tati
stic
al.
vari
abili
ty.
Rec
ogn
ize.
a.st
atis
tica
l.que
stio
n.as
.one
.th
at.a
ntic
ipat
es.v
aria
bilit
y.in
.the
.dat
a.re
late
d.to
.the
.que
stio
n.an
d.ac
cou
nts.
for.
it.in
.the
.an
swer
s.
6.SP
.1X
50 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
6St
atis
tics
.and
.P
roba
bilit
yD
evel
op.
und
erst
and
ing.
of.s
tati
stic
al.
vari
abili
ty.
Und
erst
and.
that
.a.s
et.o
f.dat
a.co
llect
ed.
to.a
nsw
er.a
.sta
tist
ical
.que
stio
n.ha
s.a.
dis
trib
utio
n.w
hic
h.ca
n.be
.des
crib
ed.b
y.it
s.ce
nter
,.spr
ead,
.and
.ove
rall.
shap
e.
6.SP
.2X
6St
atis
tics
.and
.P
roba
bilit
yD
evel
op.
und
erst
and
ing.
of.s
tati
stic
al.
vari
abili
ty.
Rec
ogn
ize.
that
.a.m
easu
re.o
f.cen
ter.
for.
a.nu
mer
ical
.dat
a.se
t.su
mm
ariz
es.a
ll.of
.it
s.va
lues
.wit
h.a.
sing
le.n
um
ber,.
wh
ile.a
.m
easu
re.o
f.var
iati
on.d
escr
ibes
.how
.its.
valu
es.v
ary.
wit
h.a.
sing
le.n
um
ber.
6.SP
.3X
6St
atis
tics
.and
.P
roba
bilit
ySu
mm
ariz
e.an
d.de
scri
be.
dis
trib
utio
ns.
Dis
play
.nu
mer
ical
.dat
a.in
.plo
ts.o
n.a.
num
ber.
line,
.incl
udin
g.do
t.pl
ots,
.h
isto
gra
ms,
.and
.box
.plo
ts.
6.SP
.4X
6St
atis
tics
.and
.P
roba
bilit
ySu
mm
ariz
e.an
d.de
scri
be.
dis
trib
utio
ns.
Sum
mar
ize.
num
eric
al.d
ata.
sets
.in.
rela
tion
.to.
thei
r.co
ntex
t,.s
uch.
as.b
y:R
epor
ting
.the
.nu
mbe
r.of
.ob
serv
atio
ns.
6.SP
.5a
X
6St
atis
tics
.and
.P
roba
bilit
ySu
mm
ariz
e.an
d.de
scri
be.
dis
trib
utio
ns.
Sum
mar
ize.
num
eric
al.d
ata.
sets
.in.
rela
tion
.to.
thei
r.co
ntex
t,.s
uch.
as.b
y:D
escr
ibin
g.th
e.na
ture
.of
.the
.att
ribu
te.u
nder
.in
vest
igat
ion,
.incl
udin
g.ho
w.it
.was
.mea
sure
d.an
d.it
s.u
nit
s.of
.mea
sure
men
t.
6.SP
.5b
X
6St
atis
tics
.and
.P
roba
bilit
ySu
mm
ariz
e.an
d.de
scri
be.
dis
trib
utio
ns.
Sum
mar
ize.
num
eric
al.d
ata.
sets
.in.
rela
tion
.to.
thei
r.co
ntex
t,.s
uch.
as.b
y:G
ivin
g.q
uant
itat
ive.
mea
sure
s.of
.cen
ter.
(med
ian.
and
/or.
mea
n).a
nd.
vari
abili
ty.(i
nter
qua
rtile
.ra
nge.
and
/or.
mea
n.ab
solu
te.d
evia
tion
),.as
.w
ell.a
s.de
scri
bing
.any
.ov
eral
l.pat
tern
.and
.any
.st
riki
ng.d
evia
tion
s.fr
om.
the.
over
all.p
atte
rn.w
ith.
refe
renc
e.to
.the
.con
text
.in
.wh
ich.
the.
data
.wer
e.ga
ther
ed.
6.SP
.5c
X
6St
atis
tics
.and
.P
roba
bilit
ySu
mm
ariz
e.an
d.de
scri
be.
dis
trib
utio
ns.
Sum
mar
ize.
num
eric
al.d
ata.
sets
.in.
rela
tion
.to.
thei
r.co
ntex
t,.s
uch.
as.b
y:R
elat
ing.
the.
choi
ce.o
f.m
easu
res.
of.c
ente
r.an
d.va
riab
ility
.to.
the.
shap
e.of
.th
e.da
ta.d
istr
ibut
ion.
and.
the.
cont
ext.
in.w
hic
h.th
e.da
ta.w
ere.
gath
ered
.
6.SP
.5d
X
51College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Ana
lyze
.pr
opor
tion
al.
rela
tion
ship
s.an
d.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.
Com
pute
.un
it.r
ates
.ass
ocia
ted.
wit
h.ra
tios
.of.f
ract
ion
s,.in
clud
ing.
rati
os.o
f.le
ngth
s,.a
reas
.and
.oth
er.q
uant
itie
s.m
easu
red.
in.li
ke.o
r.d
iffe
rent
.un
its.
7.R
P.1
XX
7R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Ana
lyze
.pr
opor
tion
al.
rela
tion
ship
s.an
d.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.
Rec
ogn
ize.
and.
repr
esen
t.pr
opor
tion
al.
rela
tion
ship
s.be
twee
n.q
uant
itie
s.D
ecid
e.w
heth
er.t
wo.
qua
ntit
ies.
are.
in.a
.pr
opor
tion
al.r
elat
ion
ship
,.e.
g.,.b
y.te
stin
g.fo
r.eq
uiv
alen
t.ra
tios
.in.a
.ta
ble.
or.g
raph
ing.
on.a
.co
ord
inat
e.pl
ane.
and.
obse
rvin
g.w
heth
er.t
he.
gra
ph.is
.a.s
trai
ght.
line.
thro
ugh.
the.
orig
in.
7.R
P.2a
XX
X
7R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Ana
lyze
.pr
opor
tion
al.
rela
tion
ship
s.an
d.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.
Rec
ogn
ize.
and.
repr
esen
t.pr
opor
tion
al.
rela
tion
ship
s.be
twee
n.q
uant
itie
s.Id
enti
fy.t
he.c
onst
ant.
of.p
ropo
rtio
nalit
y.(u
nit
.ra
te).i
n.ta
bles
,.gra
phs,
.eq
uati
ons,
.dia
gra
ms,
.and
.ve
rbal
.des
crip
tion
s.of
.pr
opor
tion
al.r
elat
ion
ship
s.
7.R
P.2b
XX
X
7R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Ana
lyze
.pr
opor
tion
al.
rela
tion
ship
s.an
d.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.
Rec
ogn
ize.
and.
repr
esen
t.pr
opor
tion
al.
rela
tion
ship
s.be
twee
n.q
uant
itie
s.R
epre
sent
.pro
port
iona
l.re
lati
onsh
ips.
by.
equa
tion
s.
7.R
P.2c
XX
X
7R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Ana
lyze
.pr
opor
tion
al.
rela
tion
ship
s.an
d.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.
Rec
ogn
ize.
and.
repr
esen
t.pr
opor
tion
al.
rela
tion
ship
s.be
twee
n.q
uant
itie
s.E
xpla
in.w
hat.
a.po
int.
(x, y
).on.
the.
gra
ph.o
f.a.
prop
orti
onal
.rel
atio
nsh
ip.
mea
ns.
in.t
erm
s.of
.the
.si
tuat
ion,
.wit
h.sp
ecia
l.at
tent
ion.
to.t
he.p
oint
s.(0
, 0).a
nd.(1
, r).w
here
.r.is
.th
e.u
nit
.rat
e.
7.R
P.2d
XX
X
52 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7R
atio
s.an
d.P
ropo
rtio
nal.
Rel
atio
nsh
ips
Ana
lyze
.pr
opor
tion
al.
rela
tion
ship
s.an
d.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.
Use
.pro
port
iona
l.rel
atio
nsh
ips.
to.s
olve
.m
ulti
step
.rat
io.a
nd.p
erce
nt.p
robl
ems.
7.R
P.3
XX
X
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.o
pera
tion
s.w
ith.
frac
tion
s.to
.add
,.su
btra
ct,.
mu
ltipl
y,.a
nd.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.a
ddit
ion.
and.
subt
ract
ion.
to.a
dd.a
nd.s
ubt
ract
.rat
iona
l.nu
mbe
rs;.r
epre
sent
.add
itio
n.an
d.su
btra
ctio
n.on
.a.h
oriz
onta
l.or.
vert
ical
.nu
mbe
r.lin
e.d
iag
ram
.
Des
crib
e.si
tuat
ion
s.in
.w
hic
h.op
posi
te.q
uant
itie
s.co
mbi
ne.t
o.m
ake.
0.
7.N
S.1a
XX
.X
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.o
pera
tion
s.w
ith.
frac
tion
s.to
.add
,.su
btra
ct,.
mu
ltipl
y,.a
nd.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.a
ddit
ion.
and.
subt
ract
ion.
to.a
dd.a
nd.s
ubt
ract
.rat
iona
l.nu
mbe
rs;.r
epre
sent
.add
itio
n.an
d.su
btra
ctio
n.on
.a.h
oriz
onta
l.or.
vert
ical
.nu
mbe
r.lin
e.d
iag
ram
.
Und
erst
and.
p.+
.q.a
s.th
e.nu
mbe
r.lo
cate
d.a.
dis
tanc
e.|q
|.fro
m.p
,.in.
the.
posi
tive
.or.
nega
tive
.d
irec
tion
.dep
end
ing.
on.
whe
ther
.q.is
.pos
itiv
e.or
.neg
ativ
e..S
how
.th
at.a
.nu
mbe
r.an
d.it
s.op
posi
te.h
ave.
a.su
m.o
f.0.
(are
.add
itiv
e.in
vers
es)..
Inte
rpre
t.su
ms.
of.r
atio
nal.
num
bers
.by.
desc
ribi
ng.
real
-wor
ld.c
onte
xts.
7.N
S.1b
XX
X.X
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.o
pera
tion
s.w
ith.
frac
tion
s.to
.add
,.su
btra
ct,.
mu
ltipl
y,.a
nd.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.a
ddit
ion.
and.
subt
ract
ion.
to.a
dd.a
nd.s
ubt
ract
.rat
iona
l.nu
mbe
rs;.r
epre
sent
.add
itio
n.an
d.su
btra
ctio
n.on
.a.h
oriz
onta
l.or.
vert
ical
.nu
mbe
r.lin
e.d
iag
ram
.
Und
erst
and.
subt
ract
ion.
of.r
atio
nal.n
um
bers
.as.
add
ing.
the.
add
itiv
e.in
vers
e,.p
.–.q
.=.p
.+.(–
q)..
Show
.tha
t.th
e.d
ista
nce.
betw
een.
two.
rati
onal
.nu
mbe
rs.o
n.th
e.nu
mbe
r.lin
e.is
.the
.abs
olut
e.va
lue.
of.t
heir
.dif
fere
nce,
.and
.ap
ply.
this
.pri
ncip
le.in
.re
al-w
orld
.con
text
s.
7.N
S.1c
XX
XX
53College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.o
pera
tion
s.w
ith.
frac
tion
s.to
.add
,.su
btra
ct,.
mu
ltipl
y,.a
nd.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.a
ddit
ion.
and.
subt
ract
ion.
to.a
dd.a
nd.s
ubt
ract
.rat
iona
l.nu
mbe
rs;.r
epre
sent
.add
itio
n.an
d.su
btra
ctio
n.on
.a.h
oriz
onta
l.or.
vert
ical
.nu
mbe
r.lin
e.d
iag
ram
.
App
ly.p
rope
rtie
s.of
.op
erat
ion
s.as
.str
ateg
ies.
to.
add.
and.
subt
ract
.rat
iona
l.nu
mbe
rs.
7.N
S.1d
XX
XX
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
Und
erst
and.
that
.m
ulti
plic
atio
n.is
.ext
ende
d.fr
om.fr
acti
ons.
to.r
atio
nal.
num
bers
.by.
req
uir
ing.
that
.ope
rati
ons.
cont
inue
.to
.sat
isfy
.the
.pro
pert
ies.
of.o
pera
tion
s,.p
arti
cula
rly.
the.
dis
trib
utiv
e.pr
oper
ty,.
lead
ing.
to.p
rodu
cts.
such
.as
.(–1)
(–1)
.=.1
.and
.the
.ru
les.
for.
mu
ltipl
ying
.si
gne
d.nu
mbe
rs..I
nter
pret
.pr
oduc
ts.o
f.rat
iona
l.nu
mbe
rs.b
y.de
scri
bing
.re
al-w
orld
.con
text
s.
7.N
S.2a
XX
XX
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
Und
erst
and.
that
.inte
gers
.ca
n.be
.div
ided
,.pro
vide
d.th
at.t
he.d
ivis
or.is
.not
.ze
ro,.a
nd.e
very
.quo
tien
t.of
.inte
gers
.(wit
h.no
n-ze
ro.d
ivis
or).i
s.a.
rati
onal
.nu
mbe
r..If
.p.a
nd.q
.are
.in
tege
rs,.t
hen.
–(p/
q).=
.(–
p)/q
.=.p
/(–q)
..Int
erpr
et.
quo
tien
ts.o
f.rat
iona
l.nu
mbe
rs.b
y.de
scri
bing
.re
al-w
orld
.con
text
s.
7.N
S.2b
XX
XX
54 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.p
rope
rtie
s.of
.op
erat
ion
s.as
.str
ateg
ies.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
7.N
S.2c
XX
XX
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
App
ly.a
nd.e
xten
d.pr
evio
us.
und
erst
and
ings
.of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
Con
vert
.a.r
atio
nal.n
um
ber.
to.a
.dec
imal
.usi
ng.lo
ng.
div
isio
n;.k
now
.tha
t.th
e.de
cim
al.fo
rm.o
f.a.r
atio
nal.
num
ber.
term
inat
es.in
.0s.
or.e
vent
ually
.rep
eats
.
7.N
S.2d
XX
XX
7T
he.N
um
ber.
Syst
emA
pply
.and
.ex
tend
.pre
viou
s.u
nder
stan
din
gs.
of.m
ulti
plic
atio
n.an
d.d
ivis
ion.
and.
of.fr
acti
ons.
to.m
ulti
ply.
and.
div
ide.
rati
onal
.nu
mbe
rs.
Solv
e.re
al-w
orld
.and
.mat
hem
atic
al.
prob
lem
s.in
volv
ing.
the.
fou
r.op
erat
ion
s.w
ith.
rati
onal
.nu
mbe
rs.
7.N
S.3
XX
XX
7E
xpre
ssio
ns.
and.
Eq
uati
ons
Use
.pro
pert
ies.
of.o
pera
tion
s.to
.gen
erat
e.eq
uiv
alen
t.ex
pres
sion
s.
App
ly.p
rope
rtie
s.of
.ope
rati
ons.
as.
stra
teg
ies.
to.a
dd,.s
ubt
ract
,.fac
tor,.
and.
expa
nd.li
near
.exp
ress
ion
s.w
ith.
rati
onal
.co
effic
ient
s.
7.E
E.1
XX
XX
7E
xpre
ssio
ns.
and.
Eq
uati
ons
Use
.pro
pert
ies.
of.o
pera
tion
s.to
.gen
erat
e.eq
uiv
alen
t.ex
pres
sion
s.
Und
erst
and.
that
.rew
riti
ng.a
n.ex
pres
sion
.in.d
iffe
rent
.form
s.in
.a.
prob
lem
.con
text
.can
.she
d.lig
ht.o
n.th
e.pr
oble
m.a
nd.h
ow.t
he.q
uant
itie
s.in
.it.
are.
rela
ted.
7.E
E.2
XX
XX
55College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7E
xpre
ssio
ns.
and.
Eq
uati
ons
Solv
e.re
al-li
fe.a
nd.
mat
hem
atic
al.
prob
lem
s.us
ing.
num
eric
al.
and.
alge
brai
c.ex
pres
sion
s.an
d.eq
uati
ons.
Solv
e.m
ulti
-ste
p.re
al-li
fe.a
nd.
mat
hem
atic
al.p
robl
ems.
pose
d.w
ith.
posi
tive
.and
.neg
ativ
e.ra
tion
al.n
um
bers
.in
.any
.form
.(who
le.n
um
bers
,.fra
ctio
ns,
.an
d.de
cim
als)
,.usi
ng.t
ools
.str
ateg
ical
ly..
App
ly.p
rope
rtie
s.of
.ope
rati
ons.
to.
calc
ula
te.w
ith.
num
bers
.in.a
ny.fo
rm;.
conv
ert.
betw
een.
form
s.as
.app
ropr
iate
;.an
d.as
sess
.the
.rea
sona
blen
ess.
of.
answ
ers.
usin
g.m
enta
l.com
puta
tion
.and
.es
tim
atio
n.st
rate
gie
s.
7.E
E.3
XX
XX
7E
xpre
ssio
ns.
and.
Eq
uati
ons
Solv
e.re
al-li
fe.a
nd.
mat
hem
atic
al.
prob
lem
s.us
ing.
num
eric
al.
and.
alge
brai
c.ex
pres
sion
s.an
d.eq
uati
ons.
Use
.var
iabl
es.t
o.re
pres
ent.
qua
ntit
ies.
in.
a.re
al-w
orld
.or.
mat
hem
atic
al.p
robl
em,.
and.
con
stru
ct.s
impl
e.eq
uati
ons.
and.
ineq
ualit
ies.
to.s
olve
.pro
blem
s.by
.re
ason
ing.
abou
t.th
e.q
uant
itie
s.
Solv
e.w
ord.
prob
lem
s.le
adin
g.to
.eq
uati
ons.
of.
the.
form
.px.
+.q
.=.r
.and
.p(
x.+
.q).=
.r,.w
here
.p,.q
,.an
d.r.
are.
spec
ific
.rat
iona
l.nu
mbe
rs..S
olve
.eq
uati
ons.
of.t
hese
.form
s.flu
entl
y..
Com
pare
.an.
alge
brai
c.so
luti
on.t
o.an
.ari
thm
etic
.so
luti
on,.i
dent
ifyi
ng.
the.
seq
uenc
e.of
.the
.op
erat
ion
s.us
ed.in
.eac
h.ap
proa
ch.
7.E
E.4
aX
XX
X
7E
xpre
ssio
ns.
and.
Eq
uati
ons
Solv
e.re
al-li
fe.a
nd.
mat
hem
atic
al.
prob
lem
s.us
ing.
num
eric
al.
and.
alge
brai
c.ex
pres
sion
s.an
d.eq
uati
ons.
Use
.var
iabl
es.t
o.re
pres
ent.
qua
ntit
ies.
in.
a.re
al-w
orld
.or.
mat
hem
atic
al.p
robl
em,.
and.
con
stru
ct.s
impl
e.eq
uati
ons.
and.
ineq
ualit
ies.
to.s
olve
.pro
blem
s.by
.re
ason
ing.
abou
t.th
e.q
uant
itie
s.
Solv
e.w
ord.
prob
lem
s.le
adin
g.to
.ineq
ualit
ies.
of.
the.
form
.px.
+.q
.>.r
.or.
px +
.q.
<.r,
.whe
re.p
,.q,.a
nd.r
.are
.sp
ecif
ic.r
atio
nal.n
um
bers
..G
raph
.the
.sol
utio
n.se
t.of
.the
.ineq
ualit
y.an
d.in
terp
ret.
it.in
.the
.con
text
.of
.the
.pro
blem
.
