Vermont State Standards for Mathematics:

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.

MK:1. Demonstrates conceptual understanding of rational numbers with respect to whole numbers by connecting oral number words and numerals (up to and including two-digit numbers to 50) to the quantities they represent using physical models and representations and shows correct sequence of cardinal numbers.

MK:2. Demonstrates understanding of the relative magnitude of numbers from 0 to 50 by ordering whole numbers; by demonstrating one-one correspondence; and by showing the relationship between whole numbers (1 more, 1 less). Apply number parameters consistent with MK:1.

MK:3. Demonstrates conceptual understanding of mathematical operations involving addition and subtraction by solving problems involving situations in which one adds to, takes from.

MK:4. Accurately solves problems in context involving addition and subtraction using whole numbers.

MK:5. Recognizes and names coins.

MK:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

MK:9. Uses attributes, composition, or decomposition to sort or classify objects using at least one attribute (e.g., color). Recognizes and names polygons (triangles, squares, rectangles) and circles in their environment.

MK:15. Identifies the appropriate standard tool used to measure length, temperature, and weight.

MK:16. Determines elapsed and accrued time as it relates to before/after and sequences of events (first, next, last), and identifies a clock and calendar as measurement tools.

MK:18. Find and name locations with simple relationships (i.e., near, far, above, below, next to).

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

MK:19. Identifies and extends to specific cases a variety of patterns including sequences of shapes, sounds, movement, colors, letters, and numbers by extending the pattern to the next one, two, or three elements.

MK:20. Demonstrates a conceptual understanding of change qualitatively (growth - student growing taller).

MK:22. Demonstrates conceptual understanding of equality by showing equivalence between two expressions (4+1=5; 2+3=5) by solving one-step equations involving whole number addition or subtraction using models or verbal explanations.

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

MK:23. Interprets a given representation (models and tally charts) through written or verbal/scribed response to answer questions related to the data, or to analyze the data to formulate conclusions.

MK:24. Analyzes patterns, trends, or distributions in data in a variety of contexts using 'more,' 'less,' or 'equal.' (e.g., 'In a plus 2 pattern, there will be more items on the fifth day than on the first day.')

MK:25. Organizes and displays data using diagrams, models, or tally charts through written or verbal/scribed response to answer questions related to the data, to analyze the data to formulate conclusions.

MK:28. In response to a teacher- or student-generated question or hypothesis, collects appropriate data and makes observations about the data through written or verbal/scribed response.

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.

MK:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

MK:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.

MK: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.

MK:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.

MK:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.

MK:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.

M1:1. Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 100 using place value (a grouping system wherein a digit's place in a number denotes its value; e.g., in 34, 3 represents 3 tens, or 30); by applying the concepts of equivalency in composing or decomposing numbers (e.g., 12 = 7 + 5); and in expanded notation (e.g., 41 = 4 tens + 1 one or 41 = 40 + 1) using models, explanations, or other representations. Shows correct sequence of ordinal and cardinal numbers and compares cardinal numbers 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 part/whole relationships of benchmark fractions with models, diagrams, or written or verbal/scribed response.

M1:2. Demonstrates understanding of the relative magnitude of numbers from 0 to 100 by ordering whole numbers; by comparing whole numbers to each other or to benchmark numbers (10, 25, 50); by showing the relationship between whole numbers (1 more, 1 less; 10 more, 10 less); or by connecting number words and numerals to the quantities they represent using models, representations, or number lines. Apply number parameters consistent with M1:1.

M1:3. Demonstrates conceptual understanding of mathematical operations involving addition and subtraction by solving problems involving situations in which one adds to, takes from, puts together, and takes apart, or adds.

M1:4. Accurately solves problems in and out of context involving addition and subtraction using whole numbers.

M1:5. Demonstrates understanding of monetary value of coins and adds coins together to a value no greater than $1.00.

M1:6. Mentally adds and subtracts whole-number facts through ten with accuracy.

M1:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

M1:8. Applies properties of numbers (odd, even, composition/decomposition [5 is the same as 2 + 3]) and operations (commutative, identity) to solve problems and to simplify computations involving whole numbers.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

M1:9. Uses attributes, composition, or decomposition to sort or classify polygons (triangles, squares, rectangles, rhombi, trapezoids, and hexagons) or objects by a combination of two non-measurable or measurable attributes. Recognizes and names polygons and circles in their environment.

