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Apply to the solution of problems.tG.G.14 Demonstrate an understanding of the relationship between geometric and algebraic representations of circles.AlgebraSeeing Structure in Expressions&Interpret the structure of expressions7Write expressions in equivalent forms to solve problems2Arithmetic with Polynomials and Rational Functions,Perform arithmetic operations on polynomials It is in the Model Mathematics III course that students integrate and apply the mathematics they have learned from their earlier courses. This course is comprised of standards selected from the high school conceptual categories, which were written to encompass the scope of content and skills to be addressed throughout grades 9 12 rather than through any single course. Therefore, the complete standard is presented in the model course, with clarifying footnotes as needed to limit the scope of the standard and indicate what is appropriate for study in this particular course. Standards that were limited in Model Mathematics I and Model Mathematics II no longer have those restrictions in Model Mathematics III. For the high school Model Mathematics III course, instructional time should focus on four critical areas: (1) apply methods from probability and statistics to draw inferences and conclusions from data; (2) expand understanding of functions to include polynomial, rational, and radical functions; (3) expand right triangle trigonometry to include general triangles; and (4) consolidate functions and geometry to create models and solve contextual problems. (1) Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data including sample surveys, experiments, and simulations and the role that randomness and careful design play in the conclusions that can be drawn.F (4) Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying functions. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. The Standards for Mathematical Practice complement the content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.QR Model Mathematics III March 2011Unmatched Standards For those MA 2011 standards that are not matched with any MA 2000 standards the MA 2000 column is empty and shaded green. There is a clarifying comment in the third column that indicates if the MA 2011 standard is new in the course or new for MA standards. There are two other categories of unmatched standards located at the end of each grade level crosswalk: (1) MA 2000 standards that match MA 2011 standards at a different grade level or course, with the best match indicated in the first column; and (2) MA 2000 standards that do not match any MA 2011 standards. We hope that you find these crosswalks useful. Please email any comments and questions to the Office of Math, Science, and Technology/Engineering at mathsciencetech@doe.mass.edu.EditReasonCourse Algebra IAdded S-MD.6 and S-MD.7.Added A-REI.6. Mathematics I4These standards are included in the Geometry course.5These standards are included in the Algebra I course.6This standard is included in the Mathematics I course.XRemoved footnote on N-RN.1 and N-RN.2 that limits rational exponents to square and cube.5The footnote is not included in the Algebra I course. Algebra IIDeleted N-RN.1 and and N-RN.2. qThese standards are no longer limited in Algebra I and therefore there is no reason to repeat them in Algebra II. ModificationsESE staff are grateful to members of the field who recommended modifications to the original Pre-K-8 crosswalk posted in January 2011. We appreciate all comments and suggestions that make these crosswalks more useful.FInterpret functions that arise in applications in terms of the context1Analyze functions using different representations< Building FunctionsCBuild a functions that models a relationship between two quantities+Build new functions from existing functions)Linear, Quadratic, and Exponential ModelsRConstruct and compare linear, quadratic, and exponential models and solve problems/Interpreting Categorical and Quantitative Data RSummarize, represent, and interpret data on a single count or measurement variableInterpret linear modelsSConstruct and compare linear, quadratic, and exponential models and solve problems.Trigonometric FunctionsBExtend the domain of trigonometric functions using the unit circlev 10.P.4 / AI.P.8 Demonstrate facility in symbolic manipulation of polynomial and rational expressions by rearranging and collecting terms; factoring (e.g., a2 b2 = (a + b)(a b), x2 + 10x + 21 = (x + 3)(x + 7), 5x4 + 10x3 5x2 = 5x2 (x2 + 2x 1)); identifying and canceling common factors in rational expressions, and applying properties of positive integer exponents.}10.N.1 / AI.N.1 Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth root of and the nth power of a positive real number; and the density of the set of rational numbers in the set of real numbers.10.P.5 / AI.P.9 Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods. \10.P.3 / AI.P.7 Add, subtract, and multiply polynomials. Divide polynomials by monomials. 12.D.2 / AII.D.1 Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data.HInterpret expressions for functions in terms of the situation they modelThe Complex Number System5Model periodic phenomena with trigonometric functions 10.G.6 / G.G.8 Use the properties of special triangles (e.g., isosceles, equilateral, 30 60 90, 45 45 90) to solve problems. 10.M.1 / G.M.1 Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles. 010.G.3 Recognize and solve problems involving angles formed by transversals of coplanar lines. Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle.12.G.3 / PC.G.3 / G.G.6 Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.10.D.1 / AI.D.1 Select, create, and interpret an appropriate graphical representation (e.g., scatter plot, table, stem-and-leaf plots, box-and-whisker plots, circle graph, line graph, and line plot) for a set of data and use appropriate statistics (e.g. mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.10.G.7 / G.G.12 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.{AI.P.4 Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.10.P.1 / AI.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonacci Numbers), linear, quadratic, and exponential functional relationships.MA.9-12.A.REI.4c Demonstrate an understanding of the equivalence of factoring, completing the square, or using the quadratic formula to solve quadratic equations. :Represent and solve equations and inequalities graphically#Prove theorems involving similarityIDefine trigonometric ratios and solve problems involving right triangles.'Apply trigonometry to general trianglesCircles+Understand and apply theorems about circles0Find arc lengths and areas of sectors of circles.Expressing Geometric Properties with EquationsPTranslate between the geometric description and the equation for a conic section#Geometric Measurement and Dimension6Explain volume formulas and use them to solve problems10.M.2 / G.M.2 Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area. 10.M.3 / G.M.3 Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume. z10.G.10 / G.G.16 Demonstrate the ability to visualize solid objects and recognize their projections and cross sections. rMA.9-12.G.MG.4 Use dimensional analysis for unit conversion to confirm that expressions and equations make sense. #Standards for Mathematical Practice7Reason quantitatively and use units to solve problems.*Creating Equations*VSummarize, represent, and interpret data on two categorical and quantitative variablesaMA 2011 Footnote: MA 2011 Mathematics II expands to include quadratic and exponential functions.7 (3) 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. Building on these earlier experiences, students analyze and explain the process of solving an equation and to justify the process used in solving a system of equations. 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. Students explore systems of equations and inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them. (4) Students prior experiences with data Is the basis for with the more formal means of assessing how a model fits data. Students use regression techniques to des< cribe 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. 7-(5) In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. Students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. (6) Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. The Standards for Mathematical Practice complement the content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.#&Model Mathematics II March 20116MA 2011 Integrated Pathway Mathematics II IntroductionFor the high school Model Mathematics II course, instructional time should focus on five critical areas: (1) extend the laws of exponents to rational exponents; (2) compare key characteristics of quadratic functions with those of linear and exponential functions; (3) create and solve equations and inequalities involving linear, exponential, and quadratic expressions; (4) extend work with probability; and (5) establish criteria for similarity of triangles based on dilations and proportional reasoning.The focus of the Model Mathematics II course is on quadratic expressions, equations, and functions; comparing their characteristics and behavior to those of linear and exponential relationships from Model Mathematics I. This course is comprised of standards selected from the high school conceptual categories, which were written to encompass the scope of content and skills to be addressed throughout grades 9 12 rather than through any single course. Therefore, the complete standard is presented in the model course, with clarifying footnotes as needed to limit the scope of the standard and indicate what is appropriate for study in this particular course. For example, the scope of Model Mathematics II is limited to quadratic expressions and functions, and some work with absolute value, step, and functions that are piecewise-defined. Therefore, although a standard may include references to logarithms or trigonometry, those functions should not be included in coursework for Model Mathematics II; they will be addressed in Model Mathematics III. 9-12.F-BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. J 9-12.F-BF.1c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Massachusetts March 20119-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* 9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.* 9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.* 9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.* B N E 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning * Specific modeling standards appear throughout the high school standards indicated in this crosswalk by an asterisk (*).eMA 2011 Footnote: MA 2011 Mathematics II limits quadratic equations to those with real coefficients.+ Traditional Pathway Algebra I IntroductionAlgebra I (MA 2000),Use polynomial identities to solve problems.The fundamental purpose of the Model Algebra I course is to formalize and extend the mathematics that students learned in the middle grades. This course is comprised of standards selected from the high school conceptual categories, which were written to encompass the scope of content and skills to be addressed throughout grades 9 12 rather than through any single course. Therefore, the complete standard is presented in the model course, with clarifying footnotes as needed to limit the scope of the standard and indicate what is appropriate for study in this particular course. For example, the scope of Model Algebra I is limited to linear, quadratic, and exponential expressions and functions as well as some work with absolute value, step, and functions that are piecewise-d< efined. Therefore, although a standard may include references to logarithms or trigonometry, those functions are not to be included in coursework for Model Algebra I; they will be addressed later in Model Algebra II. Reminders of this limitation are included as footnotes where appropriate in the Model Algebra I standards. For the high school Model Algebra I course, instructional time should focus on four critical areas: (1) deepen and extend understanding of linear and exponential relationships; (2) contrast linear and exponential relationships with each other and engage in methods for analyzing, solving, and using quadratic functions; (3) extend the laws of exponents to square and cube roots; and (4) apply linear models to data that exhibit a linear trend.9-12.F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals, where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries, end behavior; and periodicity.