Pre-K HS Mathematics Core Course Objectives

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1 Pre-K HS Mathematics Core Course Objectives The Massachusetts Department of Elementary and Secondary Education (ESE) partnered with WestEd to convene panels of expert educators to review and develop statements of essential curriculum content,, for approximately 100 different grades/subjects and courses. In conjunction with the Massachusetts Curriculum Frameworks, the CCOs were used by a team of WestEd evaluators to ensure content alignment in the identification and evaluation of example assessments suitable for use as District-Determined Measures (DDMs). Districts can utilize the CCOs themselves or the process of developing CCOs in their own work when selecting DDMs. At each meeting, educators developed highquality CCOs that met the following criteria: Scope: The CCO describes an overarching learning goal. Assess-ability: The CCO describes knowledge, skills, or abilities that are readily able to be measured. are statements that: describe different elements of core, essential content (knowledge, skills, or abilities); are pulled, created, or synthesized from a larger set of curriculum standards; and clarify key knowledge, skills, and abilities that many educators and other content experts working together agree are most critical in that content area, grade, or course. Each CCO should be high-level and represent broad enough learning goals to be taught using a wide variety of instructional tools or methods (scope), while also focused enough that students growth in learning that knowledge or skill can be measured by an assessment (assess-ability). Centrality: The CCO describes a critically important concept, skill, or ability that is central to the subject/grade or course. Relevance: The CCO represents knowledge, skills, and abilities that are consistent with Massachusetts s values and goals. After public review, WestEd s content specialists reviewed the comments submitted by educators and other stakeholders. The CCOs that emerged from this process are presented below. 1

2 Mathematics Grade: Pre-K 1 Students identify and name written numerals and connect them with quantities up to ten, using subitizing (recognizing quantities without counting) and/or counting. 2 Students listen to and say the names of numbers in meaningful contexts. 3 Students count many kinds of concrete objects and actions up to ten, using one-to-one correspondence, and accurately count as many as seven things in a scattered configuration. 4 Students compare collections of objects and describe collections using language, such as more/less than or equal to/same as, to compare and describe collections of objects. 5 Students use concrete objects (e.g., cubes, fingers, manipulatives) to model real-world addition problems (putting together) and subtraction problems (taking away) up through five. 6 Students recognize and compare measurable attributes of the length and weight of everyday objects using appropriate vocabulary (e.g., long[er], short[er], tall[er], heavy[heavier], light[er], wide[wider], narrow[er]). 7 Students sort, categorize, and classify objects by more than one attribute. 8 Students use appropriate language to identify relative positions of objects in space (e.g., beside, inside, next to, close to, above, below, and apart). course. Source documents used are as follows: Massachusetts Mathematics Curriculum Framework (2011); Learning Progressions Frameworks Designed for Use with The Common Core State Standards in Mathematics K 12 (NCIEA) (2011). 2

3 Mathematics Algebra I Grade: HS 1 Students use the properties of exponents to rewrite expressions involving radicals and rational exponents and to transform expressions for exponential functions. 2 Students interpret expressions that represent a quantity, interpret parts of an expression (terms, factors, and coefficients) in terms of context, and use the structure of an expression to rewrite it. 3 Students produce an equivalent form of a quadratic expression by factoring to reveal the zeros of the function defined; and, by completing the square, to reveal the maximum or minimum value of the function defined. 4 Students perform addition, subtraction, and multiplication of polynomials. 5 Students write, interpret, and translate among various forms of linear equations and inequalities in solving real-world problems. 6 Students explain each step in solving a simple equation and construct a viable argument to justify a solution method. 7 Students write, interpret, and translate among various forms of quadratic equations and inequalities and solve real-world problems. 8 Students solve linear equations and inequalities in one variable and solve systems of linear equations in two variables. 9 Students solve quadratic equations in one variable using completing the square, factoring, and the quadratic formula. 10 Students write and evaluate functions and interpret statements that use function notation in context. 11 Students sketch graphs of functions that model a relationship between two quantities and interpret key features of the graphs. 12 Students compare properties of two functions (i.e., linear, quadratic, and exponential) each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 3