M1:11. Identifies objects in the environment given an example of a three-dimensional shape (e.g., show a wooden cylinder and students identify common objects of the same shape).

M1:15. Selects an appropriate tool with which to measure length, temperature, weight, and volume, and uses nonstandard units for linear measurement and weight.

M1:16. Determines elapsed and accrued time as it relates to the patterns of days of the week, yesterday, today, tomorrow and tells time to the half hour.

M1:18. Find and name locations with simple relationships (i.e., near, far, above, below, next to, up, down, right, left).

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

M1:19. Identifies and extends to specific cases a variety of patterns including sequences of shapes, sounds, movement, colors, letters, and numbers by extending the pattern to the next one, two, or three elements.

M1:20. Demonstrates a conceptual understanding of linear relationships (y = kx) as a constant rate of change qualitatively (growth - student growing taller) and quantitatively (measurable growth - 2 inches each year).

M1:22. Demonstrates conceptual understanding of equality by showing equivalence between two expressions (4+1=5; 2+3=5) by solving one-step equations involving whole number addition or subtraction using models, verbal explanations, or written equations.

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

M1:23. Interprets a given representation (models, tally charts, pictographs with one-to-one correspondence, and tables) through written or verbal/scribed response to answer questions related to the data, or to analyze the data to formulate conclusions.

M1:24. Analyzes patterns, trends, or distributions in data in a variety of contexts using 'more,' 'less,' or 'equal.'

M1:25. Organizes and displays data using diagrams, models, or tally charts through written or verbal/scribed response to answer questions related to the data, to analyze the data to formulate conclusions.

M1:27. For a probability event in which the sample space may or may not contain equally likely outcomes, uses experimental probability to describe the likelihood or chance of an event (using 'more likely,' 'less likely').

M1:28. In response to a teacher- or student-generated question or hypothesis, collects appropriate data to answer the question or hypothesis being tested through written or verbal/scribed response.

M1:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

M1:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.

M1:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.

M1:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.

M1:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.

M1:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.

M2:1. Demonstrates 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.

M2:2. Demonstrates 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.

M2:3. Demonstrates conceptual understanding of mathematical operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, part- whole relationships, and comparison situations; and addition of multiple one-digit whole numbers.

M2:5. Demonstrates 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).

M2:6. Mentally adds and subtracts whole-numbers facts through twenty with accuracy.

M2:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

M2:8. Applies properties of numbers (odd, even) and operations (commutative, associative, identity) to solve problems and to simplify computations involving whole numbers.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

M2:9. Uses properties, attributes, composition, or decomposition to sort or classify polygons or objects by a combination of two or more non-measurable or measurable attributes.

M2:11. Identifies three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, or spheres) and their attributes and recognizes them in their environment.

M2:14. Demonstrates conceptual understanding of perimeter and area by using models or manipulatives to surround and cover polygons.

M2:15. Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands.

M2:16. Determines elapsed and accrued time as it relates to the patterns of days of the week, months, hours, and tells time to five minutes.

M2:18. Solves problems using a two-dimensional coordinate system (x and y axes - quadrant I) to locate and describe positions on a map.

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

M2:19. Identifies and extends 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).

M2:20. Demonstrates a conceptual understanding of linear relationships (y = kx) as a constant rate of change qualitatively (growth - student growing taller) and quantitatively (measurable growth - 2 inches each year) change.

M2:22. Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (e.g., 2 + __ = 7) (limited to one operation and limited to use addition or subtraction).

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

M2:23. Interprets 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.

M2:24. Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using 'more,' 'less,' or 'equal.'

M2:25. Organizes and displays data using diagrams, models, tally charts, or tables to answer questions related to the data, to analyze the data to formulate conclusions.

M2:26. Uses counting techniques to solve problems involving combinations using a variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, or others); (e.g., 'How many ways can you make 50 cents using nickels, dimes, and quarters?')

M2:27. For a probability event in which the sample space may or may not contain equally likely outcomes, uses experimental probability to describe the likelihood or chance of an event using 'more likely,' 'less likely,' 'equally likely,' 'certain,' or 'impossible.'

