# 05-071 C.M.R. ch. 131, § 2 - MATHEMATICS

Current through 2022-14, April 6, 2022

2.1 Number and Operations

2.1.1 Students demonstrate conceptual understanding of rational numbers with respect to: whole numbers from 0 to 199 using place value, by applying the concepts of equivalency in composing or decomposing numbers (e.g., 34 = 17 + 17; 34 = 29 + 5); and in expanded notation (e.g., 141 = 1 hundred + 4 tens + 1 one or 141 = 100 + 40 + 1) using models, explanations, or other representations; and positive fractional numbers (benchmark fractions: a/2, a/3, or a/4, where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area and set models where the denominator is equal to the number of parts in the whole using models, explanations, or other representations. (M(N&O)-2-1)

2.1.2 Students demonstrate understanding of the relative magnitude of numbers from 0 to 199 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (10, 25, 50, 75, 100, 125, 150, or 175); by demonstrating an understanding of the relation of inequality when comparing whole numbers by using "1 more", "1 less", "10 more", "10 less", "100 more", or "100 less"; or by connecting number words and numerals to the quantities they represent using models, number lines, or explanations. (M(N&O)-2-2)

2.1.3 Students demonstrate conceptual understanding of mathematical operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations; and addition of multiple one-digit whole numbers. (M(N&O)-2-3)

2.1.4 Students demonstrate understanding of monetary value by adding coins together to a value no greater than \$1.99 and representing the result in dollar notation; making change from \$1.00 or less, or recognizing equivalent coin representations of the same value (values up to \$1.99). (M(N&O)-2-5))

2.1.5 Students demonstrate conceptual understanding of rational numbers with respect to: whole numbers from 0 to 999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations; and positive fractional numbers (benchmark fractions: a/2, a/3, a/4, a/6, or a/8, where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area and set models where the number of parts in the whole is equal to the denominator; and decimals (within a context of money) as a part of 100 using models, explanations, or other representations. (M(N&O)-3-1)

2.1.6 Students demonstrate understanding of the relative magnitude of numbers from 0 to 999 by ordering whole numbers; by comparing whole numbers to benchmark whole numbers (100, 250, 500, or 750); or by comparing whole numbers to each other; and comparing or identifying equivalent positive fractional numbers (a/2, a/3, a/4 where a is a whole number greater than 0 and less than or equal to the denominator) using models, number lines, or explanations. (M(N&O)-3-2)

2.1.7 Students demonstrate conceptual understanding of mathematical operations by describing or illustrating the inverse relationship between addition and subtraction of whole numbers; and the relationship between repeated addition and multiplication using models, number lines, or explanations. (M(N&O)-3-3)

2.1.8 Students accurately solve problems involving addition and subtraction with and without regrouping; the concept of multiplication; and addition or subtraction of decimals (in the context of money). (M(N&O)-3-4)

2.1.9 Students demonstrate conceptual understanding of rational numbers with respect to: whole numbers from 0 to 999,999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations; and positive fractional numbers (benchmark fractions: a/2, a/3, a/4, a/5, a/6, a/8, or a/10, where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area, set, or linear models where the number of parts in the whole are equal to, and a multiple or factor of the denominator; and decimals as hundredths within the context of money, or tenths within the context of metric measurements (e.g., 2.3 cm) using models, explanations, or other representations. (M(N&O)-4-1)

2.1.10 Students demonstrate understanding of the relative magnitude of numbers from 0 to 999,999 by ordering or comparing whole numbers; and ordering, comparing, or identifying equivalent proper positive fractional numbers; or decimals using models, number lines, or explanations.

(M(N&O)-4-2)

2.1.11 Students demonstrate conceptual understanding of mathematical operations by describing or illustrating the relationship between repeated subtraction and division (no remainders); the inverse relationship between multiplication and division of whole numbers; or the addition or subtraction of positive fractional numbers with like denominators using models, number lines, or explanations. (M(N&O)-4-3)

2.1.12 Students accurately solve problems involving multiple operations on whole numbers or the use of the properties of factors and multiples; and addition or subtraction of decimals and positive proper fractions with like denominators. (Multiplication limited to 2 digits by 2 digits, and division limited to 1 digit divisors.) (IMPORTANT: Applies the conventions of order of operations where the left to right computations are modified only by the use of parentheses.) (M(N&O)-4-4)

