Seventh Grade Common Core State Standards

Grade 7 Overview
Ratios and Proportional Relationships
 Analyze proportional relationships and use them to solve realworld and mathematical problems.
The Number System
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Expressions and Equations
 Use properties of operations to generate equivalent expressions.
 Solve reallife and mathematical problems using numerical and algebraic expressions and equations.
Geometry
 Draw, construct and describe geometrical figures and describe the relationships between them.
 Solve reallife and mathematical problems involving angle measure, area, surface area, and volume.
Statistics and Probability
 Use random sampling to draw inferences about a population.
 Draw informal comparative inferences about two populations.
 Investigate chance processes and develop, use, and evaluate probability models.
Mathematical Practices
 Make sense of problems and persevere in solving them.
 Reason abstractly and quantitatively.
 Construct viable arguments and critique the reasoning of others.
 Model with mathematics.
 Use appropriate tools strategically.
 Attend to precision.
 Look for and make use of structure.
 Look for and express regularity in repeated reasoning.
Ratios & Proportional Relationships

Ratios & Proportional Relationships
Analyze proportional relationships and use them to solve realworld and mathematical problems.
CCSS.MATH.CONTENT.7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ^{1/2}/_{1/4} miles per hour, equivalently 2 miles per hour.CCSS.MATH.CONTENT.7.RP.A.2
Recognize and represent proportional relationships between quantities.CCSS.MATH.CONTENT.7.RP.A.2.A
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.CCSS.MATH.CONTENT.7.RP.A.2.B
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.CCSS.MATH.CONTENT.7.RP.A.2.C
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.CCSS.MATH.CONTENT.7.RP.A.2.D
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.CCSS.MATH.CONTENT.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
The Number System

The Number System
Apply and extend previous understandings of operations with fractions.
CCSS.MATH.CONTENT.7.NS.A.1
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.CCSS.MATH.CONTENT.7.NS.A.1.A
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.CCSS.MATH.CONTENT.7.NS.A.1.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 realworld contexts.CCSS.MATH.CONTENT.7.NS.A.1.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 realworld contexts.CCSS.MATH.CONTENT.7.NS.A.1.D
Apply properties of operations as strategies to add and subtract rational numbers.CCSS.MATH.CONTENT.7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.CCSS.MATH.CONTENT.7.NS.A.2.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 realworld contexts.CCSS.MATH.CONTENT.7.NS.A.2.B
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero 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 realworld contexts.CCSS.MATH.CONTENT.7.NS.A.2.C
Apply properties of operations as strategies to multiply and divide rational numbers.CCSS.MATH.CONTENT.7.NS.A.2.D
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.CCSS.MATH.CONTENT.7.NS.A.3
Solve realworld and mathematical problems involving the four operations with rational numbers.^{1}^{1} Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
Expressions & Equations

Expressions & Equations
Use properties of operations to generate equivalent expressions.
CCSS.MATH.CONTENT.7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.CCSS.MATH.CONTENT.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."Solve reallife and mathematical problems using numerical and algebraic expressions and equations.
CCSS.MATH.CONTENT.7.EE.B.3
Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), 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. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.CCSS.MATH.CONTENT.7.EE.B.4
Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.CCSS.MATH.CONTENT.7.EE.B.4.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. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?CCSS.MATH.CONTENT.7.EE.B.4.B
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Geometry

Geometry
Draw construct, and describe geometrical figures and describe the relationships between them.
CCSS.MATH.CONTENT.7.G.A.1
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.CCSS.MATH.CONTENT.7.G.A.2
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.CCSS.MATH.CONTENT.7.G.A.3
Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Solve reallife and mathematical problems involving angle measure, area, surface area, and volume.
CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.CCSS.MATH.CONTENT.7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.CCSS.MATH.CONTENT.7.G.B.6
Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Statistics & Probability

Statistics & Probability
Use random sampling to draw inferences about a population.
CCSS.MATH.CONTENT.7.SP.A.1
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.CCSS.MATH.CONTENT.7.SP.A.2
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.Draw informal comparative inferences about two populations.
CCSS.MATH.CONTENT.7.SP.B.3
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.CCSS.MATH.CONTENT.7.SP.B.4
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventhgrade science book are generally longer than the words in a chapter of a fourthgrade science book.Investigate chance processes and develop, use, and evaluate probability models.
CCSS.MATH.CONTENT.7.SP.C.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. 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.CCSS.MATH.CONTENT.7.SP.C.6
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.CCSS.MATH.CONTENT.7.SP.C.7
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.CCSS.MATH.CONTENT.7.SP.C.7.A
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.CCSS.MATH.CONTENT.7.SP.C.7.B
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land openend down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?CCSS.MATH.CONTENT.7.SP.C.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.CCSS.MATH.CONTENT.7.SP.C.8.A
Understand that, 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.CCSS.MATH.CONTENT.7.SP.C.8.B
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.CCSS.MATH.CONTENT.7.SP.C.8.C
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Eighth Grade Common Core State Standards

