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common core state standards for mathematics appendix a designing high school mathematics courses based on the common core state standards

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common core state standards for mathematics overview the common core state standards ccss for mathematics are organized by grade level in grades k8 at the high school level the standards are organized by conceptual category number and quantity algebra functions geometry modeling and probability and statistics showing the body of knowledge students should learn in each category to be college and career ready and to be prepared to study more advanced mathematics as states consider how to implement the high school standards an important consideration is how the high school ccss might be organized into courses that provide a strong foundation for post-secondary success to address this need achieve in partnership with the common core writing team has convened a group of experts including state mathematics experts teachers mathematics faculty from two and four year institutions mathematics teacher educators and workforce representatives to develop model course pathways in mathematics based on the common core state standards in considering this document there are four things important to note 1 the pathways and courses are models not mandates they illustrate possible approaches to organizing the content of the ccss into coherent and rigorous courses that lead to college and career readiness states and districts are not expected to adopt these courses as is rather they are encouraged to use these pathways and courses as a starting point for developing their own appendix a designing high school mathematics courses based on the common core state standards all college and career ready standards those without a are found in each pathway a few standards are included to increase coherence but are not necessarily expected to be addressed on high stakes assessments the course descriptions delineate the mathematics standards to be covered in a course they are not prescriptions for curriculum or pedagogy additional work will be needed to create coherent instructional programs that help students achieve these standards units within each course are intended to suggest a possible grouping of the standards into coherent blocks in this way units may also be considered critical areas or big ideas and these terms are used interchangeably throughout the document the ordering of the clusters within a unit follows the order of the standards document in most cases not the order in which they might be taught attention to ordering content within a unit will be needed as instructional programs are developed while courses are given names for organizational purposes states and districts are encouraged to carefully consider the content in each course and use names that they feel are most appropriate similarly unit titles may be adjusted by states and districts 2 3 4 5 while the focus of this document is on organizing the standards for mathematical content into model pathways to college and career readiness the content standards must also be connected to the standards for mathematical practice to ensure that the skills needed for later success are developed in particular modeling defined by a in the ccss is defined as both a conceptual category for high school mathematics and a mathematical practice and is an important avenue for motivating students to study mathematics for building their understanding of mathematics and for preparing them for future success development of the pathways into instructional programs will require careful attention to modeling and the mathematical practices assessments based on these pathways should reflect both the content and mathematical practices standards 2

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common core state standards for mathematics the pathways four model course pathways are included 1 2 3 an approach typically seen in the u.s traditional that consists of two algebra courses and a geometry course with some data probability and statistics included in each course an approach typically seen internationally integrated that consists of a sequence of three courses each of which includes number algebra geometry probability and statistics a compacted version of the traditional pathway where no content is omitted in which students would complete the content of 7th grade 8th grade and the high school algebra i course in grades 7 compacted 7th grade and 8 8th grade algebra i which will enable them to reach calculus or other college level courses by their senior year while the k-7 ccss effectively prepare students for algebra in 8th grade some standards from 8th grade have been placed in the accelerated 7th grade course to make the 8th grade algebra i course more manageable a compacted version of the integrated pathway where no content is omitted in which students would complete the content of 7th grade 8th grade and the mathematics i course in grades 7 compacted 7th grade and 8 8th grade mathematics i which will enable them to reach calculus or other college level courses by their senior year while the k-7 ccss effectively prepare students for algebra in 8th grade some standards from 8th grade have been placed in the accelerated 7th grade course to make the 8th grade mathematics i course more manageable ultimately all of these pathways are intended to significantly increase the coherence of high school mathematics 4 appendix a designing high school mathematics courses based on the common core state standards 5 the non-compacted or regular pathways assume mathematics in each year of high school and lead directly to preparedness for college and career readiness in addition to the three years of study described in the traditional and integrated pathways students should continue to take mathematics courses throughout their high school career to keep their mathematical understanding and skills fresh for use in training or course work after high school a variety of courses should be available to students reflecting a range of possible interests possible options are listed in the following chart based on a variety of inputs and factors some students may decide at an early age that they want to take calculus or other college level courses in high school these students would need to begin the study of high school content in the middle school which would lead to precalculus or advanced statistics as a junior and calculus advanced statistics or other college level options as a senior strategic use of technology is expected in all work this may include employing technological tools to assist students in forming and testing conjectures creating graphs and data displays and determining and assessing lines of fit for data geometric constructions may also be performed using geometric software as well as classical tools and technology may aid three-dimensional visualization testing with and without technological tools is recommended as has often occurred in schools and districts across the states greater resources have been allocated to accelerated pathways such as more experienced teachers and newer materials the achieve pathways group members strongly believe that each pathway should get the same attention to quality and resources including class sizes teacher assignments professional development and materials indeed these and other pathways should be avenues for students to pursue interests and aspirations the following flow chart shows how the courses in the two regular pathways are sequenced the in the chart on the following page means that calculus follows precalculus and is a fifth course in most cases more information about the compacted pathways can be found later in this appendix 3

