Standard 1: Number and Computation

The student uses numerical and computational concepts and procedures in a variety of situations.

Sixth Grade

Benchmark 1
The student demonstrates number sense for rational numbers and simple algebraic expressions in one variable in a variety of situations. Knowledge Base Indicators
  1. The student knows, explains, and uses equivalent representations for rational numbers expressed as fractions, terminating decimals, and percents; positive rational number bases with whole number exponents; time; and money.
  2. The student compares and orders:
    1. integers;
    2. fractions greater than or equal to zero,
    3. decimals greater than or equal to zero through thousandths place.
  3. The student explains the relative magnitude between whole numbers, fractions greater than or equal to zero, and decimals greater than or equal to zero.
  4. The student knows and explains numerical relationships between percents, decimals, and fractions between 0 and 1.
  5. The student uses equivalent representations for the same simple algebraic expression with understood coefficients of 1.
Application Indicators
  1. The student generates and/or solves real-world problems using equivalent representations:
    1. integers,
    2. fractions greater than or equal to zero,
    3. decimals greater than or equal to zero through thousandths place.
  2. The student determines whether or not solutions to real-world problems that involve the following are reasonable.
    1. integers,
    2. fractions greater than or equal to zero,
    3. decimals greater than or equal to zero through thousandths place.
 Benchmark 2
The student demonstrates number sense for rational numbers and simple algebraic expressions in one variable in a variety of situations. Knowledge Base Indicators
  1. The student classifies subsets of the rational number system as counting (natural) numbers, whole numbers, integers, fractions (including mixed numbers), or decimals.
  2. The student identifies prime and composite numbers and explains their meaning.
  3. The student uses and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects:
    1. commutative and associative properties of addition and multiplication (commutative-changing the order of the numbers does not change the solution; associative-changing the grouping of the numbers does not change the solution);
    2. identify properties for addition and multiplication (additive identity - zero added to any number is equal to that number; multiplicative identity - one multiplied by any number is equal to that number);
    3. symmetric property of equality;
    4. zero property of multiplication (any number multiplied by zero is zero);
    5. distributive property (distributing multiplication or division over addition or subtraction);
    6. substitution property (one name of a number can be substituted for another name of the same number);
    7. addition property of equality (adding the same number to each side of an equation results in an equivalent equation - an equation with the same solution);
    8. multiplication property of equality (for any equation, if the same number is multiplied to each side of that equation, then the new statement describes an equation equivalent to the original);
    9. additive inverse property (every number has a value known as the additive inverse and when the original number is added to that additive inverse, the answer is zero);
  4. The student recognizes and explains the need for integers.
  5. The student recognizes that the irrational number pi can be represented by an approximate rational value.
Application Indicators
  1. The student generates and/or solves real-world problems with rational numbers using the concepts of these properties to explain reasoning:
    1. commutative and associative properties for addition and multiplication.
    2. additive and multiplicative identities.
    3. symmetric property of equality.
    4. distributive property.
    5. substitution property.
    6. addition property of equality.
    7. multiplication property of equality.
    8. additive inverse property.
  2. The student analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals, or the irrational number pi and its rational approximations in solving a given real-world problem.
 Benchmark 3
The student uses computational estimation with rational numbers and irrational number pi in a variety of situations. Knowledge Base Indicators
  1. The student estimates quantities with combinations of rational numbers and/or the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology.
  2. The student uses various estimation strategies and explains how they were used to estimate rational number quantities or the irrational number pi.
  3. The student recognizes and explains the difference between an exact and an approximate answer.
  4. The student determines the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result.
Application Indicators
  1. The student adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference).
  2. The student estimates to check whether or not the result of a real-world problem using rational numbers is reasonable and makes predictions based on the information.
  3. The student selects a reasonable magnitude from given quantities based on a real-world problem and explains the reasonableness of the selection.
  4. The student determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, or appropriate technology.
 Benchmark 4
The student models, performs, and explains computation with positive rational numbers and integers in a variety of situations. Knowledge Base Indicators
  1. The student computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology.
  2. The student performs and explains these computational procedures:
    1. divides whole numbers through a two-digit divisor and a four-digit dividend and expresses the remainder as a whole number, fraction, or decimal;
    2. adds and subtracts decimals from millions place through thousandths place;
    3. multiplies and divides a four-digit number by a two-digit number using numbers from thousands place through hundredths place;
    4. multiplies and divides using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each;
    5. adds integers;
    6. adds, subtracts, and multiplies fractions (including mixed numbers) expressing answers in simplest form;
    7. finds the root of perfect whole number squares;
    8. uses basic order of operations (multiplication and division in order from left to right, then addition and subtraction in order from left to right) with whole numbers;
    9. adds, subtracts, multiplies, and divides rational numbers using concrete objects.
  3. The student recognizes, describes, and uses different representations to express the computational procedures.
  4. The student identifies, explains, and finds the prime factorization of whole numbers.
  5. The student finds prime factors, greatest common factor, multiples, and the least common multiple.
  6. The student finds a whole number percent (between 0 and 100) of a whole number.
Application Indicators
  1. The student generates and/or solves one- and two-step real-world problems with rational numbers using these computational procedures:
    1. division with whole numbers.
    2. addition, subtraction, multiplication, and division of decimals through hundredths place.
    3. addition, subtraction, and multiplication of fractions (including mixed numbers).