7.E
E.4
bX
XX
X
7G
eom
etry
Dra
w,.c
onst
ruct
,.an
d.de
scri
be.
geom
etri
cal.
figu
res.
and.
desc
ribe
.the
.re
lati
onsh
ips.
betw
een.
them
.
Solv
e.pr
oble
ms.
invo
lvin
g.sc
ale.
dra
win
gs.o
f.geo
met
ric.
figu
res,
.in
clud
ing.
com
puti
ng.a
ctua
l.len
gth
s.an
d.ar
eas.
from
.a.s
cale
.dra
win
g.an
d.re
prod
ucin
g.a.
scal
e.d
raw
ing.
at.a
.d
iffe
rent
.sca
le.
7.G
.1X
X.
56 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7G
eom
etry
Dra
w,.c
onst
ruct
,.an
d.de
scri
be.
geom
etri
cal.
figu
res.
and.
desc
ribe
.the
.re
lati
onsh
ips.
betw
een.
them
.
Dra
w.(f
reeh
and,
.wit
h.ru
ler.
and.
prot
ract
or,.a
nd.w
ith.
tech
nolo
gy)
.ge
omet
ric.
shap
es.w
ith.
giv
en.
cond
itio
ns.
.Foc
us.o
n.co
nst
ruct
ing.
tria
ngle
s.fr
om.t
hre
e.m
easu
res.
of.a
ngle
s.or
.sid
es,.n
otic
ing.
whe
n.th
e.co
ndit
ion
s.de
term
ine.
a.u
niq
ue.t
rian
gle,
.mor
e.th
an.
one.
tria
ngle
,.or.
no.t
rian
gle.
7.G
.2X
X
7G
eom
etry
Dra
w,.c
onst
ruct
,.an
d.de
scri
be.
geom
etri
cal.
figu
res.
and.
desc
ribe
.the
.re
lati
onsh
ips.
betw
een.
them
.
Des
crib
e.th
e.tw
o-d
imen
sion
al.
figu
res.
that
.res
ult.
from
.slic
ing.
thre
e-d
imen
sion
al.fi
gu
res,
.as.
in.p
lane
.se
ctio
ns.
of.r
ight
.rec
tang
ula
r.pr
ism
s.an
d.ri
ght.
rect
ang
ula
r.py
ram
ids.
7.G
.3X
X
7G
eom
etry
Solv
e.re
al-li
fe.a
nd.
mat
hem
atic
al.
prob
lem
s.in
volv
ing.
angl
e.m
easu
re,.a
rea,
.su
rfac
e.ar
ea,.a
nd.
volu
me.
Kno
w.t
he.fo
rmu
las.
for.
the.
area
.and
.ci
rcu
mfe
renc
e.of
.a.c
ircl
e.an
d.us
e.th
em.
to.s
olve
.pro
blem
s;.g
ive.
an.in
form
al.
deri
vati
on.o
f.the
.rel
atio
nsh
ip.b
etw
een.
the.
circ
um
fere
nce.
and.
area
.of.a
.cir
cle.
7.G
.4X
X
7G
eom
etry
Solv
e.re
al-li
fe.a
nd.
mat
hem
atic
al.
prob
lem
s.in
volv
ing.
angl
e.m
easu
re,.a
rea,
.su
rfac
e.ar
ea,.a
nd.
volu
me.
Use
.fact
s.ab
out.
supp
lem
enta
ry,.
com
plem
enta
ry,.v
erti
cal,.
and.
adja
cent
.an
gles
.in.a
.mu
lti-s
tep.
prob
lem
.to.
wri
te.
and.
solv
e.si
mpl
e.eq
uati
ons.
for.
an.
un
know
n.an
gle.
in.a
.fig
ure
.
7.G
.5X
X
7G
eom
etry
Solv
e.re
al-li
fe.a
nd.
mat
hem
atic
al.
prob
lem
s.in
volv
ing.
angl
e.m
easu
re,.a
rea,
.su
rfac
e.ar
ea,.a
nd.
volu
me.
Solv
e.re
al-w
orld
.and
.mat
hem
atic
al.
prob
lem
s.in
volv
ing.
area
,.vol
um
e.an
d.su
rfac
e.ar
ea.o
f.tw
o-.a
nd.t
hre
e-d
imen
sion
al.o
bjec
ts.c
ompo
sed.
of.
tria
ngle
s,.q
uad
rila
tera
ls,.p
olyg
ons,
.cu
bes,
.and
.rig
ht.p
rism
s.
7.G
.6X
XX
57College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7St
atis
tics
.and
.P
roba
bilit
yU
se.r
ando
m.
sam
plin
g.to
.dra
w.
infe
renc
es.a
bout
.a.
popu
lati
on.
Und
erst
and.
that
.sta
tist
ics.
can.
be.
used
.to.
gain
.info
rmat
ion.
abou
t.a.
popu
lati
on.b
y.ex
amin
ing.
a.sa
mpl
e.of
.th
e.po
pula
tion
;.gen
eral
izat
ion
s.ab
out.
a.po
pula
tion
.from
.a.s
ampl
e.ar
e.va
lid.
only
.if.t
he.s
ampl
e.is
.rep
rese
ntat
ive.
of.t
hat.
popu
lati
on..U
nder
stan
d.th
at.
rand
om.s
ampl
ing.
tend
s.to
.pro
duce
.re
pres
enta
tive
.sam
ples
.and
.sup
port
.va
lid.in
fere
nces
.
7.SP
.1X
7St
atis
tics
.and
.P
roba
bilit
yU
se.r
ando
m.
sam
plin
g.to
.dra
w.
infe
renc
es.a
bout
.a.
popu
lati
on.
Use
.dat
a.fr
om.a
.ran
dom
.sam
ple.
to.
dra
w.in
fere
nces
.abo
ut.a
.pop
ula
tion
.w
ith.
an.u
nkn
own.
char
acte
rist
ic.o
f.in
tere
st..G
ener
ate.
mu
ltipl
e.sa
mpl
es.(o
r.si
mu
late
d.sa
mpl
es).o
f.the
.sam
e.si
ze.
to.g
auge
.the
.var
iati
on.in
.est
imat
es.o
r.pr
edic
tion
s.
7.SP
.2X
7St
atis
tics
.and
.P
roba
bilit
yD
raw
.info
rmal
.co
mpa
rati
ve.
infe
renc
es.a
bout
.tw
o.po
pula
tion
s.
Info
rmal
ly.a
sses
s.th
e.de
gre
e.of
.vi
sual
.ove
rlap.
of.t
wo.
num
eric
al.d
ata.
dis
trib
utio
ns.
wit
h.si
mila
r.va
riab
iliti
es,.
mea
suri
ng.t
he.d
iffe
renc
e.be
twee
n.th
e.ce
nter
s.by
.exp
ress
ing.
it.a
s.a.
mu
ltipl
e.of
.a.
mea
sure
.of.v
aria
bilit
y.
7.SP
.3X
7St
atis
tics
.and
.P
roba
bilit
yD
raw
.info
rmal
.co
mpa
rati
ve.
infe
renc
es.a
bout
.tw
o.po
pula
tion
s.
Use
.mea
sure
s.of
.cen
ter.
and.
mea
sure
s.of
.var
iabi
lity.
for.
num
eric
al.d
ata.
from
.ra
ndom
.sam
ples
.to.
dra
w.in
form
al.
com
para
tive
.infe
renc
es.a
bout
.tw
o.po
pula
tion
s.
7.SP
.4X
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
Und
erst
and.
that
.the
.pro
babi
lity.
of.a
.ch
ance
.eve
nt.is
.a.n
um
ber.
betw
een.
0.an
d.1.
that
.exp
ress
es.t
he.li
kelih
ood.
of.t
he.e
vent
.occ
urr
ing.
.Lar
ger.
num
bers
.ind
icat
e.g
reat
er.li
kelih
ood.
.A
.pro
babi
lity.
near
.0.in
dic
ates
.an.
un
likel
y.ev
ent,
.a.p
roba
bilit
y.ar
ound
.1/
2.in
dic
ates
.an.
even
t.th
at.is
.nei
ther
.u
nlik
ely.
nor.
likel
y,.a
nd.a
.pro
babi
lity.
near
.1.in
dic
ates
.a.li
kely
.eve
nt.
7.SP
.5X
58 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
App
roxi
mat
e.th
e.pr
obab
ility
.of.a
.ch
ance
.eve
nt.b
y.co
llect
ing.
data
.on.
the.
chan
ce.p
roce
ss.t
hat.
prod
uces
.it.
and.
obse
rvin
g.it
s.lo
ng-r
un.
rela
tive
.fr
eque
ncy,
.and
.pre
dic
t.th
e.ap
prox
imat
e.re
lati
ve.fr
eque
ncy.
giv
en.t
he.p
roba
bilit
y.
7.SP
.6X
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
Dev
elop
.a.p
roba
bilit
y.m
odel
.and
.use
.it.
to.fi
nd.p
roba
bilit
ies.
of.e
vent
s..C
ompa
re.
prob
abili
ties
.from
.a.m
odel
.to.
obse
rved
.fr
eque
ncie
s;.if
.the
.ag
reem
ent.
is.n
ot.
good
,.exp
lain
.pos
sibl
e.so
urc
es.o
f.the
.d
iscr
epan
cy.
Dev
elop
.a.u
nifo
rm.
prob
abili
ty.m
odel
.by
.ass
ign
ing.
equa
l.pr
obab
ility
.to.
all.
outc
omes
,.and
.use
.the
.m
odel
.to.
dete
rmin
e.pr
obab
iliti
es.o
f.eve
nts.
7.SP
.7a
X
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
Dev
elop
.a.p
roba
bilit
y.m
odel
.and
.use
.it.
to.fi
nd.p
roba
bilit
ies.
of.e
vent
s..C
ompa
re.
prob
abili
ties
.from
.a.m
odel
.to.
obse
rved
.fr
eque
ncie
s;.if
.the
.ag
reem
ent.
is.n
ot.
good
,.exp
lain
.pos
sibl
e.so
urc
es.o
f.the
.d
iscr
epan
cy.
Dev
elop
.a.p
roba
bilit
y.m
odel
.(wh
ich.
may
.not
.be.
un
iform
).by.
obse
rvin
g.fr
eque
ncie
s.in
.dat
a.ge
nera
ted.
from
.a.c
hanc
e.pr
oces
s.
7.SP
.7b
X
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
Fin
d.pr
obab
iliti
es.o
f.com
pou
nd.e
vent
s.us
ing.
orga
niz
ed.li
sts,
.tab
les,
.tre
e.d
iag
ram
s,.a
nd.s
imu
lati
on.
Und
erst
and.
that
,.jus
t.as
.w
ith.
sim
ple.
even
ts,.t
he.
prob
abili
ty.o
f.a.c
ompo
und
.ev
ent.
is.t
he.fr
acti
on.o
f.ou
tcom
es.in
.the
.sam
ple.
spac
e.fo
r.w
hic
h.th
e.co
mpo
und
.eve
nt.o
ccu
rs.
7.SP
.8a
X
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
Fin
d.pr
obab
iliti
es.o
f.com
pou
nd.e
vent
s.us
ing.
orga
niz
ed.li
sts,
.tab
les,
.tre
e.d
iag
ram
s,.a
nd.s
imu
lati
on.
Rep
rese
nt.s
ampl
e.sp
aces
.fo
r.co
mpo
und
.eve
nts.
usin
g.m
etho
ds.s
uch.
as.
orga
niz
ed.li
sts,
.tab
les.
and.
tree
.dia
gra
ms.
.For
.an
.eve
nt.d
escr
ibed
.in.
ever
yday
.lang
uage
.(e.g
.,.“r
ollin
g.do
ubl
e.si
xes”
),.id
enti
fy.t
he.o
utco
mes
.in.
the.
sam
ple.
spac
e.w
hic
h.co
mpo
se.t
he.e
vent
.
7.SP
.8b
X
59College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
7St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.ch
ance
.pro
cess
es.
and.
deve
lop,
.use
,.an
d.ev
alua
te.
prob
abili
ty.
mod
els.
Fin
d.pr
obab
iliti
es.o
f.com
pou
nd.e
vent
s.us
ing.
orga
niz
ed.li
sts,
.tab
les,
.tre
e.d
iag
ram
s,.a
nd.s
imu
lati
on.
Des
ign.
and.
use.
a.si
mu
lati
on.t
o.ge
nera
te.
freq
uenc
ies.
for.
com
pou
nd.
even
ts.
7.SP
.8c
X
8T
he.N
um
ber.
Syst
emK
now
.tha
t.th
ere.
are.
num
bers
.tha
t.ar
e.no
t.ra
tion
al,.
and.
appr
oxim
ate.
them
.by.
rati
onal
.nu
mbe
rs.
Und
erst
and.
info
rmal
ly.t
hat.
ever
y.nu
mbe
r.ha
s.a.
deci
mal
.exp
ansi
on;.
the.
rati
onal
.nu
mbe
rs.a
re.t
hose
.wit
h.de
cim
al.e
xpan
sion
s.th
at.t
erm
inat
e.in
.0s.
or.e
vent
ually
.rep
eat.
..Kno
w.t
hat.
othe
r.nu
mbe
rs.a
re.c
alle
d.ir
rati
onal
.
8.N
S.1
XX
XX
8T
he.N
um
ber.
Syst
emK
now
.tha
t.th
ere.
are.
num
bers
.tha
t.ar
e.no
t.ra
tion
al,.
and.
appr
oxim
ate.
them
.by.
rati
onal
.nu
mbe
rs.
Use
.rat
iona
l.app
roxi
mat
ion
s.of
.ir
rati
onal
.nu
mbe
rs.t
o.co
mpa
re.t
he.
size
.of.i
rrat
iona
l.nu
mbe
rs,.l
ocat
e.th
em.a
ppro
xim
atel
y.on
.a.n
um
ber.
line.
dia
gra
m,.a
nd.e
stim
ate.
the.
valu
e.of
.ex
pres
sion
s.(e
.g.,.
π2 ).
8.N
S.2
XX
X
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Wor
k.w
ith.
rad
ical
s.an
d.in
tege
r.ex
pone
nts.
Kno
w.a
nd.a
pply
.the
.pro
pert
ies.
of.in
tege
r.ex
pone
nts.
to.g
ener
ate.
equ
ival
ent.
num
eric
al.e
xpre
ssio
ns.
8.E
E.1
XX
XX
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Wor
k.w
ith.
rad
ical
s.an
d.in
tege
r.ex
pone
nts.
Use
.sq
uare
.roo
t.an
d.cu
be.r
oot.
sym
bols
.to
.rep
rese
nt.s
olut
ion
s.to
.eq
uati
ons.
of.
the.
form
.x2 .=
.p.a
nd.x
3 .=.p
,.whe
re.p
.is.
a.po
siti
ve.r
atio
nal.n
um
ber..
Eva
luat
e.sq
uare
.roo
ts.o
f.sm
all.p
erfe
ct.s
qua
res.
and.
cube
.roo
ts.o
f.sm
all.p
erfe
ct.c
ube
s..
Kno
w.t
hat.
√2 .is.
irra
tion
al.
8.E
E.2
XX
XX
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Wor
k.w
ith.
rad
ical
s.an
d.in
tege
r.ex
pone
nts.
Use
.nu
mbe
rs.e
xpre
ssed
.in.t
he.fo
rm.o
f.a.
sing
le.d
igit
.tim
es.a
n.in
tege
r.po
wer
.of.
10.t
o.es
tim
ate.
very
.larg
e.or
.ver
y.sm
all.
qua
ntit
ies,
.and
.to.
expr
ess.
how
.man
y.ti
mes
.as.
muc
h.on
e.is
.tha
n.th
e.ot
her.
8.E
E.3
XX
XX
60 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Wor
k.w
ith.
rad
ical
s.an
d.in
tege
r.ex
pone
nts.
Perf
orm
.ope
rati
ons.
wit
h.nu
mbe
rs.
expr
esse
d.in
.sci
enti
fic.n
otat
ion,
.in
clud
ing.
prob
lem
s.w
here
.bot
h.de
cim
al.
and.
scie
ntif
ic.n
otat
ion.
are.
used
..Use
.sc
ient
ific
.not
atio
n.an
d.ch
oose
.un
its.
of.
appr
opri
ate.
size
.for.
mea
sure
men
ts.o
f.ve
ry.la
rge.
or.v
ery.
smal
l.qua
ntit
ies.
(e.g
.,.us
e.m
illim
eter
s.pe
r.ye
ar.fo
r.se
aflo
or.
spre
adin
g)..I
nter
pret
.sci
enti
fic.n
otat
ion.
that
.has
.bee
n.ge
nera
ted.
by.t
ech
nolo
gy.
8.E
E.4
XX
X
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Und
erst
and.
the.
con
nect
ion
s.be
twee
n.pr
opor
tion
al.
rela
tion
ship
s,.
lines
,.and
.line
ar.
equa
tion
s.
Gra
ph.p
ropo
rtio
nal.r
elat
ion
ship
s,.
inte
rpre
ting
.the
.un
it.r
ate.
as.t
he.s
lope
.of
.the
.gra
ph..C
ompa
re.t
wo.
dif
fere
nt.
prop
orti
onal
.rel
atio
nsh
ips.
repr
esen
ted.
in.d
iffe
rent
.way
s.
8.E
E.5
XX
X
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Und
erst
and.
the.
con
nect
ion
s.be
twee
n.pr
opor
tion
al.
rela
tion
ship
s,.
lines
,.and
.line
ar.
equa
tion
s.
Use
.sim
ilar.
tria
ngle
s.to
.exp
lain
.why
.th
e.sl
ope.
m.is
.the
.sam
e.be
twee
n.an
y.tw
o.d
isti
nct.
poin
ts.o
n.a.
non-
vert
ical
.lin
e.in
.the
.coo
rdin
ate.
plan
e;.d
eriv
e.th
e.eq
uati
on y
= m
x fo
r.a.
line.
thro
ugh.
the.
orig
in.a
nd.t
he.e
qua
tion
y =
mx
+ b
for.
a.lin
e.in
terc
epti
ng.t
he.v
erti
cal.a
xis.
at.b
.
8.E
E.6
XX
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Ana
lyze
.and
.so
lve.
linea
r.eq
uati
ons.
and.
pair
s.of
.si
mu
ltane
ous.
linea
r.eq
uati
ons.
Solv
e.lin
ear.
equa
tion
s.in
.one
.var
iabl
e.G
ive.
exam
ples
.of.
linea
r.eq
uati
ons.
in.
one.
vari
able
.wit
h.on
e.so
luti
on,.i
nfi
nit
ely.
man
y.so
luti
ons,
.or.
no.s
olut
ion
s..
Show
.wh
ich.
of.t
hese
.po
ssib
iliti
es.is
.the
.cas
e.by
.su
cces
sive
ly.t
ran
sfor
min
g.th
e.g
iven
.eq
uati
on.in
to.
sim
pler
.form
s,.u
ntil.
an.
equ
ival
ent.
equa
tion
.of.t
he.
form
.x.=
.a,.a
.=.a
,.or.
a.=
b
resu
lts.(w
here
.a.a
nd b
are
.d
iffe
rent
.nu
mbe
rs).
8.E
E.7
aX
XX
61College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Ana
lyze
.and
.so
lve.
linea
r.eq
uati
ons.
and.
pair
s.of
.si
mu
ltane
ous.
linea
r.eq
uati
ons.