M2:28. In response to a teacher- or student-generated question or hypothesis, collects appropriate data, organizes the data, displays/represents the data, and makes observations about the data to draw conclusions about the question or hypothesis being tested.

M2:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

M2:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.

M2:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.

M2:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.

M2:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.

M2:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.

M3:1. Demonstrates 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.

M3:2. Demonstrates 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, 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.

M3:3. Demonstrates 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.

M3:4. Accurately solves problems involving addition and subtraction with and without regrouping; the concept of multiplication; and addition or subtraction of decimals (in the context of money).

M3:6. Mentally adds and subtracts whole-numbers facts through twenty with accuracy.

M3:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

M3:8. Applies properties of numbers (odd, even) and applies the commutative and associative properties of addition to solve problems and to simplify computations.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

M3:9. Uses 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.

M3:11. Uses 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).

M3:12. Demonstrates conceptual understanding of congruency using transformations (flips and slides and turns), and shape and size of polygons.

M3:14. Demonstrates conceptual understanding of perimeter of polygons, and the area of rectangles on grids using a variety of models or manipulatives. Expresses all measures using appropriate units.

M3:15. Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands.

M3:16. Determines elapsed and accrued time to the 1/4 hour.

M3:18. Solves problems using the Cartesian coordinate system (Quadrant I) to locate coordinates and to represent data from tables.

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

M3:19. Identifies and extends 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.

M3:20. Demonstrates a conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change.

M3:22. Demonstrates 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., 2 + __ = 7) (limited to one operation and limited to use addition, subtraction, or multiplication).

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

M3:23. Interprets 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.

M3:24. Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using 'most frequent' (mode), 'least frequent,' 'largest,' or 'smallest.'

M3:25. Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M3:23. Organizes and displays data using bar graphs or tables to answer question related to the data, to analyze the data to formulate or justify conclusions, or to make predictions.

M3:26. Uses counting techniques to solve problems in context to determine possibilities using a variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, or others); (e.g., 'How many ways can you make 50 cents using nickels, dimes, and quarters?' Given a map - 'How many different ways can you go from point A to B?')

M3:27. For a probability event in which the sample space may or may not contain equally likely outcomes, determines the likelihood of the occurrence of an event (using 'more likely,' 'less likely,' or 'equally likely').

M3:28. In response to a teacher- or student-generated question or hypothesis, collects appropriate data, organizes the data, displays/represents the data, and makes observations about the data to draw conclusions about the question or hypothesis being tested.

M3:29. Uses experimental probability to describe the likelihood or chance of an event using 'more likely,' 'less likely,' 'equally likely,' 'certain,' or 'impossible.'

M3:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

M3:31. Demonstrate understanding of mathematical problem solving and communication through Connections - Demonstration of observations, applications, extensions, and generalizations.

M3:32. Demonstrate understanding of mathematical problem solving and communication through Solution - All of the work that was done to solve the problem, including the answer.

M3:33. Demonstrate understanding of mathematical problem solving and communication through Mathematical Language - The use of mathematical language in communicating the solution.

M3:34. Demonstrate understanding of mathematical problem solving and communication through Mathematical Representation - The use of mathematical representation to communicate the solution.

M3:35. Demonstrate understanding of mathematical problem solving and communication through Documentation - Presentation of the solution.

M4:1. Demonstrates 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.

M4:2. Demonstrates 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.

M4:3. Demonstrates 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.

M4:4. Accurately solves 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.)

M4:6. Mentally adds and subtracts whole numbers through twenty and multiplies whole numbers through twelve with accuracy.

M4:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

M4:8. Applies properties of numbers (odd, even, factor, multiple, remainders, composition/decomposition) to solve problems and to simplify computations.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

M4:9. Uses 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 90o as more than, less than, or equal to. Recognizes symmetrical figures and uses symmetry to identify and classify figures.

M4:11. Uses 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). Identifies components (faces, edges, and vertices) of three-dimensional shapes (cubes and rectangular prisms).

M4:12. Demonstrates 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.

M4:13. Demonstrates 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. Describes relationships using models or explanations.

M4:14. Demonstrates 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. Expresses all measures using appropriate units.

M4:15. Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands.

M4:16. Determines elapsed and accrued time to the 1/4 hour.

M4:18. Solves problems using the Cartesian coordinate system (Quadrant I) to locate coordinates and to represent data from tables.

