Tennessee Department of Education

Sixth Grade

Number and Operation

Content Standard 1.0 The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

Learning Expectations:

1.1 Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

1.2 Understand operations and how they relate to one another.

1.3 Solve problems, compute fluently, and make reasonable estimates.

Accomplishments

6.1.1 Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

    1. recognize the place value of a given digit;
    2. read, write, and represent whole numbers and decimals in expanded notation;
    3. develop understanding of equivalent number representations (i.e., fractions, decimals, percents);
    4. compare fractions, decimals, percents, and integers using the appropriate symbol (i.e., <, >, =);
    5. connect whole numbers, fractions, decimals, percents, and integers on the number line;
    6. develop meaning for percents greater than 100 and less than one;
    7. develop meaning for ratios using real-world models and/or situations;
    8. use a variety of representations for ratios (e.g., 3 to 5; 3/5; 3:5);
    9. develop and apply strategies to determine if two ratios form a proportion;
    10. develop the concept of prime and composite numbers;
    11. develop meaning for integers using real-world connections;
    12. use concrete, pictorial, and symbolic representations for integers;
    13. develop meaning for number theory concepts (i.e., divisibility, factors, multiples).

6.1.2 Understand operations and how they relate to one another.

    1. understand the meaning and effects of arithmetic operations on fractions and decimals;
    2. apply the associative and commutative properties of addition and multiplication to simplify computations with integers, fractions, and decimals;
    3. use the distributive property to simplify computations with integers, fractions, and decimals;
    4. apply order of operations when computing with whole numbers and decimals;
    5. understand and use the inverse relationships of addition and subtraction and multiplication and division to simplify computations and solve problems.

6.1.3 Solve problems, compute fluently and make reasonable estimates.

    1. select and use appropriate methods and tools for computing with whole numbers, fractions, decimals, and percents in problem-solving situations (e.g., mental computation, estimation, calculators, computers, paper and pencil);
    2. analyze procedures for computing with fractions, decimals, and integers;
    3. solve one-step real-world problems involving whole numbers, fractions, and decimals;
    4. use strategies to estimate the results of computations involving whole numbers, fractions, and decimals in real-world situations;
    5. judge the reasonableness of the results of rational number estimates and computations;
    6. recognize when an estimate is more appropriate than an exact answer in a variety of problem situations

Sixth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

6.1.spi.1. identify the place value of a given digit;

6.1.spi.2. solve one-step real-world problems involving whole numbers and decimals.

at Level 2, the student is able to

6.1.spi.3. represent numbers using a variety of models and equivalent forms (i.e., whole numbers, mixed numbers, fractions, decimals, and percents);

6.1.spi.4. connect whole numbers, mixed numbers, fractions, and decimals to locations on the number line;

6.1.spi.5. compare and order whole numbers, fractions, decimals, and percents using the appropriate symbol (<, >, =);

6.1.spi.6. identify prime and composite numbers up to 50;

6.1.spi.7. apply order of operations when computing with whole numbers (no parentheses or exponents);

6.1.spi.8. use estimation to select a reasonable solution to a computation involving whole numbers, fractions, and/or decimals;

6.1.spi.9. compute efficiently and accurately with whole numbers, fractions, and decimals .

at Level 3, the student is able to

6.1.spi.10. connect ratios to a variety of models, real-world situations, and symbolic representations;

6.1.spi.11. select a reasonable solution to a real-world division problem in which the remainder must be considered.

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

6.1.tpi.1. explain the inverse relationships of addition and subtraction and of multiplication and division.

at Level 2, the student is able to

6.1.tpi.2. read, write, and represent whole numbers and decimals in expanded notation;

6.1.tpi.3. use concrete, pictorial, and symbolic representations for integers;

6.1.tpi.4. develop meaning for number theory concepts (i.e.., divisibility, factors, multiples);

6.1.tpi.5. use strategies to estimate the results of computations involving whole numbers, fractions, and decimals in real-world situations

6.1.tpi.6. judge the reasonableness of the results of rational number estimates for computations;

6.1.tpi.7. recognize when an estimate is more appropriate than an exact answer in a variety of problem situations.

at Level 3, the student is able to

6.1.tpi.8. develop meaning for percents greater than 100 and percents less than 1;

6.1.tpi.9. develop and apply strategies to determine if two ratios form a proportion;

6.1.tpi.10. understand the meaning and effects of arithmetic operations on fractions and decimals;

6.1.tpi.11. analyze procedures for computing with fractions, decimals, and integers.

Algebra

Content Standard 2.0 The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

Learning Expectations:

2.1 Understand patterns, relations, and functions.

2.2 Represent and analyze mathematical situations and structures using algebraic symbols.

2.3 Use mathematical models to represent and understand quantitative relationships.

2.4 Analyze change in various contexts.

Accomplishments

6.2.1 Understand patterns, relations, and functions.

    1. represent, analyze, and extend geometric and numerical patterns;
    2. use tables and graphs to generalize patterns in data;
    3. apply function rules to complete tables.