Solv
e.lin
ear.
equa
tion
s.in
.one
.var
iabl
e.So
lve.
linea
r.eq
uati
ons.
wit
h.ra
tion
al.n
um
ber.
coef
ficie
nts,
.incl
udin
g.eq
uati
ons.
who
se.s
olut
ion
s.re
qu
ire.
expa
ndin
g.ex
pres
sion
s.us
ing.
the.
dis
trib
utiv
e.pr
oper
ty.a
nd.
colle
ctin
g.lik
e.te
rms.
8.E
E.7
bX
XX
X
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Ana
lyze
.and
.so
lve.
linea
r.eq
uati
ons.
and.
pair
s.of
.si
mu
ltane
ous.
linea
r.eq
uati
ons.
Ana
lyze
.and
.sol
ve.p
airs
.of.
sim
ulta
neou
s.lin
ear.
equa
tion
s.U
nder
stan
d.th
at.s
olut
ion
s.to
.a.s
yste
m.o
f.tw
o.lin
ear.
equa
tion
s.in
.tw
o.va
riab
les.
corr
espo
nd.t
o.po
ints
.of
.inte
rsec
tion
.of.t
heir
.g
raph
s,.b
ecau
se.p
oint
s.of
.in
ters
ecti
on.s
atis
fy.b
oth.
equa
tion
s.si
mu
ltane
ousl
y.
8.E
E.a
XX
X
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Ana
lyze
.and
.so
lve.
linea
r.eq
uati
ons.
and.
pair
s.of
.si
mu
ltane
ous.
linea
r.eq
uati
ons.
Ana
lyze
.and
.sol
ve.p
airs
.of.
sim
ulta
neou
s.lin
ear.
equa
tion
s.So
lve.
syst
ems.
of.t
wo.
linea
r.eq
uati
ons.
in.t
wo.
vari
able
s.al
gebr
aica
lly,.
and.
esti
mat
e.so
luti
ons.
by.
gra
phin
g.th
e.eq
uati
ons.
.So
lve.
sim
ple.
case
s.by
.in
spec
tion
.
8.E
E.b
XX
X
8E
xpre
ssio
ns.
and.
Eq
uati
ons
Ana
lyze
.and
.so
lve.
linea
r.eq
uati
ons.
and.
pair
s.of
.si
mu
ltane
ous.
linea
r.eq
uati
ons.
Ana
lyze
.and
.sol
ve.p
airs
.of.
sim
ulta
neou
s.lin
ear.
equa
tion
s.So
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.p
robl
ems.
lead
ing.
to.t
wo.
linea
r.eq
uati
ons.
in.t
wo.
vari
able
s.
8.E
E.c
XX
X
8F
unc
tion
sD
efin
e,.e
valu
ate,
.an
d.co
mpa
re.
func
tion
s.
Und
erst
and.
that
.a.fu
ncti
on.is
.a.r
ule
.th
at.a
ssig
ns.
to.e
ach.
inpu
t.ex
actl
y.on
e.ou
tput
..The
.gra
ph.o
f.a.fu
ncti
on.is
.the
.se
t.of
.ord
ered
.pai
rs.c
onsi
stin
g.of
.an.
inpu
t.an
d.th
e.co
rres
pond
ing.
outp
ut.
8.F.
1X
XX
X
8F
unc
tion
sD
efin
e,.e
valu
ate,
.an
d.co
mpa
re.
func
tion
s.
Com
pare
.pro
pert
ies.
of.t
wo.
func
tion
s.ea
ch.r
epre
sent
ed.in
.a.d
iffe
rent
.way
.(a
lgeb
raic
ally
,.gra
phic
ally
,.nu
mer
ical
ly.
in.t
able
s,.o
r.by
.ver
bal.d
escr
ipti
ons)
.
8.F.
2X
XX
.
62 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
8F
unc
tion
sD
efin
e,.e
valu
ate,
.an
d.co
mpa
re.
func
tion
s.
Inte
rpre
t.th
e.eq
uati
on y
= m
x +
b a
s.de
fin
ing.
a.lin
ear.
func
tion
,.who
se.g
raph
.is
.a.s
trai
ght.
line;
.giv
e.ex
ampl
es.o
f.fu
ncti
ons.
that
.are
.not
.line
ar.
8.F.
3X
XX
X
8F
unc
tion
sU
se.fu
ncti
ons.
to.m
odel
.re
lati
onsh
ips.
betw
een.
qua
ntit
ies.
Con
stru
ct.a
.func
tion
.to.
mod
el.a
.line
ar.
rela
tion
ship
.bet
wee
n.tw
o.q
uant
itie
s..
Det
erm
ine.
the.
rate
.of.c
hang
e.an
d.in
itia
l.val
ue.o
f.the
.func
tion
.from
.a.
desc
ript
ion.
of.a
.rel
atio
nsh
ip.o
r.fr
om.t
wo.
(x, y
).val
ues,
.incl
udin
g.re
adin
g.th
ese.
from
.a.t
able
.or.
from
.a.g
raph
..Int
erpr
et.
the.
rate
.of.c
hang
e.an
d.in
itia
l.val
ue.o
f.a.
linea
r.fu
ncti
on.in
.ter
ms.
of.t
he.s
itua
tion
.it
.mod
els,
.and
.in.t
erm
s.of
.its.
gra
ph.o
r.a.
tabl
e.of
.val
ues.
8.F.
4X
XX
X
8F
unc
tion
sU
se.fu
ncti
ons.
to.m
odel
.re
lati
onsh
ips.
betw
een.
qua
ntit
ies.
Des
crib
e.q
ualit
ativ
ely.
the.
func
tion
al.
rela
tion
ship
.bet
wee
n.tw
o.q
uant
itie
s.by
.ana
lyzi
ng.a
.gra
ph.(e
.g.,.
whe
re.t
he.
func
tion
.is.in
crea
sing
.or.
decr
easi
ng,.
linea
r.or
.non
linea
r)..S
ketc
h.a.
gra
ph.
that
.exh
ibit
s.th
e.q
ualit
ativ
e.fe
atu
res.
of.a
.func
tion
.tha
t.ha
s.be
en.d
escr
ibed
.ve
rbal
ly.
8.F.
5X
XX
.
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Ver
ify.
expe
rim
enta
lly.t
he.p
rope
rtie
s.of
.ro
tati
ons,
.ref
lect
ion
s,.a
nd.t
ran
slat
ion
s:L
ines
.are
.tak
en.t
o.lin
es,.
and.
line.
seg
men
ts.t
o.lin
e.se
gm
ents
.of.t
he.s
ame.
leng
th.
8.G
.1a
XX
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Ver
ify.
expe
rim
enta
lly.t
he.p
rope
rtie
s.of
.ro
tati
ons,
.ref
lect
ion
s,.a
nd.t
ran
slat
ion
s:A
ngle
s.ar
e.ta
ken.
to.a
ngle
s.of
.the
.sam
e.m
easu
re.
8.G
.1b
XX
63College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Ver
ify.
expe
rim
enta
lly.t
he.p
rope
rtie
s.of
.ro
tati
ons,
.ref
lect
ion
s,.a
nd.t
ran
slat
ion
s:Pa
ralle
l.lin
es.a
re.t
aken
.to.
para
llel.l
ines
.8.
G.1
cX
X
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Und
erst
and.
that
.a.t
wo-
dim
ensi
onal
.fig
ure
.is.c
ong
ruen
t.to
.ano
ther
.if.t
he.
seco
nd.c
an.b
e.ob
tain
ed.fr
om.t
he.fi
rst.
by.a
.seq
uenc
e.of
.rot
atio
ns,
.ref
lect
ion
s,.
and.
tran
slat
ion
s;.g
iven
.tw
o.co
ngru
ent.
figu
res,
.des
crib
e.a.
seq
uenc
e.th
at.
exh
ibit
s.th
e.co
ngru
ence
.bet
wee
n.th
em.
8.G
.2X
X
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Des
crib
e.th
e.ef
fect
.of.d
ilati
ons,
.tr
ansl
atio
ns,
.rot
atio
ns,
.and
.ref
lect
ion
s.on
.tw
o-d
imen
sion
al.fi
gu
res.
usin
g.co
ord
inat
es.
8.G
.3X
X
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Und
erst
and.
that
.a.t
wo-
dim
ensi
onal
.fig
ure
.is.s
imila
r.to
.ano
ther
.if.t
he.
seco
nd.c
an.b
e.ob
tain
ed.fr
om.t
he.fi
rst.
by.a
.seq
uenc
e.of
.rot
atio
ns,
.ref
lect
ion
s,.
tran
slat
ion
s,.a
nd.d
ilati
ons;
.giv
en.
two.
sim
ilar.
two-
dim
ensi
onal
.fig
ure
s,.
desc
ribe
.a.s
eque
nce.
that
.exh
ibit
s.th
e.si
mila
rity
.bet
wee
n.th
em.
8.G
.4X
X
8G
eom
etry
Und
erst
and.
cong
ruen
ce.a
nd.
sim
ilari
ty.u
sing
.ph
ysic
al.m
odel
s,.
tran
spar
enci
es,.
or.g
eom
etry
.so
ftw
are.
Use
.info
rmal
.arg
um
ents
.to.
esta
blis
h.fa
cts.
abou
t.th
e.an
gle.
sum
.and
.ext
erio
r.an
gle.
of.t
rian
gles
,.abo
ut.t
he.a
ngle
s.cr
eate
d.w
hen.
para
llel.l
ines
.are
.cut
.by
.a.t
ran
sver
sal,.
and.
the.
angl
e-an
gle.
crit
erio
n.fo
r.si
mila
rity
.of.t
rian
gles
.
8.G
.5X
X
64 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
8G
eom
etry
Und
erst
and.
and.
appl
y.th
e.P
ytha
gore
an.
The
orem
.
Exp
lain
.a.p
roof
.of.t
he.P
ytha
gore
an.
The
orem
.and
.its.
conv
erse
.8.
G.6
XX
8G
eom
etry
Und
erst
and.
and.
appl
y.th
e.P
ytha
gore
an.
The
orem
.
App
ly.t
he.P
ytha
gore
an.T
heor
em.t
o.de
term
ine.
un
know
n.si
de.le
ngth
s.in
.rig
ht.t
rian
gles
.in.r
eal-w
orld
.and
.m
athe
mat
ical
.pro
blem
s.in
.tw
o.an
d.th
ree.
dim
ensi
ons.
8.G
.7X
X
8G
eom
etry
Und
erst
and.
and.
appl
y.th
e.P
ytha
gore
an.
The
orem
.
App
ly.t
he.P
ytha
gore
an.T
heor
em.t
o.fi
nd.
the.
dis
tanc
e.be
twee
n.tw
o.po
ints
.in.a
.co
ord
inat
e.sy
stem
.
8.G
.8X
X
8G
eom
etry
Solv
e.re
al-
wor
ld.a
nd.
mat
hem
atic
al.
prob
lem
s.in
volv
ing.
volu
me.
of.c
ylin
ders
,.co
nes,
.and
.sp
here
s.
Kno
w.t
he.fo
rmu
las.
for.
the.
volu
mes
.of
.con
es,.c
ylin
ders
,.and
.sph
eres
.and
.us
e.th
em.t
o.so
lve.
real
-wor
ld.a
nd.
mat
hem
atic
al.p
robl
ems.
8.G
.9X
X
8St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.pa
tter
ns.
of.
asso
ciat
ion.
in.
biva
riat
e.da
ta.
Con
stru
ct.a
nd.in
terp
ret.
scat
ter.
plot
s.fo
r.bi
vari
ate.
mea
sure
men
t.da
ta.t
o.in
vest
igat
e.pa
tter
ns.
of.a
ssoc
iati
on.
betw
een.
two.
qua
ntit
ies.
.Des
crib
e.pa
tter
ns.
such
.as.
clus
teri
ng,.o
utlie
rs,.
posi
tive
.or.
nega
tive
.ass
ocia
tion
,.lin
ear.
asso
ciat
ion,
.and
.non
linea
r.as
soci
atio
n.
8.SP
.1X
8St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.pa
tter
ns.
of.
asso
ciat
ion.
in.
biva
riat
e.da
ta.
Kno
w.t
hat.
stra
ight
.line
s.ar
e.w
idel
y.us
ed.t
o.m
odel
.rel
atio
nsh
ips.
betw
een.
two.
qua
ntit
ativ
e.va
riab
les.
.For
.sca
tter
.pl
ots.
that
.sug
gest
.a.li
near
.ass
ocia
tion
,.in
form
ally
.fit.
a.st
raig
ht.li
ne,.a
nd.
info
rmal
ly.a
sses
s.th
e.m
odel
.fit.
by.
judg
ing.
the.
clos
enes
s.of
.the
.dat
a.po
ints
.to
.the
.line
.
8.SP
.2X
XX
65College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
8St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.pa
tter
ns.
of.
asso
ciat
ion.
in.
biva
riat
e.da
ta.
Use
.the
.eq
uati
on.o
f.a.li
near
.mod
el.
to.s
olve
.pro
blem
s.in
.the
.con
text
.of
.biv
aria
te.m
easu
rem
ent.
data
,.in
terp
reti
ng.t
he.s
lope
.and
.inte
rcep
t.
8.SP
.3X
XX
8St
atis
tics
.and
.P
roba
bilit
yIn
vest
igat
e.pa
tter
ns.
of.
asso
ciat
ion.
in.
biva
riat
e.da
ta.
Und
erst
and.
that
.pat
tern
s.of
.ass
ocia
tion
.ca
n.al
so.b
e.se
en.in
.biv
aria
te.
cate
gori
cal.d
ata.
by.d
ispl
ayin
g.fr
eque
ncie
s.an
d.re
lati
ve.fr
eque
ncie
s.in
.a.t
wo-
way
.tab
le..C
onst
ruct
.and
.in
terp
ret.
a.tw
o-w
ay.t
able
.su
mm
ariz
ing.
data
.on.
two.
cate
gori
cal.v
aria
bles
.co
llect
ed.fr
om.t
he.s
ame.
subj
ects
..Use
.re
lati
ve.fr
eque
ncie
s.ca
lcu
late
d.fo
r.ro
ws.
or.c
olu
mn
s.to
.des
crib
e.po
ssib
le.
asso
ciat
ion.
betw
een.
the.
two.
vari
able
s.
8.SP
.4X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Rea
l.N
um
ber.
Syst
em
Ext
end.
the.
prop
erti
es.o
f.ex
pone
nts.
to.r
atio
nal.
expo
nent
s.
Exp
lain
.how
.the
.def
init
ion.
of.t
he.
mea
nin
g.of
.rat
iona
l.exp
onen
ts.fo
llow
s.fr
om.e
xten
din
g.th
e.pr
oper
ties
.of.i
nteg
er.
expo
nent
s.to
.tho
se.v
alue
s,.a
llow
ing.
for.
a.no
tati
on.fo
r.ra
dic
als.
in.t
erm
s.of
.ra
tion
al.e
xpon
ents
.
N-R
N.1
XX
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Rea
l.N
um
ber.
Syst
em
Ext
end.
the.
prop
erti
es.o
f.ex
pone
nts.
to.r
atio
nal.
expo
nent
s.
Rew
rite
.exp
ress
ion
s.in
volv
ing.
rad
ical
s.an
d.ra
tion
al.e
xpon
ents
.usi
ng.t
he.
prop
erti
es.o
f.exp
onen
ts.
N-R
N.2
XX
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Rea
l.N
um
ber.
Syst
em
Use
.pro
pert
ies.
of.r
atio
nal.
and.
irra
tion
al.
num
bers
.
Exp
lain
.why
.the
.su
m.o
r.pr
oduc
t.of
.tw
o.ra
tion
al.n
um
bers
.is.r
atio
nal;.
that
.th
e.su
m.o
f.a.r
atio
nal.n
um
ber.
and.
an.
irra
tion
al.n
um
ber.
is.ir
rati
onal
;.and
.th
at.t
he.p
rodu
ct.o
f.a.n
onze
ro.r
atio
nal.
num
ber.
and.
an.ir
rati
onal
.nu
mbe
r.is
.ir
rati
onal
.
N-R
N.3
XX
..
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Qua
ntit
ies
Rea
son.
qua
ntit
ativ
ely.
and.
use.
un
its.
to.
solv
e.pr
oble
ms.
Use
.un
its.
as.a
.way
.to.
und
erst
and.
prob
lem
s.an
d.to
.gu
ide.
the.
solu
tion
.of
.mu
lti-s
tep.
prob
lem
s;.c
hoos
e.an
d.in
terp
ret.
un
its.
con
sist
ently
.in.fo
rmu
las;
.ch
oose
.and
.inte
rpre
t.th
e.sc
ale.
and.
the.
orig
in.in
.gra
phs.
and.
data
.dis
play
s.
N-Q
.1X
XX
X
66 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Qua
ntit
ies
Rea
son.
qua
ntit
ativ
ely.
and.
use.
un
its.
to.
solv
e.pr
oble
ms.
Def
ine.
appr
opri
ate.
qua
ntit
ies.
for.
the.
purp
ose.
of.d
escr
ipti
ve.m
odel
ing.
N-Q
.2X
XX
.
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Qua
ntit
ies
Rea
son.
qua
ntit
ativ
ely.
and.
use.
un
its.
to.
solv
e.pr
oble
ms.
Cho
ose.
a.le
vel.o
f.acc
ura
cy.a
ppro
pria
te.
to.li
mit
atio
ns.
on.m
easu
rem
ent.
whe
n.re
port
ing.
qua
ntit
ies.
N-Q
.3X
XX
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Perf
orm
.ar
ith
met
ic.
oper
atio
ns.
wit
h.co
mpl
ex.
num
bers
.
Kno
w.t
here
.is.a
.com
plex
.nu
mbe
r.i.s
uch.
that
.i2 .=.–
1,.a
nd.e
very
.com
plex
.nu
mbe
r.ha
s.th
e.fo
rm.a
.+.b
i.wit
h.a.
and
b re
al.
N-C
N.1
XX
.
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Perf
orm
.ar
ith
met
ic.
oper
atio
ns.
wit
h.co
mpl
ex.
num
bers
.
Use
.the
.rel
atio
n.i2 .=
.–1.
and.
the.
com
mut
ativ
e,.a
ssoc
iati
ve,.a
nd.
dis
trib
utiv
e.pr
oper
ties
.to.
add,
.su
btra
ct,.
and.
mu
ltipl
y.co
mpl
ex.n
um
bers
.
N-C
N.2
XX
.
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Perf
orm
.ar
ith
met
ic.
oper
atio
ns.
wit
h.co
mpl
ex.
num
bers
.
(+).F
ind.
the.
conj
ugat
e.of
.a.c
ompl
ex.
num
ber;.
use.
conj
ugat
es.t
o.fi
nd.m
odu
li.an
d.q
uoti
ents
.of.c
ompl
ex.n
um
bers
.
N-C
N.3
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Rep
rese
nt.
com
plex
.nu
mbe
rs.
and.
thei
r.op
erat
ion
s.on
.the
.co
mpl
ex.p
lane
.
(+).R
epre
sent
.com
plex
.nu
mbe
rs.o
n.th
e.co
mpl
ex.p
lane
.in.r
ecta
ngu
lar.
and.
pola
r.fo
rm.(i
nclu
din
g.re
al.a
nd.im
agin
ary.
num
bers
),.an
d.ex
plai
n.w
hy.t
he.
rect
ang
ula
r.an
d.po
lar.
form
s.of
.a.g
iven
.co
mpl
ex.n
um
ber.
repr
esen
t.th
e.sa
me.
num
ber.