Standard 2: Algebra

The student uses algebraic concepts and procedures in a variety of situations.

Sixth Grade

Benchmark 1
The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations. Knowledge Base Indicators
  1. The student identifies, states, and continues a pattern presented in various formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes include:
    1. counting numbers including perfect squares, and factors and multiples;
    2. positive rational numbers limited to two operations (addition, subtraction, multiplication, division) including arithmetic sequences (a sequence of numbers in which the difference of two consecutive numbers is the same);
    3. geometric figures through two attribute changes;
    4. measurements;
    5. things related to daily life.
  2. The student generates a pattern (repeating, growing).
  3. The student extends a pattern when given a rule of one or two simultaneous operational changes (addition, subtraction, multiplication, division) between consecutive terms.
  4. The student states the rule to find the next number of a pattern with one operational change (addition, subtraction, multiplication, division) to move between consecutive terms.
Application Indicators
  1. The student recognizes the same general pattern presented in different representations, [numeric (list or table), visual (picture, table, or graph), and written].
  2. The student recognizes multiple representations of the same pattern.
 Benchmark 2
The student uses variables, symbols, positive rational numbers, and algebraic expressions in one variable to solve linear equations and inequalities in a variety of situations. Knowledge Base Indicators
  1. The student explains and uses variables and/or symbols to represent unknown quantities and variable relationships.
  2. The student uses equivalent representations for the same simple algebraic expression with understood coefficients of 1.
  3. The student solves:
    1. one-step linear equations (addition, subtraction, multiplication, division) with one variable and whole number solutions.
    2. one-step linear inequalities (addition, subtraction) in one variable with whole numbers.
  4. The student explains and uses equality and inequality symbols (= ≠ < ≤ > ≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) to represent mathematical relationships with positive rational numbers.
  5. The student knows and uses the relationship between ratios, proportions, and percents and finds the missing term in simple proportions where the missing term is a whole number.
  6. The student finds the value of algebraic expressions using whole numbers.
Application Indicators
  1. The student represents real-world problems using variables and symbols to:
    1. write algebraic or numerical expressions or one-step equations (addition, subtraction, multiplication, division) with whole number solutions.
    2. write and/or solve one-step equations (addition, subtraction, multiplication, and division).
  2. The student generates real-world problems that represent simple expressions or one-step linear equations (addition, subtraction, multiplication, division) with whole number solutions.
  3. The student explains the mathematical reasoning that was used to solve a real-world problem using a one-step equation (addition, subtraction, multiplication, division).
 Benchmark 3
The student recognizes, describes, and analyzes linear relationships in a variety of situations. Knowledge Base Indicators
  1. The student recognizes linear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology.
  2. The student finds the values and determines the rule with one operation using a function table (input/output machine, T-table).
  3. The student generalizes numerical patterns up to two operations by stating the rule using words.
  4. The student uses a given function table (input/output machine, T-table) to identify, plot, and label the ordered pairs using the four quadrants of a coordinate plane.
Application Indicators
  1. The student represents a variety of mathematical relationships using written and oral descriptions of the rule, tables, graphs, and when possible, symbolic notation.
  2. The student interprets and describes the mathematical relationships of numerical, tabular, and graphical representations.
 Benchmark 4
The student generates and uses mathematical models to represent and justify mathematical relationships in a variety of situations. Knowledge Base Indicators
  1. The student knows, explains, and uses mathematical models to represent mathematical concepts, procedures, and relationships.
    Mathematical models include:
    1. process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures and mathematical relationships and to solve equations;
    2. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures.
    3. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities:
    4. factor trees to find least common multiple and greatest common factor;
    5. equations and inequalities to model numerical relationships;
    6. function tables (input/output machines, T-tables) to model numerical and algebraic relationships;
    7. two-dimensional geometric models (geoboards or dot paper) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (nets or solids) and real-world objects to model volume and to identify attributes (faces, edges, vertices, bases) of geometric shapes;
    8. tree diagrams to organize attributes and determine the number of possible combinations;
    9. graphs using concrete objects, two-and three-dimensional geometric models (spinners or number cubes) and process models (concrete objects, pictures, diagrams, or coins) to model probability.
    10. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, tables, single stem-and-leaf plots, and scatter plots to organize and display data;
    11. Venn diagrams to sort data and to show relationships.
  2. The student uses one or more mathematical models to show the relationship between two or more things.
Application Indicators
  1. The student recognizes that various mathematical models can be used to represent the same problem situation.
    Mathematical models include:
    1. process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures and mathematical relationships to represent problem situations, and to solve equations;
    2. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations;
    3. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities:
    4. factor trees to find least common multiple and greatest common factor;
    5. equations and inequalities to model numerical relationships;
    6. function tables (input/output machines, T-tables) to model numerical and algebraic relationships;
    7. two-dimensional geometric models (geoboards or dot paper) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (nets or solids) and real-world objects to model volume and to identify attributes (faces, edges, vertices, bases) of geometric shapes;
    8. scale drawings to model large and small real-world objects;
    9. tree diagrams to organize attributes and determine the number of possible combinations;
    10. two-and three-dimensional geometric models (spinners or number cubes) and process models (concrete objects, pictures, diagrams, or coins) to model probability.
    11. graphs using concrete objects, frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, line plots, charts, tables, single stem-and-leaf plots, and scatter plots to organize, display, explain, and interpret data;
    12. Venn diagrams to sort data and to show relationships.
  2. The student selects a mathematical model and justifies why some mathematical models are more accurate than other mathematical models in certain situations.