*9-12.F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*MA.9-12.F-IF.10 Given algebraic, numeric, and/or graphical representations of functions, identify the function as polynomial, rational, logarithmic, exponential, or trigonometric. 9-12.F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*9-12.F-BF.3 . Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.9-12.F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*9-12.F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*_9-12.F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. 9-12.G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.U9-12.G-CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch.). 9-12.G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.9-12.G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.9-12.G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.9-12.G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.9-12.G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.9-12.G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.9-12.G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.g9-12.G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 4 9-12.G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, "3) lies on the circle centered at the origin and containing the point (0.2). P9-12.G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 9-12.G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*h9-12.S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).* 9-12.S-ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.* 9-12.S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).* 9-12.S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.*w 9-12.N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i).F]^9-12.N.VM.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).+ 9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2).Pcdhiopqrwxyz How to read this crosswalk: The first column of this Mathematics III Crosswalk presents the 2011 Massachusetts Integrated Pathway Mathematics II course standards, including the Massachusetts additional standards. Please note that some of these standards are marked by an asterisk (*) indicating a modeling standard, as defined in < the High School Modeling Conceptual Category. The second column presents related standards from the Massachusetts 2000 Algebra I, Geometry, or Algebra II course. The third column provides informational comments highlighting differences with italicized footnotes that limit the scope of the standard for this course. If there is no appropriate MA 2000 match, the second and third columns are shaded green, with appropriate comments in the third column. This crosswalk is designed as a tool for use by districts and schools as they prepare for the implementation of the Massachusetts 2011 Standards for Mathematics. When reviewing the crosswalk, please keep in mind that the correlations between standards indicated in the crosswalk could be direct, meaning that the standards contain the same content, or could be partial, meaning that parts of the standards are related. Also note that several MA 2000 standards may be matched to one MA 2011 standard, and conversely, one MA 2000 standard could be matched to several MA 2011 standards. At the end of the Mathematics II crosswalk, MA 2000 Mathematics I standards that are unmatched are presented in two ways: (1) MA 2000 Mathematics II standards that match MA 2011standards at a different grade level, with the best match indicated in the first column; and (2) MA 2000 Mathematics II standards that do not match any MA 2011 standards.!AU9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*-Similarity, Right Triangles, and Trigonometry<Understand similarity in terms of similarity transformations.Algebra I, Geometry, and Algebra II (MA 2000)3Perform arithmetic operations with complex numbers.n12.G.4 / AII.G.3 Relate geometric and algebraic representations of lines, simple curves, and conic sections.;Represent and solve equations and inequalities graphically.9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. High School Model Pathways There are two model high school pathways: Traditional (Algebra I, Geometry, & Algebra II) and Integrated (Mathematics I, Mathematics II, & Mathematics III). Both pathways cover the standards identified as necessary for college and career readiness in the High School Conceptual Categories and prepare students for fourth year courses such as those identified in the Curriculum Framework. This crosswalk includes all three courses from each of the pathways. Crosswalks for the two fourth year courses will be released in the future. For the traditional pathway, each MA 2011 course (Algebra I, Geometry, Algebra II) is compared side-by-side with the corresponding MA 2000 course. For the integrated pathway, the content of any single MA 2011 course (Mathematics I, Mathematics II, or Mathematics II) is not restricted to any one MA 2000 course (Algebra I, Geometry, or Algebra II); therefore, the Integrated model courses may be matched to standards from any of the three MA 2000 courses.9M Format of the High School Model Course crosswalks The first column of each crosswalk presents the standards for the MA 2011 model course (Algebra I, Geometry, Algebra II, Mathematics I, Mathematics II, or Mathematics III) coded by grade level (9-12), conceptual category, domain, and standard number (see the Table below for conceptual category and domain codes). The second column presents the related MA 2000 standards with the original grade-span and/or course standard codes. The last column provides informational comments, highlighting ways that the MA 2011 standards are different from the MA 2000 standards, as well as the footnotes (in italics) that are included in the model courses in the MA 2011 standards.2Mathematical modeling is a Standard for Mathematical Practice and specific modeling standards appear throughout the high school conceptual categories. In the crosswalk, modeling standards are indicated by an asterisk (*). =zDegree of Match It is important to note that the standards in the crosswalk have varying degrees of correlation. An example of a match where the MA 2000 standard contains elements of the matching MA 2011 standard is: (MA 2000)10.P.3 /AI.P.7 Add, subtract, and multiply polynomials. Divide polynomials by monomials. ! L (MA 2011) 9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.!! MFor this example the third column includes the comment: MA 2011 specifies that polynomials form a system analogous to integers. There is not a one-to-one correspondence between the MA 2011 standards and the MA 2000 standards. In some cases several MA 2000 standards are matched to one MA 2011 standard and vice-versa.CC.9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.KUnderstand and evaluate random processes underlying statistical experimentsAI.P.2 Use properties of the real number system to judge the validity of equations and inequalities, to prove or disprove statements, and to justify every step in a sequential argument.AI.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of functions. Number and QuantityThe Real Number System9Extend the properties of exponents to rational exponents.2Use properties of rational and irrational numbers. QuantitiesNumber and QuantitiesCC.9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*,Making Inferences and Justifying Conclusions12.M.2 / PC.M.2 /G.M.5 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense.10.N.3 / AI.N.3 Find the approximate value for solutions to problems involving square roots and cube roots without the use of a calculator, e.g., 10.M.4 / G.M.4 Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements./10.P.7 / AI.P.11 Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include compound interest, and direct and inverse variation problems. Use technology when appropriate.-Experiment with transformations in the plane.9-12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* 9-12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*SUnderstand independence and conditional probability and use them to interpret data.3 Use probability to evaluate outcomes of decisions.4Arithmetic with Polynomials and Rational ExpressionsModel Mathematics I March 2011H12.P.13 / AII.P.13 / PC.P.6 Describe the translations and scale changes o< f a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af (b(x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, logarithmic, and trigonometric functions.] ^ {AI.P.4 Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.Statistics and Probability9-12.G.SRT.9 (+) Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.V9-12.G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.9-12.G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces)..9-12.G.C.1 Prove that all circles are similar.99-12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.[[9-12.G-C.4 (+) Construct a tangent line from a point outside a given circle to the circle.+9-12.G-C.5 Find arc lengths and areas of sectors of circles. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.9-12.G-GPE.1 Translate between the geometric description and the equation for a conic section. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.M9-12.G-GPE.2 Derive the equation of a parabola given a focus and a directrix. ~9-12.G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 9-12.G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. 9-12.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*12.P.1 / AII.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Pascal's Triangle.`9-12.G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* 9-12.S-CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ).*9-12.S-CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.**9-12.S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.* 9-12.S-CP.6 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model.*{ 9-12.S-CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model.*r9-12.S-CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.*9.-12.S-CP.8 (+) Apply the general Multiplication Rule in a uniform probability model P(A and B) = P(A) P(B|A)= P(B)P(A|B), and interpret the answer in terms of the model.*lmxyt9-12.S-MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).* 9-12.S-MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).*9-12.S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*X9-12.N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5(1/3) to be the cube root of 5 because we want (51/3)3 = 5(1/3x3) to hold, so (51/3)3 must equal 5.&)*+05EHIJw 9-12.N-CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i).G$[\]#^cev$9-12.A-SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.* i 9-12.A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). Qac#dh#inp#qr#sx#yz#{####;9-12.F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*L9-12.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*MFor the high school Model Mathematics I course, instructional time should focus on six critical areas, each of which is described in more detail below: (1) extend understanding of numerical manipulation to algebraic manipulation; (2) syn< thesize understanding of function; (3) deepen and extend understanding of linear relationships; (4) apply linear models to data that exhibit a linear trend; (5) establish criteria for congruence based on rigid motions; and (6) apply the Pythagorean Theorem to the coordinate plane. (1) By the end of eighth grade students have had a variety of experiences working with expressions and creating equations. Students become facile with algebraic manipulation in much the same way that they are facile with numerical manipulation. Algebraic facility includes rearranging and collecting terms, factoring, identifying and canceling common factors in rational expressions and applying properties of exponents. Students continue this work by using quantities to model and analyze situations, to interpret expressions, and by creating equations to describe situations. The fundamental purpose of Mathematics I is to formalize and extend the mathematics that students learned in the middle grades. This course is comprised of standards selected from the High School Conceptual Categories, which were written to encompass the scope of content and skills to be addressed throughout grades 9 12 rather than in any single course. Therefore, the complete standard is presented in the model course with clarifying footnotes as needed to limit the scope of the standard and indicate what is appropriate for study in this particular course. For example, the scope of Mathematics I is limited to linear and exponential expressions and functions as well as some work with absolute value, step, and functions that are piecewise-defined; therefore, although a standard may include references to quadratic, logarithmic or trigonometric functions, those functions should not be included in the work of Mathematics I students, rather they will be addressed in Mathematics II or III. The Standards for Mathematical Practice complement the content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.