N-C
N.4
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Rep
rese
nt.
com
plex
.nu
mbe
rs.
and.
thei
r.op
erat
ion
s.on
.the
.co
mpl
ex.p
lane
.
(+).R
epre
sent
.add
itio
n,.s
ubt
ract
ion,
.m
ulti
plic
atio
n,.a
nd.c
onju
gati
on.o
f.co
mpl
ex.n
um
bers
.geo
met
rica
lly.o
n.th
e.co
mpl
ex.p
lane
;.use
.pro
pert
ies.
of.t
his
.re
pres
enta
tion
.for.
com
puta
tion
.
N-C
N.5
XX
67College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Rep
rese
nt.
com
plex
.nu
mbe
rs.
and.
thei
r.op
erat
ion
s.on
.the
.co
mpl
ex.p
lane
.
(+).C
alcu
late
.the
.dis
tanc
e.be
twee
n.nu
mbe
rs.in
.the
.com
plex
.pla
ne.a
s.th
e.m
odu
lus.
of.t
he.d
iffe
renc
e,.a
nd.t
he.
mid
poin
t.of
.a.s
egm
ent.
as.t
he.a
vera
ge.o
f.th
e.nu
mbe
rs.a
t.it
s.en
dpoi
nts.
N-C
N.6
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Use
.com
plex
.nu
mbe
rs.in
.po
lyno
mia
l.id
enti
ties
.and
.eq
uati
ons.
Solv
e.q
uad
rati
c.eq
uati
ons.
wit
h.re
al.c
oeff
icie
nts.
that
.hav
e.co
mpl
ex.
solu
tion
s.
N-C
N.7
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Use
.com
plex
.nu
mbe
rs.in
.po
lyno
mia
l.id
enti
ties
.and
.eq
uati
ons.
(+).E
xten
d.po
lyno
mia
l.ide
ntit
ies.
to.t
he.
com
plex
.nu
mbe
rs.
N-C
N.8
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
The
.Com
plex
.N
um
ber.
Syst
em
Use
.com
plex
.nu
mbe
rs.in
.po
lyno
mia
l.id
enti
ties
.and
.eq
uati
ons.
(+).K
now
.the
.Fu
ndam
enta
l.The
orem
.of
.Alg
ebra
;.sho
w.t
hat.
it.is
.tru
e.fo
r.q
uad
rati
c.po
lyno
mia
ls.
N-C
N.9
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Rep
rese
nt.a
nd.
mod
el.w
ith.
vect
or.q
uant
itie
s.
(+).R
ecog
niz
e.ve
ctor
.qua
ntit
ies.
as.
havi
ng.b
oth.
mag
nit
ude.
and.
dir
ecti
on..
Rep
rese
nt.v
ecto
r.q
uant
itie
s.by
.dir
ecte
d.lin
e.se
gm
ents
,.and
.use
.app
ropr
iate
.sy
mbo
ls.fo
r.ve
ctor
s.an
d.th
eir.
mag
nit
udes
.(e.g
.,.v,
.|v|,.
||v||,
.v).
N-V
M.1
.X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Rep
rese
nt.a
nd.
mod
el.w
ith.
vect
or.q
uant
itie
s.
(+).F
ind.
the.
com
pone
nts.
of.a
.vec
tor.
by.
subt
ract
ing.
the.
coor
din
ates
.of.a
n.in
itia
l.po
int.
from
.the
.coo
rdin
ates
.of.a
.ter
min
al.
poin
t.
N-V
M.2
.X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Rep
rese
nt.a
nd.
mod
el.w
ith.
vect
or.q
uant
itie
s.
(+).S
olve
.pro
blem
s.in
volv
ing.
velo
city
.an
d.ot
her.
qua
ntit
ies.
that
.can
.be.
repr
esen
ted.
by.v
ecto
rs.
N-V
M.3
.X
68 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.ve
ctor
s.
(+).A
dd.a
nd.s
ubt
ract
.vec
tors
.A
dd.v
ecto
rs.e
nd-t
o-en
d,.
com
pone
nt-w
ise,
.and
.by.
the.
para
llelo
gra
m.r
ule
..U
nder
stan
d.th
at.t
he.
mag
nit
ude.
of.a
.su
m.o
f.tw
o.ve
ctor
s.is
.typ
ical
ly.
not.
the.
sum
.of.t
he.
mag
nit
udes
.
N-V
M.4
a.
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.ve
ctor
s.
(+).A
dd.a
nd.s
ubt
ract
.vec
tors
.G
iven
.tw
o.ve
ctor
s.in
.m
agn
itud
e.an
d.d
irec
tion
.fo
rm,.d
eter
min
e.th
e.m
agn
itud
e.an
d.d
irec
tion
.of
.the
ir.s
um
.
N-V
M.4
b.
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.ve
ctor
s.
(+).A
dd.a
nd.s
ubt
ract
.vec
tors
.U
nder
stan
d.ve
ctor
.su
btra
ctio
n.v.
–.w
.as.v.
+.(–w
),.w
here
.–w
.is.t
he.
addi
tive
.inve
rse.
of.w
,.w
ith.
the.
sam
e.m
agni
tude
.as
.w.a
nd.p
oint
ing.
in.
the.
oppo
site
.dir
ecti
on..
Rep
rese
nt.v
ecto
r.su
btra
ctio
n.g
raph
ical
ly.
by.c
onne
ctin
g.th
e.ti
ps.in
.th
e.ap
prop
riat
e.or
der,.
and.
perf
orm
.vec
tor.
subt
ract
ion.
com
pone
nt-w
ise.
N-V
M.4
c.
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.ve
ctor
s.
(+).M
ulti
ply.
a.ve
ctor
.by.
a.sc
alar
.R
epre
sent
.sca
lar.
mu
ltipl
icat
ion.
gra
phic
ally
.by
.sca
ling.
vect
ors.
and.
poss
ibly
.rev
ersi
ng.t
heir
.d
irec
tion
;.per
form
.sca
lar.
mu
ltipl
icat
ion.
com
pone
nt-
wis
e,.e
.g.,.
as.c
(vx ,.
vy ).=
.(c
vx ,.cv
y ).
N-V
M.5
a.
X
69College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.ve
ctor
s.
(+).M
ulti
ply.
a.ve
ctor
.by.
a.sc
alar
.C
ompu
te.t
he.m
agn
itud
e.of
.a.s
cala
r.m
ulti
ple.
cv.u
sing
.||cv
||.=
.|c|v
..C
ompu
te.t
he.d
irec
tion
.of.
cv.k
now
ing.
that
.whe
n.|c
|v.
≠.0,
.the
.dir
ecti
on.o
f.cv.
is.
eith
er.a
long
.v.(f
or.c
.>.0
).or.
agai
nst
.v.(f
or.c
.<.0
).
N-V
M.5
b.
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).U
se.m
atri
ces.
to.r
epre
sent
.and
.m
anip
ula
te.d
ata,
.e.g
.,.to
.rep
rese
nt.
payo
ffs.
or.in
cide
nce.
rela
tion
ship
s.in
.a.
netw
ork.
N-V
M.6
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).M
ulti
ply.
mat
rice
s.by
.sca
lars
.to.
prod
uce.
new
.mat
rice
s,.e
.g.,.
as.w
hen.
all.
of.t
he.p
ayof
fs.in
.a.g
ame.
are.
dou
bled
.
N-V
M.7
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).A
dd,.s
ubt
ract
,.and
.mu
ltipl
y.m
atri
ces.
of.a
ppro
pria
te.d
imen
sion
s.N
-VM
.8X
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).U
nder
stan
d.th
at,.u
nlik
e.m
ulti
plic
atio
n.of
.nu
mbe
rs,.m
atri
x.m
ulti
plic
atio
n.fo
r.sq
uare
.mat
rice
s.is
.not
.a.c
omm
utat
ive.
oper
atio
n,.b
ut.
still
.sat
isfie
s.th
e.as
soci
ativ
e.an
d.d
istr
ibut
ive.
prop
erti
es.
N-V
M.9
XX
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).U
nder
stan
d.th
at.t
he.z
ero.
and.
iden
tity
.mat
rice
s.pl
ay.a
.rol
e.in
.mat
rix.
add
itio
n.an
d.m
ulti
plic
atio
n.si
mila
r.to
.th
e.ro
le.o
f.0.a
nd.1
.in.t
he.r
eal.n
um
bers
..T
he.d
eter
min
ant.
of.a
.sq
uare
.mat
rix.
is.
nonz
ero.
if.an
d.on
ly.if
.the
.mat
rix.
has.
a.m
ulti
plic
ativ
e.in
vers
e.
N-V
M.1
0X
X
70 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).M
ulti
ply.
a.ve
ctor
.(reg
arde
d.as
.a.
mat
rix.
wit
h.on
e.co
lum
n).b
y.a.
mat
rix.
of.s
uit
able
.dim
ensi
ons.
to.p
rodu
ce.
anot
her.
vect
or..W
ork.
wit
h.m
atri
ces.
as.
tran
sfor
mat
ion
s.of
.vec
tors
.
N-V
M.1
1X
X
Hig
h.Sc
hool
-N
um
ber.
and.
Qua
ntit
y
Vec
tor.
and.
Mat
rix.
Qua
ntit
ies
Perf
orm
.op
erat
ion
s.on
.m
atri
ces.
and.
use.
mat
rice
s.in
.ap
plic
atio
ns.
(+).W
ork.
wit
h.2.
×.2
.mat
rice
s.as
.tr
ansf
orm
atio
ns.
of.t
he.p
lane
,.and
.in
terp
ret.
the.
abso
lute
.val
ue.o
f.the
.de
term
inan
t.in
.ter
ms.
of.a
rea.
N-V
M.1
2X
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Inte
rpre
t.th
e.st
ruct
ure
.of.
expr
essi
ons
Inte
rpre
t.ex
pres
sion
s.th
at.r
epre
sent
.a.
qua
ntit
y.in
.ter
ms.
of.it
s.co
ntex
t.In
terp
ret.
part
s.of
.an.
expr
essi
on,.s
uch.
as.t
erm
s,.
fact
ors,
.and
.coe
ffic
ient
s.
A-S
SE.1
aX
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Inte
rpre
t.th
e.st
ruct
ure
.of.
expr
essi
ons
Inte
rpre
t.ex
pres
sion
s.th
at.r
epre
sent
.a.
qua
ntit
y.in
.ter
ms.
of.it
s.co
ntex
t.In
terp
ret.
com
plic
ated
.ex
pres
sion
s.by
.vie
win
g.on
e.or
.mor
e.of
.the
ir.p
arts
.as
.a.s
ingl
e.en
tity
.
A-S
SE.1
bX
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Inte
rpre
t.th
e.st
ruct
ure
.of.
expr
essi
ons
Use
.the
.str
uctu
re.o
f.an.
expr
essi
on.t
o.id
enti
fy.w
ays.
to.r
ewri
te.it
.A
-SSE
.2X
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Wri
te.e
xpre
ssio
ns.
in.e
qu
ival
ent.
form
s.to
.sol
ve.
prob
lem
s
Cho
ose.
and.
prod
uce.
an.e
qu
ival
ent.
form
.of
.an.
expr
essi
on.t
o.re
veal
.and
.exp
lain
.pr
oper
ties
.of.t
he.q
uant
ity.
repr
esen
ted.
by.t
he.e
xpre
ssio
n.
Fact
or.a
.qua
dra
tic.
expr
essi
on.t
o.re
veal
.the
.ze
ros.
of.t
he.fu
ncti
on.it
.de
fine
s.
A-S
SE.3
aX
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Wri
te.e
xpre
ssio
ns.
in.e
qu
ival
ent.
form
s.to
.sol
ve.
prob
lem
s
Cho
ose.
and.
prod
uce.
an.e
qu
ival
ent.
form
.of
.an.
expr
essi
on.t
o.re
veal
.and
.exp
lain
.pr
oper
ties
.of.t
he.q
uant
ity.
repr
esen
ted.
by.t
he.e
xpre
ssio
n.
Com
plet
e.th
e.sq
uare
.in.
a.q
uad
rati
c.ex
pres
sion
.to
.rev
eal.t
he.m
axim
um
.or
.min
imu
m.v
alue
.of.t
he.
func
tion
.it.d
efin
es.
A-S
SE.3
bX
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Wri
te.e
xpre
ssio
ns.
in.e
qu
ival
ent.
form
s.to
.sol
ve.
prob
lem
s
Cho
ose.
and.
prod
uce.
an.e
qu
ival
ent.
form
.of
.an.
expr
essi
on.t
o.re
veal
.and
.exp
lain
.pr
oper
ties
.of.t
he.q
uant
ity.
repr
esen
ted.
by.t
he.e
xpre
ssio
n.
Use
.the
.pro
pert
ies.
of.
expo
nent
s.to
.tra
nsf
orm
.ex
pres
sion
s.fo
r.ex
pone
ntia
l.fu
ncti
ons.
A-S
SE.3
cX
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Seei
ng.
Stru
ctu
re.in
.E
xpre
ssio
ns
Wri
te.e
xpre
ssio
ns.
in.e
qu
ival
ent.
form
s.to
.sol
ve.
prob
lem
s
Der
ive.
the.
form
ula
.for.
the.
sum
.of.a
.fi
nit
e.ge
omet
ric.
seri
es.(w
hen.
the.
com
mon
.rat
io.is
.not
.1),.
and.
use.
the.
form
ula
.to.
solv
e.pr
oble
ms.
A-S
SE.4
XX
.
71College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Perf
orm
.ar
ith
met
ic.
oper
atio
ns.
on.
poly
nom
ials
Und
erst
and.
that
.pol
ynom
ials
.form
.a.
syst
em.a
nalo
gous
.to.
the.
inte
gers
,.na
mel
y,.t
hey.
are.
clos
ed.u
nder
.the
.op
erat
ion
s.of
.add
itio
n,.s
ubt
ract
ion,
.an
d.m
ulti
plic
atio
n;.a
dd,.s
ubt
ract
,.and
.m
ulti
ply.
poly
nom
ials
.
A-A
PR.1
XX
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Und
erst
and.
the.
rela
tion
ship
.be
twee
n.ze
ros.
and.
fact
ors.
of.
poly
nom
ials
Kno
w.a
nd.a
pply
.the
.Rem
aind
er.
The
orem
:.For
.a.p
olyn
omia
l.p(x
).and
.a.
num
ber.
a,.t
he.r
emai
nder
.on.
div
isio
n.by
.x.–
.a.is
.p(a
),.so
.p(a
).=.0
.if.a
nd.o
nly
.if.
(x.–
.a).i
s.a.
fact
or.o
f.p(x
).
A-A
PR.2
XX
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Und
erst
and.
the.
rela
tion
ship
.be
twee
n.ze
ros.
and.
fact
ors.
of.
poly
nom
ials
Iden
tify
.zer
os.o
f.pol
ynom
ials
.whe
n.su
itab
le.fa
ctor
izat
ion
s.ar
e.av
aila
ble,
.an
d.us
e.th
e.ze
ros.
to.c
onst
ruct
.a.r
ough
.g
raph
.of.t
he.fu
ncti
on.d
efin
ed.b
y.th
e.po
lyno
mia
l.
A-A
PR.3
XX
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Use
.pol
ynom
ial.
iden
titi
es.t
o.so
lve.
prob
lem
s
Pro
ve.p
olyn
omia
l.ide
ntit
ies.
and.
use.
them
.to.
desc
ribe
.nu
mer
ical
.re
lati
onsh
ips.
A-A
PR.4
XX
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Use
.pol
ynom
ial.
iden
titi
es.t
o.so
lve.
prob
lem
s
(+).K
now
.and
.app
ly.t
he.B
inom
ial.
The
orem
.for.
the.
expa
nsio
n.of
.(x.+
.y)n .
in.
pow
ers.
of.x
.and
y fo
r.a.
posi
tive
.inte
ger.
n,.w
here
.x.a
nd y
are
.any
.num
bers
,.wit
h.co
effic
ient
s.de
term
ined
.for.
exam
ple.
by.
Pasc
al’s
.Tri
angl
e.
A-A
PR.5
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Rew
rite
.rat
iona
l.ex
pres
sion
sR
ewri
te.s
impl
e.ra
tion
al.e
xpre
ssio
ns.
in.
dif
fere
nt.fo
rms;
.wri
te.a
(x)/
b(x)
.in.t
he.
form
.q(x
).+.r
(x)/
b(x)
,.whe
re.a
(x),.
b(x)
,.q
(x),.
and.
r(x)
.are
.pol
ynom
ials
.wit
h.th
e.de
gre
e.of
.r(x
).les
s.th
an.t
he.d
egre
e.of
.b(
x),.u
sing
.insp
ecti
on,.l
ong.
div
isio
n,.o
r,.fo
r.th
e.m
ore.
com
plic
ated
.exa
mpl
es,.a
.co
mpu
ter.
alge
bra.
syst
em.
A-A
PR.6
XX
X
72 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-A
lgeb
ra
Ari
thm
etic
.wit
h.Po
lyno
mia
ls.
and.
Rat
iona
l.E
xpre
ssio
ns
Rew
rite
.rat
iona
l.ex
pres
sion
s(+
).Und
erst
and.
that
.rat
iona
l.exp
ress
ion
s.fo
rm.a
.sys
tem
.ana
logo
us.t
o.th
e.ra
tion
al.
num
bers
,.clo
sed.
und
er.a
ddit
ion,
.su
btra
ctio
n,.m
ulti
plic
atio
n,.a
nd.d
ivis
ion.
by.a
.non
zero
.rat
iona
l.exp
ress
ion;
.add
,.su
btra
ct,.m
ulti
ply,
.and
.div
ide.
rati
onal
.ex
pres
sion
s.
A-A
PR.7
XX
.X
Hig
h.Sc
hool
-A
lgeb
ra
Cre
atin
g.E
qua
tion
sC
reat
e.eq
uati
ons.
that
.des
crib
e.nu
mbe
rs.o
r.re
lati
onsh
ips
Cre
ate.
equa
tion
s.an
d.in
equa
litie
s.in
.on
e.va
riab
le.a
nd.u
se.t
hem
.to.
solv
e.pr
oble
ms.
.Incl
ude.
equa
tion
s.ar
isin
g.fr
om.li
near
.and
.qua
dra
tic.
func
tion
s,.
and.
sim
ple.
rati
onal
.and
.exp
onen
tial
.fu
ncti
ons.
A-C
ED
.1X
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Cre
atin
g.E
qua
tion
sC
reat
e.eq
uati
ons.
that
.des
crib
e.nu
mbe
rs.o
r.re
lati
onsh
ips
Cre
ate.
equa
tion
s.in
.tw
o.or
.mor
e.va
riab
les.
to.r
epre
sent
.rel
atio
nsh
ips.
betw
een.
qua
ntit
ies;
.gra
ph.e
qua
tion
s.on
.co
ord
inat
e.ax
es.w
ith.
labe
ls.a
nd.s
cale
s.
A-C
ED
.2X
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Cre
atin
g.E
qua
tion
sC
reat
e.eq
uati
ons.
that
.des
crib
e.nu
mbe
rs.o
r.re
lati
onsh
ips
Rep
rese
nt.c
onst
rain
ts.b
y.eq
uati
ons.
or.in
equa
litie
s,.a
nd.b
y.sy
stem
s.of
.eq
uati
ons.
and
/or.
ineq
ualit
ies,
.an
d.in
terp
ret.
solu
tion
s.as
.via
ble.
or.n
onvi
able
.opt
ion
s.in
.a.m
odel
ing.
cont
ext.
