MATH CURRICULUM
Sixth Grade
March 2002

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Standard 1: The student uses numerical and computational concepts and procedures in a variety of situations. 

Benchmark 1: NUMBER SENSE - The student demonstrates number sense for rational numbers, pi, and simple algebraic expressions in one variable in a variety of situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. compares and orders decimals and fractions greater than zero and integers, and explains the relative magnitude among them.
  2. knows, explains, and uses equivalent representations for fractions and decimals.
  3. knows and explains relationships between percents and ratios.
  4. knows, explains, and uses whole number exponents.
  5. uses equivalent representations for the same simple algebraic expression such as x+x+x could be written as 3x.

Sixth Grade Application Indicators
The student:

  1. uses appropriate representations of positive rational numbers (integers, decimals, fractions, percents, ratios, numbers with whole number exponents, and scientific notation) to formulate and solve real-world problems.
  2. determines reasonableness of solutions to problems involving rational numbers greater than or equal to zero.

Benchmark 2: NUMBER SYSTEMS AND THEIR PROPERTIES - The student demonstrates an understanding of the rational number system and pi, recognizes, applies, and explains its properties, and extends these properties to algebraic expressions in one variable.

Sixth Grade Knowledge Base Indicators
The student:

  1. recognizes the need for integers.
  2. understands that counting numbers, whole numbers, fractions, mixed numbers, and decimals are rational numbers.
  3. identifies and explains prime and composite numbers.
  4. uses the following properties: commutative and associative properties for addition and multiplication; the distributive property; additive and multiplicative identities; additive inverse; multiplicative property of equality; additive property of equality; and the reflexive, transitive, and symmetric properties of equalities.

Sixth Grade Application Indicators
The student :

  1. uses properties of the rational number system to explain reasoning and to formulate and solve real-world problems.
  2. uses rational number properties to perform various computational procedures and explains how they were used.

Benchmark 3: ESTIMATION - The student uses numerical estimation with rational numbers and pi in a variety of situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. uses a variety of computational methods including mental mathematics, paper and pencil, concrete materials, and technological tools such as calculators and computers to estimate quantities involving rational numbers.
  2. estimates quantities involving rational numbers using various estimation techniques such as rounding, special numbers, clustering, truncation or compatible numbers.
  3. recognizes and explains the difference between exact and approximate values.
  4. explains the estimation strategy implemented and its impact on the result.

Sixth Grade Application Indicators
The student:

  1. uses estimation to check reasonableness of results, and makes predictions in situations involving rational numbers and pi. (rounding, special numbers, clustering, truncation, and compatible numbers)
  2. determines if a problem situation calls for an exact or approximate answer and performs the appropriate computation.

Benchmark 4: COMPUTATION - The student explains and performs computations with rational numbers, pi and first degree algebraic expressions in one variable in a variety of situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. uses a variety of computational methods including mental arithmetic (doubles and neighbors), paper and pencil, concrete materials, or technological tools such as calculators and computers.
  2. explains and performs: whole number division and expresses remainders as decimals; addition, subtraction, multiplication and division with decimals; addition and subtraction with integers; and addition, subtraction, and multiplication with fractions and mixed numerals, expressing answers in simple form.
  3. identifies, explains, and finds the greatest common factor and least common multiple of two or more whole numbers.
  4. identifies, explains, and finds the prime factorization of whole numbers. 
  5. combines like terms of simple algebraic expressions.

Sixth Grade Application Indicators
The student:

  1. uses arithmetic operations and inverse relationships to formulate and solve real-world problems involving selected rational numbers. 

Standard 2: The student uses algebraic concepts and procedures in a variety of situations.


Benchmark 1: PATTERNS - The student recognizes, describes, extends, develops, and explains the general rule of patterns from a variety of situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. identifies and continues patterns presented in a variety of formats: numeric, algebraic, visual, oral, written, kinesthetic, pictorial, tabular, graphical, or listing.
  2. creates a pattern.
  3. recognizes patterns involving simultaneous change.

A partial list of patterns to help students meet these indicators could include:
everyday life patterns appropriate for grade level across the curriculum.
numerical and geometric patterns including perfect squares, multiples, and arithmetic sequences using grade appropriate numbers.
Special Note: Working with various types of patterns is an important precursor to working with functions and sequences in high school.

Sixth Grade Application Indicators
 The student:

  1. generalizes a pattern using a written description.

Benchmark 2: VARIABLES, EQUATIONS, AND INEQUALITIES - The student uses variables, symbols, rational numbers, and simple algebraic expressions in one variable to solve linear equations and inequalities in a variety of situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. explains and uses symbols to represent unknown quantities and variable relationships.
  2. solves one-step linear equations in one variable with positive whole number solutions such as 2x=8 or x+7=12 .
  3. solves one-step whole number linear inequalities in one variable such as for what values of x is 3x > 12.
  4. finds a whole number percentage of a number such as what is 10% of 40?
  5. explains and uses the relationship among ratios, proportions, and percents.

Sixth Grade Application Indicators
 The student:

  1. uses symbols, variables, inequalities, and equations to represent problems involving variable situations and unknown quantities.
  2. formulates and solves problems involving simple linear relationships, percents, variable situations and unknown quantities.

Benchmark 3: FUNCTIONS - The student recognizes, describes, and examines constant and linear relationships in a variety situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. identifies or graphs ordered pairs on the coordinate plane using a variety of scales.
  2. uses ratios to describe relationships.

Sixth Grade Application Indicators
 The student:

  1. converts between verbal, numerical, and graphical models or concrete materials used to describe mathematical relationships.

Benchmark 4: MODELS - The student develops and uses models to represent and justify mathematical relationships found in a variety of situations.