Standard 3: Geometry

The student uses geometric concepts and procedures in a variety of situations.

Sixth Grade

Benchmark 1
The student recognizes geometric figures and compares their properties in a variety of situations. Knowledge Base Indicators
  1. The student recognizes and compares properties of plane figures and solids using concrete objects, constructions, drawings, and appropriate technology.
  2. The student recognizes and names regular and irregular polygons through 10 sides including all special types of quadrilaterals: squares, rectangles, parallelograms, rhombi, trapezoids, kites.
  3. The student names and describes the solids [prisms (rectangular and triangular)] using the terms faces, edges, vertices, and bases.
  4. The student recognizes all existing lines of symmetry in two-dimensional figures.
  5. The student recognizes and describes the attributes of similar and congruent figures.
  6. The student recognizes and uses symbols for angle (find symbol for), line(  ), line segment (  ), parallel (  ), and perpendicular (  ).
  7. classifies:
    1. angles as right, obtuse, acute, or straight;
    2. triangles as right, obtuse, acute, scalene, isosceles, or equilateral.
  8. The student identifies and defines circumference, radius, and diameter of circles and semicircles.
  9. The student recognizes that the sum of the angles of a triangle equals 180.
  10. The student determines the radius or diameter of a circle given one or the other.
Application Indicators
  1. The student solves real-world problems by applying the properties of:
    1. plane figures (regular polygons through 10 sides, circles, and semicircles) and the line(s) of symmetry.
    2. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases.
    3. intersecting, parallel, and perpendicular lines.
  2. The student decomposes geometric figures made from:
    1. regular and irregular polygons through 10 sides, circles, and semicircles.
    2. nets (two-dimensional shapes that can be folded into three-dimensional figures).
  3. The student composes geometric figures made from:
    1. regular and irregular polygons through 10 sides, circles, and semicircles;
    2. nets (two-dimensional shapes that can be folded into three-dimensional figures).
 Benchmark 2
The student estimates, measures, and uses measurement formulas in a variety of situations. Knowledge Base Indicators
  1. The student determines and uses whole number approximations (estimations) for length, width,, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure.
  2. The student selects, explains the selection of, and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area, and angle measurements.
  3. The student converts:
    1. within the customary system.
    2. within the metric system using the prefixes: kilo, hecto, deka, deci, centi, and milli.
  4. The student uses customary units of measure to the nearest sixteenth of an inch and metric units of measure to the nearest millimeter.
  5. The student recognizes and states perimeter and area formulas for squares, rectangles, and triangles.
    1. uses given measurement formulas to find perimeter and area of: squares and rectangles,
    2. figures derived from squares and/or rectangles.
  6. The student describes the composition of the metric system:
    1. meter, liter, and gram (root measures);
    2. kilo, hecto, deka, deci, centi, and milli (prefixes).
  7. The student finds the volume of rectangular prisms using concrete objects.
  8. The student estimates an approximate value of the irrational number pi.
Application Indicators
  1. The student solves real-world problems by applying these measurement formulas:
    1. perimeter of polygons using the same unit of measurement,
    2. area of squares, rectangles, and triangles using the same unit of measurement,
    3. conversions within the metric system.
  2. The student estimates to check whether or not measurements and calculations for length, width, weight, volume, temperature, time, perimeter, and area in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference).
 Benchmark 3
The student recognizes and performs transformations on two- and three- dimensional geometric figures in a variety of situations. Knowledge Base Indicators
  1. The student identifies, describes, and performs one or two transformations (reflection, rotation, translation) on a two-dimensional figure.
  2. The student reduces (contracts/shrinks) and enlarges (magnifies/grows) simple shapes with simple scale factors.
  3. The student recognizes three-dimensional figures from various perspectives (top, bottom, sides, corners).
  4. The student recognizes which figures will tessellate.
Application Indicators
  1. The student describes a transformation of a given two-dimensional figure that moves it from its initial placement (preimage) to its final placement (image).
  2. The student makes a scale drawing of a two-dimensional figure using a simple scale.
 Benchmark 4
  1. The student uses a number line (horizontal/vertical) to order integers and positive rational numbers (in both fractional and decimal form).
  2. The student organizes integer data using a T-table and plots the ordered pairs in all four quadrants of a coordinate plane (coordinate grid).
  3. The student uses all four quadrants of the coordinate plane to:
    1. identify the ordered pairs of integer values on a given graph;
    2. plot the ordered pairs of integer values.
Application Indicators
  1. The student represents, generates, and/or solves real-world problems using a number line with integer values.
  2. The student represent and/or generates real-world problems using a coordinate plane with integer values to find:
    1. the perimeter of squares and rectangles.
    2. the area of triangles, squares, and rectangles.