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common core state standards for mathematics appendix a designing high school mathematics courses based on the common core state standards some teachers and schools are effectively getting students to be college and career ready we can look to these teachers and schools to see what kinds of courses are getting results and to compare pathways courses to the mathematics taught in effective classrooms a study done by act and the education trust gives evidence to support these pathways the study looked at highpoverty schools where a high percentage of students were reaching and exceeding act s college-readiness benchmarks from these schools the most effective teachers described their courses and opened up their classrooms for observation the commonality of mathematics topics in their courses gives a picture of what it takes to get students to succeed and also provides a grounding for the pathways there were other commonalities for more detailed information about this study search for the report on course for success at www.act.org 1 implementation considerations as states districts and schools take on the work of implementing the common core state standards the model course pathways in mathematics can be a useful foundation for discussing how best to organize the high school standards into courses the pathways have been designed to be modular in nature where the modules or critical areas units are identical in nearly every manner between the two pathways but are arranged in different orders to accommodate different organizational offerings assessment developers may consider the creation of assessment modules in a similar fashion curriculum designers may create alternative model pathways with altogether different organizations of the standards some of this work is already underway in short this document is intended to contribute to the conversations around assessment and curriculum design rather than end them effectively implementing these standards will require a long-term commitment to understanding what best supports student learning and attainment of college and career readiness skills by the end of high school as well as regular revision of pathways as student learning data becomes available supporting students one of the hallmarks of the common core state standards for mathematics is the specification of content that all students must study in order to be college and career ready this college and career ready line is a minimum for all students however this does not mean that all students should progress uniformly to that goal some students progress 1 the study provides evidence that the pathways high school algebra i geometry algebra ii sequence is a reasonable and rigorous option for preparing students for college and career topics aligned almost completely between the ccss topics and topics taught in the study classrooms the starting point for the pathways high school algebra i course is slightly beyond the starting point for the study algebra i courses due to the existence of many typical algebra i topics in the 8th grade ccss therefore some of the study algebra ii topics are a part of the pathways high school algebra i course specifically using the quadratic formula a bit more with exponential functions including comparing and contrasting linear and exponential growth and the inclusion of the spread of data sets the pathways geometry course is very similar to what was done in the study geometry courses with the addition of the laws of sines and cosines and the work with conditional probability plus applications involving completing the square because that topic was part of the pathways high school algebra i course the pathways algebra ii course then matches well with what was done in the study algebra ii courses and continues a bit into what was done in the study precalculus classrooms including inverse functions the behavior of logarithmic and trigonometric functions and in statistics with the normal distribution margin of error and the differences among sample surveys experiments and observational studies all in all the topics and the order of topics is very comparable between the pathways high school algebra i geometry algebra ii sequence and the sequence found in the study courses 4