2.1.13 Students demonstrate conceptual understanding of rational numbers with respect to: whole numbers from 0 to 9,999,999 through equivalency, composition, decomposition, or place value using models, explanations, or other representations; and positive fractional numbers (proper, mixed number, and improper) (halves, fourths, eighths, thirds, sixths, twelfths, fifths, or powers of ten (10, 100, 1000)), decimals (to thousandths), or benchmark percents (10%, 25%, 50%, 75% or 100%) as a part to whole relationship in area, set, or linear models using models, explanations, or other representations*. (M(N&O)-5-1)

2.1.14 Students demonstrate understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent positive fractional numbers, decimals, or benchmark percents within number formats (fractions to fractions, decimals to decimals, or percents to percents); or integers in context using models or number lines. M(N&O)-5-2

2.1.15 Students demonstrate conceptual understanding of mathematical operations by describing or illustrating the meaning of a remainder with respect to division of whole numbers using models, explanations, or solving problems. M(N&O)-5-3

2.1.16 Students accurately solve problems involving multiple operations on whole numbers or the use of the properties of factors, multiples, prime, or composite numbers; and addition or subtraction of fractions (proper) and decimals to the hundredths place. (Division of whole numbers by up to a two-digit divisor.) (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) (M(N&O)-5-4)

2.1.17 Students demonstrate conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a : b, and a / b , where b 1 0); and rates (e.g., a out of b, 25%) using models, explanations, or other representations*. (M(N&O)-6-1)

2.1.18 Students demonstrate understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents (e.g.,33, 43), integers, or rational numbers within and across number formats (fractions, decimals, or whole number percents from 1- 100) using number lines or equality and inequality symbols.

(M(N&O)-6-2)

2.1.19 Students demonstrate conceptual understanding of mathematical operations by describing or illustrating the meaning of a power by representing the relationship between the base (whole number) and the exponent (whole number) (e.g.,33, 43); and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number, decimal, or fraction. (M(N&O)-6-3)

2.1.20 Students accurately solve problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or problems involving greatest common factor or least common multiple. (IMPORTANT: Applies the conventions of order of operations with and without parentheses.) (M(N&O)-6-4)

2.1.21 Students demonstrate conceptual understanding of rational numbers with respect to percents as a means of comparing the same or different parts of the whole when the wholes vary in magnitude (e.g., 8 girls in a classroom of 16 students compared to 8 girls in a classroom of 20 students, or 20% of 400 compared to 50% of 100); and percents as a way of expressing multiples of a number (e.g., 200% of 50) using models, explanations, or other representations*. (M(N&O)-7-1)

2.1.22 Students demonstrate understanding of the relative magnitude of numbers by ordering, comparing, or identifying equivalent rational numbers across number formats, numbers with whole number bases and whole number exponents (e.g., 33, 43), integers, absolute values, or numbers represented in scientific notation using number lines or equality and inequality symbols. (M(N&O)-7-2)

2.1.23 Students accurately solve problems involving proportional reasoning; percents involving discounts, tax, or tips; and rates. (IMPORTANT: Applies the conventions of order of operations including parentheses, brackets, or exponents.) (M(N&O)-7-4)

*Specifications for area, set, and linear models for grades 5 - 8: Fractions: The number of parts in the whole are equal to the denominator, a multiple of the denominator, or a factor of the denominator. Percents: The number of parts in the whole is equal to 100, a multiple of 100, or a factor of 100 (for grade 5); the number of parts in the whole is a multiple or a factor of the numeric value representing the whole (for grades 6-8). Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal.

2.1.24 Students demonstrate understanding of the relative magnitude of real numbers by solving problems involving ordering or comparing rational numbers, common irrational numbers (e.g., Click here to view Image, Click here to view Image ) , rational bases with integer exponents, square roots, absolute values, integers, or numbers represented in scientific notation using number lines or equality and inequality symbols. (M(N&O ;)-10-2)

2.1.25 Students accurately solve problems involving rational numbers within mathematics, across content strands, disciplines or contexts (with emphasis on, but not limited to, proportions, percents, ratios, and rates).

(M(N&O)-10-4)

2.2 Geometry and Measurement

2.2.1 Students use properties, attributes, composition, or decomposition to sort or classify polygons or objects by a combination of two or more non-measurable or measurable attributes. (M(G&M)-2-1)

2.2.2 Students demonstrate conceptual understanding of perimeter and area by using models or manipulatives to surround and cover polygons.

