Minnesota 9 th -11 th Grade MCA-II, GRAD & MCA-Modified 2003 Math Strands & Standards SUBSTRAND/STANDARD

Transcription

1 Minnesota 9 th -11 th Grade, & odified 2003 Math Strands & Standards STRAND SUBSTRAND/STANDARD 9.I Mathematical Reasoning 9. Number Sense, Computation & Operations 9.I Patterns, Functions & Algebra 9.IV Data Analysis, Statistics & Probability 9.V Spatial Sense, Geometry & Measurement 9.I.- Apply skills of mathematical representation, communication & reasoning throughout the remaining content standards. 9..A Number Sense Use real number, represented in a variety of ways, to quantify information & to solve real-world & mathematical problems. 9..B Computation & Operations Appropriately use calculators & other technologies to solve algebraic, geometric, probabilistic & statistical problems. 9.I.A Patterns & Functions Represent & analyze real-world & mathematical problems using numeric, graphic & symbolic methods for a variety of functions. 9.I.B Algebra (Algebraic Thinking) Solve simple equations & inequalities numerically, graphically, & symbolically. Use recursion to model & solve real-world & mathematical problems. 9.IV.A Data & Statistics Represent data & use various measures associated with data to draw conclusions & identify trends. Understand the effects of display distortion & measurement error on the interpretation of data. 9.IV.B Probability Use appropriate counting procedures, calculate probabilities in various ways and apply theoretical probability concepts to solve real-world and mathematical problems. 9.V.A Spatial Sense Use models to represent and understand two- and three-dimensional shapes and how various motions affect them. Recognize the relationship between different representations of the same shape. 9.V.B Geometry Apply basic theorems of plane geometry, right triangle trigonometry, coordinate geometry and a variety of visualization tools to solve real-world and mathematical problems. 9.V.C Measurement Use the interconnectedness of geometry, algebra and measurement to explore real-world and mathematical problems.

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3 9 th -11 th Grade Mathematical Reasoning Strand 2003 MN Math Standard to Benchmarks SUBSTRAND STANDARD BENCHMARK 9.I Mathematical Reasoning 9.I.- Apply skills of mathematical representation, communication & reasoning throughout the remaining content strands. 9.I.-.1 Assess the reasonableness of a solution by comparing the solution to appropriate graphical or numerical estimates or by recognizing the feasibility of solutions in a given context & rejecting extraneous solutions. 9.I.-.2 Appropriately use examples & counterexamples to make & test conjectures, justify solutions, & explain results. 9.I.-.3 Translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically & interpret the result in the original context. 9.I.-.4 Support mathematical results by explaining why the steps in a solution are valid & why a particular method is appropriate. 9.I.-.5 Determine whether or not relevant information is missing from a problem & if so, decide how to best express the results that can be obtained without that information. 9.I.-.6 Know & use the relationship that exists among a logical implication of the form if A, then B, its converse if B, then A, its inverse if not A, then not B, & its contrapositive if not B, then not A.

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5 9 th -11 th Grade Number Sense, Computation & Operations 2003 MN Math Standard to Benchmarks ( 5-8 ) *Item Type Multiple Choice () or Gridded Response (GR) *Cognitive Levels Knowing (A), Understanding (B), Application, Analysis Synthesis & Evaluation (C) * Calculator Use Item appears in calculator section of assessment (), Calculator Neutral, item appears in either section of assessment () SUBSTRAND STANDARD * BENCHMARK 9..A Number Sense 9..A Use real numbers, represented in a variety of ways, to quantify information & to solve real-world & mathematical problems. /GR C 9..A.- N/A These benchmarks are continued from grade 8 & embedded in algebra, probability & geometry in grade B Computation & Operations B Appropriately use calculators & other technologies to solve algebraic, geometric, probabilistic & statistical problems. /GR C /GR C 9..B.1 Apply the correct order of operations & grouping symbols when using calculators & other technologies. This benchmark is assessed in grade 8 It is embedded in other grade B.2 Know, use & translate calculator notational conventions to mathematical notation. Assessed only at the classroom level 9..B.3 Recognize the impact of units such as degrees & radians on calculations. This benchmark is embedded in grade 11 from Strand V