4 13 Students represent data on two quantitative variables on a scatter plot, describe how the variables are related, and use functions fitted to the data to solve problems in context. 14 Students interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of data. course. Source document used is as follows: Massachusetts Mathematics Curriculum Framework(2011). 4

5 Mathematics Algebra II Grade: HS 1 Students identify zeros of polynomials, including complex zeros of quadratic polynomials, make connections between zeros of polynomials and solutions of polynomial equations, and use the zeros of a polynomial to sketch a graph of the function defined by the polynomial. 2 Students perform arithmetic operations with complex numbers. 3 Students explain the impact of changes to a function (i.e., terms, factors, and coefficients) in terms of the context of a problem. 4 Students perform arithmetic operations on polynomials and on rational expressions. 5 Students rearrange formulas to highlight a quantity of interest, constructing a viable argument to justify their method. 6 Students create and solve equations that arise from linear, quadratic, simple rational, and simple exponential functions to model and solve problems. 7 Students solve equations and inequalities graphically. 8 Students interpret functions using key features of graphs and tables and sketch graphs showing key features of the relationship. (Key features include: domain, intercepts, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums, symmetries, end behavior, and periodicity where relevant.) 9 Students compare key features of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (Key features include: domain, intercepts, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums, symmetries, end behavior, and periodicity where relevant.) 10 Students use the unit circle to extend trigonometric functions to all real numbers and use the Pythagorean identity to find sin (θ), cos (θ), or tan (θ), given sin (θ), cos (θ), or tan (θ), and the quadrant. 11 Students find the inverse for exponential and logarithmic models. 12 Students make inferences and justify conclusions from sample surveys, experiments, and observational studies (e.g., students use data from a sample survey to estimate a population mean or proportion). 5

6 course. Source document used is as follows: Massachusetts Mathematics Curriculum Framework (2011). 6

7 Mathematics Advanced Quantitative Reasoning Grade: HS 1 Students solve problems involving velocity and other quantities that can be represented by vectors, recognizing vector quantities as having magnitude and direction and representing vectors as directed line segments in component form. 2 Students use the numeric operations of matrices, including addition, subtraction, multiplication, and scalar multiplication, demonstrating knowledge of the properties of matrices, including commutative, associative, and distributive laws, and of the existence of identities and inverses. 3 Students use matrices to represent and manipulate data; for example, through linear transformations or incidence relationships in a network. 4 Students solve systems of linear equations using matrix inverses and applying technology for matrices of dimension 3 x 3 or greater. 5 Students use the unit circle to determine geometrically the values of sine, cosine, and tangent for special angles. 6 Students choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 7 Students use inverse functions to solve trigonometric equations that arise in modeling contexts and apply the Laws of Cosines and Sines to find unknown measurements in right and non-right triangles. 8 Students derive the equations of conic sections and use the equations and their graphs to model real-world problems. 9 Students apply geometric methods and concepts to model and solve design problems and use dimensional analysis to confirm that expressions and equations make sense. 10 Students make inferences and justify conclusions from sample surveys, experiments, and observational studies, distinguishing between correlation and causation. 11 Students use combinations and permutations to compute theoretical probabilities, including conditional probability, for events that are either dependent or independent. 12 Students calculate and use expected value as a measure of center for the probability distribution of a random variable. 13 Students use probability to: weigh the possible outcomes of a decision by assigning probabilities; make fair decisions; and analyze strategies. 7

8 course. Source document used is as follows: Massachusetts Mathematics Curriculum Framework, Model Advanced Courses: Model Advanced Quantitative Reasoning (2011). 8