6.2.2 Represent and analyze mathematical situations and structures using algebraic symbols.

    1. develop an initial conceptual understanding of different uses of variables;
    2. represent mathematical statements and real-world situations using symbols;
    3. evaluate algebraic expressions for a given value of the variable;
    4. find missing addends or factors represented as variables in simple equations;
    5. model algebraic expressions using manipulatives, technology, and pencil and paper.

6.2.3 Use mathematical models to represent and understand quantitative relationships.

    1. model simple real-world problems using graphs.

6.2.4 Analyze change in various contexts.

    1. describe how changes in one quantity or variable result in changes in another

Sixth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

6.2.spi.1. extend geometric and numerical patterns.

at Level 2, the student is able to

6.2.spi.2. generalize patterns in data represented in tables;

6.2.spi.3. apply function rules;

6.2.spi.4. find missing addends or factors represented as variables in simple equations;

6.2.spi.5. extend rate charts to solve real-world problems;

6.2.spi.6. select an equation that represents a given mathematical relationship.

at Level 3, the student is able to

6.2.spi.7. evaluate algebraic expressions for a given value of the variable.

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

6.2.tpi.1. use appropriate representations to show properties of whole number operations.

at Level 2, the student is able to

6.2.tpi.2. develop a conceptual understanding of different uses of variables;

6.2.tpi.3. model algebraic expressions using manipulatives, technology, and paper and pencil;

6.2.tpi.4. model simple real-world problems using a variety of representations, including graphs.

at Level 3, the student is able to

6.2.tpi.5. describe how changes in one quantity or variable result in changes in another.

Geometry

Content Standard 3.0 The student will develop an understanding of geometric concepts and relationships as the basis for geometric modeling and reasoning to solve problems involving one-, two-, and three-dimensional figures.

Learning Expectations:

3.1 Analyze characteristics and properties of two- and three-dimensional geometric figures.

3.2 Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

3.3 Apply transformations and use symmetry to analyze mathematical situations.

3.4 Use visualization, spatial reasoning, and geometric modeling to solve problems.

Accomplishments

6.3.1 Analyze characteristics and properties of two- and three-dimensional geometric figures.

    1. describe, classify, and understand relationships among types of two-dimensional figures;
    2. compare and classify angles as acute, obtuse, right, and straight;
    3. use appropriate mathematical language to describe characteristics of lines (e.g., parallel, perpendicular, intersecting);
    4. compare and classify quadrilaterals using their defining properties;
    5. describe similarity and congruence.

6.3.2 Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

    1. plot a given set of points in Quadrant I of a coordinate system.

 6.3.3 Apply transformations and use symmetry to analyze mathematical situations.

    1. investigate, predict, and describe the results of transformations of two-dimensional figures (e.g., slides, flips, turns);
    2. describe line and rotational symmetry in two-dimensional figures;
    3. describe a motion or a series of motions that will show that two shapes are congruent.

6.3.4 Use visualization, spatial reasoning, and geometric modeling to solve problems.

    1. draw two-and three-dimensional geometric figures with specified properties, (e.g., side lengths, angle measure);
    2. build a three-dimensional object from a two-dimensional representation (net) of that object and vice versa;
    3. use visualization and spatial reasoning to solve real-world problems.

Sixth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

6.3.spi.1. identify parallel, perpendicular, and intersecting lines;

6.3.spi.2. use ordered pairs to describe given points in Quadrant I of a coordinate system.

at Level 2, the student is able to

6.3.spi.3. classify two-dimensional geometric figures using properties;

6.3.spi.4. identify the results of transformations of two-dimensional figures (i.e.., rotations/turns, flips/reflections, slides/translations);

6.3.spi.5. use spatial reasoning to identify the three-dimensional figure created from a two-dimensional representation (net) of that figure (i.e., cube, rectangular prism, pyramid, cone, or cylinder);

6.3.spi.6. classify angles as acute, obtuse, right, or straight.

at Level 3, the student is able to

6.3.spi.7. classify quadrilaterals using their defining properties.

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

6.3.tpi.1. use geometric models to illustrate and explore number concepts and properties.

at Level 2, the student is able to

6.3.tpi.2. describe line and rotational symmetry in two-dimensional figures;

6.3.tpi.3. build a three-dimensional object from a two-dimensional representation (net) of that object and vice versa.

at Level 3, the student is able to

6.3.tpi.4. describe a motion or a series of motions that will show that two shapes are congruent;

6.3.tpi.5. draw or build two-and three-dimensional geometric figures with specified properties;

6.3.tpi.6. use visualization and spatial reasoning to solve real-world problems.

Measurement

Content Standard 4.0 The student will become familiar with the units and processes of measurement in order to use a variety of tools, techniques, and formulas to determine and to estimate measurements in mathematical and real-world problems.