* U9-12.F.BF.1 Write a function that describes a relationship between two quantities.* 9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.:>BY^h~%9-12.F.BF.4 Find inverse functions. 9-12.S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.*b9-12.S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.*T9-12.S-ID.9 Interpret linear models. Distinguish between correlation and causation.*J 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning * Specific modeling standards appear throughout the high school standards indicated in this crosswalk by an asterisk (*).}MA 2011 Footnote: MA 2011 Mathematics I is limited to linear expressions and exponential expressions with integer exponents.nMA 2011 Footnote: MA 2011 Mathematics I is limited to linear and exponential equations with integer exponents.bMake inferences and justify conclusions from sample surveys, experiments and observational studies10.P.1 / AI.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonacci Numbers), linear, quadratic, and exponential functional relationships. 10.D.2 / AI.D.2 Approximate a line of best fit (trend line) given a set of data (e.g., scatter plot). Use technology when appropriate. G.G.1 Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses). Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, mid-segments); determine interior angles for regular polygons. Draw and label sets of points such as line segments, rays, and circles. Detect symmetries of geometric figures.10.G.2 / G.G.4 Draw congruent and similar figures using a compass, straightedge, protractor, and other tools such as computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments.10.G.9 / G.G.15 Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems. _G.G.2 Write simple proofs of theorems in geometric situations, such as theorems about congruent and similar figures, parallel or perpendicular lines. Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contrapositive.G.G.5 Apply congruence and similarity correspondences (e.g., ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.N-RNN-QN-CNN-VMA-SSEA-APRA-CEDA-REIF-IFF-BFF-LEF-TFG-COG-SRTG-C G-GPEG-GMDG-MGS-IDS-ICS-CPS-MD;Table: Codes for Conceptual Categories and Domains (con't)eMA 2011 Mathematics III specifies rational expressions as a system analogous to the rational numbers.wMA 2011 Footnote: MA 2011 Mathematics III should focus on rational expressions with linear and quadratic denominators.LMA 2011 Mathematics III specifies to rewrite expressions in different forms.RMA 2011 Mathematics III includes deriving the formula for finite geometric series.nMA 2011 Footnote: MA 2011 Mathematics III should expand understanding to polynomial and rational expressions.9-12.A-CED.< 4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R.*wMA 2000 Algebra I and GeometryMA 20000 Algebra I (1) Students extend the laws of exponents to rational exponents and explore distinctions between rational and irrational numbers by considering their decimal representations. Students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to the way in which extending the whole numbers to the negative numbers allows x+1 = 0 to have a solution. Students explore relationships between number systems: whole numbers, integers, rational numbers, real numbers, and complex numbers. The guiding principle is that equations with no solutions in one number system may have solutions in a larger number system. (2) 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. When quadratic equations do not have real solutions, students learn that that the graph of the related quadratic function does not cross the horizontal axis. They expand their experience with functions to include more specialized functions absolute value, step, and those that are piecewise-defined.4(3) Students begin by focusing on the structure of expressions, rewriting expressions to clarify and reveal aspects of the relationship they represent. They create and solve equations, inequalities, and systems of equations involving exponential and quadratic expressions. (4) Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. (5) Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop facility with geometric proof. They use what they know about congruence and similarity to prove theorems involving lines, angles, triangles, and other polygons. They explore a variety of formats for writing proofs. 3 9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x3) or f(x) = (x+1)/(x-1) for x `" 1. 9-12.F.LE.4 For exponential models, express as a logarithm the solution to ab(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.* LNR UV \b gh n12.P.4 / AII.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions. MA 2011 Mathematics III specifies bases 2, 10, and e. MA 2011 Footnote: MA 2011 Mathematics III should only include logarithms as solutions of exponential functions.3 v9-12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. z9-12.F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* 9-12.G.SRT.9 (+) Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. V9-12.G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. 9-12.G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). 9-12.G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 9-12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* 12.M.2 / PC.M.2 /G.M.5 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense.9-12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* 9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* nMA 2011 Footnote: The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.8Create equations that describe numbers or relationships.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*_ 9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R.*x< MA.9-12.F.IF.8c Translate between different representations of functions and relations: graphs, equations, point sets, and tables. 9-12.F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. I12.M.1 / PC.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular problems involving angular velocity and acceleration.(Prove and apply trigonometric identities9-12.F.TF.8 Prove the Pythagorean identity (sin A)2 + (cos A)2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.45?@ 12.G.2 / AII.G.2 / PC.G.1 Derive and apply basic trigonometric identities (e.g., sin2q + cos2q = 1, tan2q + 1 = sec2q) and the laws of sines and cosines. UV^_jkwxSSummarize, represent, and interpret data on a single count or measurement variable.GMA 2000 Algebra II Standards Matched at other grades/courses in MA 2011 CC.9-12.S.CP.6 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model.**This standard is new in MA 2011 Algebra I9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.* 9-12.A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.*] j9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.MA 2011 Footnote: It is sufficient in Algebra I to recognize when roots are not real; writing complex roots are included in Algebra II.!MA.9-12.N-Q.3a Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure.* V9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.*Z9-12.A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.* 9-12.A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*9-12.A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*79-12.A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.* 9-12.A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 9-12.A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. _ dMA.9-12.A-REI.3a Solve linear equations and inequalities in one variable involving absolute value. 9-12.A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 9-12.A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). W9-12.A-REI.12 Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. I N (2) The structural similarities between the system of polynomials and the system of integers are developed. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials and make connections between zeros of polynomials and solutions of polynomial equations. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of ration !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{|}~al numbers. This critical area also includes and exploration of the Fundamental Theorem of Algebra. (3) Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. This discussion of general triangles open up the idea of trigonometry applied beyond the right triangle that is, at least to obtuse angles. Students build on this idea to develop the notion of radian measure for angles and extend the domain of the trigonometric functions to all real numbers. They apply this knowledge to model simple periodic phenomena.)Traditional Pathway Geometry IntroductionModel Geometry January 2011Geometry standards MA 2000 Quantity 6Reason quantitatively and use units to solve problems.T9-12.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.* This standard new to Geometryk9-12.N.Q.3< Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*Geometry MA 20009-12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.9-12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.9-12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.9-12.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in two and three dimensions.12.P.6 / AII.P.6 Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.n12.P.4 / AII.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions. BBuild a function that models a relationship between two quantitiesCreating Equations7Create equations that describe numbers or relationships)Reasoning with Equations and InequalitiesPUnderstand solving equations as a process of reasoning and explain the reasoning0Solve equations and inequalities in one variableSolve systems of equations12.P.7 / AII.P.7 Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems.Interpreting Functions>Understand the concept of a function and use function notation 9-12.N-CN.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. !Z9-12.N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.i9-12.N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.9-12.A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.89-12.A-REI.4 Solve quadratic equations in one variable. 9-12.A-REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from this form. A9-12.A-REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a +/- bi for real numbers a and b.BC %' 9: ?@ 9-12.A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2+y2 = 3.$$##9-12.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. $9-12.F-BF.4 Find inverse functions. 9-12.F-TF.8 Prove the Pythagorean identity (sin A)2 + (cos A)2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.-4#5?#@DXe hov F9-12.G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. .j9-12.G-SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. J9-12.G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.~9-12.G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 9-12.G-SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. .9-12.G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 9-12.G-SRT.6 Define trigonometric ratios and solve problems involving right triangles. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 9-12.S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ).*9-12.S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.*hUse the rules of probability to compute probabilities of compound events in a uniform probability model. 9-12.S.CP.6 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model.*{ 9-12.S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model.*r9-12.S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.*EMA 2000 Geometry Standards Matched at other grades/courses in MA 2011j9-12.N.VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.;MA 2000 Geometry Standards Not Matched by MA 2011 StandardsNo match in MA 2011j9-12.G.SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle simil< arity. .M9-12.G.GPE.2 Derive the equation of a parabola given a focus and a directrix. 12.G.3 / G.G.18 Use the notion of vectors to solve problems. Describe addition of vectors and multiplication of a vector by a scalar, both symbolically and pictorially. Use vector methods to obtain geometric results. ();MA 2011 Traditional Pathway Model Algebra II IntroductionAlgebra II (MA 2000)'CC.9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.sMA 2011 Footnote for F.BF.3 & 4: MA 2011 Mathematics II expands to include quadratic and absolute value functions.CC.9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MA.9-12.A.APR.1a Divide polynomials. 