# MATHEMATICS Grade 6 Standard: Number, Number Sense and Operations

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1 Standard: Number, Number Sense and Operations Number and Number C. Develop meaning for percents including percents greater than 1. Describe what it means to find a specific percent of a number, Systems 100 and less than 1. using real-life examples. Number and Number Systems Meaning of Operations Number and Number Systems D. Use models and pictures to relate concepts of ratio, proportion and percent. E. Use order of operations, including use of parenthesis and exponents to solve multi-step problems and verify and interpret the results. G. Apply and explain the use of prime factorizations, common factors, and common multiples in problem situations. 2. Construct and use models and pictures to relate concepts of ratio, proportion and percent, including percents less than 1 and greater than Explain why a number is referred to as being rational and recognize that the expression can mean a parts of size each, a divided by b, or the ratio of a two b. 2. Create and use models and pictures to relate concepts of ratio, proportion and percent, including percents less than 1 and greater than Give examples of how ratios are used to represent comparisons (e.g., part-to-part, part-to-whole, whole-to-part). 1. Use the order of operations, including the use of exponents, decimals and rational numbers, to simplify numerical expressions. 1. Decompose and recompose whole numbers using factors and exponents (e.g., 32 = 2 x 2 x 2 x 2 x 2 =25) and explain why squared means second power and cubed means third power.

2 Standard: Number, Number Sense and Operations Number and Number Systems G. Apply and explain the use of prime factorizations, common factors, and common multiples in problem situations. 2. Find and use the prime factorization of composite numbers. For example: Use the prime factorization to recognize the greatest common factor (GCF) Use the prime factorization to recognize the least common multiple (LCM) Apply the prime factorization to solve problems and Meaning of Operations H. Use and analyze the steps in standards and non-standard algorithms for computing with fractions, decimals and integers. explain solutions 1. Represent multiplication and division situations involving fractions and decimals with models and visual representations (e.g., show with pattern blocks what it means to take 2 2/3 1/6. Meaning of Operations Computation and Estimation I. Use a variety of strategies, including proportional reasoning to estimate, compute, solve and explain solutions to problems involving integers, fractions, decimals and percents. 2. Develop and analyze algorithms for multiplying and dividing with fractions and decimals and demonstrate fluency in their use. 1. Use simple expressions involving integers to represent and solve problems (e.g., if a running back loses 15 yards on the first carry but gains 8 yards on the second carry, what is the net gain/loss?). 2. Perform fraction and decimal computations and justify their solutions (e.g., using manipulatives, diagrams and mathematical reasoning. 3. Estimate reasonable solutions to problem situations involving fractions and decimals. 4. Use proportional reasoning, ratios and percents to represent problem situations and determine the reasonableness of solutions.

3 Standard: Number, Number Sense and Operations Computation and Estimation I. Use a variety of strategies, including proportional reasoning to estimate, compute, solve and explain solutions to problems involving integers, fractions, decimals and percents. 5. Determine the percent of a number and solve related problems (e.g., find the percent markdown if the original prices was \$140, and the sale price is \$100). 6. Recognize that a quotient may be larger than the dividend when the divisor is a fraction (e.g., 6 ½ = 12).

4 Standard: Measurement Use Measurement Techniques and Tools Use Measurement Techniques and Tools Use Measurement Techniques and Tools Measurement Units Use Measurement Techniques and Tools C. Identify appropriate tools and apply appropriate techniques for measuring angles, perimeter or circumference and area of triangles, quadrilaterals, circles, and composite shapes, and surface area and volume of prisms and cylinders. E. Use problem solving techniques and technology as needed to solve problems involving length, weight, perimeter, area, volume, time and temperature. F. Analyze and explain what happens to area and perimeter or surface area and volume when the dimensions of an object are changed. G. Understand and demonstrate the independence of perimeter and area for two-dimensional shapes and of surface area and volume for three-dimensional shapes. 1. Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of sectors (e.g., 1/2 circle, 2/3 circle, 1/3 circle, ¼ circle). 2. Estimate perimeter or circumference and area for circles, triangles and quadrilaterals, and surface area and volume for prisms and cylinders by: Estimating lengths using string or links, areas using tiles or grid, and volumes using cubes Measuring attributes (diameter, side lengths, or heights) and using established formulas for circles, triangles, rectangles, parallelograms and rectangular prisms 1. Determine which measure (perimeter, area, surface area and volume) matches the context for a problem situation (e.g., perimeter is the context for fencing a garden, surface are is the context for painting a room). 1. Describe what happens to the perimeter and area of a twodimensional shape when the measurements of the shape are changes (e.g., length of sides are doubles). 1. Understand and describe the difference between surface area and volume. 2. Assess the difference between perimeter and area, and demonstrate that two shapes may have the same perimeter, but different areas or may have the same area, but different perimeters.

5 Standard: Measurement Use Measurement Techniques and Tools A. Select appropriate units to measure angles, circumference, surface area, mass and volume using: U.S. customary units (e.g., degrees, square feet, pounds, and other units as appropriate) Metric units (e.g., square meters, kilograms, and other units as appropriate 1. Determine which measure (perimeter, area, surface area and volume) matches the context for a problem situation (e.g., perimeter is the context for fencing a garden, surface are is the context for painting a room).

