# data[].attributes.body
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

# data[].attributes.body
Apply the formulas for the volumes and surface areas of cones, cylinders and spheres and use them to solve real-world and mathematical problems.

# data[].attributes.body
Use similar triangles to explain why the slope, m, is the same between any two distinct points on a non-vertical line in the coordinate plane; know the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

# data[].attributes.body
Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms.

# data[].attributes.body
Use numbers expressed in the form of a single digit times an integer power of 10 (Scientific Notation) to estimate very large or very small quantities and express how many times larger or smaller one is than the other.

# data[].attributes.body
Analyze and solve a system of two linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously; understand that a system of two linear equation may have one solution, no solution, or infinitely many solutions.
b. Solve systems of two linear equations in two variables algebraically by using substitution where at least one equation contains at least one variable whose coefficient is 1 and by inspection for simple cases.
c. Solve real-world and mathematical problems leading to two linear equations in two variables.

# data[].attributes.body
Use graphs to represent functions.
a. Describe qualitatively the functional relationship between two quantities by analyzing a graph.
b. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

# data[].attributes.body
Use variables to represent two quantities in a real-world problem that change in relationship to one another;
a. Appropriately recognize one quantity as the dependent variable and the other as the independent variable.
b. 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.
c. Analyze the relationship between the dependent and independent variables using graphs and tables and relate these to the equation.

# data[].attributes.body
Solve real world problems involving multiplication of fractions and mixed numbers.

# data[].attributes.body
Identify and gather data for statistical questions focused on both categorical and numerical data. Select an appropriate data display (bar graph, pictograph, dot plot). Make observations from the graph about the questions posed.

# data[].attributes.body
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.

# data[].attributes.body
Use percents to solve mathematical and real-world problems.
a. Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, a part, and a percent, given two of these.
b. Use proportional relationships to solve multistep ratio and percent problems.

# data[].attributes.body
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number and compute such quotients.
b. Interpret division of a whole number by a unit fraction and compute such quotients.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

# data[].attributes.body
Solve word problems involving addition and subtraction of fractions.
a. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators.
b. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

# data[].attributes.body
Fluently add, subtract, multiply, and divide multi-digit decimals using an algorithm for each operation.

# data[].attributes.body
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

# data[].attributes.body
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes, using appropriate range and intervals, to represent points on the line and in the plane that include negative numbers and coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize 0 is its own opposite and the opposite of a negative number is a positive, and the opposite of a negative number is the number itself, such as –(–3) = 3.
b. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
c. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize the similarity between whole numbers, their negative opposites and their positions on a number line, ordered pairs differ only by signs and their locations on one or both axes.

# data[].attributes.body
Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions.

# data[].attributes.body
Find the volume of a right rectangular prism with rational number edge lengths. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with rational number edge lengths in the context of solving real-world and mathematical problems.

# data[].attributes.body
Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that perfect squares and perfect cubes are rational.

# data[].attributes.body
Fluently divide multi-digit numbers using an algorithm.
a. Convert a rational number to a decimal using long division.
b. Know that the decimal form of a rational number terminates in 0s or eventually repeats.

# data[].attributes.body
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.

# data[].attributes.body
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

# data[].attributes.body
Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
d. Distinguish comparisons of absolute value from statements about order.

# data[].attributes.body
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b =(n×a)/(n×b) to the effect of multiplying a/b by 1.

# data[].attributes.body
Fluently multiply multi-digit whole numbers (not to exceed four-digit by two-digit multiplication) using an algorithm.

# data[].attributes.body
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.

# data[].attributes.body
Efficiently add and subtract fractions with unlike denominators(including mixed numbers) by…
● using reasoning strategies, such as counting up on a number line or creating visual fraction models
● finding common denominators

# data[].attributes.body
Divide up to four-digit dividends by two-digit divisors.
a. Find whole-number quotients of whole numbers with up to four dividends and two-digit divisors using…
● strategies based on place value
● the properties of operations
● the relationship between multiplication and division
b. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

# data[].attributes.body
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
b. Find the area of a rectangle with fractional side lengths by tiling it with squares of the appropriate unit fraction side lengths and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangle sand represent fraction products as rectangular areas.