9-12.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x).EIvw| 9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2).Qcdhiopqrwxyzw 9-12.N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i).G]^H 9-12.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)YZ59-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.* 9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R.*v612.P.8 / AII.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic functions; expressions involving absolute values; and simple rational expressions. DUnderstand the relationship between zeros and factors of polynomialsProve geometric theorems12.P.2 / AII.P.2 Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the general term and sum recursively and explicitly. 12.P.11 / AII.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G(m1m2)/d2) variation, and periodic processes.=>BC[\]^abo12.P.3 / AII.P.3 Demonstrate an understanding of the binomial theorem and use it in the solution of problems.12.P.10 / AII.P.10 Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Also use technology where appropriate. Describe the relationships among the methods. MA 2011 Footnote: MA 2011 Mathematics I should master this skill for linear equations and inequalities and learn as a general principle to be expanded in Mathematics II and III.yMA 2011 Footnote: MA 2011 Mathematics I is limited to linear inequalities and exponential equations of the form 2x=1/16.rsSMA 2011 Footnote: MA 2011 Mathematics I is limited to systems of linear equations.MA 2011 Footnote for A.REI.10, 11, 1& 2: MA 2011 Mathematics I should master this skill for linear equations and inequalities and learn as a general principle to be expanded in Mathematics II and III.MA 2011 Footnote for F.IF.1, 2, &3: MA 2011 Mathematics I should focus on linear and exponential functions with integer domains and on arithmetic and geometric sequences.MA 2011 Footnote for F.IF.7, 9, & 10: MA 2011 Mathematics I is limited to linear and exponential functions with integer domains.hMA 2011 Footnote for F.LE.1, 2, & 3: MA 2011 Mathematics I is limited to linear and exponential models.MA 2011 Footnote for G.CO.6, 7, & 8: MA 2011 Mathematics I should use rigid motions as a familiar starting point to develop geometric proof. MA 2011 Footnote for G.GPE.4, 5, &7: MA 2011 Mathematics I should include the distance formula and relate it to the Pythagorean Theorem.MA 2011 Footnote for S.ID.5 & 6: MA 2011 Mathematics I should focus on linear applications; learn as a general principle to be expanded in Mathematics II and III.G.G.5 Apply congruence and similarity correspondences (e.g., ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.GMA 2000 High School Course Standards Matched at other grades in MA 20119-12.N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*S9-12.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.* l9-12.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*10.G.4 / G.G.5 Apply congruence and similarity correspondences (e.g., ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.t12.N.2 / AII.N.2 Simplify numerical expressions with powers and roots, including fractional and negative exponents.m12.G.1 / AII.G.1 Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems./Understand congruence in terms of rigid motions10.P.1 / AI.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonacci Numbers), linear, quadratic, and exponential functional relationships.9-12.S.ID.4 Summarize, represent,< and interpret data on a single count or measurement variable. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.*9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.*9-12.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.*9-12.S.IC.4 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.*9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.*.9-12.S.IC.6 Evaluate reports based on data.*12.M.2 / G.M.5 / PC.M.2 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. 9-12.F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.+9-12.F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.* 9-12.F-LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.*{ How to read this crosswalk: The first column of this Mathematics I Crosswalk presents the 2011 Massachusetts Integrated Pathway Mathematics I course standards, including the Massachusetts additional standards. Please note that some of these standards are marked by an asterisk (*) indicating a modeling standard, as defined in the High School Modeling Conceptual Category. The second column presents related standards from the Massachusetts 2000 Algebra I, Geometry, or Algebra II course. The third column provides informational comments highlighting differences with italicized footnotes that limit the scope of the standard for this course. If there is no appropriate MA 2000 match, the second and third columns are shaded green, with any appropriate comments in the third column. This crosswalk is designed as a tool for use by districts and schools as they prepare for the implementation of the Massachusetts 2011 Standards for Mathematics. When reviewing the crosswalk, please keep in mind that the correlations between standards indicated in the crosswalk could be direct, meaning that the standards contain the same content, or could be partial, meaning that parts of the standards are related. Also note that several MA 2000 standards may be matched to one MA 2011 standard, and conversely, one MA 2000 standard could be matched to several MA 2011 standards. At the end of the Mathematics I crosswalk, MA 2000 Mathematics I standards that are unmatched are presented in two ways: (1) MA 2000 Mathematics I standards that match MA 2011standards at a different grade level, with the best match indicated in the first column; and (2) MA 2000 Mathematics I standards that do not match any MA 2011 standards.!> R\(2) In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. Students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. 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. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. 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. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.10.G.9 / G.G.15 Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems. _9-12.F-IF.1 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).78V^ How to read this crosswalk: The first column of this Algebra I Crosswalk presents the 2011 Massachusetts Traditional Pathway Model Algebra I course standards, including the Massachusetts additional standards. Please note that some of these standards are marked by an asterisk (*) indicating a modeling standard, as defined in the High School Modeling Conceptual Category. The second column presents related standards from the Massachusetts 2000 Algebra I course. The third column provides informational comments highlighting differences with italicized footnotes that limit the scope of the standard for this course. If there is no appropriate MA 2000 Algebra I match, the second and third columns are shaded green, with any appropriate comments in the third column. This crosswalk is designed as a tool for use by districts and schools as they prepare for the implementation of the Massachusetts 2011 Standards for Mathematics. When reviewing the crosswalk, please keep in mind that the correlations between standards indicated in the crosswalk could be direct, meaning that the standards contain the same content, or could be partial, meaning that parts of the standards are related. Also note that several MA 2000 standards may be matched to one MA 2011 standard, and conversely, one MA 2000 standard could be matched to several MA 2011 standards. At the end of the Algebra I crosswalk, MA 2000 Algebra I standards that are unmatched are presented in two ways: (1) MA 2000 Algebra I standards that match MA 2011 standards at a different grade level, with the best match indicated in the first column; and (2) MA 2000 Algebra I standards that do not match any MA 2011 standards.!$8 10.P.2 / AI.P.5 / G.G.11 Demonstrate an understanding of the relationship between various representations of a line. Determine a line s slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or geometric description of the line, e.g., by using the "point-slope" or "slope / y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.5MA 2011 Integrated Pathway Mathematics I Introduction< <MA 2000 Algebra I Standards Not Matched by MA 2011 StandardsZMA 2011 additional standard requiring recognition of equivalent ways to present functions.9-12.F.LE.4 For exponential models, express as a logarithm the solution to ab(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.*LNRUV\bghn12.P.4 / AII.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions. 12.M.2 / PC.M.2 /G.M.5 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. 10.N.2 / AI.N.2 Simplify numerical expressions, including those involving positive integer exponents or the absolute value, e.g., 3(24 1) = 45, 4|3 5| + 6 = 14; apply such simplifications in the solution of problems. 10.P.1 / AI.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonacci Numbers), linear, quadratic, and exponential functional relationships.10.P.8 / AI.P.12 Solve everyday problems that can be modeled using systems of linear equations or inequalities. Apply algebraic and graphical methods to the solution. Use technology when appropriate. Include mixture, rate, and work problems. )Reasoning with equations and inequalitiesiUse the rules of probability to compute probabilities of compound events in a uniform probability model. 9-12.S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).*t9-12.S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).* 12.N.1/ AII.N.1 Define complex numbers (e.g., a + bi) and operations on them, in particular, addition, subtraction, multiplication, and division. Relate the system of complex numbers to the systems of real and rational numbers.8 9612.P.8 / AII.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic functions; expressions involving absolute values; and simple rational expressions.:12.P.8 / AII.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic functions; expressions involving the absolute values; and simple rational expressions.i9-12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.8Write expressions in equivalent forms to solve problems.'Interpret the structure of expressions.CC.9-12.A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*-Perform arithmetic operations on polynomials.9-12.A.APR.3 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.Rewrite rational expressions. 9-12.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 y2)2 + (2xy)2 can be used to generate Pythagorean triples. ]9-12.A.APR.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.RUnderstand solving equations as a process of reasoning and explain the reasoning. >9-12.A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. LMQRVbs{9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* !]ejsKOW[GInterpret functions that arise in applications in terms of the context. Functions4Conditional Probability and the Rules of ProbabilityLG.G.3 Apply formulas for a rectangular coordinate system to prove theorems.G.G.10 Apply the triangle inequality and other inequalities associated with triangles (e.g., the longest side is opposite the greatest angle) to prove theorems and solve problems.9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomials, rational, absolute value, exponential, and logarithmic functions.* W9-12.A.REI.12 Represent and solve equations and inequalities graphically. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. I _9-12.F.IF.1 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).;9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals, where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries, end behavior; and periodicity.*12.D.2 / AII.D.1 Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data. 10.N.2 / AI.N.2 Simplify numerical expressions, including those involving positive integer exponents or the absolute value, e.g., 3(24 1) = 45, 4< |3 5| + 6 = 14; apply such simplifications in the solution of problems. GeometryAI.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent and independent variables of functions.10.M.3 / G.M.3 Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume. Make geometric constructions9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*<9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*c9-12.G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. p9-12.G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.