(M(G&M)-2-6)

2.2.3 Students measure and use units of measures appropriately and consistently, and make conversions within systems when solving problems across the content strands. (M(G&M)-2-7)

2.2.4 Students use properties or attributes of angles (number of angles) or sides (number of sides or length of sides) or composition or decomposition of shapes to identify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or circles. (M(G&M)-3-1)

2.2.5 Students demonstrate conceptual understanding of perimeter of polygons, and the area of rectangles on grids using a variety of models or manipulatives. Express all measures using appropriate units. (M(G&M)-3-6)

2.2.6 Students measure and use units of measures appropriately and consistently, and make conversions within systems when solving problems across the content strands. (M(G&M)-3-7)

2.2.7 Students use properties or attributes of angles (number of angles) or sides (number of sides, length of sides, parallelism, or perpendicularity) to identify, describe, or distinguish among triangles, squares, rectangles, rhombi, trapezoids, hexagons, or octagons; or classify angles relative to 90° as more than, less than, or equal to. (M(G&M)-4-1)

2.2.8 Students use properties or attributes (shape of bases or number of lateral faces) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, or spheres). (M(G&M)-4-3)

2.2.9 Students demonstrate conceptual understanding of congruency by matching congruent figures using reflections, translations, or rotations (flips, slides, or turns), or as the result of composing or decomposing shapes using models or explanations. (M(G&M)-4-4)

2.2.10 Students demonstrate conceptual understanding of similarity by applying scales on maps, or applying characteristics of similar figures (same shape but not necessarily the same size) to identify similar figures, or to solve problems involving similar figures. Describe relationships using models orsc explanations. (M(G&M)-4-5)

2.2.11 Students demonstrate conceptual understanding of perimeter of polygons, and the area of rectangles, polygons or irregular shapes on grids using a variety of models, manipulatives, or formulas. Express all measures using appropriate units. (M(G&M)-4-6)

2.2.12 Students measure and use units of measures appropriately and consistently, and make conversions within systems when solving problems across the content strands. (M(G&M)-4-7)

2.2.13 Students use properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify, or distinguish among different types of triangles (right, acute, obtuse, equiangular, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms). (M(G&M)-5-1)

2.2.14 Students use properties or attributes (shape of bases, number of lateral faces, or number of bases) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones). (M(G&M)-5-3)

2.2.15 Students demonstrate conceptual understanding of perimeter of polygons, and the area of rectangles or right triangles through models, manipulatives, or formulas, the area of polygons or irregular figures on grids, and volume of rectangular prisms (cubes) using a variety of models, manipulatives, or formulas. Express all measures using appropriate units. (M(G&M)-5-6)

2.2.16 Students measure and use units of measures appropriately and consistently, and make conversions within systems when solving problems across the content strands. (M(G&M)-5-7)

2.2.17 Students use properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify, or distinguish among different types of triangles (right, acute, obtuse, equiangular, scalene, isosceles, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms). (M(G&M)-6-1)

2.2.18 Students use properties or attributes (shape of bases, number of lateral faces, number of bases, number of edges, or number of vertices) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones). (M(G&M)-6-3)

2.2.19 Students demonstrate conceptual understanding of similarity by describing the proportional effect on the linear dimensions of polygons or circles when scaling up or down while preserving the angles of polygons, or by solving related problems (including applying scales on maps). Describe effects using models or explanations. (M(G&M)-6-5)

2.2.20 Students demonstrate conceptual understanding of perimeter of polygons, the area of quadrilaterals or triangles, and the volume of rectangular prisms by using models, formulas, or by solving problems; and demonstrate understanding of the relationships of circle measures (radius to diameter and diameter to circumference) by solving related problems. Express all measures using appropriate units. (M(G&M)-6-6)

2.2.21 Students measure and use units of measures appropriately and consistently, and make conversions within systems when solving problems across the content strands. (M(G&M)-6-7)

2.2.22 Students use properties of angle relationships resulting from two or three intersecting lines (adjacent angles, vertical angles, straight angles, or angle relationships formed by two non-parallel lines cut by a transversal), or two parallel lines cut by a transversal to solve problems. (M(G&M)-7-1)

2.2.23 Students apply theorems or relationships (triangle inequality or sum of the measures of interior angles of regular polygons) to solve problems. (M(G&M)-7-2)

2.2.24 Students apply the concepts of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations.

(M(G&M)-7-4)

2.2.25 Students apply concepts of similarity by solving problems involving scaling up or down and their impact on angle measures, linear dimensions and areas of polygons, and circles when the linear dimensions are multiplied by a constant factor. Describe effects using models or explanations.