6 9..B.4 Recognize that applying an inverse function with a calculator may lead to extraneous or incomplete solutions. Assessed only at the classroom level 9..B.5 Understand the limitations of calculators such as missing or additional features on graphs due to viewing parameters or misleading representations of zero or very large numbers. Assessed only at the classroom level 9..B.6 Understand that use of a calculator requires appropriate mathematical reasoning & does not replace the need for mental computation. Assessed only at the classroom level

7 B.G7 Students will apply the correct order of operations to simplify & evaluate numeric expressions. Items must use positive rational numbers Items will use addition, subtraction, multiplication, division & grouping symbols only Fraction denominators are limited to 2, 3, 4, 5, 6, 8, 10 & 12 Subtraction cannot be a mixed number minus a mixed number requiring 1 7 regrouping (e.g., 3 1 is not 4 8 acceptable) Multiplication may be expressed as a raised dot, x, or ( ) (e.g., 5 6, 5 6, 5(6) ) Division may be expressed using division symbol or fraction bar (e.g., 6 2 or 6 2 ) Division must have a whole number divisor For multiplication & division, mixed numbers must be expressed as improper fractions No nested grouping symbols are allowed (e.g., (100 / 2) is not allowed) Items may require the identification of the correct order of operations shown (calculation not required) Items may require integer approximations of square roots of positive numbers Items may include exponents Problems may include context

8 B.G8 Students will use rational numbers in complex ways to solve multi-step realworld & mathematical problems. Rationals are limited to positive rationals Non-integer rationals will be represented in decimal form Items may require integer approximations of square roots of positive integers Squares must be less than or equal to B.G9 Students will use fractions, decimals & percents in multiple representations for estimation & computation to solve realworld & mathematical problems. Fraction denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25 & 100. Restrictions on denominators apply to problem and answer options (e.g., 1/3 + 1/5 is not allowed; 1/15 1/3 is not allowed) Items may include positive & negative fractions, decimals & percents

9 B.G10 Students will use proportional reasoning to solve real-world & mathematical problems. Items may involve: rates scale drawings & maps similar figures ratio unit pricing showing how changing one or more dimensions affect change in area Shapes are limited to circles, parallelograms & triangles Items are limited to two-dimensional figures Pictures or diagrams may be used but are note required Similarity may be shown using similarity symbol ( ) or using markings on figures Items may include context

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11 9 th -11 th Grade Patterns, Functions & Algebra 2003 MN Math Standard to Benchmarks ( ) ( ) (odified ) *Item Type Multiple Choice () or Gridded Response (GR) *Cognitive Levels Knowing (A), Understanding (B), Application, Analysis Synthesis & Evaluation (C) * Calculator Use Item appears in calculator section of assessment (), Calculator Neutral, item appears in either section of assessment () SUBSTRAND STANDARD * BENCHMARK 9.I.A Patterns & Functions I.A Represent & analyze real-world & mathematical problems using numeric, graphic & symbolic methods for a variety of functions. C / 9.I.A.1 Students will know the numeric, graphic & symbolic properties of linear, step, absolute value & quadratic functions. / Items may include rates of change, intercepts, maxima & minima Items may include intersection between two graphs Step functions must model real-world situations Step functions will not be represented symbolically Items may not include step or absolute value functions Items may include rates of change & intercepts Items that assess quadratics are limited to graphical properties Increments of x & y axes must be integers

12 C / 9.I.A.2 Students will model exponential growth & decay. / Models may be numeric, graphic & symbolic When calculation is required, exponents must be integers Items may have real-world context (e.g., bacterial growth, half-life, compound interest) Models may be numeric or graphic Items may have real-world context (e.g., bacterial growth, half-life, compound interest) C 9.I.A.3 Students will analyze the effects of coefficient changes on linear & quadratic functions & their graphs. / Changes to coefficients in 2 ax bx c are limited to a & c