A-C
ED
.3X
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Cre
atin
g.E
qua
tion
sC
reat
e.eq
uati
ons.
that
.des
crib
e.nu
mbe
rs.o
r.re
lati
onsh
ips
Rea
rran
ge.fo
rmu
las.
to.h
igh
light
.a.
qua
ntit
y.of
.inte
rest
,.usi
ng.t
he.s
ame.
reas
onin
g.as
.in.s
olvi
ng.e
qua
tion
s.
A-C
ED
.4X
XX
X
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Und
erst
and.
solv
ing.
equa
tion
s.as
.a.p
roce
ss.o
f.re
ason
ing.
and.
expl
ain.
the.
reas
onin
g
Exp
lain
.eac
h.st
ep.in
.sol
ving
.a.s
impl
e.eq
uati
on.a
s.fo
llow
ing.
from
.the
.eq
ualit
y.of
.nu
mbe
rs.a
sser
ted.
at.t
he.p
revi
ous.
step
,.sta
rtin
g.fr
om.t
he.a
ssu
mpt
ion.
that
.th
e.or
igin
al.e
qua
tion
.has
.a.s
olut
ion.
.C
onst
ruct
.a.v
iabl
e.ar
gu
men
t.to
.just
ify.
a.so
luti
on.m
etho
d.
A-R
EI.1
XX
XX
73College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Und
erst
and.
solv
ing.
equa
tion
s.as
.a.p
roce
ss.o
f.re
ason
ing.
and.
expl
ain.
the.
reas
onin
g
Solv
e.si
mpl
e.ra
tion
al.a
nd.r
adic
al.
equa
tion
s.in
.one
.var
iabl
e,.a
nd.g
ive.
exam
ples
.sho
win
g.ho
w.e
xtra
neou
s.so
luti
ons.
may
.ari
se.
A-R
EI.2
XX
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.eq
uati
ons.
and.
ineq
ualit
ies.
in.o
ne.v
aria
ble
Solv
e.lin
ear.
equa
tion
s.an
d.in
equa
litie
s.in
.one
.var
iabl
e,.in
clud
ing.
equa
tion
s.w
ith.
coef
ficie
nts.
repr
esen
ted.
by.le
tter
s.
A-R
EI.3
XX
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.eq
uati
ons.
and.
ineq
ualit
ies.
in.o
ne.v
aria
ble
Solv
e.q
uad
rati
c.eq
uati
ons.
in.o
ne.
vari
able
.U
se.t
he.m
etho
d.of
.co
mpl
etin
g.th
e.sq
uare
.to.
tran
sfor
m.a
ny.q
uad
rati
c.eq
uati
on.in
.x.in
to.a
n.eq
uati
on.o
f.the
.form
.(x
.–.p
)2 .=.q
.tha
t.ha
s.th
e.sa
me.
solu
tion
s..D
eriv
e.th
e.q
uad
rati
c.fo
rmu
la.fr
om.
this
.form
.
A-R
EI.4
aX
X
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.eq
uati
ons.
and.
ineq
ualit
ies.
in.o
ne.v
aria
ble
Solv
e.q
uad
rati
c.eq
uati
ons.
in.o
ne.
vari
able
.So
lve.
qua
dra
tic.
equa
tion
s.by
.insp
ecti
on.(e
.g.,.
for.
x2 .=
.49)
,.tak
ing.
squa
re.r
oots
,.co
mpl
etin
g.th
e.sq
uare
,.th
e.q
uad
rati
c.fo
rmu
la.a
nd.
fact
orin
g,.a
s.ap
prop
riat
e.to
.the
.init
ial.f
orm
.of.t
he.
equa
tion
..Rec
ogn
ize.
whe
n.th
e.q
uad
rati
c.fo
rmu
la.
giv
es.c
ompl
ex.s
olut
ion
s.an
d.w
rite
.the
m.a
s.a.
±.b
i.fo
r.re
al.n
um
bers
.a.a
nd.b
.
A-R
EI.4
bX
X
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.sy
stem
s.of
.eq
uati
ons
Pro
ve.t
hat,
.giv
en.a
.sys
tem
.of.t
wo.
equa
tion
s.in
.tw
o.va
riab
les,
.rep
laci
ng.
one.
equa
tion
.by.
the.
sum
.of.t
hat.
equa
tion
.and
.a.m
ulti
ple.
of.t
he.o
ther
.pr
oduc
es.a
.sys
tem
.wit
h.th
e.sa
me.
solu
tion
s.
A-R
EI.5
XX
74 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.sy
stem
s.of
.eq
uati
ons
Solv
e.sy
stem
s.of
.line
ar.e
qua
tion
s.ex
actl
y.an
d.ap
prox
imat
ely.
(e.g
.,.w
ith.
gra
phs)
,.foc
usin
g.on
.pai
rs.o
f.lin
ear.
equa
tion
s.in
.tw
o.va
riab
les.
A-R
EI.6
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.sy
stem
s.of
.eq
uati
ons
Solv
e.a.
sim
ple.
syst
em.c
onsi
stin
g.of
.a.
linea
r.eq
uati
on.a
nd.a
.qua
dra
tic.
equa
tion
.in.t
wo.
vari
able
s.al
gebr
aica
lly.
and.
gra
phic
ally
.
A-R
EI.7
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.sy
stem
s.of
.eq
uati
ons
(+).R
epre
sent
.a.s
yste
m.o
f.lin
ear.
equa
tion
s.as
.a.s
ingl
e.m
atri
x.eq
uati
on.
in.a
.vec
tor.
vari
able
.
A-R
EI.8
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Solv
e.sy
stem
s.of
.eq
uati
ons
(+).F
ind.
the.
inve
rse.
of.a
.mat
rix.
if.it
.ex
ists
.and
.use
.it.t
o.so
lve.
syst
ems.
of.
linea
r.eq
uati
ons.
(usi
ng.t
ech
nolo
gy.
for.
mat
rice
s.of
.dim
ensi
on.3
.×.3
.or.
gre
ater
).
A-R
EI.9
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Rep
rese
nt.a
nd.
solv
e.eq
uati
ons.
and.
ineq
ualit
ies.
gra
phic
ally
Und
erst
and.
that
.the
.gra
ph.o
f.an.
equa
tion
.in.t
wo.
vari
able
s.is
.the
.se
t.of
.all.
its.
solu
tion
s.pl
otte
d.in
.the
.co
ord
inat
e.pl
ane,
.oft
en.fo
rmin
g.a.
curv
e.(w
hic
h.co
uld
.be.
a.lin
e).
A-R
EI.1
0X
XX
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Rep
rese
nt.a
nd.
solv
e.eq
uati
ons.
and.
ineq
ualit
ies.
gra
phic
ally
Exp
lain
.why
.the
.x-c
oord
inat
es.
of.t
he.p
oint
s.w
here
.the
.gra
phs.
of.
the.
equa
tion
s y
=.f
(x).a
nd y
=.g
(x).
inte
rsec
t.ar
e.th
e.so
luti
ons.
of.t
he.
equa
tion
.f(x
).=.g
(x);.
find
.the
.sol
utio
ns.
appr
oxim
atel
y,.e
.g.,.
usin
g.te
chno
log
y.to
.gra
ph.t
he.fu
ncti
ons,
.mak
e.ta
bles
.of.v
alue
s,.o
r.fi
nd.s
ucce
ssiv
e.ap
prox
imat
ion
s..In
clud
e.ca
ses.
whe
re.
f(x)
.and
/or.
g(x
).are
.line
ar,.p
olyn
omia
l,.ra
tion
al,.a
bsol
ute.
valu
e,.e
xpon
enti
al,.
and.
loga
rith
mic
.func
tion
s.
A-R
EI.1
1X
XX
75College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-A
lgeb
ra
Rea
son
ing.
wit
h.E
qua
tion
s.an
d.In
equa
litie
s
Rep
rese
nt.a
nd.
solv
e.eq
uati
ons.
and.
ineq
ualit
ies.
gra
phic
ally
Gra
ph.t
he.s
olut
ion
s.to
.a.li
near
.in
equa
lity.
in.t
wo.
vari
able
s.as
.a.h
alf-
plan
e.(e
xclu
din
g.th
e.bo
und
ary.
in.t
he.
case
.of.a
.str
ict.
ineq
ualit
y),.a
nd.g
raph
.th
e.so
luti
on.s
et.t
o.a.
syst
em.o
f.lin
ear.
ineq
ualit
ies.
in.t
wo.
vari
able
s.as
.the
.in
ters
ecti
on.o
f.the
.cor
resp
ond
ing.
half-
plan
es.
A-R
EI.1
2X
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sU
nder
stan
d.th
e.co
ncep
t.of
.a.
func
tion
.and
.use
.fu
ncti
on.n
otat
ion
Und
erst
and.
that
.a.fu
ncti
on.fr
om.o
ne.
set.
(cal
led.
the.
dom
ain)
.to.
anot
her.
set.
(cal
led.
the.
rang
e).a
ssig
ns.
to.e
ach.
elem
ent.
of.t
he.d
omai
n.ex
actl
y.on
e.el
emen
t.of
.the
.ran
ge..I
f.f.is
.a.fu
ncti
on.
and.
x.is
.an.
elem
ent.
of.it
s.do
mai
n,.
then
.f(x
).den
otes
.the
.out
put.
of.f.
corr
espo
ndin
g.to
.the
.inpu
t.x.
.The
.gra
ph.
of.f.
is.t
he.g
raph
.of.t
he.e
qua
tion
y =
.f(x
).
F-IF
.1X
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sU
nder
stan
d.th
e.co
ncep
t.of
.a.
func
tion
.and
.use
.fu
ncti
on.n
otat
ion
Use
.func
tion
.not
atio
n,.e
valu
ate.
func
tion
s.fo
r.in
puts
.in.t
heir
.dom
ain
s,.
and.
inte
rpre
t.st
atem
ents
.tha
t.us
e.fu
ncti
on.n
otat
ion.
in.t
erm
s.of
.a.c
onte
xt.
F-IF
.2X
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sU
nder
stan
d.th
e.co
ncep
t.of
.a.
func
tion
.and
.use
.fu
ncti
on.n
otat
ion
Rec
ogn
ize.
that
.seq
uenc
es.a
re.
func
tion
s,.s
omet
imes
.def
ined
.re
curs
ivel
y,.w
hose
.dom
ain.
is.a
.su
bset
.of
.the
.inte
gers
.
F-IF
.3X
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sIn
terp
ret.
func
tion
s.th
at.a
rise
.in.
appl
icat
ion
s.in
.ter
ms.
of.t
he.
cont
ext
For.
a.fu
ncti
on.t
hat.
mod
els.
a.re
lati
onsh
ip.b
etw
een.
two.
qua
ntit
ies,
.in
terp
ret.
key.
feat
ure
s.of
.gra
phs.
and.
tabl
es.in
.ter
ms.
of.t
he.q
uant
itie
s,.a
nd.
sket
ch.g
raph
s.sh
owin
g.ke
y.fe
atu
res.
giv
en.a
.ver
bal.d
escr
ipti
on.o
f.the
.re
lati
onsh
ip..K
ey.fe
atu
res.
incl
ude:
.in
terc
epts
;.int
erva
ls.w
here
.the
.func
tion
.is
.incr
easi
ng,.d
ecre
asin
g,.p
osit
ive,
.or
.neg
ativ
e;.r
elat
ive.
max
imu
ms.
and.
min
imu
ms;
.sym
met
ries
;.end
.beh
avio
r;.
and.
peri
odic
ity.
F-IF
.4X
XX
.
76 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sIn
terp
ret.
func
tion
s.th
at.a
rise
.in.
appl
icat
ion
s.in
.ter
ms.
of.t
he.
cont
ext
Rel
ate.
the.
dom
ain.
of.a
.func
tion
.to.
its.
gra
ph.a
nd,.w
here
.app
licab
le,.t
o.th
e.q
uant
itat
ive.
rela
tion
ship
.it.d
escr
ibes
.
F-IF
.5X
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sIn
terp
ret.
func
tion
s.th
at.a
rise
.in.
appl
icat
ion
s.in
.ter
ms.
of.t
he.
cont
ext
Cal
cula
te.a
nd.in
terp
ret.
the.
aver
age.
rate
.of.c
hang
e.of
.a.fu
ncti
on.(p
rese
nted
.sy
mbo
lical
ly.o
r.as
.a.t
able
).ove
r.a.
spec
ifie
d.in
terv
al..E
stim
ate.
the.
rate
.of.
chan
ge.fr
om.a
.gra
ph.
F-IF
.6X
XX
.
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Gra
ph.fu
ncti
ons.
expr
esse
d.sy
mbo
lical
ly.
and.
show
.key
.feat
ure
s.of
.the
.gra
ph,.
by.h
and.
in.s
impl
e.ca
ses.
and.
usin
g.te
chno
log
y.fo
r.m
ore.
com
plic
ated
.cas
es.
Gra
ph.li
near
.and
.q
uad
rati
c.fu
ncti
ons.
and.
show
.inte
rcep
ts,.m
axim
a,.
and.
min
ima.
F-IF
.7a
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Gra
ph.fu
ncti
ons.
expr
esse
d.sy
mbo
lical
ly.
and.
show
.key
.feat
ure
s.of
.the
.gra
ph,.
by.h
and.
in.s
impl
e.ca
ses.
and.
usin
g.te
chno
log
y.fo
r.m
ore.
com
plic
ated
.cas
es.
Gra
ph.s
qua
re.r
oot,
.cu
be.
root
,.and
.pie
cew
ise-
defi
ned.
func
tion
s,.
incl
udin
g.st
ep.fu
ncti
ons.
and.
abso
lute
.val
ue.
func
tion
s.
F-IF
.7b
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Gra
ph.fu
ncti
ons.
expr
esse
d.sy
mbo
lical
ly.
and.
show
.key
.feat
ure
s.of
.the
.gra
ph,.
by.h
and.
in.s
impl
e.ca
ses.
and.
usin
g.te
chno
log
y.fo
r.m
ore.
com
plic
ated
.cas
es.
Gra
ph.p
olyn
omia
l.fu
ncti
ons,
.iden
tify
ing.
zero
s.w
hen.
suit
able
.fa
ctor
izat
ion
s.ar
e.av
aila
ble,
.and
.sho
win
g.en
d.be
havi
or.
F-IF
.7c
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Gra
ph.fu
ncti
ons.
expr
esse
d.sy
mbo
lical
ly.
and.
show
.key
.feat
ure
s.of
.the
.gra
ph,.
by.h
and.
in.s
impl
e.ca
ses.
and.
usin
g.te
chno
log
y.fo
r.m
ore.
com
plic
ated
.cas
es.
(+).G
raph
.rat
iona
l.fu
ncti
ons,
.iden
tify
ing.
zero
s.an
d.as
ympt
otes
.w
hen.
suit
able
.fa
ctor
izat
ion
s.ar
e.av
aila
ble,
.and
.sho
win
g.en
d.be
havi
or.
F-IF
.7d
XX
.
77College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Gra
ph.fu
ncti
ons.
expr
esse
d.sy
mbo
lical
ly.
and.
show
.key
.feat
ure
s.of
.the
.gra
ph,.
by.h
and.
in.s
impl
e.ca
ses.
and.
usin
g.te
chno
log
y.fo
r.m
ore.
com
plic
ated
.cas
es.
Gra
ph.e
xpon
enti
al.a
nd.
loga
rith
mic
.func
tion
s,.
show
ing.
inte
rcep
ts.
and.
end.
beha
vior
,.and
.tr
igon
omet
ric.
func
tion
s,.
show
ing.
peri
od,.m
idlin
e,.
and.
ampl
itud
e.
F-IF
.7e
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Wri
te.a
.func
tion
.def
ined
.by.
an.
expr
essi
on.in
.dif
fere
nt.b
ut.e
qu
ival
ent.
form
s.to
.rev
eal.a
nd.e
xpla
in.d
iffe
rent
.pr
oper
ties
.of.t
he.fu
ncti
on.
Use
.the
.pro
cess
.of.
fact
orin
g.an
d.co
mpl
etin
g.th
e.sq
uare
.in.a
.qua
dra
tic.
func
tion
.to.
show
.zer
os,.
extr
eme.
valu
es,.a
nd.
sym
met
ry.o
f.the
.gra
ph,.
and.
inte
rpre
t.th
ese.
in.
term
s.of
.a.c
onte
xt.
F-IF
.8a
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Wri
te.a
.func
tion
.def
ined
.by.
an.
expr
essi
on.in
.dif
fere
nt.b
ut.e
qu
ival
ent.
form
s.to
.rev
eal.a
nd.e
xpla
in.d
iffe
rent
.pr
oper
ties
.of.t
he.fu
ncti
on.
Use
.the
.pro
pert
ies.
of.
expo
nent
s.to
.inte
rpre
t.ex
pres
sion
s.fo
r.ex
pone
ntia
l.fu
ncti
ons.
F-IF
.8b
XX
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Inte
rpre
ting
.F
unc
tion
sA
naly
ze.
func
tion
s.us
ing.
dif
fere
nt.
repr
esen
tati
ons
Com
pare
.pro
pert
ies.
of.t
wo.
func
tion
s.ea
ch.r
epre
sent
ed.in
.a.d
iffe
rent
.way
.(a
lgeb
raic
ally
,.gra
phic
ally
,.nu
mer
ical
ly.
in.t
able
s,.o
r.by
.ver
bal.d
escr
ipti
ons)
.
F-IF
.9X
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.a.fu
ncti
on.
that
.mod
els.
a.re
lati
onsh
ip.
betw
een.
two.
qua
ntit
ies
Wri
te.a
.func
tion
.tha
t.de
scri
bes.
a.re
lati
onsh
ip.b
etw
een.
two.
qua
ntit
ies.
Det
erm
ine.
an.e
xplic
it.
expr
essi
on,.a
.rec
urs
ive.
proc
ess,
.or.
step
s.fo
r.ca
lcu
lati
on.fr
om.a
.con
text
.
F-B
F.1a
XX
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.a.fu
ncti
on.
that
.mod
els.
a.re
lati
onsh
ip.
betw
een.
two.
qua
ntit
ies
Wri
te.a
.func
tion
.tha
t.de
scri
bes.
a.re
lati
onsh
ip.b
etw
een.
two.
qua
ntit
ies.
Com
bine
.sta
ndar
d.fu
ncti
on.t
ypes
.usi
ng.
arit
hm
etic
.ope
rati
ons.
F-B
F.1b
XX
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.a.fu
ncti
on.
that
.mod
els.
a.re
lati
onsh
ip.
betw
een.
two.
qua
ntit
ies
Wri
te.a
.func
tion
.tha
t.de
scri
bes.
a.re
lati
onsh
ip.b
etw
een.
two.
qua
ntit
ies.
(+).C
ompo
se.fu
ncti
ons.
F-B
F.1c
XX
..
78 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.a.fu
ncti
on.
that
.mod
els.
a.re
lati
onsh
ip.
betw
een.
two.
qua
ntit
ies
Wri
te.a
rith
met
ic.a
nd.g
eom
etri
c.se
que
nces
.bot
h.re
curs
ivel
y.an
d.w
ith.
an.e
xplic
it.fo
rmu
la,.u
se.t
hem
.to.
mod
el.
situ
atio
ns,
.and
.tra
nsl
ate.
betw
een.
the.
two.
form
s.