@This standard is new to MA 2011 high school mathematics courses.gMA 2011 Footnote: MA 2011 Algebra I limits to interpreting linear, quadratic and exponential functions.MA 2011 Footnote: In MA 2011 Algebra I, only linear, exponential, quadratic, absolute value, step, and piecewise defined functions are included in this cluster..These standards are new in MA 2011 Algebra I.MMA 2011 Footnote for (b): Do not include square root and cube root functions./MA 2011 Algebra I includes combining functions.cMA 2011 Footnote: Functions are limited to linear, quadratic, and exponential in MA 2011 Algebra I.MA 2011 Footnote: In Algebra I, identify linear and exponential sequences that are defined recursively, continue the study of sequences in Algebra II.+This standard is new in MA 2011 Algebra I.`MA 2011 Footnote: Finding inverse functions is limited to linear functions in MA 2011 Algebra I.MA 2011 Algebra I requires distinguishing between situations modeled with linear and exponential functions and proving how linear and exponential functions grow.6MA 2011 Algebra I relates sequences to function types.MA 2011 Algebra I explicitly requires contextual interpretation of function parameters. MA 2011 Footnote: Limit exponential function to the form f(x) = bx + k. X dMA 2011 Algebra I limits required graphical representations to dot plots, histograms, and box plots..MA 2011 Algebra I includes standard deviation.3MA 2011 Algebra I includes accounting for outliers.5MA 2011 Algebra I requires fitting functions to data.> 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning * Specific modeling standards appear throughout the high school standards indicated in this crosswalk by an asterisk (*).F 9-12.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)YZCMA 2011 Geometry requires knowing definitions of geometric figures.nMA 2011 Geometry specifies methods for performing transformations and describes transformations as functions. ZMA 2011 Geometry separates standards regarding similarity from those regarding congruence.DMA 2011 Geometry includes developing definitions of transformations.9-12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.;MA 2011 Geometry specifies using definition of congruence. 5MA 2011 Geometry requires application of definitions.dMA 2011 Geometry lists the criteria for triangle congruence and relates congruence to rigid motions.5MA 2011 Geometry specifies proof of lines and angles.?MA 2011 Geometry specifies figures and attributes to be proven.4MA 2011 Geometry identifies specific constructions .:MA 2011 Geometry identifies specific figures to construct.EMA 2011 Geometry requires verification of the properties of dilation.7MA 2011 Geometry relates similarity to transformations.OMA 2011 Geometry requires using properties of similarity to establish criteria.\MA 2011 Geometry makes direct connection between similar triangles and trigonometric ratios.QMA 2011 Geometry specifically connects sine and cosine with complementary angles.YMA 2011 Geometry specifically connects trigonometric ratios with the Pythagorean theorem.-These standards are new in MA 2011 Geometry.=MA 2011 Geometry requires proof of similarity of all circles.MA 2011 (+) standards.`MA 2011 (+) standards; MA 2011 Geometry specifies using auxiliary lines and using trigonometry.8MA 2011 Geometry specifies more geometric relationships.IMA 2011 Geometry specifies construction of tangent. MA 2011 (+) standard.KMA 2011 Geometry includes arc lengths, area of sectors, and radian measure.2MA 2011 Geometry derives the equation of a circle.6MA 2011 Geometry requires proof of the slope criteria.AMA 2011 Geometry includes broader range of partitioning segments.LMA 2011 Geometry specifies using the distance formula to compute perimeters.=MA 2011 Geometry specifies "informal argument" for formulas. v9-12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.z9-12.F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* 9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*12.P.12 / AII.P.12 Identify maximum and minimum values of functions in simple situations. Apply to the solution< of problems. 12.P.7 / AII.P.7 Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems.10.N.4 / AI.N.4 Use estimation to judge the reasonableness of computations and of solutions to problems involving real numbers.10.G.8 / AI.P.6 / G.G.13 Find linear equations that represent lines either perpendicular or parallel to a given line through a point, e.g., by using the "point-slope" form of the equation. c8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. F10.G.11 / G.G.17 Use vertex edge graphs to model and solve problems. v10.G.5 / G.G.7 Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean theorem. 12.D.6 / AII.D.2 Use combinatorics (e.g., fundamental counting principle, permutations, and combinations) to solve problems, in particular, to compute probabilities of compound events. Use technology as appropriate. GMA 2011 Mathematics II asks to explain similarity using transformations=MA 2011 Mathematics II specifies more geometric relationshipsFMA 2011 Mathematics II includes broader range of partitioning segments#Using Probability to Make Decisions2Use probability to evaluate outcomes of decisions.hMA 2011 Mathematics II requires identifying and solving for non-real roots using the quadratic formula. 9-12.F-BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x3) or f(x) = (x+1)/(x-1) for x `" 1. #10.G.7 / G.G.12 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.lMA 2011 additional standard requires understanding that methods to solve quadratic equations are equivalent.PMA 2011 additional standard requires description of effect of approximate error.3Table: Codes for Conceptual Categories and DomainsConceptual CategoryDomainCodeNumber and Quantity (N)Quantities Vector and Matrix QuantitiesAlgebra (A) Functions (F)Geometry (G)Modeling with GeometryStatistics and .Interpreting Categorical and Quantitative DataProbability (S)6only partially matches to the following 2011 standard:9-12.F-BF.1 Write a function that describes a relationship between two quantities.* 9-12.F-BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. \ iV9-12.G-SRT.1 Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: 9-12.G-SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. MA 2011 Mathematics II specifies construction of inscribed and circumscribed figures. MA 2011 additional standard derives the formula relating sums of interior and exterior angles to number of sides of a polygon.5 9-12.G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, "3) lies on the circle centered at the origin and containing the point (0.2). PMA 2011 Mathematics III extends polynomial identities to the complex numbers. MA 2011 Footnote: MA 2011 Mathematics III is limited to polynomials with real coefficients.N uMA 2011 Footnote: MA 2011 Mathematics III should expand previous understanding to include higher degree polynomials.FMA 2011 Mathematics III includes rotating 2-D objects to generate 3-D.9MA 2011 Mathematics III specifies using geometric models.SMA 2011 Mathematics III addresses relationship between density and volume directly.@MA 2011 Mathematics III match with MA 2000 Precalculus standard.LMA 2011 Mathematics III requires relating parts and parameters to the model.oMA 2011 Mathematics III focuses on using key features to guide selection of appropriate type of model function.>MA 2011 Mathematics III relates domain to graph of a function.YMA 2011 Mathematics III specifies key features to be used to interpret graphs and tables.@MA 2011 specifies to include examples with extraneous solutions.cMA 2011Mathematics III specifies to represent constraints and interpret solutions as viable or not.:MA 2011 Mathematics I relates sequences to function types.3MA 2011 Mathematics I requires knowing definitions.GMA 2011 Mathematics I specifies methods for performing transformations.`MA 2011 Mathematics I separates standards regarding similarity from those regarding congruence. 6MA 2011 Mathematics I includes developing definitions.@MA 2011 Mathematics I specifies using definition of congruence. :MA 2011 Mathematics I requires application of definitions.AMA 2011 Mathematics I lists the criteria for triangle congruence.MA 2011 Mathematics I lists constructions. MA 2011 Footnote for G.CO.12 & 13: MA 2011 Mathematics I should formalize proof focusing on the explanation of the process.47MA 2011 Mathematics I lists constructions by category. 8MA 2011 Mathematics I requires prove the slope criteria.QMA 2011 Mathematics I specifies using the distance formula to compute perimeters.hMA 2011 Mathematics I limits required graphical representations to dot plots, histograms, and box plots.2MA 2011 Mathematics I includes standard deviation.7MA 2011 Mathematics I includes accounting for outliers.9MA 2011 Mathematics I requires fitting functions to data.JMA 2011 Mathematics I requires using residuals to assess fit of functions./Vertex edge graphs are not included in MA 2011.VThis standard is new to MA 2011 high school mathematics courses. MA 2011 (+) standard.BMA 2011 Mathematics II specifies "informal argument" for formulas.8MA 2011 Mathematics II derives the equation of a circle.OMA 2011 Mathematics II specifies construction of tangent. MA 2011 (+) standard.QMA 2011 Mathematics II includes arc lengths, area of sectors, and radian measure.4MA 2011 Mathematics II requires proof of similarity.PMA 2011 specifically connects trigonometric ratios with the Pythagorean theorem.WMA 2011 Mathematics II specifically connects sine and cosine with complementary angles.bMA 2011 Mathematics II makes direct connection between similar triangles and trigonometric ratios.VMA 2011 Mathematics II requires using properties to establish criteria for similarity.hMA 2011 Footnote: Focus on validity underlying reasoning and using a variety of ways of writing proofs.7MA 2011 Mathematics II verifies properties of dilation.EMA 2011 Mathematics II specifies figures and attributes to be proven.MA 2011 Footnote: Focus on validity underlying reasoning and using a variety of ways of writing proofs. MA 2011 Mathematics II specifies proofs of lines an< d angles.j KMA 2011 Mathematics II requires relating parts and parameters to the model.nMA 2011 Mathematics II focuses on using key features to guide selection of appropriate type of model function.4This standard is new in MA 2011 high school courses.>MA 2011 Mathematics II relates domain to graph of a function. >QMA 2011 Footnote: MA 2011 Mathematics II expands to include quadratic functions.XMA 2011 Mathematics II specifies key features to be used to interpret graphs and tables.SMA 2011 Footnote: Mathematics II should expand to include linear/quadratic systems.?This standard is new to MA 2011high school mathematics courses.cMA 2011 Footnote: MA 2011 Mathematics II expands to include quadratic and exponential expressions.High School Model Mathematics Course Crosswalk Introduction: Comparing the 2011 MA Integrated Pathway Model Mathematics Course Standards to the 2000 MA High School Course Standards >! On December 21, 2010, the Board of Elementary and Secondary Education adopted the 2011 Massachusetts Curriculum Framework for Mathematics, Grades Pre-Kindergarten to 12: Incorporating the Common Core State Standards for Mathematics. RX !The High School Mathematics Course Crosswalk is intended to assist districts and schools to align curriculum, instruction, and assessments to the new Massachusetts 2011 mathematics standards (MA 2011). For each course, the crosswalk presents the MA 2011 standards side-by-side with the 2000 Massachusetts standards for mathematics (MA 2000). Each course crosswalk begins with a brief How to Read this Crosswalk note, followed by the model course introduction, the eight Standards for Mathematical Practice, and then the crosswalk in table form. MA 2011 Mathematics I explicitly requires contextual interpretation of function parameters. MA 2011 Footnote: MA 2011 Mathematics I is limited to linear functions and exponential functions of the form f(x) = bx + k. d WMA 2011 Mathematics I requires interpretation of slope and intercept of a linear model.PMA 2011 Mathematics I includes distinguishing between correlation and causation.bMA 2011 Footnote: MA 2011 Mathematics II should link to data from simulations and/or experiments.MA 2011 Footnote: MA 2011 Mathematics II should introduce these concepts and apply counting rules. MA 2011(+) standard.w5MA 2000 standard matched to MA 2011 Grade 8 standard.12.D.5 / PC.D.5 Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities. 