Minnesota 9 th -11 th Grade 2003 Math Strands & Standards

Transcription

1 Minnesota 9 th -11 th Grade 2003 Math Strands & Standards Mathematical Reasoning Number Sense, Computation & Operations Patterns, Functions & Algebra Data Analysis, Statistics & Probability Spatial Sense, Geometry & Measurement Apply skills of mathematical representation, communication & reasoning throughout the remaining three content strands. Number Sense Use real numbers, represented in a variety of ways, to quantify information & to solve real-world & mathematical problems. Computation & Operation Appropriately use calculators & other technologies to solve algebraic, geometric, probabilistic & statistical problems. Patterns & Functions Represent & analyze real-world & mathematical problems using numeric, graphic & symbolic methods for a variety of functions. Algebra (Algebraic Thinking) Solve simple equations & inequalities numerically, graphically, & symbolically. Use recursion to model & solve real-world & mathematical problems. 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. Probability Use appropriate counting procedures, calculate probabilities in various ways and apply theoretical probability concepts to solve real-world and mathematical problems. 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. Geometry Apply basic theorems of plane geometry, right triangle trigonometry, coordinate geometry and a variety of visualization tools to solve realworld and mathematical problems. Measurement Use the interconnectedness of geometry, algebra and measurement to explore real-world and mathematical problems.

2 9 th -11 th Grade Mathematical Reasoning 2003 MN Math Standard to Benchmarks Apply skills of mathematical representation, communication & reasoning throughout the remaining three 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 and rejecting extraneous solutions. 9.I.-.2 Appropriately use examples and counterexamples to make and test conjectures, justify solutions, and explain results. 9.I.-.3 Translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically and interpret the result in the original context. 9.I.-.4 Support mathematical results by explaining why the steps in a solution are valid and why a particular solution method is appropriate. 9.I.-.5 Determine whether or not relevant information is missing from a problem and if so, decide how to best express the results that can be obtained without that information. 9.I.-.6 Know and 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, and its contrapositive if not B, then not A.

3 Computation & Operation Number Sense 9 th -11 th Grade Number Sense, Computation & Operations 2003 MN Math Standard to Benchmarks With & Test Specs ( 20% embedded) ( 5-8 items) Use real numbers, represented in a variety of ways, to quantify information & to solve real-world & mathematical problems. /CR/GR 9.II.A.- N/A These benchmarks are continued from grade 8 & embedded in algebra, probability & geometry in grade 11. /CR/GR 9.II.B.1 Apply the correct order of operations and grouping symbols when using calculators and other technologies. This benchmark is assessed in Grade 8 It is embedded in other Grade 11 items 9.II.B.2 Know, use and translate calculator notational conventions to mathematical notation. Assessed only at the classroom level 9.II.B.3 Recognize the impact of units such as degrees and radians on calculations. This benchmark is embedded in Grade 11 items from Strand V 9.II.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.II.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. Appropriately use calculators & other technologies to solve algebraic, geometric, probabilistic & statistical problems. /CR/GR

4 Assessed only at the classroom level 9.II.B.6 Understand that use of a calculator requires appropriate mathematical reasoning and does not replace the need for mental computation. Assessed only at the classroom level 9.II.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 1 7 mixed number requiring regrouping (e.g., is not acceptable) Multiplication may be expressed as raised dot, x, or ( ) (e.g., 56, 5 6, 5(6) ). Division may be expressed using division symbol or fraction bar (e.g., 6 2 or 6 2 ). Division items must have a whole number divisor For multiplication & division items, mixed numbers must be expressed as improper fractions No nested grouping symbols are allowed (e.g., 3[292 (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 2-3 items 9.II.B.G8 Students will use rational numbers in complex ways to solve multi-step real-world & mathematical problems. Rationals are limited to positive rationals Non-integer rationals will be represented in decimal

5 form Items may require integer approximations of square roots of positive integers Squares must be less than or equal to items 9.II.B.G9 Students will use fractions, decimals & percents in multiple representations for estimation & computation to solve real-world & 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.II.B.G10 Students will use proportional reasoning to solve realworld & 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 not required Similarity may be shown using similarity symbol ( ) or using markings on figures Items may include context 2-3 items