Sixth Grade Knowledge Base Indicators
 The student:

  1. creates mathematical models to show the relationship between two or more things.

    The student knows, explains, and uses mathematical models such as:
    the number line to model the relationship between rational numbers and rational number operations.
    Venn diagrams to show the relationships between characteristics of two or more sets.
    pictorial representations of addition and subtraction of rational numbers with regrouping.
    equations and inequalities to model numerical relationships. manipulatives or pictures to model computational procedures. three-dimensional geometric models, graphs, tables, and charts to describe, interpret or analyze data. diagrams or pictures to represent problem situations.

Sixth Grade Application Indicators
 The student:

  1. recognizes that the same situation can be represented in more than one way.
  2. explains why some models are more useful than others in certain situations.

Standard 3: The student uses geometric concepts and procedures in a variety of situations.

Benchmark 1: GEOMETRIC FIGURES AND THEIR PROPERTIES - The student recognizes, applies, and compares properties of geometric figures in a variety of situations.

Sixth Grade Knowledge Base Indicators
 The student:

  1. recognizes all geometric figures listed in previous benchmarks and indicators plus the following figures, rays and angles as right, acute, obtuse, and straight.
  2. recognizes triangles as scalene, isosceles, and equilateral.
  3. recognizes regular and irregular polygons.
  4. recognizes similar and congruent triangles and quadrilaterals.
  5. uses symbols for angle, line, line segment, and ray.
  6. uses appropriate technology, manipulatives, constructions or drawings to recognize, apply or compare geometric figures.
  7. recognizes parallel lines.

Sixth Grade Application Indicators
 The student:

  1. formulates and solves real-world problems by applying properties of geometric figures.

Benchmark 2: MEASUREMENT AND ESTIMATION - The student estimates, measures, and uses measurement formulas in a variety of situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. uses various estimation techniques to determine whole number approximations of length, volume, area, perimeter, weight, capacity, temperature and time.
  2. recognizes and states measurement formulas for areas and perimeters of squares and rectangles.
  3. applies given measurement formulas including area and perimeter to two-dimensional figures.
  4. applies various measurement techniques and uses appropriate tools, units of measure, and degrees of accuracy to find accurate rational number representations for length, area, perimeter, temperature and time.
  5. performs conversions within standard measurement systems.
  6. performs conversions within the metric measurement system such as millimeters to meters.
  7. uses standard units of measurement up to the nearest sixteenth of an inch, metric measurements up to the nearest millimeter.
  8. selects, explains the selection, or uses measurement tools, units of measure, and degrees of accuracy appropriate to the given situation.

Sixth Grade Application Indicators
 The student:

  1. formulates and solves real-world problems by applying measurements and measurement formulas.
  2. uses estimation to check reasonableness of measurements and calculations and/or adjusts original measurements based on additional 
    information (estimation from a frame of reference).

Benchmark 3: TRANSFORMATIONAL GEOMETRY - The student recognizes and performs transformations of two-dimensional figures in a variety situations.

Sixth Grade Knowledge Base Indicators
The student:

  1. recognizes, explains and performs up to two transformations on two-dimensional figures.
  2. recognizes or draws three-dimensional shapes as they would appear from a variety of visual perspectives (top, bottom, side).

Sixth Grade Application Indicators
 The student:

  1. describes and draws two dimensional shapes as they would appear after undergoing two specified transformations without actually performing the transformations with concrete objects.

Benchmark 4: GEOMETRY FROM AN ALGEBRAIC PERSPECTIVE - The student will analyze two-dimensional geometry using a coordinate system in a variety of situations.

Sixth Grade Knowledge Base Indicators
 The student:

  1. graphs or identifies points on the coordinate plane in the first quadrant.
  2. uses T-tables.
  3. recognizes and plots ordered pairs on the coordinate plane.
  4. explains relationships among mixed numbers and decimals as they appear on a number line.

Sixth Grade Application Indicators
 The student:

  1. represents, formulates and solves distance problems by using a number line.
  2. represents, formulates and solves area and perimeter problems on the coordinate plane.

Standard 4: The student uses concepts and procedures of data analysis in a variety of situations.

Benchmark 1: PROBABILITY - The student uses probability to generate convincing arguments, draw conclusions, and make decisions in a variety of situations.

Sixth Grade Knowledge Base Indicators
 The student:

  1. lists all possible outcomes for a compound event composed of two independent events in an organized way and recognizes whether an outcome is certain, impossible, likely or unlikely.
  2. describes probability of simple events using fractions.

Sixth Grade Application Indicators
 The student:

  1. conducts experiments and/or simulations, records the results in charts, tables, or graphs, and uses the results to draw conclusions and make predictions.
  2. compares what should happen (theoretical/expected results) with what did happen (experimental/empirical results) in a simple experiment.

Benchmark 2: STATISTICS - The student generates, organizes, and interprets rational number and other data in a variety of situations. The student applies measures of central tendency when drawing conclusions from the data.

Sixth Grade Knowledge Base Indicators
The student:

  1. organizes, displays and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized and accurate manner including the use of correct titles, labels, and intervals or categories. Specific formats for the displays of rational number data and other types of data include:
    frequency distributions.
    stem-and-leaf plots.
    scatterplots.
    bar, line, and circle graphs.
    Venn diagrams or other pictorial displays.
    charts and tables.
  2. determines the mean, mode and range for a whole number data set. 
  3. determines the median for a rational number data set containing an odd number of data points.

Sixth Grade Application Indicators
 The student:

  1. uses data analysis to make accurate inferences, decisions/predictions, and to develop convincing arguments from data displayed in a variety of formats. (see listing of graphical displays)
  2. recognizes or explains the affects of scale and/or interval changes on graphs of whole number data sets. 
  3. explains advantages and disadvantages of using various display formats for a specific data set.