9-12.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.* 9-12.S.IC.4 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.* 9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.* .9-12.S.IC.6 Evaluate reports based on data.* %9-12.F.BF.4 Find inverse functions. '9-12.F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic functions and an algebraic expressions for another, say which has the larger maximum. MA additional standard. Recognizing that a function can be presented in a variety of ways assists in assessing how to approach solving a problem.l9-12.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MA.9-12.G-C.3a Derive the formula for the relationship between the number of sides and sums of the interior and sums of the exterior angles of polygons and apply to the solution of problems. MA 2011 Footnote: MA 2011 Mathematics III should emphasize the selection of appropriate function model; expand to include rational functions, square and cube root functions.MA 2011 Footnote F.IF.7, 8, & 9: MA 2011 Mathematics III should expand previous understanding to include rational and radical function; focus on using key features to guide selection of appropriate type of function model.9-12.S.ID.4 Summarize, represent, and interpret data on a single count or measurement variable. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.* 12.D.2 / AII.D.1 Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data. 9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.* j12.D.1 / PC.D.1 Design surveys and apply random sampling techniques to avoid bias in the data collection. 9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the model?* 9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*[9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.* 9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.* MA 2011 Footnotes for (e): In Algebra I, it is sufficient to graph exponential functions showing intercepts. Showing end behavior of exponential functions and graphing logarithmic and trigonometric functions is not part of Algebra I.(This standard new to MA 2011 Algebra I. 9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*v9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.*9-12.F.LE.1a Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.*9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*h9-12.S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).* 9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (inte< rquartile range, standard deviation) of two or more different data sets.* 9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).* 9-12.S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trendsy9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.*9-12.S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.* \9-12.S.ID.6b Informally assess the fit of a function by plotting and analyzing residuals.* \ 9-12.S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.*b9-12.S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.*T9-12.S.ID.9 Interpret linear models. Distinguish between correlation and causation.*KMA 2011 Algebra I includes distinguishing between correlation and causationIMA 2000 Algebra I Standards Matched at Other Grades or Courses in MA 20119-12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). +MA 2000 standard Matched to MA 2011 Grade 8 9-12.A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). Pcdhiopqrwxyz9-12.A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. _ dMA.9-12.A.REI.3a Solve linear equations and inequalities in one variable involving absolute value. 89-12.A.REI.4 Solve quadratic equations in one variable. 9-12.A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from this form. 9-12.A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 9-12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x2 + y2 = 3.9-12.F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n>=1. 9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.* MA.9/12.F.IF.10 Given algebraic, numeric, and/or graphical representations of functions, identify the function as polynomial, rational, logarithmic, exponential, or trigonometric. 9-12.F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. I$9-12.F.BF.4 Find inverse functions. 9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x3) or f(x) = (x+1)/(x-1) for x `" 1. _9-12.F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. Z9-12.S.ID.6c Fit a linear function for a scatter plot that suggests a linear association.* F9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. .9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.J9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.9-12.G.SRT.6 Define trigonometric ratios and solve problems involving right triangles. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.+9-12.G.C.5 Find arc lengths and areas of sectors of circles. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.9-12.G.GPE.1 Translate between the geometric description and the equation for a conic section. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.[MA 2011 Footnote: MA 2011 Mathematics II should include formulas involving quadratic terms.cMA 2011 Footnote: MA 2011 Mathematics II is limited to quadratic equations with real coefficients.MA 2011 Footnote for F.IF.7, 8, & 10: MA 2011 Mathematics II is limited to linear, exponential, quadratic, piecewise-defined, and absolute value functions.SMA 2011 Footnote: MA 2011 Mathematics II limits use of radians to unit of measure.JMA 2011 Footnote: MA 2011 Mathematics II includes simple circle theorems.MA 2011 Mathematics II includes the definition of the imaginary number i MA 2011 Footnote: MA 2011 Mathematics II limits i2 as highest power of i.GI z{ | 12.P.11 / AII.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the sol< ution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G(m1m2)/d2) variation, and periodic processes.'=%>B%CDU\%]^%_b&cv 10.P.4 / AI.P.8 Demonstrate facility in symbolic manipulation of polynomial and rational expressions by rearranging and collecting terms; factoring (e.g., a2 b2 = (a + b)(a b), x2 + 10x + 21 = (x + 3)(x + 7), 5x4 + 10x3 5x2 = 5x2 (x2 + 2x 1)); identifying and canceling common factors in rational expressions, and applying properties of positive integer exponents.&&&&&&&&12.G.2 / AII.G.2 / PC.G.1 Derive and apply basic trigonometric identities (e.g., sin2q + cos2q = 1, tan2q + 1 = sec2q) and the laws of sines and cosines. RU&V^&_j&kw&xy"Algebra I and Geometry (MA 2000)m9-12.N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.+ c9-12.N-RN.3 Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 9-12.N-CN.1 Know there is a complex number i such that i2 = "1, and every complex number has the form a + bi with a and b real. +,-".89:gmstyz&9-12.S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.* 9-12.S-ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function~ suggested by the context. Emphasize linear, quadratic, and exponential models.* 9-12.S-ID.6b Informally assess the fit of a function by plotting and analyzing residuals.* 9-12.S-ID.6c Fit a linear function for a scatter plot that suggests a linear association.*P \ 4This standard is new to MA 2011 high school courses6MA 2000 standard matched to MA 2011 Grade 8 standard.There is no close MA 2011 match to this 2000 9/10 and Algebra I standard. MA 2011 requires estimation and mental computation beginning in grade 4.9MA 2000 standard matched to MA 2011 Algebra II standard.6MA 2000 standard matched to MA 2011 Geometry standard9-12.G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MA.9-12.G-CO.11a Prove theorems about polygons. Theorems include measures of interior and exterior angles, properties of inscribed polygons. 3 3bMA 2011 Mathematics II specifies figures and attributes to be proven. MA additional standard. d8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. 9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.* 9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.* 9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.*] j 6mMA 2011 Mathematics III specifies polynomials as a system analogous to the integers. MA additional standard.MA 2011 additional standard. 9MA 2011 requires evaluation of reports based on the data.MThese standards are new to MA 2011 high school courses. MA 2011 (+) standard.ZMA 2011(+) standard. MA 2011 Mathematics III requires proof of Laws of Sines and Cosines.NMA 2011 (+) standard. MA 2011 Mathematics III specifies using auxiliary lines.5This standard is new to MA 2011 high school courses.HMA 2011 (+) standard. This standard new to MA 2011 high school courses MA 2011 (+) standardh How to read this crosswalk: The first column of this Mathematics III Crosswalk presents the 2011 Massachusetts Integrated Pathway Mathematics III course standards, including the Massachusetts additional standards. Please note that some of these standards are marked by an asterisk (*) indicating a modeling standard, as defined in the High School Modeling Conceptual Category. The second column presents related standards from the Massachusetts 2000 Algebra I, Geometry, or Algebra II course. The third column provides informational comments highlighting differences with italicized footnotes that limit the scope of the standard for this course. If there is no appropriate MA 2000 match, the second and third columns are shaded green, with appropriate comments in the third column. This crosswalk is designed as a tool for use by districts and schools as they prepare for the implementation of the Massachusetts 2011 Standards for Mathematics. When reviewing the crosswalk, please keep in mind that the correlations between standards indicated in the crosswalk could be direct, meaning that the standards contain the same content, or could be partial, meaning that parts of the standards are related. Also note that several MA 2000 standards may be matched to one MA 2011 standard, and conversely, one MA 2000 standard could be matched to several MA 2011 standards.!BV2MA 2000 Standards Not Matched by MA 2011 StandardsMA 2011 Mathematics II specifies polynomials as a system analogous to the integers. MA 2011 Footnote: MA 2011 Mathematics II focuses on adding and multiplying polynomial expressions, factoring expressions to identify and collect like terms, and applying the distributive property.T 9-12.N.CN.1 Know there is a complex number i such that i2 = "1, and every complex number has the form a + bi with a and b real. +-89:gmstyz9-12.F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. J9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0. 5. Would a result of 5 tails in a row cause you to question the model?*MA 2011 Footnote A.CED.1, 2, 3, & 4: MA 2011 Mathematics III should expand previous understanding to include simple root functions. 12.G.2 / AII.G.2 / PC.G.1 Derive and apply basic trigonometric identities (e.g., sin2q + cos2q = 1, tan2q + 1 = sec2q) and the laws of sines and cosines. UV^_klxyn12.G.4 / AII.G.3 Relate geometric and algebraic representations of lines, simple curves, and conic sections. 10.P.6 / AI.P.10 Solve equations and inequalities including those involving absolute value of linear expressions (e.g., |x 2| < > 5) and apply to the solution of problems. 10.P.5 / AI.P.9 Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods.12.P.6 / AII.P.6 Given algebraic, numeric, and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.Comment12.P.6 / AII.P.6 Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.n12.P.4 / AII.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions. d12.P.5 / AII.P.5 Perform operations on functions, including composition. Find inverses of functions. CongruenceMVisualize relationships between two-dimensional and three-dimensional objectsModeling with Geometry /Apply geometric concepts in modeling situations@Use coordinates to prove simple geometric theorems algebraically3 9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,"3) lies on the circle centered at the origin and containing the point (0.2). Pl9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MA.9-12.G.C.3a Derive the formula for the relationship between the number of sides and sums of the interior and sums of the exterior angles of polygons and apply to the solution of problems. MA 2011 Geometry specifies construction of inscribed and circumscribed figures. MA 2011 additional standard requires deriving the formula relating sums of interior and exterior angles to number of sides of a polygon.X9-12.G.SRT.1 Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: 9-12.G.SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. 