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common core state standards for mathematics more slowly than others these students will require additional support and the following strategies consistent with response to intervention practices may be helpful · · · · · creating a school-wide community of support for students providing students a math support class during the school day after-school tutoring extended class time or blocking of classes in mathematics and additional instruction during the summer watered-down courses which leave students uninspired to learn unable to catch up to their peers and unready for success in postsecondary courses or for entry into many skilled professions upon graduation from high school are neither necessary nor desirable the results of not providing students the necessary supports they need to succeed in high school are well-documented too often after graduation such students attempt to continue their education at 2or 4-year postsecondary institutions only to find they must take remedial courses spending time and money mastering high school level skills that they should have already acquired this in turn has been documented to indicate a greater chance of these students not meeting their postsecondary goals whether a certificate program two or fouryear degree as a result in the workplace many career pathways and advancement may be denied to them to ensure students graduate fully prepared those who enter high school underprepared for high school mathematics courses must receive the support they need to get back on course and graduate ready for life after high school furthermore research shows that allowing low-achieving students to take low-level courses is not a recipe for academic success kifer 1993 the research strongly suggests that the goal for districts should not be to stretch the high school mathematics standards over all four years rather the goal should be to provide support so that all students can reach the college and career ready line by the end of the eleventh grade ending their high school career with one of several high-quality mathematical courses that allows students the opportunity to deepen their understanding of the college and career-ready standards with the common core state standards initiative comes an unprecedented ability for schools districts and states to collaborate while this is certainly the case with respect to assessments and professional development programs it is also true for strategies to support struggling and accelerated students the model course pathways in mathematics are intended to launch the conversation and give encouragement to all educators to collaborate for the benefit of our states children appendix a designing high school mathematics courses based on the common core state standards 5

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common core state standards for mathematics how to read the pathways each pathway consists of two parts the first is a chart that shows an overview of the pathway organized by course and by conceptual category algebra functions geometry etc these charts show which clusters and standards appear in which course see page 5 of the ccss for definitions of clusters and standards for example in the chart below the three standards n.q.1 2 3 associated with the cluster reason quantitatively and use units to solve problems are found in course 1 this cluster is found under the domain quantities in the number and quantity conceptual category all high school standards in the ccss are located in at least one of the courses in this chart courses domain appendix a designing high school mathematics courses based on the common core state standards clusters notes and standards conceptual category 6

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common core state standards for mathematics the second part of the pathways shows the clusters and standards as they appear in the courses each course contains the following components · · · an introduction to the course and a list of the units in the course unit titles and unit overviews see below units that show the cluster titles associated standards and instructional notes below it is important to note that the units or critical areas are intended to convey coherent groupings of content the clusters and standards within units are ordered as they are in the common core state standards and are not intended to convey an instructional order considerations regarding constraints extensions and connections are found in the instructional notes the instructional notes are a critical attribute of the courses and should not be overlooked for example one will see that standards such as a.ced.1 and a.ced.2 are repeated in multiple courses yet their emphases change from one course to the next these changes are seen only in the instructional notes making the notes an indispensable component of the pathways unit title and overview appendix a designing high school mathematics courses based on the common core state standards standards associated with cluster cluster instructional note 7

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common core state standards for mathematics overview of the traditional pathway for the common core state mathematics standards this table shows the domains and clusters in each course in the traditional pathway the standards from each cluster included in that course are listed below each cluster for each course limits and focus for the clusters are shown in italics domains high school algebra i · xtend the properties e of exponents to rational exponents geometry algebra ii fourth courses the real number system n.rn.1 2 · se properties of u rational and irrational numbers n.rn.3 · eason quantitatively r and use units to solve problems appendix a designing high school mathematics courses based on the common core state standards quantities foundation for work with expressions equations and functions n.q.1 2 3 · erform arithmetic p operations with complex numbers n.cn.1 2 · erform arithmetic p operations with complex numbers n.cn.3 · epresent complex r numbers and their operations on the complex plane n.cn.4 5 6 · epresent and model r with vector quantities n.vm.1 2 3 · perform operations on vectors number and quantity the complex number system · se complex numbers u in polynomial identities and equations polynomials with real coefficients n.cn.7 8 9 vector quantities and matrices n.vm.4a 4b 4c 5a 5b · perform operations on matrices and use matrices in applications n.vm.6 7 8 9 10 11 12 the standards in this column are those in the common core state standards that are not included in any of the traditional pathway courses they would be used in additional courses developed to follow algebra ii 8

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common core state standards for mathematics domains high school algebra i · nterpret the structure i of expressions linear exponential quadratic geometry algebra ii · nterpret the structure i of expressions polynomial and rational a.sse.1a 1b 2 · rite expressions in w equivalent forms to solve problems a.sse.4 fourth courses seeing structure in expressions a.sse.1a 1b 2 · rite expressions in w equivalent forms to solve problems quadratic and exponential a.sse.3a 3b 3c · erform arithmetic p operations on polynomials linear and quadratic a.apr.1 · erform arithmetic p operations on polynomials beyond quadratic a.apr.1 · nderstand the u relationship between zeros and factors of polynomials a.apr.2 3 · se polynomial u identities to solve problems a.apr.4 5 · ewrite rational r expressions linear and quadratic denominators a.apr.6 7 appendix a designing high school mathematics courses based on the common core state standards algebra arithmetic with polynomials and rational expressions · reate equations that c describe numbers or relationships creating equations linear quadratic and exponential integer inputs only for a.ced.3 linear only a.ced.1 2 3 4 · reate equations that c describe numbers or relationships equations using all available types of expressions including simple root functions a.ced.1 2 3 4 9