6 Standard: Geometry and Spatial Sense Characteristics and Properties D. Identify, describe and classify types of line pairs, angles, twodimensional figures and three-dimensional objects using their properties. 1. Classify and describe two-dimensional and three-dimensional geometric figures and objects by using their properties (e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides). Read, write and apply. 2. Identify standard language to define geometric vocabulary: vertex, face, altitude, diagonal, isosceles, equilateral, acute, obtuse, supplementary angles, complementary angles, vertical angles and other vocabulary as appropriate. 3. Use multiple classification criteria to classify triangles (e.g., right scalene triangle). Transformations and Symmetry Characteristics and Properties Spatial Relationships E. Use proportions to express relationships among corresponding parts of similar figures. F. Describe and use the concepts of congruence, similarity and symmetry to solve problems. J. Apply properties of equality and proportionality to solve problems involving congruent or similar figures (e.g., create a scale drawing). G. Describe and use properties of triangles to solve problems involving angle measures and side lengths of right triangles. H. Predict and describe results (size, position, orientation) of transformations of two-dimensional figures. 4. Identify and define relationships between planes (e.g., parallel, perpendicular and intersecting). 1. Draw similar figures that model proportional relationships (e.g., model similar figures with a 1 to 2 relationship by sketching two of the same figure, one with corresponding sides twice the length of the other. 1. Use multiple classification criteria to classify triangles (e.g., right scalene triangle). 1. Predict and describe sizes, positions and orientations of twodimensional shapes after transformations such as reflections, rotations, translations and dilations. Demonstrate these concepts on the coordinate plane.

7 Standard: Geometry and Spatial Sense Visualization and Geometric Models I. Identify and draw three-dimensional objects from different views (top, side, front and perspective). 1. Build three-dimensional objects built with cubes and sketch the two-dimensional representations of each side (e.g., projection sets).

8 Standard: Patterns, Functions and Algebra Use Patterns, Relations and Functions A. Describe extend and determine the rule for patterns and relationships occurring in numeric patterns, computation, geometry, graphs and other applications. B. Represent, analyze and generalize a variety of patterns and functions with tables, graphs, words and symbolic rules. 1. Represent and analyze patterns, rules and functions, using physical materials, tables and graphs. 2. Use words and symbols to describe numerical and geometric patterns, rules and functions. E. Use rules and variables to describe patterns, functions and other relationships. C. Use variables to create and solve equations and inequalities representing problem situations. D. Use symbolic algebra to represent and explain mathematical relationships. F. Use representations, such as tables, graphs and equations, to model situations and to solve problems, especially those that involve linear relationships. 1. Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations. 1. Recognize and generate equivalent forms of algebraic expressions, and explain how the commutative, associative and distributive properties can be used to generate equivalent forms; (e.g., perimeter as 2(1 + w) or w. 1. Produce and interpret graphs that represent the relationship between two variables. G. Write, simplify and evaluate algebraic expressions. 1. Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations. H. Solve linear equations and inequalities symbolically, graphically 1. Solve simple linear equations and inequalities using physical and numerically. models, paper and pencil, tables and graphs.

9 Standard: Patterns, Functions and Algebra J. Use formulas in problem-solving situations. 1. Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of sectors. 2. Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations. K. Graph linear equations and inequalities. 1. Solve simple linear equations and inequalities using physical models, paper and pencil, tables and graphs. Analyze Change Analyze Change L. Analyze functional relationships and explain how a change in one quantity results in a change in the other. M. Approximate and interpret rates of change from graphical and numerical data. 2. Produce and interpret graphs that represent the relationship between two variables. 1. Identify and describe situations with constant or varying rates of change and compare them. 1. Use technology to analyze change; e.g., use computer applications or graphing calculators to display and interpret rate of change.

10 Standard: Data Analysis and Probability Data Collection A. Read, create and use line graphs, histograms, circle graphs, boxand-whisker plots, stem-and-leaf plots and other representations when appropriate. Statistical Methods Data Collection Data Collection Statistical Methods Statistical Methods Probability B. Interpret data by looking for patterns and relationships, draw and justify conclusions and answer related questions. D. Compare increasingly complex displays of data, such as multiple sets of data on the same graph. E. Collect, organize, display and interpret data for a specific purpose or need. F. Determine and use the range, mean, median and mode to analyze and compare data and explain what each indicates about the data. G. Evaluate conjectures and predictions based upon data presented in tables and graphs, and identify misuses of statistical data and display. K. Make and justify predictions based on experimental and theoretical probabilities. 1. Read, construct and interpret line graphs, circle graphs, histograms and stem-and-leaf plot. 1. Describe the frequency distribution of a set of data, as shown in a histogram or frequency table, by general appearance or shape; e.g., number of modes, middle of data and level symmetry, outliers. 1. Compare representations of the same data in different types of graphs, such as a bar graph and circle graph. 1. Select, create and use graphical representations that are appropriate for the type of data collected. 1. Analyze the different information provided by measures of center (mean, mode and median) and measures of spread (range). 1. Make logical inferences from statistical data. 1. Design an experiment to test a theoretical probability and explain how the results may vary.

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