9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MA.9-12.G.CO.11a Prove theorems about polygons. Theorems include measures of interior and exterior angles, properties of inscribed polygons. 3 9MA 2011 Geometry specifies figures and attributes to be proven. MA additional standard: proof of properties for shapes other than triangles and parallelograms should be specified.9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MA.9-12.A.APR.1a Divide polynomials. 9-12.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x).EIv}CC.9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*'9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.* 9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.* MA 2011 additional standard.$9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 9-12.F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x3) or f(x) = (x+1)/(x-1) for x `" 1. _MA 2011 Algebra II specifies to represent constraints and interpret solutions as viable or not.. For the high school Model Algebra II course, instructional time should focus on four critical areas: (1) relate arithmetic of rational expressions to arithmetic of rational numbers; (2) expand understandings of functions and graphing to include trigonometric functions; (3) synthesize and generalize functions and extend understanding of exponential functions to logarithmic functions; and (4) relate data display and summary statistics to probability and explore a variety of data collection methods. (1) A central theme of this Model Algebra II course is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Students explore the structural similarities between the system of polynomials and the system of integers. They draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Connections are made between multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The Fundamental Theorem of Algebra is examined. '. (2) Building on their previous work with functions and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. (3) Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the und< erlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. (4) Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data including sample surveys, experiments, and simulations and the role that randomness and careful design play in the conclusions that can be drawn.*%*)Model Algebra II March 20112These standards are new in MA 2011 (+) standard.9-12.F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* 9-12.F-IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. 9-12.F-IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.* 9-12.F-IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = 1.02t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth and decay. MA.9-12.F-IF.8c Translate between different representations of functions and relations: graphs, equations, point sets, and tables. b * 7MA 2011 Integrated Pathway Mathematics III IntroductionModel Algebra I (March 2011)Massachusetts March 2011*9-12.S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.*9-12.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*9.-12.S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model P(A and B) = P(A) P(B|A)= P(B)P(A|B), and interpret the answer in terms of the model. *lmxy How to read this crosswalk: The first column of this Algebra II Crosswalk presents the 2011 Massachusetts Traditional Pathway Algebra II course including the Massachusetts additional standards. Please note that some of these standards are marked by an asterisk (*) indicating a modeling standard, as defined in the High School Modeling Conceptual Category. The second column presents related standards from the Massachusetts 2000 Algebra II course. The third column provides informational comments highlighting differences and italicized footnotes that limit the scope of the standard for this course. If there is no appropriate MA 2000 Algebra II match, the second and third columns are shaded green, with appropriate comments in the third column. This crosswalk is designed as a tool for use by districts and schools as they prepare for the 2012-13 implementation of the Massachusetts 2011 Standards for Mathematics. When reviewing the crosswalk, please keep in mind that the correlations between standards indicated in the crosswalk could be direct, meaning that the standards contain the same content, or could be partial, meaning that parts of the standards are related. Also note that several MA 2000 standards may be matched to one MA 2011 standard, and conversely, one MA 2000 standard could be matched to several MA 2011 standards. At the end of this crosswalk MA 2000 Algebra II standards that are unmatched by any MA 2011 high school standards are listed.#+9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.aMA 2011 Geometry Footnote: Proving the converse of theorems should be included when appropriate. (4) Building upon prior students prior experiences with data, students explore a 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. The Standards for Mathematical Practice complement the content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.(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. In this course students analyze and explain the process of solving an equation and justify the process used in solving a system of equations. 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. (2) In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. Students will learn function notation and develop the concepts of domain and range. T< hey focus on linear, quadratic, and exponential functions, including sequences, and also explore absolute value, step, and piecewise-defined functions; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and 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.P (3) Students extend the laws of exponents to rational exponents involving square and cube roots and apply this new understanding of number; they 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. Students become facile with algebraic manipulation, including rearranging and collecting terms, factoring, identifying and canceling common factors in rational expressions. 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.> 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics* 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning * Specific modeling standards appear throughout the high school standards indicated in this crosswalk by an asterisk (*).79-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.* 9-12.A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. ?9-12.A.REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b.BC !" #% 78 => $9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.9-12.F.BF.3 . Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.09-12.S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.* 10.P.2 / AI.P.5 / G.G.11 Demonstrate an understanding of the relationship between various representations of a line. Determine a line s slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or geometric description of the line, e.g., by using the "point-slope" or "slope / y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.@This standard is new to MA 2011high school mathematics courses.OMA 2011 Algebra I specifies use of units in problem solving and interpretation.+This standard is new in MA 2011 Algebra I.bMA 2011 Footnote: MA 2011 Algebra I is limited to linear, quadratic, and exponential expressions.MA 2011 Algebra I includes identifying structure of expressions and rewriting them. MA 2011 Footnote: MA 2011 Algebra I is limited to linear, quadratic, and exponential expressions.U VMA 2011 Algebra I includes using the properties of exponents in exponential functions.=MA 2011 Algebra I includes finding the max/min of quadratics.MA 2011 specifies that polynomials form a system analogous to integers MA 2011 Footnote: For Algebra I, focus on adding and multiplying polynomial expressions, factor or expand polynomial expressions to identify and collect like terms, apply the distributive property.G eMA 2011 Footnote: Create linear, quadratic, exponential (with integer domain) equations in Algebra I.MA 2011 Footnote: Create linear, quadratic, exponential (with integer domain) equations in Algebra I. MA 2011 Footnote: Equations and inequalities in this standard shoul< d be limited to linear.ZMA 2011 additional standard specifies equations and inequalities involving absolute value.}MA 2011 Algebra I requires recognizing non-real roots using the quadratic formula but does not require writing complex roots.1MA 2011 Algebra I requires proof of the solution.MA 2011 Footnote: Algebra I does not include the study of conic equations; include quadratic equations typically included in Algebra I.MA 2011 Footnote: In MA 2011 Algebra I, functions are limited to linear, absolute value, and exponential functions for this standard. MA 2011 Algebra I requires explanation of points of intersection on graphs. MA 2011 Footnote: In MA 2011 Algebra I, functions are limited to linear, absolute value, and exponential functions for this standard.NpMA 2011 Algebra I requires graphing solution of systems of linear inequalities as the corresponding half-planes.MA 2011 Algebra I includes interpreting the key features of graphs and tables. MA 2011 Footnote: In MA 2011 Algebra I, functions are limited to linear, absolute value, and exponential functions for this standard.PMA 2011 explicitly requires relating domain of function to graph. MA 2011 Footnote: MA 2011 Algebra I limits to interpreting linear, quadratic and exponential functions.C 9-12.N.CN.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. !?MA 2011 Algebra II does not include division of complex numbers;Use complex numbers in polynomial identities and equations.Z9-12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.HMA 2011 Algebra II extends polynomial identities to the complex numbers.*This standard is new in MA 2011 Algebra IIVector Quantities and Matrices,Represent and model with vector quantities. @Perform operations on matrices and use matrices in applications.9-12.N.VM.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. q12.P.9 / AII.P.9 Use matrices to solve systems of linear equations. Apply to the solution of everyday problems.P9-12.N.VM.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.9-12.N.VM.12 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. W9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.*Z9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients.* 9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.* ?9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9-12.F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = 1.02t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth and decay. MA.9-12.F.IF.8c Translate between different representations of functions and relations: graphs, equations, point sets, and tables. t 12>?QT_c MA 2011 Algebra I specifies interpreting zeros, extreme values, and symmetry of a quadratic graph in context. MA 2011 Algebra I specifies using properties of exponents to interpret expressions. MA 2011 Footnote: In MA 2011 Algebra I, only linear, exponential, quadratic, absolute value, step, and piecewise defined functions are included in this cluster. MA 2011 additional standard requires translation between different representations of functions.f 9-12.F.BF.1 Write a function that describes a relationship between two quantities.* 9-12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. [ g9-12.F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.* 9-12.F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*{ `Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include logarithmic, polynomial, rational, and radical functions in the Model Algebra II course. This course is comprised of standards selected from the high school conceptual categories, which were written to encompass the scope of content and skills to be addressed throughout grades 9 12 rather than through any single course. Therefore, the complete standard is presented in the model course, with clarifying footnotes as needed to limit the scope of the standard and indicate what is appropriate for study in this particular course. Standards that were limited in Model Algebra I no longer have those restrictions in Model Algebra II. Students work closely with the expressions that define the functions, are facile with algebraic manipulations of expressions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Standards for Mathematical Practice complement the content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.Q9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. 