Vermont State Standards for Mathematics: Grade 8
VT.7.6. Mathematical Understanding: Arithmetic, Number, and Operation Concepts: Students understand arithmetic in computation, and they select and use, in appropriate situations, mental arithmetic, pencil and paper, calculator, and computer.
M8:1. Demonstrates conceptual understanding of rational numbers with respect to percents as a way of describing change (percent increase and decrease) using explanations, models, or other representations.
M8:2. Demonstrates understanding of the relative magnitude of numbers by ordering or comparing rational numbers, common irrational numbers, numbers with whole-number or fractional bases and whole-number exponents, square roots, absolute values, integers, or numbers represented in scientific notation using number lines or equality and inequality symbols.
M8:4. Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates); and squares, cubes and taking square or cube roots.
M8:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.
M8:8. Applies properties of numbers (greatest common factor [GCF], least common multiple [LCM], prime factorization, divisibility, inverses, and identities), and commutative, distributive, and associative properties of operations to solve problems and to simplify computations.
VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.
M8:9. Models situations geometrically. Uses properties and attributes of lines, angles, and two- and three-dimensional shapes) to formulate and solve problems.
M8:10. Applies the Pythagorean Theorem to find a missing side of a right triangle, or in problem-solving situations and solves problems by applying the Triangle Inequality Theorem to determine if three line segments with given lengths form a triangle, and the sum of the angles in a convex polygon of any number of sides.
M8:13. Applies concepts of similarity to determine the impact of scaling on the volume or surface area of three-dimensional figures when linear dimensions are multiplied by a constant factor; to determine the length of sides of similar triangles, or to solve problems involving growth and rate and makes scale drawings.
M8:14. Demonstrates conceptual understanding of surface area or volume by solving problems involving surface area and volume of rectangular prisms, cylinders, or pyramids. Expresses all measures using appropriate units.
M8:15. Measures and uses units of measures appropriately and consistently when solving problems across the content strands. Makes conversions within or across systems.
M8:17. Sketches a variety of three-dimensional objects using orthogonal views (projections and isometric views), or constructs or accurately represents angle bisector, perpendicular bisector, congruent segments and regular polygons. Draws nets of three-dimensional shapes.
VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.
M8:19. Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear relationship (nonrecursive explicit equation); generalizes a linear relationship to find a specific case; generalizes a nonlinear relationship using words or symbols; or generalizes a common nonlinear relationship to find a specific case.
M8:20. Demonstrates 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; informally and formally determining slopes and intercepts represented in graphs, tables, or problem situations; or describing the meaning of slope and intercept in context; and distinguishes between linear relationships (constant rates of change) and nonlinear relationships (varying rates of change) represented in tables, graphs, equations, or problem situations; or describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant and varying rates of change.
M8:21. Demonstrates conceptual understanding of algebraic expressions by evaluating and simplifying (including those with square roots, whole-number exponents, or rational numbers); or by evaluating an expression within an equation.
M8:22. Demonstrates 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 formulas for a variable requiring one transformation (e.g., d = rt; d/r = t); by solving multistep linear equations with integer coefficients; by showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution; and by informally solving problems involving systems of linear equations in a context.
VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.
M8:23. Interprets a given representation (line graphs, scatter plots, histograms, or box-and-whisker plots) to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems.
M8:24. Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode), dispersion (range or variation), outliers, quartile values, or estimated line of best fit to analyze situations, or to solve problems; and evaluates the sample from which the statistics were developed (bias, random, or nonrandom).
M8:25. Organizes and displays data using scatter plots to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems; or Identifies representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M8:23.
M8:26. Uses 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, or others).
M8:27. For a probability event in which the sample space may or may not contain equally likely outcomes, determines the possible outcomes by either sample space (organized list, table, tree model, area model) or Fundamental Counting Principle and determines the theoretical probability of that event as a ratio of favorable outcomes to possible outcomes. Expresses the ratio as a fraction, decimal, or percent.
M8:28. In response to a teacher- or student-generated question, makes a hypothesis, collects appropriate data, organizes the data, appropriately displays/represents numerical and/or categorical data, analyzes the data to draw conclusions about the questions or hypothesis being tested, and when appropriate to make predictions, asks new questions, or makes connection to real-world situations.
M8:29. Compares and contrasts theoretical and experimental probabilities of compound events using fractions, decimals, or percents; and uses theoretical or experimental probabilities to determine the fairness of a game.
VT.7.10. Mathematical Problem Solving: Applications: Students use concrete, formal, and informal strategies to solve mathematical problems, apply the process of mathematical modeling, and extend and generalize mathematical concepts. Students apply mathematics as they solve scientific and technological problems or work with technological systems.
M8:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.
M8:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.
M8:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.
M8:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.
M8:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.
M8:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.