Learning Expectations:

4.1 Understand measurable attributes of objects and the units, systems, and processes of measurement.

4.2 Apply appropriate techniques, tools, and formulas to determine measurements.

Accomplishments

6.4.1 Understand measurable attributes of objects and the units, systems, and processes of measurement.

    1. understand both metric and customary systems of measurement;
    2. understand relationships among units and convert from one unit to another within the same system;
    3. understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

6.4.2 Apply appropriate techniques, tools, and formulas to determine measurements.

    1. use a variety of strategies to estimate length, perimeter, circumference, area, and volume;
    2. use a variety of manipulatives to develop formulas to determine the circumference of circles;
    3. use formulas to determine the area of triangles and parallelograms;
    4. use a variety of manipulatives to develop formulas to determine the area of trapezoids and circles;
    5. explore surface area and volume of selected prisms and cylinders using models and manipulatives;
    6. solve problems involving measurement using ratio and proportion;
    7. use scales to read maps;
    8. recognize the need for measurement precision

Sixth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

6.4.spi.1. use strategies to estimate perimeter and area of rectangles ;

6.4.spi.2. solve real-world problems involving elapsed time.

Level 2, the student is able to

6.4.spi.3. convert from one unit to another within the same system

6.4.spi.4. select units of appropriate size and type to measure angles, perimeter, area, capacity, volume, and weight;

6.4.spi.5. apply formulas to determine the area of rectangles and triangles;

6.4.spi.6. use scales to read maps;

6.4.tpi.7. solve real-world problems involving perimeter and area of rectangles;

6.4.spi.8. determine the distance between two points on the x- or the y-axis in Quadrant I.

at Level 3, the student is able to

6.4.spi.9. solve problems involving ratios and proportions.

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

6.4.tpi.1. describe the need for precision in measurement.

at Level 2, the student is able to

6.4.tpi.2. use a variety of manipulatives to develop the formula for the circumference of a circle;

6.4.tpi.3. understand and use both metric and customary systems of measurement.

at Level 3, the student is able to

6.4.tpi.4. use a variety of strategies to estimate length, perimeter, circumference, area, and volume;

6.4.tpi.5. use a variety of manipulatives to develop formulas for area of circles and trapezoids;

6.4.tpi.6. use models and manipulatives to explore the volume of selected prisms and cylinders.

Data Analysis & Probability

Content Standard 5.0 The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

Learning Expectations:

5.1 Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer questions.

5.2 Select and use appropriate statistical methods to analyze data.

5.3 Develop and evaluate inferences and predictions that are based on data.

5.4 Understand and apply basic concepts of probability.

Accomplishments

6.5.1 Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer questions.

    1. formulate questions, design studies, and collect real-world data;
    2. understand how data-collection methods affect the nature of the data set;
    3. examine various representations of data to evaluate how accurately the data is depicted;
    4. construct, interpret, and use single-bar and single-line graphs to answer questions and solve real-world problems.

6.5.2 Select and use appropriate statistical methods to analyze data.

    1. find, use, and interpret measures of center and spread (e.g., mean, median, mode, interquartile range);
    2. discuss and understand the relationship between data sets and their graphical representations.

6.5.3 Develop and evaluate inferences and predictions that are based on data.

    1. make conjectures and predictions based on data;
    2. explain the importance of sample size in investigations;
    3. conduct a survey using random sampling;
    4. determine whether or not a sample is biased;
    5. make conjectures to formulate new questions for future studies.

6.5.4 Understand and apply basic concepts of probability.

    1. model situations by devising and carrying out experiments and simulations;
    2. make and test conjectures about the results of experiments and simulations;
    3. determine all possible outcomes of a simple event

Sixth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

6.5.spi.1. determine the mode of a data set.

at Level 2, the student is able to

6.5.spi.2. interpret bar and line graphs to answer questions and solve real-world problems;

6.5.spi.3. determine the mean of a data set;

6.5.spi.4. determine the median from a stem-and-leaf-plot;

6.5.spi.5. determine if a sample is biased;

6.5.spi.6. represent the likelihood of an event using a number from 0 – 1.

at Level 3, the student is able to

6.5.spi.7. connect data sets and their graphical representations (i.e., bar graphs, circle, graphs, and stem-and-leaf plots);

6.5.spi.8. make conjectures and predictions based on data;

6.5.spi.9. use a tree diagram or organized list to determine all possible outcomes of a simple compound event.

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

6.5.tpi.1. conduct a survey using random sampling.

at Level 2, the student is able to

6.5.tpi.2. find and interpret measures of center and spread (mean, median, mode, interquartile range);

6.5.tpi.3. model situations by devising and carrying out experiments and simulations;

6.5.tpi.4. determine an appropriate sample to test a hypothesis;

6.6.tpi.5. conduct simple experiments to compare theoretical and experimental probabilities.

at Level 3, the student is able to

6.5.tpi.6. explain the importance of sample size in surveys and research;

6.5.tpi.7. make conjectures from data to formulate new questions for further investigation.