Core Content Connectors by Common Core State Standards: Mathematics 8th Grade

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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 real-world and mathematical problems involving volume of cylinders, cones and spheres.

Statistics and Probability

• Investigate patterns of association in bivariate data.

The Number System	8.NS
Know that there are numbers that are not rational, and approximate them by rational numbers.
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.
CCCs linked to 8.NS.1	8.NO.1k1 Identify π as an irrational number.
8.NO.1k2 Round irrational numbers to the hundredths place.
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., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
CCCs linked to 8.NS.2	8.NO.1k3 Use approximations of irrational numbers to locate them on a number line.
8.NO.1k3 Use approximations of irrational numbers to locate them on a number line.

Expressions and Equations	8.EE
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
CCCs linked to 8.EE.1	8.SE.1f5 Use properties of integer exponents to produce equivalent expressions.
2. Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = 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.
CCCs linked to 8.EE.2	None
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. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
CCCs linked to 8.EE.3	8.NO.1i1 Convert a number expressed in scientific notation up to 10,000.
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 technology.
CCCs linked to 8.EE.4	8.NO.1j1 Perform operations with numbers expressed in scientific notation.
Understand the connections between proportional relationships, lines, and linear equations.
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 distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
CCCs linked to 8.EE.5	8.PRF.1e2 Represent proportional relationships on a line graph.
6. 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; 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.
CCCs linked to 8.EE.6	None
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. 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 collecting like terms.
CCCs linked to 8.EE.7	8.PRF.1g3 Solve linear equations with 1 variable.
8. Analyze and solve pairs of simultaneous 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.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
CCCs linked to 8.EE.8	8.PRF.1g4 Solve systems of two linear equations in two variables and graph the results.
8.PRG.1g5 Solve real world and mathematical problems leading to two linear equations in two variables.

Functions	8.F
Define, evaluate, and compare functions.
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.
CCCs linked to 8.F.1	8.PRG.2e1 Distinguish between functions and non-functions, using equations, graphs, or tables.
2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
CCCs linked to 8.F.2	8.PRG.2e5 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
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. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
CCCs linked to 8.F.3	8.PRF.2c1 Given two graphs, describe the function as linear and not linear.
Use functions to model relationships between quantities.
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.
CCCs linked to 8.F.4	8.PRF.2e2 Identify the rate of change (slope) and initial value (y-intercept) from graphs.
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.
CCCs linked to 8.F.5	8.PRF.2c1 Given two graphs, describe the function as linear and not linear.
	8.PRF.2e3 Given a verbal description of a situation, create or identify a graph to model the situation.
	8.PRF.2e4 Given a graph of a situation, generate a description of the situation.
	8.PRF.1f2 Describe or select the relationship between the two quantities Given a line graph of a situation.

Geometry	8.G
Understand congruence and similarity using physical models, transparencies, or geometry software.
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
CCCs linked to 8.G.1	8.GM.1f1 Recognize a rotation, reflection, or translation of a figure.
H.GM.1d1 Use the reflections, rotations, or translations in the coordinate plane to solve problems with right angles.
2. 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.
CCCs linked to 8.G.2	8.GM.1g1 Recognize congruent and similar figures.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
CCCs linked to 8.G.3	8.GM.1f2 Identify a rotation, reflection, or translation of a plane figure when given coordinates.
4. 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.
CCCs linked to 8.G.4	8.GM.1g1 Recognize congruent and similar figures.
8.ME.1e1 Describe the changes in surface area, area, and volume when the figure is changed in some way (e.g., scale drawings).
8.ME.1e2 Compare area and volume of similar figures.
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 angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so
CCCs linked to 8.G.5	8.GM.1i4 Use angle relationships to find the value of a missing angle.
Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
CCCs linked to 8.G.6	None
7. Apply the Pythagorean Theorem to determine unknown side length in right triangles in real-world and mathematical problems in two and three dimensions.
CCCs linked to 8.G.7	8.ME.2f1 Apply the Pythagorean theorem to determine lengths/distances in real-world situations.
8.GM.1j1 Find the hypotenuse of a two-dimensional right triangle (Pythagorean Theorem).
8.GM.1j2 Find the missing side lengths of a two-dimensional right triangle (Pythagorean Theorem).
H.GM.1a1 Find the hypotenuse of a two-dimensional right triangle (Pythagorean Theorem).
H.GM.1a2 Find the missing side lengths of a two-dimensional right triangle (Pythagorean Theorem).
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
CCCs linked to 8.G.8	H.GM.1a3 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
CCCs linked to 8.G.9	8.ME2d2 Apply the formula to find the volume of 3-dimensional shapes (i.e., cubes, spheres, and cylinders).

Statistics and Probability	8.SP
Investigate patterns of association in bivariate data.
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.
CCCs linked to 8.SP.1	8.DPS.1g2 Graph data using line graphs, histograms, or box plots.
8.DPS.1h1 Graph bivariate data using scatter plots and identify possible associations between the variables.
8.DPS.1i3 Using box plots and scatter plots, identify data points that appear to be outliers.
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.
CCCs linked to 8.SP.2	8.DPS.2g1 Distinguish between a linear and non-linear association when analyzing bivariate data on a scatter plot
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
CCCs linked to 8.SP.3	8.DPS.2g2 Interpret the slope and the y-intercept of a line in the context of a problem.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
CCCs linked to 8.SP.4	8.DPS.1k2 Analyze displays of bivariate data to develop or select appropriate claims about those data.
8.DPS.1f3 Construct a two-way table summarizing data on two categorical variables collected from the same subjects; identify possible association between the two variables.