(M(G&M)-7-5)

2.2.26 Students demonstrate conceptual understanding of the area of circles or the area or perimeter of composite figures (quadrilaterals, triangles, or parts of circles), and the surface area of rectangular prisms, or volume of rectangular prisms, triangular prisms, or cylinders using models, formulas, or by solving related problems. Express all measures using appropriate units.

(M(G&M)-7-6)

2.2.27 Students make and defend conjectures, construct geometric arguments, use geometric properties, or use theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem). (M(G&M)-10-2)

2.2.28 Students apply the concepts of congruency by solving problems on or off a coordinate plane involving reflections, translations, or rotations; or solve problems using congruency involving problems within mathematics or across disciplines or contexts. (M(G&M)-10-4)

2.2.29 Students apply concepts of similarity by solving problems within mathematics or across disciplines or contexts. (M(G&M)-10-5)

2.2.30 Students solve problems involving perimeter, circumference, or area of two-dimensional figures (including composite figures) or surface area or volume of three-dimensional figures (including composite figures) within mathematics or across disciplines or contexts. (M(G&M)-10-6 M)

2.2.31 Students use units of measure appropriately and consistently when solving problems across content strands; make conversions within or across systems and make decisions concerning an appropriate degree of accuracy in problem situations involving measurement in other GLEs. (M(G&M)-10-7)

2.2.32 Students solve problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope. (M(G&M)-10-9)

2.3 FUNCTIONS AND ALGEBRA

2.3.1 Students identify and extend to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences by extending the pattern to the next element, or finding a missing element (e.g., 2, 4, 6, ___, 10). (M(F& A)-2-1)

2. 3.2 Students demonstrate conceptual understanding of equality by finding the value that will make an open sentence true (e.g.,

(limited to one operation and limited to use addition or subtraction) (M(F& A)-2-4)

2. 3.3 Students identify and extend to specific cases a variety of patterns (linear and non-numeric) represented in models, tables, or sequences by extending the pattern to the next one, two, or three elements, or finding missing elements. (M(F& A)-3-1)

2.3.4 Students demonstrate conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions; or by finding the value that will make an open sentence true (e.g.,

(limited to one operation and limited to use addition, subtraction, or multiplication) (M(F& A)-3-4)

2.3.5 Students identify and extend to specific cases a variety of patterns (linear and nonlinear) represented in models, tables or sequences; and write a rule in words orsc symbols to find the next case. (M(F& A)-4-1)

2.3.6 Students demonstrate conceptual understanding of algebraic expressions by using letters or symbols to represent unknown quantities to write simple linear algebraic expressions involving any one of the four operations; or by evaluating simple linear algebraic expressions using whole numbers. (M(F& A)-4-3)

2.3.6 Students demonstrate conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions, by simplifying numerical expressions where left to right computations may be modified only by the use of parentheses [e.g., 14 - (2 X 5)] (expressions consistent with the parameters of M(F& A)-4-3), and by solving one-step linear equations of the form ax = c, x +- b = c, where a, b, and c are whole numbers with a [NOT EQUAL TO] 0. (M(F& A)-4-4)

2.3.7 Students identify and extend to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, or in problem situations; and write a rule in words orsc symbols for finding specific cases of a linear relationship. (M(F& A)-5-1)

2.3.8 Students demonstrate conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving any two of the four operations; or by evaluating linear algebraic expressions using whole numbers. (M(F& A)-5-3)

2.3.9 Students demonstrate conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions (expressions consistent with the parameters of M(F&

A)-5-3), by solving one-step linear equations of the form ax = c, x +- b = c, or x/a = c, where a, b, and c are whole numbers with a [NOT EQUAL TO] 0; or by determining which values of a replacement set make the equation (multi-step of the form ax +- b = c where a, b, and c are whole numbers with a [NOT EQUAL TO] 0) a true statement (e.g., 2x + 3 = 11, {x: x = 2, 3, 4, 5}). (M(F& A)-5-4)