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

M4:19. Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables or sequences; and writes a rule in words or symbols to find the next case.

M4:20. Demonstrates a conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change.

M4:21. Demonstrates 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.

M4:22. Demonstrates 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(FandA)-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.

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

M4:23. Interprets 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.

M4:24. Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (median or mode), or range.

M4:25. Organizes and displays data using line plots, bar graphs, tally charts and frequency charts, or tables to answer question related to the data, to analyze the data to formulate or justify conclusions, or to make predictions.

M4:26. Uses counting techniques to solve problems in context involving combinations or simple permutations (e.g., given a map, determines the number of paths from point A to point B) using a variety of strategies (e.g., organized lists, tables, tree diagrams, or others).

M4:27. For a probability event in which the sample space may or may not contain equally likely outcomes, determines the theoretical probability of an event and expresses the result as part to whole (e.g., two out of five).

M4:28. In response to a teacher- or student-generated question or hypothesis, collects appropriate data, organizes the data, displays/represents the data, analyzes the data to draw conclusions about the questions or hypothesis being tested.

M4:29. Uses experimental probability, records the outcomes, and describes the likelihood of an event as a value from 0 through 1 (for events that are certain to occur) written as either a ratio or as part to whole (e.g., 7 out of 10).

M4:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

M4:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.

M4:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.

M4:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.

M4:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.

M4:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.

M5:1. Demonstrates 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; 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.

M5:2. Demonstrates 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.

M5:3. Demonstrates 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.

M5:4. Accurately solves 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.)

M5:6. Mentally multiplies and divides whole numbers through twelve with accuracy.

M5:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

M5:8. Applies properties of numbers (odd, even, factor, multiple, prime, composite, divisibility, remainders, composition/decomposition) to solve problems and to simplify computations.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

M5:9. 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, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms).

M5:11. Uses 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).

M5:12. Demonstrates 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.

M5:13. Demonstrates conceptual understanding of similarity by describing the proportional effect on the linear dimensions of polygons 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.

M5:14. Demonstrates 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. Expresses all measures using appropriate units.

M5:15. Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands.

M5:16. Determines elapsed and accrued time to the nearest minute.

M5:18. 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.

M5:19. Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, or in problem situations; and writes a rule in words or symbols for finding specific cases of a linear relationship.

M5:20. Demonstrates a conceptual understanding of linear relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change.

M5:21. Demonstrates 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.

M5:22. 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)-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 (multistep 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., 2 x + 3 = 11, (x: x = 2, 3, 4, 5)).

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

M5:23. Interprets 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.

M5:24. 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 range to analyze situations, or to solve problems.

M5:25. Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M5:23. Organizes and displays data using line plots, bar graphs, tally charts and frequency charts, or tables to answer question related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems.

M5:26. Uses counting techniques to solve problems in context involving combinations using a variety of strategies (e.g., organized lists, tables, tree diagrams, or others); or determines the possible outcomes for a sample space that contains equally likely outcomes.

M5:27. 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 and expresses the result as a fraction.

M5:28. In response to a teacher- or student-generated question or 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 connections to real-world situations.

M5:29. Uses experimental probability, evaluates the possible outcomes, and describes the likelihood or chance of an event as a ratio of actual times the event occurred to the number of trials written as either a ratio or as part to whole.

M5:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

M5:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.

M5:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.

M5:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.

M5:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.

M5:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.

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.

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.

M7:1. Demonstrates 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.

M7:2. Demonstrates 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., 3^3, 4^3), integers, absolute values, or numbers represented in scientific notation using number lines or equality and inequality symbols.

M7:3. Demonstrates conceptual understanding of operations with integers, exponents, and square roots of perfect square numbers and non-perfect square numbers using models, diagrams, or explanations.

M7:4. Accurately solves problems involving proportional reasoning; percents involving discounts, tax, or tips; and rates. And accurately solves problems involving integers, raising numbers to whole-number powers, and determining square roots of perfect square numbers and non-perfect square numbers.

M7:7. Estimates and evaluates the reasonableness of solutions appropriate to grade level.

M7:8. Applies properties of numbers (greatest common factor [GCF], least common multiple [LCM], composition/decomposition, divisibility, prime factorization, inverses, and identities), and commutative, distributive, and associative properties of operations, and exponents using powers of ten and scientific notation to solve problems and to simplify computations.