F-B
F.2
XX
.
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.new
.fu
ncti
ons.
from
.exi
stin
g.fu
ncti
ons
Iden
tify
.the
.effe
ct.o
n.th
e.g
raph
.of.
repl
acin
g.f(
x).b
y.f(
x).+
.k,.k
.f(x)
,.f(k
x),.
and.
f(x.
+.k
).for
.spe
cific
.val
ues.
of.k
.(bot
h.po
siti
ve.a
nd.n
egat
ive)
;.fin
d.th
e.va
lue.
of.k
.giv
en.t
he.g
raph
s..E
xper
imen
t.w
ith.
case
s.an
d.ill
ustr
ate.
an.e
xpla
nati
on.
of.t
he.e
ffect
s.on
.the
.gra
ph.u
sing
.te
chno
log
y..In
clud
e.re
cog
nizi
ng.e
ven.
and.
odd.
func
tion
s.fr
om.t
heir
.gra
phs.
and.
alge
brai
c.ex
pres
sion
s.fo
r.th
em.
F-B
F.3
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.new
.fu
ncti
ons.
from
.exi
stin
g.fu
ncti
ons
Fin
d.in
vers
e.fu
ncti
ons.
Solv
e.an
.eq
uati
on.o
f.the
.fo
rm.f
(x).=
.c.fo
r.a.
sim
ple.
func
tion
.f.th
at.h
as.a
n.in
vers
e.an
d.w
rite
.an.
expr
essi
on.fo
r.th
e.in
vers
e.
F-B
F.4a
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.new
.fu
ncti
ons.
from
.exi
stin
g.fu
ncti
ons
Fin
d.in
vers
e.fu
ncti
ons.
(+).V
erif
y.by
.com
posi
tion
.th
at.o
ne.fu
ncti
on.is
.the
.in
vers
e.of
.ano
ther
.
F-B
F.4b
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.new
.fu
ncti
ons.
from
.exi
stin
g.fu
ncti
ons
Fin
d.in
vers
e.fu
ncti
ons.
(+).R
ead.
valu
es.o
f.an.
inve
rse.
func
tion
.from
.a.
gra
ph.o
r.a.
tabl
e,.g
iven
.th
at.t
he.fu
ncti
on.h
as.a
n.in
vers
e.
F-B
F.4c
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.new
.fu
ncti
ons.
from
.exi
stin
g.fu
ncti
ons
Fin
d.in
vers
e.fu
ncti
ons.
(+).P
rodu
ce.a
n.in
vert
ible
.fu
ncti
on.fr
om.a
.non
-in
vert
ible
.func
tion
.by.
rest
rict
ing.
the.
dom
ain.
F-B
F.4d
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Bu
ildin
g.F
unc
tion
sB
uild
.new
.fu
ncti
ons.
from
.exi
stin
g.fu
ncti
ons
(+).U
nder
stan
d.th
e.in
vers
e.re
lati
onsh
ip.
betw
een.
expo
nent
s.an
d.lo
gari
thm
s.an
d.us
e.th
is.r
elat
ion
ship
.to.
solv
e.pr
oble
ms.
invo
lvin
g.lo
gari
thm
s.an
d.ex
pone
nts.
F-B
F.5
XX
X
79College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els
Con
stru
ct.a
nd.
com
pare
.line
ar,.
qua
dra
tic,
.and
.ex
pone
ntia
l.m
odel
s.an
d.so
lve.
prob
lem
s
Dis
ting
uis
h.be
twee
n.si
tuat
ion
s.th
at.
can.
be.m
odel
ed.w
ith.
linea
r.fu
ncti
ons.
and.
wit
h.ex
pone
ntia
l.fu
ncti
ons.
Pro
ve.t
hat.
linea
r.fu
ncti
ons.
gro
w.b
y.eq
ual.d
iffe
renc
es.
over
.eq
ual.i
nter
vals
,.and
.th
at.e
xpon
enti
al.fu
ncti
ons.
gro
w.b
y.eq
ual.f
acto
rs.o
ver.
equa
l.int
erva
ls.
F-L
E.1
aX
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els
Con
stru
ct.a
nd.
com
pare
.line
ar,.
qua
dra
tic,
.and
.ex
pone
ntia
l.m
odel
s.an
d.so
lve.
prob
lem
s
Dis
ting
uis
h.be
twee
n.si
tuat
ion
s.th
at.
can.
be.m
odel
ed.w
ith.
linea
r.fu
ncti
ons.
and.
wit
h.ex
pone
ntia
l.fu
ncti
ons.
Rec
ogn
ize.
situ
atio
ns.
in.w
hic
h.on
e.q
uant
ity.
chan
ges.
at.a
.con
stan
t.ra
te.
per.
un
it.in
terv
al.r
elat
ive.
to.
anot
her.
F-L
E.1
bX
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els
Con
stru
ct.a
nd.
com
pare
.line
ar,.
qua
dra
tic,
.and
.ex
pone
ntia
l.m
odel
s.an
d.so
lve.
prob
lem
s
Dis
ting
uis
h.be
twee
n.si
tuat
ion
s.th
at.
can.
be.m
odel
ed.w
ith.
linea
r.fu
ncti
ons.
and.
wit
h.ex
pone
ntia
l.fu
ncti
ons.
Rec
ogn
ize.
situ
atio
ns.
in.w
hic
h.a.
qua
ntit
y.g
row
s.or
.dec
ays.
by.a
.co
nst
ant.
perc
ent.
rate
.per
.u
nit
.inte
rval
.rel
ativ
e.to
.an
othe
r.
F-L
E.1
cX
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els
Con
stru
ct.a
nd.
com
pare
.line
ar,.
qua
dra
tic,
.and
.ex
pone
ntia
l.m
odel
s.an
d.so
lve.
prob
lem
s
Con
stru
ct.li
near
.and
.exp
onen
tial
.fu
ncti
ons,
.incl
udin
g.ar
ith
met
ic.a
nd.
geom
etri
c.se
que
nces
,.giv
en.a
.gra
ph,.
a.de
scri
ptio
n.of
.a.r
elat
ion
ship
,.or.
two.
inpu
t-ou
tput
.pai
rs.(i
nclu
de.r
ead
ing.
thes
e.fr
om.a
.tab
le).
F-L
E.2
XX
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els
Con
stru
ct.a
nd.
com
pare
.line
ar,.
qua
dra
tic,
.and
.ex
pone
ntia
l.m
odel
s.an
d.so
lve.
prob
lem
s
Obs
erve
.usi
ng.g
raph
s.an
d.ta
bles
.tha
t.a.
qua
ntit
y.in
crea
sing
.exp
onen
tial
ly.
even
tual
ly.e
xcee
ds.a
.qua
ntit
y.in
crea
sing
.line
arly
,.qua
dra
tica
lly,.
or.(m
ore.
gene
rally
).as.
a.po
lyno
mia
l.fu
ncti
on.
F-L
E.3
XX
X
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els
Con
stru
ct.a
nd.
com
pare
.line
ar,.
qua
dra
tic,
.and
.ex
pone
ntia
l.m
odel
s.an
d.so
lve.
prob
lem
s
For.
expo
nent
ial.m
odel
s,.e
xpre
ss.a
s.a.
loga
rith
m.t
he.s
olut
ion.
to.a
bct.=
.d.w
here
.a,
.c,.a
nd.d
.are
.nu
mbe
rs.a
nd.t
he.b
ase
b is
.2,.1
0,.o
r.e;
.eva
luat
e.th
e.lo
gari
thm
.us
ing.
tech
nolo
gy.
F-L
E.4
XX
XX
80 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-Fu
ncti
ons
Lin
ear,.
Qua
dra
tic,
.and
.E
xpon
enti
al.
Mod
els*
Inte
rpre
t.ex
pres
sion
s.fo
r.fu
ncti
ons.
in.t
erm
s.of
.the
.si
tuat
ion.
they
.m
odel
Inte
rpre
t.th
e.pa
ram
eter
s.in
.a.li
near
.or
.exp
onen
tial
.func
tion
.in.t
erm
s.of
.a.
cont
ext.
F-L
E.5
XX
X.
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Ext
end.
the.
dom
ain.
of.
trig
onom
etri
c.fu
ncti
ons.
usin
g.th
e.u
nit
.cir
cle
Und
erst
and.
rad
ian.
mea
sure
.of.a
n.an
gle.
as.t
he.le
ngth
.of.t
he.a
rc.o
n.th
e.u
nit
.ci
rcle
.su
bten
ded.
by.t
he.a
ngle
.
F-T
F.1
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Ext
end.
the.
dom
ain.
of.
trig
onom
etri
c.fu
ncti
ons.
usin
g.th
e.u
nit
.cir
cle
Exp
lain
.how
.the
.un
it.c
ircl
e.in
.th
e.co
ord
inat
e.pl
ane.
enab
les.
the.
exte
nsi
on.o
f.tri
gono
met
ric.
func
tion
s.to
.all.
real
.nu
mbe
rs,.i
nter
pret
ed.a
s.ra
dia
n.m
easu
res.
of.a
ngle
s.tr
aver
sed.
cou
nter
cloc
kwis
e.ar
ound
.the
.un
it.
circ
le.
F-T
F.2
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Ext
end.
the.
dom
ain.
of.
trig
onom
etri
c.fu
ncti
ons.
usin
g.th
e.u
nit
.cir
cle
(+).U
se.s
peci
al.t
rian
gles
.to.
dete
rmin
e.ge
omet
rica
lly.t
he.v
alue
s.of
.sin
e,.c
osin
e,.
tang
ent.
for.
π/3,
.π/4
.and
.π/6
,.and
.use
.th
e.u
nit
.cir
cle.
to.e
xpre
ss.t
he.v
alue
s.of
.si
ne,.c
osin
e,.a
nd.t
ange
nt.fo
r.π–
x,.π
+x,.
and.
2π–x
.in.t
erm
s.of
.the
ir.v
alue
s.fo
r.x,
.w
here
.x.is
.any
.rea
l.nu
mbe
r.
F-T
F.3
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Ext
end.
the.
dom
ain.
of.
trig
onom
etri
c.fu
ncti
ons.
usin
g.th
e.u
nit
.cir
cle
(+).U
se.t
he.u
nit
.cir
cle.
to.e
xpla
in.
sym
met
ry.(o
dd.a
nd.e
ven)
.and
.pe
riod
icit
y.of
.tri
gono
met
ric.
func
tion
s.
F-T
F.4
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Mod
el.p
erio
dic
.ph
enom
ena.
wit
h.tr
igon
omet
ric.
func
tion
s
Cho
ose.
trig
onom
etri
c.fu
ncti
ons.
to.
mod
el.p
erio
dic
.phe
nom
ena.
wit
h.sp
ecif
ied.
ampl
itud
e,.fr
eque
ncy,
.and
.m
idlin
e.
F-T
F.5
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Mod
el.p
erio
dic
.ph
enom
ena.
wit
h.tr
igon
omet
ric.
func
tion
s
(+).U
nder
stan
d.th
at.r
estr
icti
ng.a
.tr
igon
omet
ric.
func
tion
.to.
a.do
mai
n.on
.w
hic
h.it
.is.a
lway
s.in
crea
sing
.or.
alw
ays.
decr
easi
ng.a
llow
s.it
s.in
vers
e.to
.be.
con
stru
cted
.
F-T
F.6
XX
81College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Mod
el.p
erio
dic
.ph
enom
ena.
wit
h.tr
igon
omet
ric.
func
tion
s
(+).U
se.in
vers
e.fu
ncti
ons.
to.s
olve
.tr
igon
omet
ric.
equa
tion
s.th
at.a
rise
.in
.mod
elin
g.co
ntex
ts;.e
valu
ate.
the.
solu
tion
s.us
ing.
tech
nolo
gy,
.and
.in
terp
ret.
them
.in.t
erm
s.of
.the
.con
text
.
F-T
F.7
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Pro
ve.a
nd.a
pply
.tr
igon
omet
ric.
iden
titi
es
Pro
ve.t
he.P
ytha
gore
an.id
enti
ty.
sin
2 (θ).+
.cos
2 (θ).=
.1.a
nd.u
se.it
.to.
calc
ula
te.t
rigo
nom
etri
c.ra
tios
.
F-T
F.8
XX
Hig
h.Sc
hool
-Fu
ncti
ons
Tri
gono
met
ric.
Fu
ncti
ons
Pro
ve.a
nd.a
pply
.tr
igon
omet
ric.
iden
titi
es
(+).P
rove
.the
.add
itio
n.an
d.su
btra
ctio
n.fo
rmu
las.
for.
sine
,.cos
ine,
.and
.tan
gent
.an
d.us
e.th
em.t
o.so
lve.
prob
lem
s.
F-T
F.9
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Exp
erim
ent.
wit
h.tr
ansf
orm
atio
ns.
in.t
he.p
lane
Kno
w.p
reci
se.d
efin
itio
ns.
of.a
ngle
,.ci
rcle
,.per
pend
icu
lar.
line,
.par
alle
l.lin
e,.a
nd.li
ne.s
egm
ent,
.bas
ed.o
n.th
e.u
ndef
ined
.not
ion
s.of
.poi
nt,.l
ine,
.d
ista
nce.
alon
g.a.
line,
.and
.dis
tanc
e.ar
ound
.a.c
ircu
lar.
arc.
G-C
O.1
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Exp
erim
ent.
wit
h.tr
ansf
orm
atio
ns.
in.t
he.p
lane
Rep
rese
nt.t
ran
sfor
mat
ion
s.in
.the
.pl
ane.
usin
g,.e
.g.,.
tran
spar
enci
es.
and.
geom
etry
.sof
twar
e;.d
escr
ibe.
tran
sfor
mat
ion
s.as
.func
tion
s.th
at.
take
.poi
nts.
in.t
he.p
lane
.as.
inpu
ts.a
nd.
giv
e.ot
her.
poin
ts.a
s.ou
tput
s..C
ompa
re.
tran
sfor
mat
ion
s.th
at.p
rese
rve.
dis
tanc
e.an
d.an
gle.
to.t
hose
.tha
t.do
.not
.(e.g
.,.tr
ansl
atio
n.ve
rsus
.hor
izon
tal.s
tret
ch).
G-C
O.2
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Exp
erim
ent.
wit
h.tr
ansf
orm
atio
ns.
in.t
he.p
lane
Giv
en.a
.rec
tang
le,.p
aral
lelo
gra
m,.
trap
ezoi
d,.o
r.re
gu
lar.
poly
gon,
.des
crib
e.th
e.ro
tati
ons.
and.
refle
ctio
ns.
that
.car
ry.
it.o
nto.
itse
lf.
G-C
O.3
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Exp
erim
ent.
wit
h.tr
ansf
orm
atio
ns.
in.t
he.p
lane
Dev
elop
.def
init
ion
s.of
.rot
atio
ns,
.re
flect
ion
s,.a
nd.t
ran
slat
ion
s.in
.ter
ms.
of.a
ngle
s,.c
ircl
es,.p
erpe
ndic
ula
r.lin
es,.
para
llel.l
ines
,.and
.line
.seg
men
ts.
G-C
O.4
XX
82 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Exp
erim
ent.
wit
h.tr
ansf
orm
atio
ns.
in.t
he.p
lane
Giv
en.a
.geo
met
ric.
figu
re.a
nd.a
.rot
atio
n,.
refle
ctio
n,.o
r.tr
ansl
atio
n,.d
raw
.the
.tr
ansf
orm
ed.fi
gu
re.u
sing
,.e.g
.,.g
raph
.pa
per,.
trac
ing.
pape
r,.or
.geo
met
ry.
soft
war
e..S
peci
fy.a
.seq
uenc
e.of
.tr
ansf
orm
atio
ns.
that
.will
.car
ry.a
.giv
en.
figu
re.o
nto.
anot
her.
G-C
O.5
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Und
erst
and.
cong
ruen
ce.in
.te
rms.
of.r
igid
.m
otio
ns
Use
.geo
met
ric.
desc
ript
ion
s.of
.rig
id.
mot
ion
s.to
.tra
nsf
orm
.fig
ure
s.an
d.to
.pre
dic
t.th
e.ef
fect
.of.a
.giv
en.r
igid
.m
otio
n.on
.a.g
iven
.fig
ure
;.giv
en.
two.
figu
res,
.use
.the
.def
init
ion.
of.
cong
ruen
ce.in
.ter
ms.
of.r
igid
.mot
ion
s.to
.de
cide
.if.t
hey.
are.
cong
ruen
t.
G-C
O.6
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Und
erst
and.
cong
ruen
ce.in
.te
rms.
of.r
igid
.m
otio
ns
Use
.the
.def
init
ion.
of.c
ong
ruen
ce.in
.te
rms.
of.r
igid
.mot
ion
s.to
.sho
w.t
hat.
two.
tria
ngle
s.ar
e.co
ngru
ent.
if.an
d.on
ly.if
.cor
resp
ond
ing.
pair
s.of
.sid
es.
and.
corr
espo
ndin
g.pa
irs.
of.a
ngle
s.ar
e.co
ngru
ent.
G-C
O.7
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Und
erst
and.
cong
ruen
ce.in
.te
rms.
of.r
igid
.m
otio
ns
Exp
lain
.how
.the
.cri
teri
a.fo
r.tr
iang
le.
cong
ruen
ce.(A
SA,.S
AS,
.and
.SSS
).fol
low
.fr
om.t
he.d
efin
itio
n.of
.con
gru
ence
.in.
term
s.of
.rig
id.m
otio
ns.
G-C
O.8
XX
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Pro
ve.g
eom
etri
c.th
eore
ms
Pro
ve.t
heor
ems.
abou
t.lin
es.a
nd.a
ngle
s..
The
orem
s in
clud
e: v
erti
cal a
ngle
s ar
e co
ngru
ent;
whe
n a
tran
sver
sal c
ross
es
para
llel l
ines
, alte
rnat
e in
teri
or a
ngle
s ar
e co
ngru
ent
and
corr
espo
ndin
g an
gles
are
con
gru
ent;
poin
ts o
n a
perp
end
icu
lar
bise
ctor
of a
line
seg
men
t ar
e ex
actly
tho
se e
qu
idis
tant
from
the
se
gm
ent’s
end
poin
ts.
G-C
O.9
XX
83College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Pro
ve.g
eom
etri
c.th
eore
ms
Pro
ve.t
heor
ems.
abou
t.tr
iang
les.
.T
heor
ems
incl
ude:
mea
sure
s of
inte
rior
an
gles
of a
tri
angl
e su
m t
o 18
0°;
base
ang
les
of is
osce
les
tria
ngle
s ar
e co
ngru
ent;
the
seg
men
t jo
inin
g m
idpo
ints
of t
wo
side
s of
a t
rian
gle
is
para
llel t
o th
e th
ird
side
and
hal
f the
le
ngth
; the
med
ians
of a
tri
angl
e m
eet
at a
poi
nt.
G-C
O.1
0X
X
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Pro
ve.g
eom
etri
c.th
eore
ms
Pro
ve.t
heor
ems.
abou
t.pa
ralle
log
ram
s..
The
orem
s in
clud
e: o
ppos
ite
side
s ar
e co
ngru
ent,
opp
osit
e an
gles
ar
e co
ngru
ent,
the
dia
gona
ls o
f a
para
llelo
gra
m b
isec
t ea
ch o
ther
, an
d co
nver
sely
, rec
tang
les
are
para
llelo
gra
ms
wit
h co
ngru
ent
dia
gona
ls.