9 Mathematics Geometry Grade: HS 1 Students use geometric descriptions of rigid motions to transform figures and to determine if geometric figures are congruent. 2 Students use trigonometric ratios and the Pythagorean Theorem to find unknown measurements in right triangles. 3 Students use the rectangular coordinate system to verify geometric relationships, including properties of special polygons and slopes of parallel and perpendicular lines. 4 Students solve problems involving volume of right and oblique solids. 5 Students use the definitions of congruence and similarity to determine if figures are congruent or similar, and to solve problems involving missing measurements. 6 Students apply geometric concepts in modeling situations, which include representing real-world objects as geometric solids, solving optimization problems, and using dimensional analysis for unit conversions. 7 Students interpret a circle diagram to solve problems involving arc measures, central angles, inscribed angles, exterior angles, and angles formed by intersecting chords. 8 Students use proportional reasoning to determine exact or approximate values for the length of an arc intercepted by an angle or the area of a sector of a circle. 9 Students use the distance formula, Pythagorean Theorem, or given information to determine the equations of circles and parabolas and to translate between the graphical and algebraic forms of circles and parabolas. 10 Students use congruence and similarity criteria to solve problems and prove relationships in geometric figures. course. Source document used is as follows: Massachusetts Mathematics Curriculum Framework, Model Traditional Pathway: Model Geometry (2011). 9

10 Mathematics Pre-Calculus Grade: HS 1 Students identify the key features of polynomial, rational, logarithmic, and trigonometric functions, and use these to sketch the graphs of these functions. (The key features, as they apply, include end behaviors, zeros and y-intercepts, asymptotes, bounds, extrema, domain, range, period, and intervals for which the function is increasing or decreasing, and values for which it is positive and negative.) 2 Students model periodic relationships from multiple representations using appropriate trigonometric functions and use those models to answer a variety of questions about the phenomena. 3 Students use inverse trigonometric relationships to solve trigonometric equations that arise in practical problems, including modeling contexts and non-right triangles. 4 Students use the inverse relationship between exponential and logarithmic functions to solve logarithmic and exponential equations that arise in practical problems. 5 Students apply odd/even, sum/difference, and co-function identities to determine a desired trigonometric value when given another function s value and quadrant, as well as to prove identity statements. 6 Students translate between the geometric descriptions and the equations of hyperbolas and ellipses. 7 Students solve problems involving position or velocity graphically and numerically using vector representations and operations. 8 Students perform arithmetic operations with complex numbers, represent complex numbers and their operations on the complex plane, and use complex numbers in polynomial identities and equations. course. Source document used is as follows: Massachusetts Mathematics Curriculum Framework, Model Advanced Courses: Model Precalculus (2011). 10

11 Mathematics Probability and Statistics Grade: HS 1 Students analyze quantitative univariate data sets and compare distributions of two or more quantitative univariate data sets, using shape and measures of center and spread. 2 Students create a linear model that represents the association between quantitative bivariate data sets and interpret the slope, y-intercept, and r-value (correlation). 3 Students analyze and describe nonlinear and linear associations between quantitative bivariate data. 4 Students use the mean and standard deviation of a normally distributed data set to calculate z-scores in order to make comparisons, using technology to determine probabilities by estimating area under the curve. 5 Students identify a sample that is representative of the population, recognizing the purpose of random sampling and distinguishing between a sample statistic and a population parameter. 6 Students explain how randomization relates to surveys, experiments, and observational studies. 7 Students make inferences and draw conclusions from collected data by using sample statistics to make inferences about a population. 8 Students use data from a two-way table to decide if events are independent. 9 Students find probabilities, including conditional probabilities, of independent or dependent compound events using the addition and multiplication rules of probability. 10 Students use simulations to estimate probabilities and check the results of a datagenerated process. 11 Students make inferences and justify conclusions from sample surveys, experiments, and observational studies. 12 Students make inferences and justify conclusions from two sets of data, distinguishing between correlation, association, and causation. 13 Students use combinations and permutations to compute theoretical probabilities, including conditional probability, for events that are either dependent or independent. 14 Students calculate and use expected value as a measure of center for the probability distribution of a random variable. 11

12 15 Students apply probability to real-world situations by using theoretical probability and empirical probability to evaluate outcomes of decisions. course. Source document used is as follows: Massachusetts Mathematics Curriculum Framework, Conceptual Category: Statistics and Probability (2011). 12

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