VT.7.7. Mathematical Understanding: Geometric and Measurement Concepts: Students use geometric and measurement concepts.

M7:9. Uses properties of angle relationships resulting from two or three intersecting lines (adjacent angles, vertical angles, straight angles, or angle relationships formed by two nonparallel lines cut by a transversal), or two parallel lines cut by a transversal to solve problems.

M7:10. Applies theorems or relationships (triangle inequality or sum of the measures of interior angles of regular polygons) to solve problems.

M7:11. Applies the properties of number of vertices, number of edges, faces, and types of angles, symmetry, to identify and distinguish among three-dimensional shapes (rectangular prisms, triangular prisms, pyramids, cubes) and uses properties to solve problems involving three-dimensional shapes.

M7:12. Applies the concepts of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations.

M7:13. Applies 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. Describes effects using models or explanations.

M7:14. Demonstrates 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. Expresses all measures using appropriate units.

M7:15. Measures and uses units of measures appropriately and consistently when solving problems across the content strands. Makes conversions within systems.

M7:17. Sketches three-dimensional solids and the nets of prisms, cylinders, and pyramids.

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

M7:19. 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 using words and symbols; generalizes a linear relationship to find a specific case; or writes an expression or equation using words or symbols to express the generalization of a nonlinear relationship.

M7:20. 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, by describing the meaning of slope in concrete situations, or informally determining the slope of a line from a table or graph; and distinguishes between constant and varying rates of change in concrete situations represented in tables or graphs; 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 rates of change.

M7:21. Demonstrates 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 = 5 x3 - 2).

M7:22. 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 multistep 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.

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

M7:23. Interprets 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.

M7:24. Analyzes 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 evaluates the sample from which the statistics were developed (bias).

M7:25. Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M7:23. Organizes and displays data using line graphs or histograms, bar graphs, tables, frequency tables, line plots, 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.

M7:26. Uses counting techniques to solve problems in context involving combinations using a variety of strategies (e.g., organized lists, tables, tree diagrams, area models, Fundamental Counting Principle, or others); or determines the possible outcomes for a sample space that may or may not contain equally likely outcomes.

M7:27. For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of a simple event or an event in a problem-solving situation.

M7:28. 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.

M7:29. Compares and contrasts theoretical and experimental probabilities of events; and uses theoretical or experimental probabilities to determine the fairness of a game. Represents probabilities using fractions, decimals, or percents.

M7:30. Demonstrate understanding of mathematical problem solving and communication through approach and reasoning - the reasoning, strategies, and skills used to solve the problem.

M7:31. Demonstrate understanding of mathematical problem solving and communication through connections - demonstration of observations, applications, extensions, and generalizations.

M7:32. Demonstrate understanding of mathematical problem solving and communication through solution - all of the work that was done to solve the problem, including the answer.

M7:33. Demonstrate understanding of mathematical problem solving and communication through mathematical language - the use of mathematical language in communicating the solution.

M7:34. Demonstrate understanding of mathematical problem solving and communication through mathematical representation - the use of mathematical representation to communicate the solution.

M7:35. Demonstrate understanding of mathematical problem solving and communication through documentation - presentation of the solution.

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.

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.

MHS:1. Accurately solves problems involving conceptual understanding and magnitude of real numbers, or simple vectors.

MHS:4. Accurately solves problems involving proportional reasoning or percents involving the effect of changing the base, rate, or percentage (the three cases of percent), or variations on order of finding percentages (10% off followed by 5% off), and compound interest.

MHS:7. Estimates and evaluates the reasonableness of numerical computations and solutions, including those carried out with technology.

MHS:8. Applies properties of numbers (greatest common factor [GCF], least common multiple [LCM], prime factorization, inverses, and identities), or 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.

MHS:9. Models situations geometrically to solve problems connecting to other areas of mathematics or to other disciplines (i.e., diagrams, coordinate systems, transformations).

MHS:11. Uses the attributes, geometric properties, or theorems involving lines, polygons and circles (e.g., parallel, perpendicular, bisectors, diagonals, radii, diameters, central angles, arc length excluding radians), the Pythagorean Theorem, Triangle Inequality Theorem to solve mathematical situations or problems in context.