Grade 8 Overview
The Number System
 Know that there are numbers that are not rational, and approximate them by rational numbers.
Expressions and Equations
 Work with radicals and integer exponents.
 Understand the connections between proportional relationships, lines, and linear equations.
 Analyze and solve linear equations and pairs of simultaneous linear equations.
Functions
 Define, evaluate, and compare functions.
 Use functions to model relationships between quantities.
Geometry
 Understand congruence and similarity using physical models, transparencies, or geometry software.
 Understand and apply the Pythagorean Theorem.
 Solve realworld and mathematical problems involving volume of cylinders, cones and spheres.
Statistics and Probability
 Investigate patterns of association in bivariate data.
Mathematical Practices
 Make sense of problems and persevere in solving them.
 Reason abstractly and quantitatively.
 Construct viable arguments and critique the reasoning of others.
 Model with mathematics.
 Use appropriate tools strategically.
 Attend to precision.
 Look for and make use of structure.
 Look for and express regularity in repeated reasoning.
Standards

Statistics & Probability
Investigate patterns of association in bivariate data.
CCSS.MATH.CONTENT.8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.CCSS.MATH.CONTENT.8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.CCSS.MATH.CONTENT.8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.CCSS.MATH.CONTENT.8.SP.A.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. 
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
CCSS.MATH.CONTENT.8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:CCSS.MATH.CONTENT.8.G.A.1.A
Lines are taken to lines, and line segments to line segments of the same length.CCSS.MATH.CONTENT.8.G.A.1.B
Angles are taken to angles of the same measure.CCSS.MATH.CONTENT.8.G.A.1.C
Parallel lines are taken to parallel lines.CCSS.MATH.CONTENT.8.G.A.2
Understand that a twodimensional 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.CCSS.MATH.CONTENT.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.CCSS.MATH.CONTENT.8.G.A.4
Understand that a twodimensional 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 twodimensional figures, describe a sequence that exhibits the similarity between them.CCSS.MATH.CONTENT.8.G.A.5
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 angleangle criterion for similarity of triangles.Understand and apply the Pythagorean Theorem.
CCSS.MATH.CONTENT.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.CCSS.MATH.CONTENT.8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.CCSS.MATH.CONTENT.8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Solve realworld and mathematical problems involving volume of cylinders, cones, and spheres.
CCSS.MATH.CONTENT.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems. 
Functions
Define, evaluate, and compare functions.
CCSS.MATH.CONTENT.8.F.A.1
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.^{1}CCSS.MATH.CONTENT.8.F.A.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).CCSS.MATH.CONTENT.8.F.A.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.Use functions to model relationships between quantities.
CCSS.MATH.CONTENT.8.F.B.4
Construct a function to model a linear relationship between two quantities. 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. 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.CCSS.MATH.CONTENT.8.F.B.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.^{1} Function notation is not required for Grade 8.

Expressions & Equations
Expressions and Equations Work with radicals and integer exponents.
CCSS.MATH.CONTENT.8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^{2} × 3^{5} = 3^{3} = 1/3^{3} = 1/27.CCSS.MATH.CONTENT.8.EE.A.2
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 √2 is irrational.CCSS.MATH.CONTENT.8.EE.A.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.CCSS.MATH.CONTENT.8.EE.A.4
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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technologyUnderstand the connections between proportional relationships, lines, and linear equations.
CCSS.MATH.CONTENT.8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distancetime graph to a distancetime equation to determine which of two moving objects has greater speed.CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive 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.Analyze and solve linear equations and pairs of simultaneous linear equations.
CCSS.MATH.CONTENT.8.EE.C.7
Solve linear equations in one variable.CCSS.MATH.CONTENT.8.EE.C.7.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).CCSS.MATH.CONTENT.8.EE.C.7.B
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.CCSS.MATH.CONTENT.8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.CCSS.MATH.CONTENT.8.EE.C.8.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.CCSS.MATH.CONTENT.8.EE.C.8.B
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.CCSS.MATH.CONTENT.8.EE.C.8.C
Solve realworld and mathematical problems leading to two linear equations in two variables. 
The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
CCSS.MATH.CONTENT.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.CCSS.MATH.CONTENT.8.NS.A.2
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 (e.g., π^{2}).