2.3.10 Students identify and extend to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations; or write a rule in words or symbols for finding specific cases of a linear relationship; or write a rule in words orsc symbols for finding specific cases of a nonlinear relationship; and write an expression orsc equation using words orsc symbols to express the generalization of a linear relationship (e.g., twice the term number plus 1 orsc 2n + 1). (M(F& A)-6-1)

2.3.11 Students demonstrate conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by constructing or interpreting graphs of real occurrences and describing the slope of linear relationships (faster, slower, greater, or smaller) in a variety of problem situations; and describe how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change. (M(F& A)-6-2)

2.3.12 Students demonstrate conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving two or more of the four operations; or by evaluating linear algebraic expressions (including those with more than one variable); or by evaluating an expression within an equation (e.g., determine the value of y when x = 4 given y = 3x - 2). (M(F& A)-6-3)

2.3.13 Students demonstrate conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions (expressions consistent with the parameters of M(F& A)-6-3), solving multi-step linear equations of the form ax +- b = c, where a, b, and c are whole numbers with a [NOT EQUAL TO] 0. (M(F& A)-6-4)

2.3.14 Students identify and extend to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations; and generalize a linear relationship using words and symbols; generalizes a linear relationship to find a specific case; or write an expression orsc equation using words orsc symbols to express the generalization of a nonlinear relationship. (M(F& A)-7-1)

2.3.15 Students demonstrate conceptual understanding of linear relationships (y = kx; y = mx + b) as a constant rate of change by solving problems involving the relationship between slope and rate of change, by describing the meaning of slope in concrete situations, or informally determining the slope of a line from a table or graph; and distinguish between constant and varying rates of change in concrete situations represented in tables or graphs; or describe how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change. (M(F& A)-7-2)

2.3.16 Students demonstrate conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating an expression within an equation (e.g., determine the value of y when x = 4 given y = 5x3 - 2). (M(F& A)-7-3)

2.3.17 Students demonstrate conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of the equations being solved at this grade level) using models or different representations of the expressions, solving multi-step linear equations of the form ax +- b = c with a [NOT EQUAL TO] 0, ax +- b = cx +- d with a, c [NOT EQUAL TO] 0, and (x/a) +- b = c with a [NOT EQUAL TO] 0, where a, b, c and d are whole numbers; or by translating a problem-solving situation into an equation consistent with the parameters of the type of equations being solved for this grade level. (M(F& A)-7-4)

2.3.18 Students identify, extend, and generalize a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs in problem solving situations. (M(F& A)-10-1)

2.3.19 Students demonstrate conceptual understanding of linear and nonlinear functions and relations (including characteristics of classes of functions) through an analysis of constant, variable, or average rates of change, intercepts, domain, range, maximum and minimum values, increasing and decreasing intervals and rates of change (e.g., the height is increasing at a decreasing rate); describe how change in the value of one variable relates to change in the value of a second variable; or works between and among different representations of functions and relations (e.g., graphs, tables, equations, function notation). (M(F& A)-10-2)

2.3.20 Students demonstrate conceptual understanding of algebraic expressions by solving problems involving algebraic expressions, by simplifying expressions (e.g., simplifying polynomial or rational expressions, or expressions involving integer exponents, square roots, or absolute values), by evaluating expressions, or by translating problem situations into algebraic expressions. (M(F& A)-10-3)

2.3.21 Students demonstrate conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or providing the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (M(F& A)-10-4)

2.4 DATA, Statistics, and Probability

2.4.1 Students interpret a given representation (pictographs with one-to-one correspondence, line plots, tally charts, or tables) to answer questions related to the data, or to analyze the data to formulate conclusions. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-2-2.) (M(DSP)-2-1)

2.4.2 Students analyze patterns, trends, or distributions in data in a variety of contexts by determining or using more, less, or equal. (M(DSP)-2-2)

2.4.3 Students use counting techniques to solve problems involving combinations using a variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, orsc others); (e.g., How many ways can you make 50 cents using nickels, dimes, and quarters") (M(DSP)-2-4)

2.4.4 Students interpret a given representation (line plots, tally charts, tables, or bar graphs) to answer questions related to the data, to analyze the data to formulate conclusions, or to make predictions. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-3-2.) (M(DSP)-3-1)