13 /GR C / 9.I.A.4 Students will apply basic concepts of linear, quadratic & exponential expressions or equations in real-world problems. / Exponents must be integers Items will be limited to linear & exponential Expressions & equations must be able to be solved numerically. Table or graph required Items will not require expressions or equations to be solved symbolically A 9.I.A.5 Students will distinguish functions from other relations using graphic & symbolic methods. / Not more than 10 increments on either side of axes

14 9.I.A.G6 Students will generate a table of values from a formula or equation. Students will graph the result of a formula or linear equations in ordered pair format on a grid. x & y axes may have different scales do not require students to graph or generate a table of a non-linear relation formulas will only have unknowns to the first degree may not require generating a linear equation from a table of values may include real-world context (e.g., converting temperature) given a continuous (i.e., individual points not indicated) linear graph, student will generate a table of values linear equations will be given in y mx b form 9.I.A.G7 Students will translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically & interpret the result in the original context. may include real-world context

15 9.I.B Algebra (Algebraic Thinking) I.B Solve simple equations & inequalities numerically, graphically & symbolically. Use recursion to model & solve real-world & mathematical problems. A / 9.I.B.1 Students will translate among equivalent forms of expressions. / Items may include: simplifying algebraic expressions involving nested pairs of parentheses & brackets simplifying rational expressions factoring a common monomial term from an expression applying associative, commutative & distributive laws A simplified expression should contain, at most, four terms with, at most, two variables per term Items may include: simplifying algebraic expressions involving nested pairs of parentheses & brackets simplifying rational expressions factoring a common monomial term from an expression applying associative, commutative & distributive laws When applying distributive law, expressions may not contain 2 binomials Expressions may not include nested pairs of parentheses or brackets A simplified expression should contain at most two terms with at most one variable per term

16 / 9.I.B.2 Students will understand the relationship between absolute value & distance on the number line. Students will graph simple expressions involving absolute value. / At most, one absolute value on each side of the equation or inequality Absolute values will be in the form of x b c; x b c; x b c; x b c; (e.g., x 3 6 or x 2 5) 9.I.B.3 Students will find equations of a line. / Items will provide two points on the line, a point & the slope of the line or the slope & y-intercept of the line All answer options will be given in the same form within a item, either slope-intercept ( y mx b ) or standard form ( ax by c ) Items may require the student to generate the equation from the graph or identify the graph given the equation Items will provide the slope & y- intercept of the line, when graph is not provided Equations must be presented in slopeintercept form

17 C I.B.4 Students will translate among equivalent forms of linear equations & inequalities. / Translating may require simplification (e.g., (2x 2) 2( x 4) y translates to y 4x 6) Equivalent forms may be slopeintercept, standard or two-point All answer options will be given in the same form within a item, either slope-intercept ( y mx b ) or standard form ( ax by c ) 9.I.B.5 Students will use a variety of models to represent functions & patterns in realworld & mathematical problems. / Models may include equations, inequalities, algebraic formulas, written statements, tables, graphs or spreadsheets of linear, quadratic, exponential, absolute value & step functions. Step functions must model real-world situations Step functions will not be represented symbolically

18 A /GR / 9.I.B.6 Students will apply the laws of exponents to perform operations on expressions with integer exponents. / A simplified expression should contain, at most, two variables Multiplication & division operations should only be performed on monomials Items may include scientific notation with appropriate treatment of significant digits 9.I.B.7 Students will solve linear equations & inequalities in one variable with numeric, graphic & symbolic methods. / Forms of the linear equations or inequalities are not limited (e.g., 4( x 5) 3x 6( x 10) is acceptable) Items may include context Items may include at most one application of the distributive property Items will not include inequalities Forms of the linear equations are limited to at most a binomial equaling a binomial Items must have a numeric solution

19 /GR / A I.B.8 Students will determine solutions to quadratic equations in one variable with numeric, graphic & symbolic methods. / All solutions are real Solutions determined from a graph will be integer solutions Items may include context All solutions are integer Coefficient on second-degree term will always be 1 9.I.B.9 Students will use appropriate terminology & mathematical notation to define & represent recursion. / x is the initial term in the sequence 1 x is the next term n 1 The term a n is also included in appropriate terminology Items require only addition & th multiplication to find the n term (arithmetic & geometric only) 9.I.B.9 & 9.I.B.10 will not both have 0 in the same administration