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common core state standards for mathematics domains high school algebra i · nderstand solving u equations as a process of reasoning and explain the reasoning master linear learn as general principle a.rei.1 · olve equations and s inequalities in one variable linear inequalities literal that are linear in the variables being solved for quadratics with real solutions a.rei.3 4a 4b · olve systems of s equations linear-linear and linearquadratic a.rei.5 6 7 · epresent and r solve equations and inequalities graphically linear and exponential learn as general principle a.rei.10 11 12 · nderstand the u concept of a function and use function notation learn as general principle focus on linear and exponential and on arithmetic and geometric sequences f.if.1 2 3 geometry algebra ii · nderstand solving u equations as a process of reasoning and explain the reasoning simple radical and rational a.rei.2 · epresent and r solve equations and inequalities graphically combine polynomial rational radical absolute value and exponential functions fourth courses · olve systems of s equations a.rei.8 9 reasoning with equations and inequalities algebra appendix a designing high school mathematics courses based on the common core state standards a.rei.11 a · nterpret functions that · nalyze functions i using different arise in applications in representations terms of a context emphasize selection of appropriate models f.if.4 5 6 · nalyze functions a using different representations focus on using key features to guide selection of appropriate type of model function f.if.7b 7c 7e 8 9 logarithmic and trigonometric functions f.if.7d functions interpreting functions · nterpret functions that i arise in applications in terms of a context linear exponential and quadratic f.if.4 5 6 · nalyze functions a using different representations linear exponential quadratic absolute value step piecewisedefined f.if.7a 7b 7e 8a 8b 9 10

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common core state standards for mathematics domains high school algebra i · uild a function that b models a relationship between two quantities for f.bf.1 2 linear exponential and quadratic f.bf.1a 1b 2 geometry algebra ii · uild a function that b models a relationship between two quantities include all types of functions studied f.bf.1b · uild new functions b from existing functions include simple radical rational and exponential functions emphasize common effect of each transformation across function types f.bf.3 4a · onstruct and c compare linear quadratic and exponential models and solve problems logarithms as solutions for exponentials f.le.4 fourth courses · uild a function that b models a relationship between two quantities f.bf.1c · uild new functions b from existing functions f.bf.4b 4c 4d 5 building functions · uild new functions b from existing functions linear exponential quadratic and absolute value for f.bf.4a linear only f.bf.3 4a appendix a designing high school mathematics courses based on the common core state standards functions · onstruct and c compare linear quadratic and exponential models and solve problems linear quadratic and exponential models f.le.1a 1b 1c 2 3 · nterpret expressions i for functions in terms of the situation they model linear and exponential of form fx bx+k f.le.5 · xtend the domain e of trigonometric functions using the unit circle f.tf.1 2 · odel periodic m phenomena with trigonometric functions f.tf.5 · rove and apply p trigonometric identities f.tf.8 · xtend the domain e of trigonometric functions using the unit circle f.tf.3 4 · odel periodic m phenomena with trigonometric functions f.tf 6 7 · rove and apply p trigonometric identities f.tf 9 trigonometric functions 11

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common core state standards for mathematics domains high school algebra i geometry · xperiment with e transformations in the plane g.co.1 2 3 4 5 · nderstand u congruence in terms of rigid motions build on rigid motions as a familiar starting point for development of concept of geometric proof algebra ii fourth courses congruence g.co.6 7 8 appendix a designing high school mathematics courses based on the common core state standards · rove geometric p theorems focus on validity of underlying reasoning while using variety of ways of writing proofs geometry g.co.9 10 11 · ake geometric m constructions formalize and explain processes g.co.12 13 · nderstand similarity u in terms of similarity transformations g.srt.1a 1b 2 3 · rove theorems p involving similarity similarity right triangles and trigonometry g.srt.4 5 · efine trigonometric d ratios and solve problems involving right triangles g.srt.6 7 8 · pply trigonometry to a general triangles g.srt.9 10 11 12