2.4.5 Students analyze patterns, trends, or distributions in data in a variety of contexts by determining or using most frequent (mode), least frequent, largest, or smallest. (M(DSP)-3-2)

2.4.6 Students identify or describe representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP)-3-1. (M(DSP)-3-3)

2.4.7 For a probability event in which the sample space may or may not contain equally likely outcomes, students determine the likelihood of the occurrence of an event (using "more likely", "less likely", or "equally likely"). (M(DSP)-3-5)

2.4.8 Students interpret a given representation (line plots, tables, bar graphs, pictographs, or circle graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-4-2.) (M(DSP)-4-1)

2.4.9 Students analyze patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (median or mode), or range. (M(DSP)-4-2)

2.4.10 Students use counting techniques to solve problems in context involving combinations or simple permutations (e.g., Given a map - Determine the number of paths from point A to point B.) using a variety of strategies (e.g., organized lists, tables, tree diagrams, or others). (M(DSP)-4-4)

2.4.11 For a probability event in which the sample space may or may not contain equally likely outcomes, students determine the theoretical probability of an event and express the result as part to whole (e.g., two out of five).

(M(DSP)-4-5)

2.4.12 Students interpret a given representation (tables, bar graphs, circle graphs, or line graphs) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-5-2.) (M(DSP)-5-1)

2.4.13 Students analyze patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode) or range to analyze situations, or to solve problems. (M(DSP)-5-2)

2.4.14 Students identify or describe representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP)-5-1. (M(DSP)-5-3)

2.4.15 For a probability event in which the sample space may or may not contain equally likely outcomes, students determine the experimental or theoretical probability of an event and express the result as a fraction. (M(DSP)-5-5)

2.4.15 Students interpret a given representation (circle graphs, line graphs, or stem-and-leaf plots) to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-6-2.) (M(DSP)-6-1)

2.4.16 Students analyze patterns, trends or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode) or dispersion (range) to analyze situations, or to solve problems. (M(DSP)-6-2)

2.4.17 Students use counting techniques to solve problems in context involving combinations or simple permutations using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, or others). (M(DSP)-6-4)

2.4.18 For a probability event in which the sample space may or may not contain equally likely outcomes, students determine the experimental or theoretical probability of an event in a problem-solving situation.

(M(DSP)-6-5)

2.4.19 Students interpret a given representation (circle graphs, scatter plots that represent discrete linear relationships, or histograms) to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-7-2.)

(M(DSP)-7-1)

2.4.20 Students analyze patterns, trends, or distributions in data in a variety of contexts by solving problems using measures of central tendency (mean, median, or mode), dispersion (range or variation), or outliers to analyze situations to determine their effect on mean, median, or mode; and evaluate the sample from which the statistics were developed (bias). (M(DSP)-7-2)

2.4.21 Students interpret or describe representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP)-7-1. (M(DSP)-7-3)

2.4.22 For a probability event in which the sample space may or may not contain equally likely outcomes, students determine the experimental or theoretical probability of an event in a problem-solving situation. (M(DSP)-7-5)

2.4.23 Students Interpret a given representation(s) (e.g., box-and-whisker plots, scatter plots, bar graphs, line graphs, circle graphs, histograms, frequency charts) to make observations, to answer questions, to analyze the data to formulate or justify conclusions, critique conclusions, make predictions, or to solve problems within mathematics or across disciplines or contexts (e.g., media, workplace, social and environmental situations). (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-10-2.) (M(DSP)-10-1)

2.4.24 Students analyze patterns, trends, or distributions in data in a variety of contexts by determining, using, or analyzing measures of central tendency (mean, median, or mode), dispersion (range or variation), outliers, quartile values, estimated line of best fit, regression line, or correlation (strong positive, strong negative, or no correlation) to solve problems; and solve problems involving conceptual understanding of the sample from which the statistics were developed. (M(DSP)-10-2)

2.4.25 Students Identify or describe representations or elements of representations that best display a given set of data or situation, consistent with the representations required in (M(DSP)-10-3)

2.4.26 Students use counting techniques to solve problems in context involving combinations or permutations using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, orsc others). (M(DSP)-10-4)

2.4.27 Students solve problems involving experimental or theoretical probability. (M(DSP)-10-5)

### Notes

05-071 C.M.R. ch. 131, § 2