G-C
O.1
1X
X
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Mak
e.ge
omet
ric.
con
stru
ctio
ns
Mak
e.fo
rmal
.geo
met
ric.
con
stru
ctio
ns.
wit
h.a.
vari
ety.
of.t
ools
.and
.met
hods
.(c
ompa
ss.a
nd.s
trai
ghte
dge,
.str
ing,
.re
flect
ive.
devi
ces,
.pap
er.fo
ldin
g,.
dyna
mic
.geo
met
ric.
soft
war
e,.e
tc.).
.C
opyi
ng a
seg
men
t; co
pyin
g an
ang
le;
bise
ctin
g a
seg
men
t; bi
sect
ing
an
angl
e; c
onst
ruct
ing
perp
end
icu
lar
lines
, in
clud
ing
the
perp
end
icu
lar
bise
ctor
of
a lin
e se
gm
ent;
and
cons
truc
ting
a li
ne
para
llel t
o a
giv
en li
ne t
hro
ugh
a po
int
not
on t
he li
ne.
G-C
O.1
2X
X
Hig
h.Sc
hool
-G
eom
etry
Con
gru
ence
Mak
e.ge
omet
ric.
con
stru
ctio
ns
Con
stru
ct.a
n.eq
uila
tera
l.tri
angl
e,.
a.sq
uare
,.and
.a.r
egu
lar.
hexa
gon.
insc
ribe
d.in
.a.c
ircl
e.
G-C
O.1
3X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Und
erst
and.
sim
ilari
ty.
in.t
erm
s.of
.si
mila
rity
.tr
ansf
orm
atio
ns
Ver
ify.
expe
rim
enta
lly.t
he.p
rope
rtie
s.of
.d
ilati
ons.
giv
en.b
y.a.
cent
er.a
nd.a
.sca
le.
fact
or:
A.d
ilati
on.t
akes
.a.li
ne.
not.
pass
ing.
thro
ugh.
the.
cent
er.o
f.the
.dila
tion
.to.
a.pa
ralle
l.lin
e,.a
nd.le
aves
.a.
line.
pass
ing.
thro
ugh.
the.
cent
er.u
ncha
nged
.
G-S
RT.
1aX
X
84 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Und
erst
and.
sim
ilari
ty.
in.t
erm
s.of
.si
mila
rity
.tr
ansf
orm
atio
ns
Ver
ify.
expe
rim
enta
lly.t
he.p
rope
rtie
s.of
.d
ilati
ons.
giv
en.b
y.a.
cent
er.a
nd.a
.sca
le.
fact
or:
The
.dila
tion
.of.a
.line
.se
gm
ent.
is.lo
nger
.or.
shor
ter.
in.t
he.r
atio
.giv
en.
by.t
he.s
cale
.fact
or.
G-S
RT.
1bX
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Und
erst
and.
sim
ilari
ty.
in.t
erm
s.of
.si
mila
rity
.tr
ansf
orm
atio
ns
Giv
en.t
wo.
figu
res,
.use
.the
.def
init
ion.
of.s
imila
rity
.in.t
erm
s.of
.sim
ilari
ty.
tran
sfor
mat
ion
s.to
.dec
ide.
if.th
ey.
are.
sim
ilar;.
expl
ain.
usin
g.si
mila
rity
.tr
ansf
orm
atio
ns.
the.
mea
nin
g.of
.si
mila
rity
.for.
tria
ngle
s.as
.the
.eq
ualit
y.of
.all.
corr
espo
ndin
g.pa
irs.
of.a
ngle
s.an
d.th
e.pr
opor
tion
alit
y.of
.all.
corr
espo
ndin
g.pa
irs.
of.s
ides
.
G-S
RT.
2X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Und
erst
and.
sim
ilari
ty.
in.t
erm
s.of
.si
mila
rity
.tr
ansf
orm
atio
ns
Use
.the
.pro
pert
ies.
of.s
imila
rity
.tr
ansf
orm
atio
ns.
to.e
stab
lish.
the.
AA
.cr
iter
ion.
for.
two.
tria
ngle
s.to
.be.
sim
ilar.
G-S
RT.
3X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Pro
ve.t
heor
ems.
invo
lvin
g.si
mila
rity
Pro
ve.t
heor
ems.
abou
t.tr
iang
les.
.T
heor
ems
incl
ude:
a li
ne p
aral
lel t
o on
e si
de o
f a t
rian
gle
div
ides
the
oth
er
two
prop
orti
onal
ly, a
nd c
onve
rsel
y; t
he
Pyt
hago
rean
The
orem
pro
ved
usin
g tr
iang
le s
imila
rity
.
G-S
RT.
4X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Pro
ve.t
heor
ems.
invo
lvin
g.si
mila
rity
Use
.con
gru
ence
.and
.sim
ilari
ty.c
rite
ria.
for.
tria
ngle
s.to
.sol
ve.p
robl
ems.
and.
to.p
rove
.rel
atio
nsh
ips.
in.g
eom
etri
c.fig
ure
s.
G-S
RT.
5X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Def
ine.
trig
onom
etri
c.ra
tios
.and
.so
lve.
prob
lem
s.in
volv
ing.
righ
t.tr
iang
les
Und
erst
and.
that
.by.
sim
ilari
ty,.s
ide.
rati
os.in
.rig
ht.t
rian
gles
.are
.pro
pert
ies.
of.t
he.a
ngle
s.in
.the
.tri
angl
e,.le
adin
g.to
.de
fin
itio
ns.
of.t
rigo
nom
etri
c.ra
tios
.for.
acut
e.an
gles
.
G-S
RT.
6X
X
85College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Def
ine.
trig
onom
etri
c.ra
tios
.and
.so
lve.
prob
lem
s.in
volv
ing.
righ
t.tr
iang
les
Exp
lain
.and
.use
.the
.rel
atio
nsh
ip.
betw
een.
the.
sine
.and
.cos
ine.
of.
com
plem
enta
ry.a
ngle
s.
G-S
RT.
7X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
Def
ine.
trig
onom
etri
c.ra
tios
.and
.so
lve.
prob
lem
s.in
volv
ing.
righ
t.tr
iang
les
Use
.tri
gono
met
ric.
rati
os.a
nd.t
he.
Pyt
hago
rean
.The
orem
.to.
solv
e.ri
ght.
tria
ngle
s.in
.app
lied.
prob
lem
s.
G-S
RT.
8X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
App
ly.
trig
onom
etry
.to.
gene
ral.t
rian
gles
(+).D
eriv
e.th
e.fo
rmu
la.A
.=.1
/2.a
b.si
n(C
).for
.the
.are
a.of
.a.t
rian
gle.
by.
dra
win
g.an
.au
xilia
ry.li
ne.fr
om.a
.ver
tex.
perp
end
icu
lar.
to.t
he.o
ppos
ite.
side
.
G-S
RT.
9X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
App
ly.
trig
onom
etry
.to.
gene
ral.t
rian
gles
(+).P
rove
.the
.Law
s.of
.Sin
es.a
nd.C
osin
es.
and.
use.
them
.to.
solv
e.pr
oble
ms.
G-S
RT.
10X
X
Hig
h.Sc
hool
-G
eom
etry
Sim
ilari
ty,.
Rig
ht.
Tri
angl
es,.a
nd.
Tri
gono
met
ry
App
ly.
trig
onom
etry
.to.
gene
ral.t
rian
gles
(+).U
nder
stan
d.an
d.ap
ply.
the.
Law
.of.
Sine
s.an
d.th
e.L
aw.o
f.Cos
ines
.to.
find
.u
nkn
own.
mea
sure
men
ts.in
.rig
ht.a
nd.
non-
righ
t.tr
iang
les.
(e.g
.,.su
rvey
ing.
prob
lem
s,.r
esu
ltant
.forc
es).
G-S
RT.
11X
X
Hig
h.Sc
hool
-G
eom
etry
Cir
cles
Und
erst
and.
and.
appl
y.th
eore
ms.
abou
t.ci
rcle
s
Pro
ve.t
hat.
all.c
ircl
es.a
re.s
imila
r.G
-C.1
XX
Hig
h.Sc
hool
-G
eom
etry
Cir
cles
Und
erst
and.
and.
appl
y.th
eore
ms.
abou
t.ci
rcle
s
Iden
tify
.and
.des
crib
e.re
lati
onsh
ips.
amon
g.in
scri
bed.
angl
es,.r
adii,
.and
.ch
ords
..Inc
lude
.the
.rel
atio
nsh
ip.
betw
een.
cent
ral,.
insc
ribe
d,.a
nd.
circ
um
scri
bed.
angl
es;.i
nsc
ribe
d.an
gles
.on
.a.d
iam
eter
.are
.rig
ht.a
ngle
s;.t
he.
rad
ius.
of.a
.cir
cle.
is.p
erpe
ndic
ula
r.to
.the
.ta
ngen
t.w
here
.the
.rad
ius.
inte
rsec
ts.t
he.
circ
le.
G-C
.2X
X
86 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Cir
cles
Und
erst
and.
and.
appl
y.th
eore
ms.
abou
t.ci
rcle
s
Con
stru
ct.t
he.in
scri
bed.
and.
circ
um
scri
bed.
circ
les.
of.a
.tri
angl
e,.
and.
prov
e.pr
oper
ties
.of.a
ngle
s.fo
r.a.
qua
dri
late
ral.i
nsc
ribe
d.in
.a.c
ircl
e.
G-C
.3X
X
Hig
h.Sc
hool
-G
eom
etry
Cir
cles
Und
erst
and.
and.
appl
y.th
eore
ms.
abou
t.ci
rcle
s
(+).C
onst
ruct
.a.t
ange
nt.li
ne.fr
om.a
.poi
nt.
outs
ide.
a.g
iven
.cir
cle.
to.t
he.c
ircl
e.G
-C.4
XX
Hig
h.Sc
hool
-G
eom
etry
Cir
cles
Fin
d.ar
c.le
ngth
s.an
d.ar
eas.
of.
sect
ors.
of.c
ircl
es
Der
ive.
usin
g.si
mila
rity
.the
.fact
.tha
t.th
e.le
ngth
.of.t
he.a
rc.in
terc
epte
d.by
.an
.ang
le.is
.pro
port
iona
l.to.
the.
rad
ius,
.an
d.de
fine
.the
.rad
ian.
mea
sure
.of.t
he.
angl
e.as
.the
.con
stan
t.of
.pro
port
iona
lity;
.de
rive
.the
.form
ula
.for.
the.
area
.of.a
.se
ctor
.
G-C
.5X
X
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Tra
nsl
ate.
betw
een.
the.
geom
etri
c.de
scri
ptio
n.an
d.th
e.eq
uati
on.fo
r.a.
con
ic.s
ecti
on
Der
ive.
the.
equa
tion
.of.a
.cir
cle.
of.
giv
en.c
ente
r.an
d.ra
diu
s.us
ing.
the.
Pyt
hago
rean
.The
orem
;.com
plet
e.th
e.sq
uare
.to.
find
.the
.cen
ter.
and.
rad
ius.
of.a
.ci
rcle
.giv
en.b
y.an
.eq
uati
on.
G-G
PE.1
XX
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Tra
nsl
ate.
betw
een.
the.
geom
etri
c.de
scri
ptio
n.an
d.th
e.eq
uati
on.fo
r.a.
con
ic.s
ecti
on
Der
ive.
the.
equa
tion
.of.a
.par
abol
a.g
iven
.a.
focu
s.an
d.d
irec
trix
.G
-GPE
.2X
X
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Tra
nsl
ate.
betw
een.
the.
geom
etri
c.de
scri
ptio
n.an
d.th
e.eq
uati
on.fo
r.a.
con
ic.s
ecti
on
(+).D
eriv
e.th
e.eq
uati
ons.
of.e
llips
es.a
nd.
hype
rbol
as.g
iven
.the
.foci
,.usi
ng.t
he.fa
ct.
that
.the
.su
m.o
r.d
iffe
renc
e.of
.dis
tanc
es.
from
.the
.foci
.is.c
onst
ant.
G-G
PE.3
XX
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Use
.coo
rdin
ates
.to
.pro
ve.s
impl
e.ge
omet
ric.
theo
rem
s.al
gebr
aica
lly
Use
.coo
rdin
ates
.to.
prov
e.si
mpl
e.ge
omet
ric.
theo
rem
s.al
gebr
aica
lly.
G-G
PE.4
XX
87College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Use
.coo
rdin
ates
.to
.pro
ve.s
impl
e.ge
omet
ric.
theo
rem
s.al
gebr
aica
lly
Pro
ve.t
he.s
lope
.cri
teri
a.fo
r.pa
ralle
l.an
d.pe
rpen
dic
ula
r.lin
es.a
nd.u
se.t
hem
.to
.sol
ve.g
eom
etri
c.pr
oble
ms.
(e.g
.,.fi
nd.t
he.e
qua
tion
.of.a
.line
.par
alle
l.or
.per
pend
icu
lar.
to.a
.giv
en.li
ne.t
hat.
pass
es.t
hro
ugh.
a.g
iven
.poi
nt).
G-G
PE.5
XX
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Use
.coo
rdin
ates
.to
.pro
ve.s
impl
e.ge
omet
ric.
theo
rem
s.al
gebr
aica
lly
Fin
d.th
e.po
int.
on.a
.dir
ecte
d.lin
e.se
gm
ent.
betw
een.
two.
giv
en.p
oint
s.th
at.
part
itio
ns.
the.
seg
men
t.in
.a.g
iven
.rat
io.
G-G
PE.6
XX
Hig
h.Sc
hool
-G
eom
etry
Exp
ress
ing.
Geo
met
ric.
Pro
pert
ies.
Eq
uati
ons
Use
.coo
rdin
ates
.to
.pro
ve.s
impl
e.ge
omet
ric.
theo
rem
s.al
gebr
aica
lly
Use
.coo
rdin
ates
.to.
com
pute
.per
imet
ers.
of.p
olyg
ons.
and.
area
s.of
.tri
angl
es.a
nd.
rect
angl
es,.e
.g.,.
usin
g.th
e.d
ista
nce.
form
ula
.
G-G
PE.7
XX
Hig
h.Sc
hool
-G
eom
etry
Geo
met
ric.
Mea
sure
men
t.an
d.D
imen
sion
Exp
lain
.vol
um
e.fo
rmu
las.
and.
use.
them
.to.
solv
e.pr
oble
ms
Giv
e.an
.info
rmal
.arg
um
ent.
for.
the.
form
ula
s.fo
r.th
e.ci
rcu
mfe
renc
e.of
.a.
circ
le,.a
rea.
of.a
.cir
cle,
.vol
um
e.of
.a.c
ylin
der,.
pyra
mid
,.and
.con
e..
Use
dis
sect
ion
arg
um
ents
, Cav
alie
ri’s
pr
inci
ple,
and
info
rmal
lim
it ar
gu
men
ts.
G-G
MD
.1X
X
Hig
h.Sc
hool
-G
eom
etry
Geo
met
ric.
Mea
sure
men
t.an
d.D
imen
sion
Exp
lain
.vol
um
e.fo
rmu
las.
and.
use.
them
.to.
solv
e.pr
oble
ms
(+).G
ive.
an.in
form
al.a
rgu
men
t.us
ing.
Cav
alie
ri’s
.pri
ncip
le.fo
r.th
e.fo
rmu
las.
for.
the.
volu
me.
of.a
.sph
ere.
and.
othe
r.so
lid.
figu
res.
G-G
MD
.2X
X
Hig
h.Sc
hool
-G
eom
etry
Geo
met
ric.
Mea
sure
men
t.an
d.D
imen
sion
Exp
lain
.vol
um
e.fo
rmu
las.
and.
use.
them
.to.
solv
e.pr
oble
ms
Use
.vol
um
e.fo
rmu
las.
for.
cylin
ders
,.py
ram
ids,
.con
es,.a
nd.s
pher
es.t
o.so
lve.
prob
lem
s.
G-G
MD
.3X
X
Hig
h.Sc
hool
-G
eom
etry
Geo
met
ric.
Mea
sure
men
t.an
d.D
imen
sion
Vis
ualiz
e.re
lati
onsh
ips.
betw
een.
two-
dim
ensi
onal
.an
d.th
ree-
dim
ensi
onal
.ob
ject
s
Iden
tify
.the
.sha
pes.
of.t
wo-
dim
ensi
onal
.cr
oss-
sect
ion
s.of
.th
ree-
dim
ensi
onal
.ob
ject
s,.a
nd.id
enti
fy.t
hre
e-d
imen
sion
al.
obje
cts.
gene
rate
d.by
.rot
atio
ns.
of.t
wo-
dim
ensi
onal
.obj
ects
.
G-G
MD
.4X
X
88 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-G
eom
etry
Mod
elin
g.w
ith.
Geo
met
ryA
pply
.geo
met
ric.
conc
epts
.in.
mod
elin
g.si
tuat
ion
s
Use
.geo
met
ric.
shap
es,.t
heir
.mea
sure
s,.
and.
thei
r.pr
oper
ties
.to.
desc
ribe
.obj
ects
.(e
.g.,.
mod
elin
g.a.
tree
.tru
nk.
or.a
.hu
man
.to
rso.
as.a
.cyl
inde
r).
G-M
G.1
XX
Hig
h.Sc
hool
-G
eom
etry
Mod
elin
g.w
ith.
Geo
met
ryA
pply
.geo
met
ric.
conc
epts
.in.
mod
elin
g.si
tuat
ion
s
App
ly.c
once
pts.
of.d
ensi
ty.b
ased
.on.
area
.and
.vol
um
e.in
.mod
elin
g.si
tuat
ion
s.(e
.g.,.
pers
ons.
per.
squa
re.m
ile,.B
TU
s.pe
r.cu
bic.
foot
).
G-M
G.2
XX
Hig
h.Sc
hool
-G
eom
etry
Mod
elin
g.w
ith.
Geo
met
ryA
pply
.geo
met
ric.
conc
epts
.in.
mod
elin
g.si
tuat
ion
s
App
ly.g
eom
etri
c.m
etho
ds.t
o.so
lve.
desi
gn.
prob
lem
s.(e
.g.,.
desi
gn
ing.
an.
obje
ct.o
r.st
ruct
ure
.to.
sati
sfy.
phys
ical
.co
nst
rain
ts.o
r.m
inim
ize.
cost
;.wor
king
.w
ith.
typo
gra
phic
.gri
d.sy
stem
s.ba
sed.
on.r
atio
s).
G-M
G.3
XX
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
a.si
ngle
.cou
nt.
or.m
easu
rem
ent.
vari
able
.
Rep
rese
nt.d
ata.
wit
h.pl
ots.
on.t
he.r
eal.
num
ber.
line.
(dot
.plo
ts,.h
isto
gra
ms,
.and
.bo
x.pl
ots)
.
S-ID
.1X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
a.si
ngle
.cou
nt.
or.m
easu
rem
ent.
vari
able
.
Use
.sta
tist
ics.
appr
opri
ate.
to.t
he.s
hape
.of
.the
.dat
a.d
istr
ibut
ion.
to.c
ompa
re.
cent
er.(m
edia
n,.m
ean)
.and
.spr
ead.
(inte
rqua
rtile
.ran
ge,.s
tand
ard.
devi
atio
n).
of.t
wo.
or.m
ore.
dif
fere
nt.d
ata.
sets
.
