## Mathematics

### Domain - Ratios and Proportional Relationships, Grade 6

Understand ratio concepts and use ratio reasoning to solve problems.

Math.6.RP.A.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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Understand ratio concepts and use ratio reasoning to solve problems.

Math.6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

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Understand ratio concepts and use ratio reasoning to solve problems.

Math.6.RP.A.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.”

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### Domain - The Number System, Grade 6

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Math.6.NS.A.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

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Compute fluently with multi-digit numbers and find common factors and multiples.

Math.6.NS.B.4: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2)..

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Compute fluently with multi-digit numbers and find common factors and multiples.

Math.6.NS.B.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

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Compute fluently with multi-digit numbers and find common factors and multiples.

Math.6.NS.B.2: Fluently divide multi-digit numbers using the standard algorithm.

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Apply and extend previous understandings of numbers to the system of rational numbers.

Math.6.NS.C.7: Understand ordering and absolute value of rational numbers.

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Apply and extend previous understandings of numbers to the system of rational numbers.

Math.6.NS.C.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

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Apply and extend previous understandings of numbers to the system of rational numbers.

Math.6.NS.C.6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

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Apply and extend previous understandings of numbers to the system of rational numbers.

Math.6.NS.C.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

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### Domain - Expressions and Equations, Grade 6

Apply and extend previous understandings of arithmetic to algebraic expressions.

Math.6.EE.A.1: Write and evaluate numerical expressions involving whole-number exponents.

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Apply and extend previous understandings of arithmetic to algebraic expressions.

Math.6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

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Apply and extend previous understandings of arithmetic to algebraic expressions.

Math.6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for..

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Apply and extend previous understandings of arithmetic to algebraic expressions.

Math.6.EE.A.2: Write, read, and evaluate expressions in which letters stand for numbers.

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Reason about and solve one-variable equations and inequalities.

Math.6.EE.B.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

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Reason about and solve one-variable equations and inequalities.

Math.6.EE.B.6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

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Reason about and solve one-variable equations and inequalities.

Math.6.EE.B.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

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Reason about and solve one-variable equations and inequalities.

Math.6.EE.B.5: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

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Represent and analyze quantitative relationships between dependent and independent variables.

Math.6.EE.C.9: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

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### Domain - Geometry, Grade 6

Solve real-world and mathematical problems involving area, surface area, and volume.

Math.6.G.A.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these technique in the context of solving real-world and mathematical problems.

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Solve real-world and mathematical problems involving area, surface area, and volume.

Math.6.G.A.2: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

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Solve real-world and mathematical problems involving area, surface area, and volume.

Math.6.G.A.4: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

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### Domain - Statistics and Probability, Grade 6

Develop understanding of statistical variability.

Math.6.SP.A.3: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

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Develop understanding of statistical variability.

Math.6.SP.A.2: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

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Develop understanding of statistical variability.

Math.6.SP.A.1: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

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Summarize and describe distributions.

Math.6.SP.B.5: Summarize numerical data sets in relation to their context, such as by:

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Summarize and describe distributions.

Math.6.SP.B.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

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