6 Patterns & Functions (7-9 items) 9 th -11 th Grade Patterns, Functions & Algebra 2003 MN Math Standard to Benchmarks With & Test Specs ( items) ( 3-6 items) Represent & analyze real-world & mathematical problems using numeric, graphic & symbolic methods for a variety of functions. /CR 9.III.A.1 Know the numeric, graphic and symbolic properties of linear, step, absolute value and quadratic functions. Graphic properties may include rates of change, intercepts, maxima and minima. 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 Students will know the numeric, graphic & symbolic properties of linear, step, absolute value & quadratic functions. 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

7 9.III.A.2 Model exponential growth and decay, numerically, graphically and symbolically, using exponential functions with integer inputs. 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) Students will model exponential growth & decay. Models may be numeric or graphic Items may have real-world context (e.g., bacterial growth, half-life, compound interest) 9.III.A.3 Analyze the effects of coefficient changes on linear and quadratic functions and their graphs. Changes to coefficients in ax2 + bx + c are limited to a and c. 9.III.A.4 Apply basic concepts of linear, quadratic and exponential expressions or equations in real-world problems such as loans, investments and the path of a projectile. Exponents must be integers Students will apply basic concepts of linear, quadratic & exponential expressions or equations in real-world problems. 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.III.A.5 Distinguish functions from other relations using graphic and symbolic methods. Not more than 10 increments on either side of axes

8 Algebra (Algebraic Thinking) (12-14 items) 9.III.A.G6 Students will generate a table of values from a formula or equation. Students will graph the result of a formula or linear equation in ordered pair format on a grid. x & y axes may have different scales Items do not require students to graph or generate a table of a non-linear relation Formulas will only have unknowns to the first degree Items may not require generating a linear equation from a table of values Items may include real-world context (e.g., converting temperature) Given a continuous (i.e., individual points not indicated) linear graph, students will generate a table of values Linear equations will be given in y mx b form 9.III.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. Items may include real-world context Appropriately use calculators & other technologies to solve algebraic, geometric, probabilistic & statistical problems. A 9.III.B.1 Translate among equivalent forms of expressions, such as, simplify algebraic expressions involving nested pairs of parentheses and brackets, simplify rational expressions, factor a common term from an expression and apply associative, commutative and distributive laws. 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 Students will translate among equivalent forms of expressions. Items may include:

9 simplifying algebraic expressions simplifying rational expressions factoring a common monomial term from an expression applying associative, commutative & distributive laws When applying distributive law, expression 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 AB 9.III.B.2 Understand the relationship between absolute value and distance on the number line and graph simple expressions involving absolute value such as, x - 3 = 6 or x + 2 < 5. 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, x b c (e.g., x 3 6 or x 2 5). /CR AB 9.III.B.3 Find equations of a line given two points on the line, a point and the slope of the line or the slope and the y-intercept of the 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 ) CR items must represent real-world contexts Students will find equations of a line. 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 slope-intercept form

10 9.III.B.4 Translate among equivalent forms of linear equations and inequalities. Translating may require simplification (e.g., (2x 2) 2( x 4) y translates to y 4x 6) Equivalent forms may be slope-intercept, 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 ) /CR A /CR/GR AB 9.III.B.5 Use a variety of models such as equations, inequalities, algebraic formulas, written statements, tables and graphs or spreadsheets to represent functions and patterns in realworld and 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 2-3 items 9.III.B.6 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 with monomials Items may include scientific notation with appropriate treatment of significant digits 9.III.B.7 Solve linear equations and inequalities in one variable with numeric, graphic and 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 Students will solve linear equations & inequalities in one

11 variable with numeric, graphic & symbolic methods. 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 /CR/GR AB 9.III.B.8 Find real solutions to quadratic equations in one variable with numeric, graphic and symbolic methods. All solutions are real Solutions determined from a graph will be integer solutions Items may include context Students will determine solutions to quadratic equations in one variable with numeric, graphic & symbolic methods. All solutions are integers Coefficient on second-degree term will always be 1 A /CR AB 9.III.B.9 Use appropriate terminology and mathematical notation to define and represent recursion. x1 is the initial term in the sequence & x n 1 is the next term The term a n is also included in appropriate terminology Items require only addition & multiplication to find the th n term (arithmetic & geometric only) 9.III.B.9 & 9.III.B.10 will not both have 0 items in the same administration 0-1 items 9.III.B.10 Create and use recursive formulas to model and solve realworld and mathematical problems. Progressions are limited to arithmetic & geometric Items will not require identification past tenth term 9.III.B.10 & 9.III.B.9 will not both have 0 items in the same administration