4FMA 2011 Algebra I requires using residuals to assess fit of functions.SMA 2011 Algebra I requires interpretation of slope and intercept of a linear model.There is no MA 2011 match to this 2000 9/10 and Algebra I standard. MA 2011 requires estimation and mental computation beginning in grade 4.9-12.F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defin< ed recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n>=1. 9-12.A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomials, rational, absolute value, exponential, and logarithmic functions.* ]ejrLPW\;9-12.F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*$9-12.F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.9-12.A-SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.* jMA 2011Mathematics II extends polynomial identities to the complex numbers. MA 2011 (+) standard. MA 2011 Footnote: MA 2011 Mathematics II limits quadratic equations to those with real coefficients.b DMA 2011 Mathematics II does not include division of complex numbers.KMA 2011 Mathematics II requires explanation of rules of rational exponents.9-12.A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* 9-12.A-SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines.* 9-12.A-SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.* 9-12.A-SSE.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as [1.15(1/12)](12t) H" 1.012(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.* 9-12.F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* 9-12.F-IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.* 9-12.F-IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.* 9-12.F-IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. continued from previous pageSMA 2011 Mathematics I specifies use of units in problem solving and interpretation./This standard is new to MA high school courses.7These standards are new to MA 2011 high school courses.jMA 2011 Footnote: MA 2011 Mathematics I is limited to linear equations and inequalities in this standard. 5MA 2011 Mathematics I requires proof of the solution.OMA 2011 Mathematics I requires explanation of points of intersection on graphs.uMA 2011 Mathematics I requires graphing solution of systems of linear inequalities as the corresponding half-planes. MA 2011 Mathematics I includes interpreting the key features of graphs and tables. MA 2011 Footnote for F.IF.4, 5, & 6: MA 2011 Mathematics I should focus on linear and exponential functions with integer domains.\MA 2011 Mathematics I explicitly requires relating domain of function to graph. MA 2011 Footnote for F.IF.4, 5, & 6: MA 2011 Mathematics I should focus on linear and exponential functions with integer domains.Y4This standard is new to MA 2011 high school courses.yFootnote for F.IF.4, 5, & 6: MA 2011 Mathematics I should focus on linear and exponential functions with integer domains.9-12.F-BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. J9-12.F-BF.1 Write a function that describes a relationship between two quantities.* 9-12.F-BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. Z hoMA 2011 Mathematics I requires distinguishing between situations modeled with linear and exponential functions.lMA 2011 Footnote: MA 2011 Mathematics I is limited to linear and exponential functions with integer domains.MA 2011 Footnote: MA 2011 Mathematics I is limited to linear and exponential functions; focus on vertical translations for exponential functions.3MA 2011 Mathematics I includes combining functions.g9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. ~9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 10.G.7 / G.G.12 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.c9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. p9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems.[9-12.G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle.10.G.8 / AI.P.6 / G.G.13 Find linear equations that represent lines either perpendicular or parallel to a given line through a point, e.g., by using the "point-slope" form of the equation. ~9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* 9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. 9-12.G.GMD.2 (+) Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures.`9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* 'The fundamental purpose of the Model Geometry course is to formalize and extend students geometric experiences from the middle grades. This course is comprised of standards selected from the high school conceptual categories, which were written to encompass the scope of content and skills< to be addressed throughout grades 9 12 rather than through any single course. Therefore, the complete standard is presented in the model course, with clarifying footnotes as needed to limit the scope of the standard and indicate what is appropriate for study in this particular course. In this high school Model Geometry course, students explore more complex geometric situations and deepen their explanations of geometric relationships, presenting and hearing formal mathematical arguments. Important differences exist between this course and the historical approach taken in geometry classes. For example, transformations are emphasized in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found on page 92. For the high school Model Geometry course, instructional time should focus on six critical areas: (1) establish criteria for congruence of triangles based on rigid motions; (2) establish criteria for similarity of triangles based on dilations and proportional reasoning; (3) informally develop explanations of circumference, area, and volume formulas; (4) apply the Pythagorean Theorem to the coordinate plan; (5) prove basic geometric theorems; and (6) extend work with probability. (1) Students have prior experience with drawing triangles based on given measurements, performing rigid motions including translations, reflections, and rotations, and have used these to develop notions about what it means for two objects to be congruent. In this course, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems using a variety of formats including deductive and inductive reasoning and proof by contradiction and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.'(5) Students prove basic theorems about circles, with particular attention to perpendicularity and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles. (6) Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. The Standards for Mathematical Practice complement the content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.Model Geometry March 2011xMA 2000 specified simplifing absolute value; MA 2011 includes work with absolute value beginning in grade 6 and grade 7.MA 2011 Algebra I requires explanation of rules of rational exponents. MA 2000 specified simplifing absolute value; MA 2011 includes work with absolute value beginning in grade 6 and grade 7.-MA 2011 focuses on approximating measurement.L9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9-12.F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = 1.02t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. MA.9-12.F.IF.8c Translate between different representations of functions and relations: graphs, equations, point sets, and tables. y :;GHTWbf MA 2011 Footnote for F.BF.3 & 4: MA 2011 Mathematics III should expand previous understanding to include simple radical, rational and exponential functions; emphasize common effect of each transformation across function types.12.M.1 / PC.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular problems involving angular velocity and acceleration.AMA 2011 additional standard requires use of dimensional analysis.mMA 2011 Algebra II specifies polynomials as a system analogous to the integers. MA 2011 additional standard. 9-12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* 9-12.A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines.* 9-12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.* < 9-12.A.SSE.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as [1.15(1/12)](12t) H" 1.012(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.* "#CIJOW] (2) Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. (3) Students experience with three-dimensional objects is extended to include informal explanations of circumference, area, and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. (4) Building on their work with the Pythagorean theorem in 8th grade to find distances, students use the rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. F 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics* 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning * Specific modeling standards appear throughout the high school standards indicated in this crosswalk by an asterisk (*).!MA.9-12.N.Q.3a Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure.* U9-12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch.). `MA 2011 Geometry specifies Cavalieri's Principle and requires "argument". MA 2011 (+) standard. bNote: MA 2011 Grade 8 requires students to know volume formulas for cylinders, cones, and spheres.@MA 2011 Geometry includes rotating 2-D objects to generate 3-D.2MA 2011 Geometry specifies using geometric models.LMA 2011 Geometry addresses relationship between density and volume directly.*This standard is new in MA 2011 Geometry.CThese standards are new to MA 2011 high school m !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|~athematics courses.MA 2011 (+) standard.CThese standards are new in MA 2011 Geometry. MA 2011 (+) standard.Included in 2011 Geometry.+This standard is new in MA 2011 Algebra II.0MA 2011 Algebra II specifies bases 2, 10, and e..GMA 2011 Algebra II requires relating parts and parameters to the model.jMA 2011 Algebra II focuses on using key features to guide selection of appropriate type of model function.9MA 2011 Algebra II relates domain to graph of a function.TMA 2011 Algebra II specifies key features to be used to interpret graphs and tables.MMA 2011 Algebra II specifies inclusion of examples with extraneous solutions.`MA 2011 Algebra II specifies rational expressions as a system analogous to the rational numbers.GMA 2011 Algebra II specifies to rewrite expressions in different forms.MMA 2011 Algebra II includes deriving the formula for finite geometric series.EMA 2011 Algebra II includes the definition of the imaginary number i.C79-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.*AThis standard is new to MA 2011 high school mathematics courses.cMA 2011 Footnote for this cluster: Foundation for work with expressions, equations, and functions. How to read this crosswalk: The first column of this Geometry Crosswalk presents the 2011 Massachusetts Traditional Pathway model Geometry course standards, including the Massachusetts additional standards. Please note that some of these standards are marked by an asterisk (*) indicating a modeling standard, as defined in the High School Modeling Conceptual Category. The second column presents related standards from the Massachusetts 2000 Geometry course. The third column provides informational comments highlighting differences with italicized footnotes that limit the scope of the standard for this course. If there is no appropriate MA 2000 Geometry match, the second and third columns are shaded green, with any appropriate comments in the third column. This crosswalk is designed as a tool for use by districts< and schools as they prepare for the implementation of the Massachusetts 2011 Standards for Mathematics. When reviewing the crosswalk, please keep in mind that the correlations between standards indicated in the crosswalk could be direct, meaning that the standards contain the same content, or could be partial, meaning that parts of the standards are related. Also note that several MA 2000 standards may be matched to one MA 2011 standard, and conversely, one MA 2000 standard could be matched to several MA 2011 standards. At the end of the Geometry crosswalk, MA 2000 Geometry standards that are unmatched are presented in two ways: (1) MA 2000 Geometry standards that match MA 2011 standards at a different grade or course, with the best match indicated in the first column; and (2) MA 2000 Geometry standards that do not match any MA 2011 standards.!!5z9-12.F-LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.* 9-12.F-LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.* |)th> Ttx00 ,hp5L%>?EHkWNoV']r<{sփjKt1C<;jDsBqz!P(.9@Kn>'-M63`8&= HMUP7 S\/a_aWcZ3hndr ss]*{}Ifv*;xFpԦ/=tV ָ 5l1 *=0q yEi5Q=EaI-OXQY4gtov #z~ҞQkccVB g2ɀ.e dMbP?_*+%DA&L&P&CDepartment of Elementary and Secondary Education&RJune 2011&?'?(?)?MC odXXLetter"cXX??&U}}I}} }}} . F@ @ @ @ @ ~ @ @ @ Y Z [ 1 9 ; O``,,,,`OO``,,,,`O @ # ) @ C E D l((((($DDD::D:::$DD:::D::::: ! 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