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### 6.1 Number Sense, Properties, and Operations

- 6.1.1 Quantities can be expressed and compared using ratios and rates
- 6.1.1.a Apply the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
- 6.1.1.b Apply 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.
- 6.1.1.c Use ratio and rate reasoning to solve real-world and mathematical problems.
- 6.1.1.c.i. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane.
- 6.1.1.c.ii. Use tables to compare ratios.
- 6.1.1.c.iii. Solve unit rate problems including those involving unit pricing and constant speed.
- 6.1.1.c.iv. Find a percent of a quantity as a rate per 100.
- 6.1.1.c.v. Solve problems involving finding the whole, given a part and the percent.
- 6.1.1.c.vi. Use common fractions and percents to calculate parts of whole numbers in problem situations including comparisons of savings rates at different financial institutions.
- 6.1.1.c.vii. Express the comparison of two whole number quantities using differences, part-to-part ratios, and part-to-whole ratios in real contexts, including investing and saving.
- 6.1.1.c.viii. Use ratio reasoning to convert measurement units.

- 6.1.2 Formulate, represent, and use algorithms with positive rational numbers with flexibility, accuracy, and efficiency
- 6.1.2.a Fluently divide multi-digit numbers using the standard algorithm.
- 6.1.2.b Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
- 6.1.2.c Find the greatest common factor of two whole numbers less than or equal to 100.
- 6.1.2.d Find the least common multiple of two whole numbers less than or equal to 12.
- 6.1.2.e 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.
- 6.1.2.f Interpret and model quotients of fractions through the creation of story contexts.
- 6.1.2.g Compute quotients of fractions.
- 6.1.2.h Solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

- 6.1.3 In the real number system, rational numbers have a unique location on the number line and in space
- 6.1.3.a Explain why positive and negative numbers are used together to describe quantities having opposite directions or values.
- 6.1.3.a.i. Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

- 6.1.3.b Use number line diagrams and coordinate axes to represent points on the line and in the plane with negative number coordinates.
- 6.1.3.b.i. Describe a rational number as a point on the number line.
- 6.1.3.b.ii. Use opposite signs of numbers to indicate locations on opposite sides of 0 on the number line.
- 6.1.3.b.iii. Identify that the opposite of the opposite of a number is the number itself.
- 6.1.3.b.iv. Explain when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
- 6.1.3.b.v. Find and position integers and other rational numbers on a horizontal or vertical number line diagram.
- 6.1.3.b.vi. Find and position pairs of integers and other rational numbers on a coordinate plane.

- 6.1.3.c Order and find absolute value of rational numbers.
- 6.1.3.c.i. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
- 6.1.3.c.ii. Write, interpret, and explain statements of order for rational numbers in real-world contexts.
- 6.1.3.c.iii. Define the absolute value of a rational number as its distance from 0 on the number line and interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
- 6.1.3.c.iv. Distinguish comparisons of absolute value from statements about order.

- 6.1.3.d 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.

- 6.1.3.a Explain why positive and negative numbers are used together to describe quantities having opposite directions or values.

### 6.2 Patterns, Functions, and Algebraic Structures

- 6.2.1 Algebraic expressions can be used to generalize properties of arithmetic
- 6.2.1.a Write and evaluate numerical expressions involving whole-number exponents.
- 6.2.1.b Write, read, and evaluate expressions in which letters stand for numbers.
- 6.2.1.b.i. Write expressions that record operations with numbers and with letters standing for numbers.
- 6.2.1.b.ii. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient) and describe one or more parts of an expression as a single entity.
- 6.2.1.b.iii. Evaluate expressions at specific values of their variables including expressions that arise from formulas used in real-world problems.
- 6.2.1.b.iv. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

- 6.2.1.c Apply the properties of operations to generate equivalent expressions.
- 6.2.1.d Identify when two expressions are equivalent.

- 6.2.2 Variables are used to represent unknown quantities within equations and inequalities
- 6.2.2.a Describe 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?
- 6.2.2.b Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
- 6.2.2.c Use variables to represent numbers and write expressions when solving a real-world or mathematical problem.
- 6.2.2.c.i. Recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

- 6.2.2.d 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.
- 6.2.2.e Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem.
- 6.2.2.f Show that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
- 6.2.2.g Represent and analyze quantitative relationships between dependent and independent variables.
- 6.2.2.g.i. Use variables to represent two quantities in a real-world problem that change in relationship to one another.
- 6.2.2.g.ii. 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.
- 6.2.2.g.iii. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

### 6.3 Data Analysis, Statistics, and Probability

- 6.3.1 Visual displays and summary statistics of one-variable data condense the information in data sets into usable knowledge
- 6.3.1.a Identify a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
- 6.3.1.b Demonstrate 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.
- 6.3.1.c Explain 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.
- 6.3.1.d Summarize and describe distributions.
- 6.3.1.d.i. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
- 6.3.1.d.ii. Summarize numerical data sets in relation to their context.
- 6.3.1.d.ii.1 Reporting the number of observations.
- 6.3.1.d.ii.2 Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
- 6.3.1.d.ii.3 Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- 6.3.1.d.ii.4 Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

### 6.4 Shape, Dimension, and Geometric Relationships

- 6.4.1 Objects in space and their parts and attributes can be measured and analyzed
- 6.4.1.a Develop and apply formulas and procedures for area of plane figures.
- 6.4.1.a.i. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
- 6.4.1.a.ii. Apply these techniques in the context of solving real-world and mathematical problems.

- 6.4.1.b Develop and apply formulas and procedures for volume of regular prisms.
- 6.4.1.b.i. 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.
- 6.4.1.b.ii. Show that volume is the same as multiplying the edge lengths of a rectangular prism.
- 6.4.1.b.iii. 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.

- 6.4.1.c Draw polygons in the coordinate plan to solve real-world and mathematical problems.
- 6.4.1.c.i. Draw polygons in the coordinate plane given coordinates for the vertices.
- 6.4.1.c.ii. Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.

- 6.4.1.d Develop and apply formulas and procedures for the surface area.
- 6.4.1.d.i. Represent three-dimensional figures using nets made up of rectangles and triangles.
- 6.4.1.d.ii. Use nets to find the surface area of figures.
- 6.4.1.d.iii. Apply techniques for finding surface area in the context of solving real-world and mathematical problems.

- 6.4.1.a Develop and apply formulas and procedures for area of plane figures.