12 0-1 items 9.III.B.11 Solve systems of two linear equations and inequalities with two variables using numeric, graphic and symbolic methods. Inequalities will only be solved graphically Items may include context /CR/GR Students will solve systems of two linear equations & inequalities with 2 variables using numeric, graphic & symbolic methods. 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, slope-intercept & table format Systems are consistent & independent (i.e., solution is one ordered pair) /CR 9.III.B.12 Understand how slopes can be used to determine whether lines are parallel or perpendicular. Given a line and a point not on the line, find the equations for the lines passing through that point and parallel or perpendicular to the given line. 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 & perpendicular to a given line Items may include context Students will understand how slopes can be used to determine whether lines are parallel or perpendicular & determine equations for parallel lines & perpendicular lines. Items may not require students to determine the equation of a line Equations in problem & answer options must be in slope-

13 intercept form 9.III.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

14 Data & Statistics (5-9 items) 9 th -11 th Grade Data Analysis, Statistics & Probability 2003 MN Math Standard to Benchmarks With & Test Specs ( 8-14 items) ( 3-5 items) 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. AB 9.IV.A.1 Construct and analyze circle graphs, bar graphs, histograms, box-and-whisker plots, scatter plots and tables, and demonstrate the strengths and weaknesses of each format by choosing appropriately among them for a given situation. Data may be displayed using charts, line plots, single bar graphs, circle graphs, pictographs, Venn diagrams or tally charts. Graphics may have at most five data categories. Total number on graph or chart will not exceed 100. Scales are in increments of 1, 2 or 5. Bar graphs may be vertical or horizontal. Circle graphs may have at most four sectors. Pictograph keys must be less than or equal to five. 3-4 items

15 /CR 9.IV.A.2 Use measures of central tendency and variability, such as, mean, median, maximum, minimum, range, standard deviation, quartile and percentile, to describe, compare and draw conclusions about sets of data. Data may be displayed using charts, line plots, single bar graphs, pictographs or tally charts. Graphics may have at most five data categories. Total number on graph will not exceed 30. Scales are increments of 1, 2 or 5. Bar graphs may be vertical or horizontal. Pictograph keys must be less than or equal to five. In a CR item, students may be asked to interpret their graph or chart. Measures may be mean, median, maximum, minimum, range, standard deviation, quartile, percentile, mode or interquartile range (IQR). 3-4 items 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, quartile, mode or interquartile range (IQR). The interquartile range may be referred to conceptually, but the terminology interquartile range will not be used. 9.IV.A.3 Determine an approximate best-fit line from a given scatter plot and use the line to draw conclusions. Students will determine approximate line of best-fit & use the line to draw conclusions. 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 Know the influence of outliers on various measures and representations of data about real-world and mathematical problems.

16 Probability (3-5 items) 9.IV.A.5 Understand the relationship between correlation and causation. 9.IV.A.6 Interpret data credibility in the context of measurement error and display distortion. 9.IV.A.7 Compare outcomes of voting methods such as majority, plurality, ranked by preference, run-off and pair-wise comparison. 9.IV.A.G8 Students will analyze histograms, bar graphs, circle graphs, stem-and-leaf plots & box-and-whisker plots. Graphics may have at most ten data categories. Circle graphs may have at 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 2-3 items 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 Select and apply appropriate counting procedures to solve real-world and mathematical problems, including probability problems. Items may involve computing probabilities. Items may include combinations & permutations. Use appropriate counting procedures, calculate probabilities in various ways & apply theoretical probability concepts

17 to solve real-world & mathematical problems. Assessed only at the classroom level. Students will select & apply appropriate counting procedures to solve real-world & mathematical problems. Items may involve determining sample space &/or computing probabilities. Items may not include formulas. Solutions may have at most 24 possibilities. 9.IV.B.2 Use area, trees, unions and intersections to calculate probabilities and relate the results to mutual exclusiveness, independence and conditional probabilities, in real-world and 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. Assessed only at the classroom level. Students will calculate probabilities & relate the results in real-world & mathematical problems. Items may involve independent events. Items will not involve conditional probability. 9.IV.B.3 Use probability models, including area and binomial models, in real-world and mathematical problems. Models may include area & binomial models. Binomial probabilities will involve at most 4 events. Assessed only at the classroom level. Students will use probability models in real-world & mathematical problems. Binomial probabilities will involve at most 3 events.