Standard 2:  Data

The student uses concepts and procedures of data analysis in a variety of situations.

Sixth Grade

Benchmark 1
The student applies the concepts of probability to draw conclusions and to make predictions and decisions including the use of concrete objects in a variety of situations.
  1. The student recognizes that all probabilities range from zero (impossible) through one (certain) and can be written as a fraction, decimal, or a percent.
  2. The student lists all possible outcomes of an experiment or simulation with a compound event composed of two independent events in a clear and organized way.
  3. The student recognizes whether an outcome in a compound event in an experiment or simulation is impossible, certain, likely, unlikely, or equally likely.
  4. The student represents the probability of a simple event in an experiment or simulation using fractions and decimals.
Application Indicators
  1. The student conducts an experiment or simulations with a compound event composed of two independent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions about the events and make predictions about future events.
  2. The student analyzes the results of a given experiment or simulation of a compound event composed of two independent events to draw conclusions and make predictions in a variety of real-world situations.
  3. The student compares what should happen (theoretical probability/expected results) with what did happen (empirical probability/experimental results) in an experiment or simulation with a compound event composed of two independent events.
 Benchmark 2
The student collects, organizes, displays, and explains numerical (rational numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency. Knowledge Base Indicators
  1. The student organizes, displays, and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays:
    1. graphs using concrete objects;
    2. frequency tables and line plots;
    3. bar, line, and circle graphs;
    4. Venn diagrams or other pictorial displays;
    5. charts and tables;
    6. single stem-and leaf plots;
    7. scatter plots;
  2. The student selects and justifies the choice of data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, or purposeful sampling) in a given situation.
  3. The student uses sampling to collect data and describe the results.
  4. The student determines mean, median, mode, and range for:
    1. a whole number data set,
    2. a decimal data set with decimals greater than or equal to zero.
Application Indicators
  1. The student uses data analysis (mean, median, mode, range) of a whole number data set or a decimal data set with decimals greater than or equal to zero to make reasonable inferences, predictions, and decisions and to develop convincing arguments from these data displays:
    1. graphs using concrete objects;
    2. frequency tables and line plots;
    3. bar, line, and circle graphs;
    4. Venn diagrams or other pictorial displays;
    5. charts and tables;
    6. single stem-and-leaf plots.
  2. The student explains advantages and disadvantages of various data displays for a given data set.
  3. The student recognizes and explains the effects of scale and/or interval changes on graphs of whole number data sets.