# data[].attributes.body
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.

# data[].attributes.body
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.

# data[].attributes.body
Solve real-world and mathematical problems involving the four operations with rational numbers.

# data[].attributes.body
Understand the concept of a unit rate a/b associated with a ratio a:bwith b ≠ 0 and use rate language in the context of a ratio relationship.

# data[].attributes.body
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then – (p/q) = (–p)/q = p/ (–q). Interpret quotients of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.

# data[].attributes.body
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

# data[].attributes.body
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

# data[].attributes.body
Understand that rewriting an expression in different forms in a problem context can clarify the problem and how the quantities in it are related.

# data[].attributes.body
Understand properties of linear functions.
a. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.
b. Identify and give examples of functions that are not linear.

# data[].attributes.body
Use variables to represent quantities in a real-world or mathematical problem and construct equations and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Graph the solution set of the equality and interpret it in the context of the problem.
b. Solve word problems leading to inequalities of the form px+q > r, px+ q < r, px + q ≥ r, px+q ≤ r ; where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in context of the problem.

# data[].attributes.body
Write, read and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers.
b. Identify parts of an expression using mathematical terms (sums, term, product, factor, quotient, coefficient); view one or more parts of an expression in a single entity.
c. Evaluate expressions for specific values of their variables, including values that are non-negative rational numbers. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

# data[].attributes.body
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.

# data[].attributes.body
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions that include symbols.

# data[].attributes.body
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

# data[].attributes.body
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.

# data[].attributes.body
Apply the properties of operations to generate equivalent expressions.

# data[].attributes.body
Solve real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.

# data[].attributes.body
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

# data[].attributes.body
Write simple expressions with numbers and interpret numerical expressions without evaluating them.

# data[].attributes.body
Generate numerical patterns for situations.
a. Generate a rule for growing patterns, identifying the relationship between corresponding terms (x, y).
b. Generate patterns using one or two given rules (x, y).
c. Use tables, ordered pairs, and graphs to represent the relationship between the quantities.

# data[].attributes.body
Write and evaluate numerical expressions involving whole-number exponents.

# data[].attributes.body
Know and apply the properties of integer exponents to generate equivalent numerical expressions.

# data[].attributes.body
Identify when two expressions are equivalent, when the two expressions name the same number regardless of which value is substituted into them.

# data[].attributes.body
Write an inequality of the form x > c, x < c, x ≥ c, or x ≤ c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of these forms have infinitely many solutions; represent solutions of such inequalities on vertical and horizontal number lines.

# data[].attributes.body
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.

# data[].attributes.body
Draw (freehand, with ruler and protractor and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

# data[].attributes.body
Find the area of right triangles, other triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and quadrilaterals; apply these techniques in the context of solving real-world and mathematical problems.

# data[].attributes.body
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations. Given two congruent figures, describe a sequence that exhibits the congruence between them.

# data[].attributes.body
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

# data[].attributes.body
Solve problems involving area of two-dimensional objects and surface area and volume of three-dimensional objects.
a. Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles, quadrilaterals and other polygons.
b. Solve real-world and mathematical problems involving volume and surface area, using nets as needed, of three-dimensional objects including cubes, pyramids and right prisms.

# data[].attributes.body
Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category.

# data[].attributes.body
Apply properties of supplementary, complementary, vertical and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

# data[].attributes.body
Use formulas for area and circumference of circles and their relationships.
a. Apply the formulas for the area and circumference of a circle to solve real-world and mathematical problems.
b. Explore and understand the relationship between the radius, diameter, circumference and area of a circle.

# data[].attributes.body
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation.

# data[].attributes.body
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations and dilations. Given two similar two dimensional figures, describe a sequence that exhibits the similarity between them.

# data[].attributes.body
Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

# data[].attributes.body
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal and the angle-angle criterion for similarity of triangles.

# data[].attributes.body
Classify two-dimensional figures in a hierarchy based on properties.

# data[].attributes.body
Verify experimentally the properties of rotations, reflections and translations:
● Lines are congruent to lines.
● Line segments are congruent to line segments of the same length.
● Angles are congruent to angles of the same measure.
● Parallel lines are congruent to parallel lines.

# data[].attributes.body
Use a pair perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin)arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis and the second number indicates how far to travel in the direction of the second.