18 9.IV.B.4 For simple probability models, determine the expected values of random variables. Assessed only at the classroom level. 9.IV.B.5 Know the effect of sample size on experimental and simulation probabilities. Assessed only at the classroom level. 9.IV.B.6 Use a variety of experimental, simulation and theoretical methods to calculate probabilities. Assessed only at the classroom level.

19 Geometry (11-13 items) Spatial Sense () 9 th -11 th Grade Spatial Sense, Geometry & Measurement 2003 MN Math Standard to Benchmarks With & Test Specs ( items) ( 0-2 items) 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. AB 9.V.A.1 Use models and visualization to understand and represent three-dimensional objects and 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 and cones. No 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 and triangular pyramids. Students will use models & visualization to understand & represent various three-dimensional objects & their cross sections from different perspectives. Shapes are limited to prisms, pyramids, cylinders and cones. Prisms will have a base with at most four sides. Pyramids will have a base with at most four sides. 9.V.B.1 Know and use theorems about triangles and parallel lines in elementary geometry to justify facts about various geometrical figures and solve real-world and mathematical problems. These theorems include criteria for two triangles to be congruent or similar and facts about parallel lines cut by a transversal. Apply basic theorems of plane geometry, right triangle trigonometry, coordinate geometry and a variety of visualization tools to /CR

20 solve real-world and 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 and mathematical problems involving other plane figures. Students will know & use theorems about triangles & parallel lines in elementary geometry to justify facts about various geometrical figures & solve real-world & mathematical problems. Items will require knowledge & use of theorems. Items will not require students to use theorems for justification. Items must include context or diagram. /CR 9.V.B.2 Know and use theorems about circles to justify geometrical facts and solve real-world and mathematical problems. These theorems include the relationships involving tangent lines and radii, the relationship between inscribed and central angles and the relationship between the measure of a central angle and arc length. Theorems may include the relationship involving tangent lines and radii. Theorems may include the relationship between inscribed and central angles. Theorems may include the relationship between the measure of the central angle and the length of the related arc. Items may involve the application of these theorems to solve real-world and mathematical problems. /CR/GR AB 9.V.B.3 Know and use properties of two- and three-dimensional figures to solve real-world and mathematical problems such as: finding area, perimeter, volume and surface area; applying direct or indirect methods of measurement; the Pythagorean theorem and its

21 converse; and properties of and triangles. Use 3.14 as an approximation for π. Situations may include finding area, perimeter, volume and surface area. Situations may include applying direct or indirect methods of measurement. Situations may include applying the Pythagorean theorem and its converse. Situations may include properties of and triangles. 2-3 items Students will use properties of two- & threedimensional figures to solve real-world & mathematical problems. Situations may include applying the Pythagorean theorem but not its converse. Situations will not include properties of and triangles. Limits on shapes in V.C.G.2 apply. /CR 2-3 items 9.V.B.4 Apply the basic concepts of right triangle trigonometry including sine, cosine and tangent to solve real-world and mathematical problems. Concepts may include sine, cosine and tangent. Items will not require the use of the reciprocals or inverses of sine, cosine and 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. Students will apply the basic concepts of right triangle trigonometry to determine unknown sides or unknown angles when solving real-world & mathematical problems. Items must include diagram.

22 /CR 9.V.B.5 Use coordinate geometry to represent and examine geometric concepts such as the distance between two points, the midpoint of a line segment, the slope of a line and the slopes of parallel and perpendicular lines. 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. /CR 9.V.B.6 Use numeric, graphic and symbolic representations of transformations such as reflections, translations and change of scale in one, two and three dimensions to solve real-world and mathematical problems. Transformations may include rotations, reflections, translations and change of scale. /CR AB 9.V.B.7 Perform basic constructions with a straightedge and 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 and angle bisector. Constructions are best assessed in the classroom. /CR AB 0-1 items 9.V.B.8 Draw accurate representations of planar figures using a variety of tools. This benchmark is best assessed in the classroom. 0-1 items

23 Measurement () /CR/GR Measurements will be provided with the item. Items may include context. Use the interconnectedness of geometry, algebra and measurement to explore real-world and mathematical problems. 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 units 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. 0-1 items 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.

24 Given the diameter or radius, items may require students to determine area or circumference. Given the circumference, items may require students to determine radius, diameter or area. Radii must be greater than 2. Grade 11 Formula Sheet will be provided. 0-1 items