S-ID
.2X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
a.si
ngle
.cou
nt.
or.m
easu
rem
ent.
vari
able
.
Inte
rpre
t.d
iffe
renc
es.in
.sha
pe,.c
ente
r,.an
d.sp
read
.in.t
he.c
onte
xt.o
f.the
.dat
a.se
ts,.a
ccou
ntin
g.fo
r.po
ssib
le.e
ffect
s.of
.ex
trem
e.da
ta.p
oint
s.(o
utlie
rs).
S-ID
.3X
89College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
a.si
ngle
.cou
nt.
or.m
easu
rem
ent.
vari
able
.
Use
.the
.mea
n.an
d.st
anda
rd.d
evia
tion
.of
.a.d
ata.
set.
to.fi
t.it
.to.
a.no
rmal
.d
istr
ibut
ion.
and.
to.e
stim
ate.
popu
lati
on.
perc
enta
ges.
.Rec
ogn
ize.
that
.the
re.a
re.
data
.set
s.fo
r.w
hic
h.su
ch.a
.pro
cedu
re.
is.n
ot.a
ppro
pria
te..U
se.c
alcu
lato
rs,.
spre
adsh
eets
,.and
.tab
les.
to.e
stim
ate.
area
s.u
nder
.the
.nor
mal
.cu
rve.
S-ID
.4X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
two.
cate
gori
cal.
and.
qua
ntit
ativ
e.va
riab
les.
Sum
mar
ize.
cate
gori
cal.d
ata.
for.
two.
cate
gori
es.in
.tw
o-w
ay.fr
eque
ncy.
tabl
es..I
nter
pret
.rel
ativ
e.fr
eque
ncie
s.in
.the
.con
text
.of.t
he.d
ata.
(incl
udin
g.jo
int,
.mar
gin
al,.a
nd.c
ond
itio
nal.r
elat
ive.
freq
uenc
ies)
..Rec
ogn
ize.
poss
ible
.as
soci
atio
ns.
and.
tren
ds.in
.the
.dat
a.
S-ID
.5X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
two.
cate
gori
cal.
and.
qua
ntit
ativ
e.va
riab
les.
Rep
rese
nt.d
ata.
on.t
wo.
qua
ntit
ativ
e.va
riab
les.
on.a
.sca
tter
.plo
t,.a
nd.d
escr
ibe.
how
.the
.var
iabl
es.a
re.r
elat
ed.
Fit
.a.fu
ncti
on.t
o.th
e.da
ta;.u
se.fu
ncti
ons.
fitte
d.to
.dat
a.to
.sol
ve.
prob
lem
s.in
.the
.con
text
.of
.the
.dat
a..U
se g
iven
fu
ncti
ons
or c
hoos
e a
func
tion
sug
gest
ed b
y th
e co
ntex
t. E
mph
asiz
e lin
ear,
qua
dra
tic,
and
exp
onen
tial
m
odel
s.
S-ID
.6a
XX
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
two.
cate
gori
cal.
and.
qua
ntit
ativ
e.va
riab
les.
Rep
rese
nt.d
ata.
on.t
wo.
qua
ntit
ativ
e.va
riab
les.
on.a
.sca
tter
.plo
t,.a
nd.d
escr
ibe.
how
.the
.var
iabl
es.a
re.r
elat
ed.
Info
rmal
ly.a
sses
s.th
e.fit
.of.
a.fu
ncti
on.b
y.pl
otti
ng.a
nd.
anal
yzin
g.re
sidu
als.
S-ID
.6b
XX
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Sum
mar
ize,
.re
pres
ent,
.and
.in
terp
ret.
data
.on.
two.
cate
gori
cal.
and.
qua
ntit
ativ
e.va
riab
les.
Rep
rese
nt.d
ata.
on.t
wo.
qua
ntit
ativ
e.va
riab
les.
on.a
.sca
tter
.plo
t,.a
nd.d
escr
ibe.
how
.the
.var
iabl
es.a
re.r
elat
ed.
Fit
.a.li
near
.func
tion
.for.
a.sc
atte
r.pl
ot.t
hat.
sugg
ests
.a.
linea
r.as
soci
atio
n.
S-ID
.6c
XX
X
90 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Inte
rpre
t.lin
ear.
mod
els
Inte
rpre
t.th
e.sl
ope.
(rat
e.of
.cha
nge)
.and
.th
e.in
terc
ept.
(con
stan
t.te
rm).o
f.a.li
near
.m
odel
.in.t
he.c
onte
xt.o
f.the
.dat
a.
S-ID
.7X
XX
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Inte
rpre
t.lin
ear.
mod
els
Com
pute
.(usi
ng.t
ech
nolo
gy)
.and
.in
terp
ret.
the.
corr
elat
ion.
coef
ficie
nt.o
f.a.
linea
r.fit
.
S-ID
.8X
XX
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Inte
rpre
ting
.C
ateg
oric
al.a
nd.
Qua
ntit
ativ
e.D
ata
Inte
rpre
t.lin
ear.
mod
els
Dis
ting
uis
h.be
twee
n.co
rrel
atio
n.an
d.ca
usat
ion.
S-ID
.9X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Mak
ing.
Infe
renc
es.
and.
Just
ifyi
ng.
Con
clus
ion
s
Und
erst
and.
and.
eval
uate
.ran
dom
.pr
oces
ses.
und
erly
ing.
stat
isti
cal.
expe
rim
ents
Und
erst
and.
stat
isti
cs.a
s.a.
proc
ess.
for.
mak
ing.
infe
renc
es.a
bout
.pop
ula
tion
.pa
ram
eter
s.ba
sed.
on.a
.ran
dom
.sam
ple.
from
.tha
t.po
pula
tion
.
S-IC
.1X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Mak
ing.
Infe
renc
es.
and.
Just
ifyi
ng.
Con
clus
ion
s
Und
erst
and.
and.
eval
uate
.ran
dom
.pr
oces
ses.
und
erly
ing.
stat
isti
cal.
expe
rim
ents
Dec
ide.
if.a.
spec
ifie
d.m
odel
.is.
con
sist
ent.
wit
h.re
sults
.from
.a.g
iven
.da
ta-g
ener
atin
g.pr
oces
s,.e
.g.,.
usin
g.si
mu
lati
on.
S-IC
.2X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Mak
ing.
Infe
renc
es.
and.
Just
ifyi
ng.
Con
clus
ion
s
Mak
e.in
fere
nces
.an
d.ju
stif
y.co
nclu
sion
s.fr
om.
sam
ple.
surv
eys,
.ex
peri
men
ts,.a
nd.
obse
rvat
iona
l.st
udie
s
Rec
ogn
ize.
the.
purp
oses
.of.a
nd.
dif
fere
nces
.am
ong.
sam
ple.
surv
eys,
.ex
peri
men
ts,.a
nd.o
bser
vati
onal
.stu
die
s;.
expl
ain.
how
.ran
dom
izat
ion.
rela
tes.
to.
each
.
S-IC
.3X
91College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Mak
ing.
Infe
renc
es.
and.
Just
ifyi
ng.
Con
clus
ion
s
Mak
e.in
fere
nces
.an
d.ju
stif
y.co
nclu
sion
s.fr
om.
sam
ple.
surv
eys,
.ex
peri
men
ts,.a
nd.
obse
rvat
iona
l.st
udie
s
Use
.dat
a.fr
om.a
.sam
ple.
surv
ey.
to.e
stim
ate.
a.po
pula
tion
.mea
n.or
.pr
opor
tion
;.dev
elop
.a.m
arg
in.o
f.err
or.
thro
ugh.
the.
use.
of.s
imu
lati
on.m
odel
s.fo
r.ra
ndom
.sam
plin
g.
S-IC
.4X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Mak
ing.
Infe
renc
es.
and.
Just
ifyi
ng.
Con
clus
ion
s
Mak
e.in
fere
nces
.an
d.ju
stif
y.co
nclu
sion
s.fr
om.
sam
ple.
surv
eys,
.ex
peri
men
ts,.a
nd.
obse
rvat
iona
l.st
udie
s
Use
.dat
a.fr
om.a
.ran
dom
ized
.exp
erim
ent.
to.c
ompa
re.t
wo.
trea
tmen
ts;.u
se.
sim
ula
tion
s.to
.dec
ide.
if.d
iffe
renc
es.
betw
een.
para
met
ers.
are.
sig
nif
ican
t.
S-IC
.5X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Mak
ing.
Infe
renc
es.
and.
Just
ifyi
ng.
Con
clus
ion
s
Mak
e.in
fere
nces
.an
d.ju
stif
y.co
nclu
sion
s.fr
om.
sam
ple.
surv
eys,
.ex
peri
men
ts,.a
nd.
obse
rvat
iona
l.st
udie
s
Eva
luat
e.re
port
s.ba
sed.
on.d
ata.
S-IC
.6X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Und
erst
and.
inde
pend
ence
.an
d.co
ndit
iona
l.pr
obab
ility
.and
.us
e.th
em.t
o.in
terp
ret.
data
Des
crib
e.ev
ents
.as.
subs
ets.
of.a
.sam
ple.
spac
e.(t
he.s
et.o
f.out
com
es).u
sing
.ch
arac
teri
stic
s.(o
r.ca
tego
ries
).of.t
he.
outc
omes
,.or.
as.u
nio
ns,
.inte
rsec
tion
s,.
or.c
ompl
emen
ts.o
f.oth
er.e
vent
s.(“
or,”
.“a
nd,”
.“no
t”).
S-C
P.1
X.
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Und
erst
and.
inde
pend
ence
.an
d.co
ndit
iona
l.pr
obab
ility
.and
.us
e.th
em.t
o.in
terp
ret.
data
Und
erst
and.
that
.tw
o.ev
ents
.A.a
nd.
B.a
re.in
depe
nden
t.if.
the.
prob
abili
ty.
of.A
.and
.B.o
ccu
rrin
g.to
geth
er.is
.the
.pr
oduc
t.of
.the
ir.p
roba
bilit
ies,
.and
.use
.th
is.c
hara
cter
izat
ion.
to.d
eter
min
e.if
.th
ey.a
re.in
depe
nden
t.
S-C
P.2
X
92 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Und
erst
and.
inde
pend
ence
.an
d.co
ndit
iona
l.pr
obab
ility
.and
.us
e.th
em.t
o.in
terp
ret.
data
Und
erst
and.
the.
cond
itio
nal.p
roba
bilit
y.of
.A.g
iven
.B.a
s.P
(A.a
nd.B
)/P
(B),.
and.
inte
rpre
t.in
depe
nden
ce.o
f.A.a
nd.B
.as.
sayi
ng.t
hat.
the.
cond
itio
nal.p
roba
bilit
y.of
.A.g
iven
.B.is
.the
.sam
e.as
.the
.pr
obab
ility
.of.A
,.and
.the
.con
dit
iona
l.pr
obab
ility
.of.B
.giv
en.A
.is.t
he.s
ame.
as.
the.
prob
abili
ty.o
f.B.
S-C
P.3
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Und
erst
and.
inde
pend
ence
.an
d.co
ndit
iona
l.pr
obab
ility
.and
.us
e.th
em.t
o.in
terp
ret.
data
Con
stru
ct.a
nd.in
terp
ret.
two-
way
.fr
eque
ncy.
tabl
es.o
f.dat
a.w
hen.
two.
cate
gori
es.a
re.a
ssoc
iate
d.w
ith.
each
.ob
ject
.bei
ng.c
lass
ifie
d..U
se.t
he.t
wo-
way
.tab
le.a
s.a.
sam
ple.
spac
e.to
.dec
ide.
if.ev
ents
.are
.inde
pend
ent.
and.
to.
appr
oxim
ate.
cond
itio
nal.p
roba
bilit
ies.
S-C
P.4
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Und
erst
and.
inde
pend
ence
.an
d.co
ndit
iona
l.pr
obab
ility
.and
.us
e.th
em.t
o.in
terp
ret.
data
Rec
ogn
ize.
and.
expl
ain.
the.
conc
epts
.of
.con
dit
iona
l.pro
babi
lity.
and.
inde
pend
ence
.in.e
very
day.
lang
uage
.an
d.ev
eryd
ay.s
itua
tion
s.
S-C
P.5
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Use
.the
.ru
les.
of.p
roba
bilit
y.to
.com
pute
.pr
obab
iliti
es.o
f.co
mpo
und
.eve
nts.
in.a
.un
iform
.pr
obab
ility
.mod
el
Fin
d.th
e.co
ndit
iona
l.pro
babi
lity.
of.A
.g
iven
.B.a
s.th
e.fr
acti
on.o
f.B’s
.out
com
es.
that
.als
o.be
long
.to.
A,.a
nd.in
terp
ret.
the.
answ
er.in
.ter
ms.
of.t
he.m
odel
.
S-C
P.6
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Use
.the
.ru
les.
of.p
roba
bilit
y.to
.com
pute
.pr
obab
iliti
es.o
f.co
mpo
und
.eve
nts.
in.a
.un
iform
.pr
obab
ility
.mod
el
App
ly.t
he.A
ddit
ion.
Ru
le,.
P(A
.or.
B).=
.P(A
).+.P
(B).–
.P(A
.and
.B),.
and.
inte
rpre
t.th
e.an
swer
.in.t
erm
s.of
.th
e.m
odel
.
S-C
P.7
X
93College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Use
.the
.ru
les.
of.p
roba
bilit
y.to
.com
pute
.pr
obab
iliti
es.o
f.co
mpo
und
.eve
nts.
in.a
.un
iform
.pr
obab
ility
.mod
el
(+).A
pply
.the
.gen
eral
.Mu
ltipl
icat
ion.
Ru
le.in
.a.u
nifo
rm.p
roba
bilit
y.m
odel
,.P
(A.a
nd.B
).=.P
(A)P
(B|A
).=.P
(B)P
(A|B
),.an
d.in
terp
ret.
the.
answ
er.in
.ter
ms.
of.
the.
mod
el.
S-C
P.8
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Con
dit
iona
l.P
roba
bilit
y.an
d.th
e.R
ule
s.of
.P
roba
bilit
y
Use
.the
.ru
les.
of.p
roba
bilit
y.to
.com
pute
.pr
obab
iliti
es.o
f.co
mpo
und
.eve
nts.
in.a
.un
iform
.pr
obab
ility
.mod
el
(+).U
se.p
erm
utat
ion
s.an
d.co
mbi
nati
ons.
to.c
ompu
te.p
roba
bilit
ies.
of.c
ompo
und
.ev
ents
.and
.sol
ve.p
robl
ems.
S-C
P.9
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Cal
cula
te.
expe
cted
.val
ues.
and.
use.
them
.to.
solv
e.pr
oble
ms
(+).D
efin
e.a.
rand
om.v
aria
ble.
for.
a.q
uant
ity.
of.in
tere
st.b
y.as
sig
nin
g.a.
num
eric
al.v
alue
.to.
each
.eve
nt.in
.a.
sam
ple.
spac
e;.g
raph
.the
.cor
resp
ond
ing.
prob
abili
ty.d
istr
ibut
ion.
usin
g.th
e.sa
me.
gra
phic
al.d
ispl
ays.
as.fo
r.da
ta.
dis
trib
utio
ns.
S-M
D.1
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Cal
cula
te.
expe
cted
.val
ues.
and.
use.
them
.to.
solv
e.pr
oble
ms
(+).C
alcu
late
.the
.exp
ecte
d.va
lue.
of.a
.ra
ndom
.var
iabl
e;.in
terp
ret.
it.a
s.th
e.m
ean.
of.t
he.p
roba
bilit
y.d
istr
ibut
ion.
S-M
D.2
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Cal
cula
te.
expe
cted
.val
ues.
and.
use.
them
.to.
solv
e.pr
oble
ms
(+).D
evel
op.a
.pro
babi
lity.
dis
trib
utio
n.fo
r.a.
rand
om.v
aria
ble.
defi
ned.
for.
a.sa
mpl
e.sp
ace.
in.w
hic
h.th
eore
tica
l.pro
babi
litie
s.ca
n.be
.cal
cula
ted;
.find
.the
.exp
ecte
d.va
lue.
S-M
D.3
X
Hig
h.Sc
hool
-St
atis
tics.
and.
Prob
abili
ty
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Cal
cula
te.
expe
cted
.val
ues.
and.
use.
them
.to.
solv
e.pr
oble
ms
(+).D
evel
op.a
.pro
babi
lity.
dis
trib
utio
n.fo
r.a.
rand
om.v
aria
ble.
defi
ned.
for.
a.sa
mpl
e.sp
ace.
in.w
hic
h.pr
obab
iliti
es.a
re.
assi
gne
d.em
piri
cally
;.fin
d.th
e.ex
pect
ed.
valu
e.
S-M
D.4
X
94 College Board Research Reports
Appendixes
Com
mon
Cor
e S
tate
Sta
nd
ard
sA
P C
ours
e
Gra
de
Dom
ain
Clu
ster
Sta
nd
ard
Sk
ill
Cod
e
AP
C
alcu
lus
AB
AP
C
alcu
lus
BC
AP
S
tati
stic
s A
P
Com
pu
ter
Sci
ence
Hig
h.Sc
hool
-St
atis
tics
.an
d.Pr
obab
ility
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Use
.pro
babi
lity.
to.e
valu
ate.
outc
omes
.of.
deci
sion
s
(+).W
eigh
.the
.pos
sibl
e.ou
tcom
es.o
f.a.
deci
sion
.by.
assi
gn
ing.
prob
abili
ties
.to.
payo
ff.va
lues
.and
.find
ing.
expe
cted
.va
lues
.
Fin
d.th
e.ex
pect
ed.p
ayof
f.fo
r.a.
gam
e.of
.cha
nce.
S-M
D.5
aX
Hig
h.Sc
hool
-St
atis
tics
.an
d.Pr
obab
ility
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Use
.pro
babi
lity.
to.e
valu
ate.
outc
omes
.of.
deci
sion
s
(+).W
eigh
.the
.pos
sibl
e.ou
tcom
es.o
f.a.
deci
sion
.by.
assi
gn
ing.
prob
abili
ties
.to.
payo
ff.va
lues
.and
.find
ing.
expe
cted
.va
lues
.
Eva
luat
e.an
d.co
mpa
re.
stra
teg
ies.
on.t
he.b
asis
.of.
expe
cted
.val
ues.
S-M
D.5
bX
Hig
h.Sc
hool
-St
atis
tics
.an
d.Pr
obab
ility
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Use
.pro
babi
lity.
to.e
valu
ate.
outc
omes
.of.
deci
sion
s
(+).U
se.p
roba
bilit
ies.
to.m
ake.
fair
.de
cisi
ons.
(e.g
.,.d
raw
ing.
by.lo
ts,.u
sing
.a.
rand
om.n
um
ber.
gene
rato
r).
S-M
D.6
X
Hig
h.Sc
hool
-St
atis
tics
.an
d.Pr
obab
ility
Usi
ng.
Pro
babi
lity.
to.
Mak
e.D
ecis
ion
s
Use
.pro
babi
lity.
to.e
valu
ate.
outc
omes
.of.
deci
sion
s
(+).A
naly
ze.d
ecis
ion
s.an
d.st
rate
gie
s.us
ing.
prob
abili
ty.c
once
pts.
(e.g
.,.pr
oduc
t.te
stin
g,.m
edic
al.t
esti
ng,.p
ulli
ng.a
.ho
ckey
.goa
lie.a
t.th
e.en
d.of
.a.g
ame)
.
S-M
D.7
X
ConneCtion11b-3443