MHS:13. Applies concepts of similarity, congruency or right triangle trigonometry to determine length or angle measures and to solve problems involving scale.

MHS:14. Demonstrates conceptual understanding of perimeter, circumference, or area of two-dimensional figures or composites of two-dimensional figures or surface area or volume of three-dimensional figures or composites of three-dimensional figures in problem-solving situations and uses appropriate units of measure and expresses formulas for the perimeter, and area of two-dimensional figures or composites of two-dimensional figures or surface area or volume of three-dimensional figures or composites of three-dimensional figures.

MHS:15. Measures and uses units of measures appropriately and consistently when solving problems across the content strands. Makes conversions within or across systems and makes decisions concerning an appropriate degree of accuracy in problem situations involving measurement. Uses measurement conversion strategies, such as unit/dimensional analysis or uses quotient measures, such as speed and density, that give per unit amounts, or uses product measures, such as person hours to solve problems.

MHS:17. Constructs or accurately represents congruent angles, perpendicular lines, equilateral or isosceles triangles, triangle given the side segments, or inscribe or circumscribe a figure.

VT.7.8. Mathematical Understanding: Function and Algebra Concepts: Students use function and algebra concepts.

MHS:19. Solves and models problems by formulating, extending, or generalizing linear and common nonlinear functions/relations.) And makes connections among representations of functions/relations (equations, tables, graphs, symbolic notation, text).

MHS:20. Demonstrates conceptual understanding of linear relationships and linear and nonlinear functions (including f(x) = ax2, f(x) = ax3, absolute value function, exponential growth) through analysis of intercepts, domain, range and constant and variable rates of change in mathematical and contextual situations.

MHS:21. Demonstrates conceptual understanding of algebraic expressions by evaluating, simplifying, or writing algebraic expressions; and writes equivalent forms of algebraic expressions or formulas (d = rt maps into r = d/t or solves a multivariable equation or formula for one variable in terms of the others).

MHS:22. Demonstrates conceptual understanding of equality by solving linear equations, systems of two linear equations, or problems using tables, graphs, algebraic manipulation, or technology. Demonstrates conceptual understanding of inequality by solving linear inequalities, comparing values of systems of linear functions, using tables, graphs, algebraic manipulation, or technology.

VT.7.9. Mathematical Understanding: Statistics and Probability Concepts: Students use statistics and probability concepts.

MHS:23. Interprets a given representation(s) (box-and-whisker or scatter plots, histograms, frequency charts) to make observations, to answer questions or justify conclusions, to make predictions, or to solve problems.

MHS:24. Analyzes patterns, trends, or distributions in single variable and two variable 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 regression line or correlation (high, low/positive, negative) to analyze situations, or to solve problems; and evaluates the sample from which the statistics were developed (bias, random, or nonrandom).

MHS:25. Organizes and displays data using scatter plots, histograms, or frequency distributions to answer questions related to the data, to analyze the data to formulate or justify conclusions, 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 MHS:23.

MHS:26. Uses combinations, arrangements or permutations to solve problems or to determine theoretical probability and experimental probability.

MHS:27. For a probability event chooses an appropriate probability model/simulations and uses it to estimate a theoretical probability for a chance event and uses the concept of a probability distribution to determine whether an event is rare or reasonably likely.

MHS:28. In response to a question, designs investigations, considers how data-collection methods affect the nature of the data set (i.e., sample size, bias, randomization, control group), collects data using observations, surveys and experiments, purposes and justifies conclusions and predictions based on the data.

MHS:29. Compares and contrasts theoretical and experimental probabilities of events; and determines and/or interprets the expected outcome of an event.

MHS:30. Demonstrate understanding of mathematical problem solving and communication by approach and reasoning - the strategies and skills used to solve the problem, and the reasoning that supports the approach.

MHS:31. Demonstrate understanding of mathematical problem solving and communication by execution - the answer and the mathematical work that supports it.

MHS:32. Demonstrate understanding of mathematical problem solving and communication by observations and extensions - demonstration of observation, connections, application, extensions, and generalizations.

MHS:33. Demonstrate understanding of mathematical problem solving and communication by mathematical communication - the use of mathematical vocabulary and representation to communicate the solution.

MHS:34. Demonstrate understanding of mathematical problem solving and communication by presentation - effective communication of how the problem was solved, and of the reasoning used.