# data[].attributes.body
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

# data[].attributes.body
Measure volumes by counting unit cubic cm, cubic in, cubic ft. and improvised units.

# data[].attributes.body
Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
c. Determining quantitative measures of center (median and/or mean) to describe distribution of numerical data.
d. Describing distributions of numerical data qualitatively relating to shape (using terms such as cluster, mode(s), gap, symmetric, uniform, skewed-left, skewed-right and the presence of outliers)and quantitatively relating to spread/variability (using terms such as range and interquartile range).
e. Relating the choice of measures of center and variability to the shape of the data distribution.

# data[].attributes.body
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes and use the model to determine probabilities of events.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

# data[].attributes.body
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number to describe a typical value, while a measure of variation describes how the values in the distribution vary.

# data[].attributes.body
Apply the four-step investigative process for statistical reasoning.
a. Formulate Questions: Formulate a statistical question as one that anticipates variability and can be answered with data.
b. Collect Data: Design and use a plan to collect appropriate data to answer a statistical question.
c. Analyze Data: Select appropriate graphical methods and numerical measures to analyze data by displaying variability within a group, comparing individual to individual, and comparing individual to group.
d. Interpret Results: Draw logical conclusions and make generalizations from the data based on the original question.

# data[].attributes.body
Create displays, including circle graphs (pie charts), scaled pictographs and bar graphs, to compare and analyze distributions of categorical data from both matching and different-sized samples.

# data[].attributes.body
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

# data[].attributes.body
Understand that a set of numerical data collected to answer a statistical question has a distribution which can be described by its center, spread and overall shape.

# data[].attributes.body
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

# data[].attributes.body
Know that lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a line and informally assess the model fit by judging the closeness of the data points to the line.

# data[].attributes.body
Calculate and use measures of center (mean and median)and measures of variability (interquartile range when comparing medians and mean absolute deviation when comparing means) for numerical data from random samples to draw informal comparative inferences about two populations.

# data[].attributes.body
Describe the degree of visual overlap (and separation) from the graphical representations of two numerical data distributions (boxplots, dot plots) with similar variabilities with similar contexts (same variable), measuring the difference between the centers (medians or means) by expressing this difference as a multiple of a measure of variability (interquartile range when comparing medians or the mean absolute deviation when comparing means).

# data[].attributes.body
Construct and interpret scatter plots for bivariate numerical data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association and nonlinear association.

# data[].attributes.body
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency and predict the approximate relative frequency given the probability.

# data[].attributes.body
Convert among different size measurement units (mass, weight, liquid volume, length, time) within one system of units (metric system, U.S. standard system and time).

# data[].attributes.body
Use the equation of a linear model to solve problems in the context of bivariate numerical data, interpreting the slope and intercept.

# data[].attributes.body
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

# data[].attributes.body
Find probabilities of compound events using organized lists, tables, tree diagrams and simulation.
a. Explain just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events described in everyday language using methods such as organized lists, tables and tree diagrams.
c. Design and use a simulation to generate frequencies for compound events.

# data[].attributes.body
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes.
b. Apply the formulas V= l x w x h and V = B x h for rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

# data[].attributes.body
Display the distribution of numerical data in plots on a number line, including dot plots, histograms and box plots.

# data[].attributes.body
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
a. Generate multiple samples of categorical data of the same size to gauge the variation in estimates or predictions.
b. Generate multiple samples (or simulated samples) of numerical data to gauge the variation in estimates or predictions.
c. Gauge how far off an estimate or prediction might be related to a population character of interest.

# data[].attributes.body
Describe the probability of a chance event is a number between 0 and 1, which tells how likely the event is, from impossible(0) to certain (1). A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely and a probability near 1 indicates a likely event.

# data[].attributes.body
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

# data[].attributes.body
Classify three-dimensional figures including cubes, prisms, pyramids, cones and spheres.

# data[].attributes.body
Explain a proof of the Pythagorean Theorem and its converse.

# data[].attributes.body
Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.

# data[].attributes.body
Recognize that in a multi-digit number, a digit in one place represents10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

# data[].attributes.body
Use place value understanding to round decimals to any place.

# data[].attributes.body
Construct a function to model a linear relationship between two quantities.
a. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.
b. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.