20 /GR C / 9.I.B.10 Students will create & use recursive formulas to model & solve real-world & mathematical problems. / Progressions are limited to arithmetic & geometric Items will not require identification past tenth term 9.I.B.10 & 9.I.B.9 will not both have 0 in the same administration 9.I.B.11 Students will solve systems of two linear equations & inequalities with 2 variables using numeric, graphic & symbolic methods. / Inequalities will only be solved graphically Items may include context Items may include at most one application of the distributive property Items will not include inequalities Forms of linear equations are limited to at most a binomial equaling a binomial Systems may be represented using graph, slop-intercept & table format Systems are consistent & independent (i.e., solution is one ordered pair)

21 C / 9.I.B.12 Students will understand how slopes can be used to determine when lines are parallel or perpendicular & determine equations for parallel lines & perpendicular lines. / Items may provide a line & a point not on that line Items may require students to determine the equation of the line passing through a given point & parallel to a given line Items may require students to determine the equation of the line passing through a given point Items may include context Items may not require students to determine the equation of a line Equations in problem & answer options must be in slope-intercept form 9.I.B.G13 Students will use formulas with more than one variable to solve real-world & mathematical problems. Formulas must be from a real-world context & may include powers (e.g., area, volume, I prt or d rt ) Items may contain formulas with at most four variables Roots are limited to square roots Formula notation may not include subscripts Formulas must be included within the item

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23 9 th -11 th Grade Data Analysis, Statistics & Probability 2003 MN Math Standard to Benchmarks ( ) ( 8-14 ) (odified ) *Item Type Multiple Choice () or Gridded Response (GR) *Cognitive Levels Knowing (A), Understanding (B), Application, Analysis Synthesis & Evaluation (C) * Calculator Use Item appears in calculator section of assessment (), Calculator Neutral, item appears in either section of assessment () SUBSTRAND STANDARD * BENCHMARK 9.IV.A Data & Statistics IV.A Represent data & use various measures associated with data to draw conclusions & identify trends. Understand the effects of display distortion & measurement error on the interpretation of data. 9.IV.A.1 Students will analyze graphs & demonstrate understanding of the strengths & weaknesses of each format by choosing appropriately among them for a given situation. / Items may contain circle graphs, bar graphs, histograms, box-&-whisker plots, scatter plots, tables & stem-&- leaf plots Circle graphs may have, at most, eight sectors Scales are in increments appropriate to the application

24 7-8 / 9.IV.A.2 Students will use measures of central tendency & variability to describe, compare & draw conclusions about sets of data. / Measures may be: mean median maximum minimum range standard deviation quartile percentile mode interquartile range (IQR) Measures may be: mean median maximum minimum range quartile mode interquartile range (IQR) The interquartile range may be referred to conceptually, but the terminology interquartile range will not be used

25 / 9.IV.A.3 Students will determine approximate line of best-fit & use the line to draw conclusions. / Items will provide a scatter plot (coordinates of points on scatter plot are integers) or data set Equations are limited to linear equations only Items will provide a scatter plot (coordinates of points on scatter plot are integers) Line of best fit may be provided & asked to draw conclusions 9.IV.A.4 Students will know the influence of outliers on various measures & representations of data about realworld & mathematical problems. / Items may require students to understand that the median is resistant to outliers Items may require students to understand that outliers affect the mean Given a mathematical definition, may require students to verify that a data point is an outlier

26 9.IV.A.5 Students will distinguish between correlation & causation. / Items may provide several statements about correlation & causation of a situation & require the student to select the correct statement (e.g., high correlation does not guarantee causation) Items will not require calculation of correlation coefficients 9.IV.A.6 Students will interpret data credibility in the context of measurement error & display distortion. / Items will assess either measurement error or display distortion, but not both in the same item Items may address the effect of sample size on measurement error