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common core state standards for mathematics domains high school algebra i geometry · nderstand and u apply theorems about circles g.c.1 2 3 4 algebra ii fourth courses circles · ind arc lengths and f areas of sectors of circles radian introduced only as unit of measure g.c.5 · ranslate between the t geometric description and the equation for a conic section g.gpe.1 2 · ranslate between the t geometric description and the equation for a conic section appendix a designing high school mathematics courses based on the common core state standards g.gpe.3 geometry expressing geometric properties with equations · se coordinates u to prove simple geometric theorems algebraically include distance formula relate to pythagorean theorem g.gpe 4 5 6 7 · xplain volume e formulas and use them to solve problems · xplain volume e formulas and use them to solve problems g.gmd.2 geometric measurement and dimension g.gmd.1 3 · isualize the relation v between twodimensional and threedimensional objects g.gmd.4 · pply geometric a concepts in modeling situations g.mg.1 2 3 · ummarize represent s and interpret data on a single count or measurement variable s.id.1 2 3 · ummarize represent s and interpret data on a single count or measurement variable s.id.4 modeling with geometry statistics and probability interpreting categorical and quantitative data · ummarize represent s and interpret data on two categorical and quantitative variables linear focus discuss general principle s.id.5 6a 6b 6c · nterpret linear models i s.id.7 8 9 13

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common core state standards for mathematics domains high school algebra i geometry algebra ii · nderstand and u evaluate random processes underlying statistical experiments fourth courses making inferences and justifying conclusions s.ic.1 2 · ake inferences and m justify conclusions from sample surveys experiments and observational studies s.ic.3 4 5 6 · nderstand u independence and conditional probability and use them to interpret data link to data from simulations or experiments s.cp.1 2 3 4 5 · se the rules of u probability to compute probabilities of compound events in a uniform probability model s.cp.6 7 8 9 · se probability to u evaluate outcomes of decisions introductory apply counting rules s.md.6 7 · se probability to u evaluate outcomes of decisions include more complex situations s.md.6 7 · alculate expected c values and use them to solve problems s.md.1 2 3 4 · se probability to u evaluate outcomes of decisions s.md 5a 5b statistics and probability appendix a designing high school mathematics courses based on the common core state standards conditional probability and the rules of probability using probability to make decisions 14

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common core state standards for mathematics traditional pathway high school algebra i the fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades because it is built on the middle grades standards this is a more ambitious version of algebra i than has generally been offered the critical areas called units deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend and students engage in methods for analyzing solving and using quadratic functions the mathematical practice standards apply throughout each course and together with the content standards prescribe that students experience mathematics as a coherent useful and logical subject that makes use of their ability to make sense of problem situations critical area 1 by the end of eighth grade students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables now students analyze and explain the process of solving an equation students develop fluency writing interpreting and translating between various forms of linear equations and inequalities and using them to solve problems they master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations critical area 2 in earlier grades students define evaluate and compare functions and use them to model relationships between quantities in this unit students will learn function notation and develop the concepts of domain and range they explore many examples of functions including sequences they interpret functions given graphically numerically symbolically and verbally translate between representations and understand the limitations of various representations students build on and informally extend their understanding of integer exponents to consider exponential functions they compare and contrast linear and exponential functions distinguishing between additive and multiplicative change students explore systems of equations and inequalities and they find and interpret their solutions they interpret arithmetic sequences as linear functions and geometric sequences as exponential functions critical area 3 this unit builds upon prior students prior experiences with data providing students with more formal means of assessing how a model fits data students use regression techniques to describe approximately linear relationships between quantities they use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models with linear models they look at residuals to analyze the goodness of fit critical area 4 in this unit students build on their knowledge from unit 2 where they extended the laws of exponents to rational exponents students apply this new understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions they create and solve equations inequalities and systems of equations involving quadratic expressions critical area 5 in this unit students consider quadratic functions comparing the key characteristics of quadratic functions to those of linear and exponential functions they select from among these functions to model phenomena students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions in particular they identify the real solutions of a quadratic equation as the zeros of a related quadratic function students expand their experience with functions to include more specialized functions absolute value step and those that are piecewise-defined appendix a designing high school mathematics courses based on the common core state standards 15

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