Vermont State Standards for Mathematics: Grade 6
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.
M6:1. Demonstrates 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 not equal to 0); and rates (e.g., a out of b, 25%) using models, explanations, or other representations. Demonstrates conceptual understanding of proportional reasoning, and fluently moves between equivalent representations of commonly used fractions and decimals.
M6:2. Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole-number bases and whole-number exponents (e.g., 3 to the power of 3 , 4 to the power of 3), integers, or rational numbers within and across number formats (fractions, decimals, or whole-number percents from 1 to 100) using number lines or equality and inequality symbols.
M6:3. Demonstrates 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., 3 to the power of 3 , 4 to the power of 3); and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number, decimal, or fraction.
M6:4. Accurately solves 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.
M6:6. Mentally multiplies and divides whole numbers through twelve with accuracy.
M6:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.
M6:8. Applies properties of numbers (factor, multiple, prime, composite, greatest common factor [GCF], least common multiple [LCM], composition/decomposition), divisibility, remainders), and commutative 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.
M6:9. Uses 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).
M6:11. Uses 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).
M6:12. Demonstrates congruency using the results of combining and subdividing shapes (e.g., rectangle into two triangles), by using transformations (flips, slides, and turns), and by using the properties of angles, and length of segments.
M6:13. Demonstrates 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). Describes effects using models or explanations. And applies concepts of similarity using constant of proportionality/scale factor to make larger and smaller scale drawings.
M6:14. Demonstrates 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 demonstrates understanding of the relationships of circle measures (radius to diameter and diameter to circumference) by solving related problems. Expresses all measures using appropriate units.
M6:15. Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands.
M6:18. Solves problems using the Cartesian coordinate system (all quadrants) to locate coordinates and to represent data from tables.
VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.
M6:19. Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations; or writes a rule in words or symbols for finding specific cases of a linear relationship; or writes a rule in words or symbols for finding specific cases of a nonlinear relationship; and writes an expression or equation using words or symbols to express the generalization of a linear relationship (e.g., twice the term number plus 1 or 2n + 1).
M6:20. Demonstrates 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 describes 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.
M6:21. Demonstrates 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 and consistent with order of operations expected at this grade level; 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 = 3 x - 2).
M6:22. Demonstrates 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(FandA)-6-3), solving multistep linear equations of the form ax +/-b = c, where a, b, and c are whole numbers with a not equal to 0.
VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.
M6:23. Interprets 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.
M6:24. Analyzes patterns, trends or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode) or dispersion (range) to analyze situations, or to solve problems.
M6:25. Organizes and displays data using bar graphs, tables, frequency tables, line plots, circle graphs, and stem-and-leaf plots to answer question related to the data, to analyze the data to formulate or justify conclusions, or to make predictions.
M6:26. Uses 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).
M6:27. For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem solving situation.
M6:28. 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 makes predictions, asks new questions, or makes connection to real-world situations.
M6:29. Uses experimental probability to make and test conjectures or design fair games. Represent probabilities using fractions, decimals, or percents.
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.
M6:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.
M6:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.
M6:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.
M6:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.
M6:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.
M6:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.