27 2-3 9.IV.A.7 Students will compare outcomes of voting methods. / Voting methods may include: majority plurality ranked by preference run-off comparison pair-wise comparison 9.IV.A.G8 Students will analyze histograms, bar graphs, circle graphs, stem-&-leaf plots & box-&-whisker plots. Graphics may have at most ten data categories Circle graphs may have a most eight sectors Scales are in increments appropriate to the application Histogram intervals must be consistent Items may involve: Reading & interpreting Identifying trends & patterns & make predictions Solve problems using information presented in the graph

28 9.IV.B Probability IV.B Use appropriate counting procedures, calculate probabilities in various ways & apply theoretical probability concepts to solve real-world & mathematical problems. /GR / 9.IV.A.G9 Students will understand the meaning of & be able to compute minimum, maximum, range, median, mean & mode of a data set. At most twenty numbers in the data set Numbers used are less than 300 Items may ask which values (mean, median, mode, range) best describes a data set in context & identify justification Items may require the calculation of quartiles The interquartile range may be referred to conceptually, but the terminology interquartile range will not be used 9.IV.B.1 Students will select & apply appropriate counting procedures to solve real-world & mathematical problems. / Items may involve computing probabilities Items may include combinations & permutations Items may involve determining sample space &/or computing probabilities Items may not include formulas Solutions may have at most 24 possibilities

29 /GR C / C / 9.IV.B.2 Students will calculate probabilities & relate the results in real-world & mathematical problems. / Items may use area, trees, unions & intersections to calculate probabilities Items may involve both the concept of mutually exclusive events or not mutually exclusive events Items may involve independent or dependent events Items may involve conditional probability Items may involve independent events Items will not involve conditional probability 9.IV.B.3 Students will use probability models in real-world & mathematical problems. / Models may include area & binomial models Binomial probabilities will involve, at most, 4 events Binomial probabilities will involve at most 3 events

30 /GR 9.IV.B.4 Students will determine the expected values of random variables for simple probability models. / Sample spaces will include, at most, four possible outcomes Probabilities for each outcome may be given or may have to be computed 9.IV.B.5 Students will know the effect of sample size on experimental & simulation probabilities. / Items may require the application of the Law of Large Numbers Items will note require the application of the Central Limit Theorem Items may require the interpretation of confidence intervals, but will not require the calculation of confidence intervals

31 /GR C 9.IV.B.6 Students will calculate probabilities. / Items use a variety of experimental, simulation & theoretical methods

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33 9 th -11 th Grade Spatial Sense, Geometry & Measurement 2003 MN Math Standard to Benchmarks ( ) ( 6-11 ) (odified 9-12 ) *Item Type Multiple Choice () or Gridded Response (GR) *Cognitive Levels Knowing (A), Understanding (B), Application, Analysis Synthesis & Evaluation (C) *Calculator Use Item appears in calculator section of assessment (), Calculator Neutral, item appears in either section of assessment () SUBSTRAND STANDARD * BENCHMARK 9.V.A Spatial Sense 9.V.A Use models to represent & understand two- & three-dimensional shapes & how various motions affect them. Recognize the relationship between different representations of the same shape. / 9.V.A.1 Students will use models & visualization to understand & represent various three-dimensional objects & their cross sections from different perspectives. / Items are limited to top view, side view, front view or net Shapes are limited to polyhedra, combinations of polyhedra, cylinders & cones Not figures will be oblique All visible sides of views are clearly labeled Prisms will have a base with, at most, six sides Pyramids will have a base with, at most, six sides Cross sections are limited to rectangular prisms, cones, cylinders, rectangular pyramids & triangular pyramids Shapes are limited to prisms, pyramids, cylinders & cones Prisms will have a base with at most four sides Pyramids will have a base with at most four sides

34 9.V.B Geometry V.B Apply basic theorems of plane geometry, right triangle trigonometry, coordinate geometry & a variety of visualization tools to solve real-world & mathematical problems. C / C 9.V.B.1 Students will know & use theorems about triangles & parallel lines in elementary geometry to justify facts about various geometric figures & solve real-world & mathematical problems. / Theorems may include criteria for two triangles to be congruent or similar Theorems may include facts about angles formed by parallel lines cut by a transversal Items may involve the application of these theorems to solve real-world & mathematical problems involving other plane figures Items will require knowledge & use of theorems Items will not require students to use theorems for justification Items must include context or diagram 9.V.B.2 Students will know & use theorems about circles to justify geometrical facts & solve real-world & mathematical problems. / Theorems may include the relationship involving tangent lines & radii Theorems may include the relationship between inscribed & central angles Theorems may include the relationship between the measure of the central angle & the length of the related arc Items may involve the application of these theorems to solve real-world 7 mathematical problems

35 /GR / C / 9.V.B.3 Students will use properties of two- & three-dimensional figures to solve realworld & mathematical problems. / Use 3.14 as an approximation for Situations may include finding area, perimeter, volume & surface area Situations may include applying direct or indirect methods of measurement Situations may include applying the Pythagorean theorem & its converse Situations may include properties of & triangles Situations may include applying the Pythagorean theorem but not its converse Situations will not include properties of & triangles Limits on shapes in 9.V.C.G2 apply 9.V.B.4 Students will apply the basic concepts of right triangle trigonometry to determine unknown sides or unknown angles when solving real-world & mathematical problems. / Concepts may include sine, cosine & tangent Items will not require the use of the reciprocals or inverses of sine, cosine & tangent Items will provide a table of decimal approximations of three trigonometry values for each angle given in the item or students may use trigonometry values from a calculator Items must include diagram

36 C 9.V.B.5 Students will use coordinate geometry. / Concepts may include distance between two points or midpoint of a line segment Concepts may include slope of a line, slopes of parallel lines or slopes of perpendicular lines C 9.V.B.6 Students will use numeric, graphic & symbolic representations of transformations to solve real-world & mathematical problems. / Transformations may include rotations, reflections translations & change of scale

37 V.B.7 Students will perform basic constructions with a straightedge & compass. / Items may require analysis or justification of the steps in a construction Items may provide construction diagrams for midpoint of a line segment, perpendicular bisector of a line segment, the perpendicular to a line through a point not on the line, the perpendicular to a line through a point on the line & angle bisector Constructions are best assessed in the classroom V.B.8 Students will draw accurate representations of planar figures using a variety of tools. / This benchmark is best assessed in the classroom 0-1

38 /GR C 9.V.C Measurement 9.V.C Use the interconnectedness of geometry, algebra & measurement to explore real-world & mathematical problems V.C.1 Students will demonstrate an understanding of the interconnectedness of geometry, algebra & measurement. / Measurements will be provided with the item Items may include context 9.V.C.G1 Students will make calculations involving time, length, area, volume, weight & mass choosing appropriate unites to calculate, measure & record. Appropriate U.S. customary unties are inches, feet, yards, miles, fluid ounces, cups, pints, quarts, gallons, ounces & pounds Metric prefixes may include milli, centi & kilo Items may not combine systems within one item Time units are years, months, weeks, days, hours, minutes & seconds Time problems may include reading arrival/departure schedules Items may include measurement conversions

39 0-1 9.V.C.G2 Students will use formulas to solve real-world & mathematical problems. Items may include determining the surface area or volume of shapes Shapes are limited to cubes, prisms & cylinders Pictures or diagrams may be used but are not required The radius or diameter is supplied for cylinders Answer options may be left in terms of (e.g., 7 ) Non-rectangular prisms must provide the area of the base Items may include calculating perimeter & area of two-dimensional figures obtained by putting together triangles, parallelograms & sectors of circles to solve real-world & mathematical problems. Items must provide a picture or diagram Items may include calculating the radius, diameter, circumference & area of a circle Given the diameter or radius, may require students to determine area or circumference Given the circumference, may require students to determine radius, diameter or area Radii must be greater than 2 Grade 11 Formula Sheet will be provided (See copy of grade 11 formula sheet in Grade 11 Item Sampler)