Grade 8

The Number System

Expressions and Equations

Functions

Geometry

Statistics and Probability

Understand that there are irrational numbers, and approximate them using rational numbers.

Work with radicals and integer exponents.

Understand the connections between proportional relationships, lines, and linear equations.

Analyze and solve linear equations, inequalities, and pairs of simultaneous linear equations.

Define, evaluate, and compare functions.

Use functions to model relationships between quantities. 

Understand congruence and similarity.

Understand and apply the Pythagorean Theorem.

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

Investigate patterns of association in bivariate data.

Investigate chance processes and develop, use, and evaluate probability models.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion. Know that numbers whose decimal expansions do not terminate in zeros or in a repeating sequence of fixed digits are called irrational. No CCC developed for this standard.

Use rational approximations of irrational numbers to compare the size of irrational numbers. Locate them approximately on a number line diagram, and estimate their values.

Understand that given any two distinct rational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b. Given any two distinct irrational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b.

Understand and apply the properties of integer exponents to generate equivalent numerical expressions.

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. Know that √2 is irrational. a. Evaluate square roots of perfect squares less than or equal to 225. b. Evaluate cube roots of perfect cubes less than or equal to 1000. 

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 express how many times larger or smaller one is than the other.

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.

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.  

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 (0, b).

Fluently solve linear equations and inequalities in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solution. 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 and inequalities with rational number coefficients, including solutions that require expanding expressions using the distributive property and collecting like terms.

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 including cases of no solution and infinite number of solutions. Solve simple cases by inspection. c. Solve mathematical problems and problems in real-world context leading to two linear equations in two variables.

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. (Function notation is not required in Grade 8.)

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.

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 in not linear because its graph contains the points (1,1), (2,4), and (3,9) which are not on a straight line.

Given a description of a situation, generate 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 a graph. Track how the values of the two quantities change together. Interpret the rate of change and initial value of a linear function in terms of the situation it models, its graph, or its table of values.

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.

Verify experimentally the properties of rotations, reflections, and translations. Properties include: lines are taken to lines, line segments are taken to line segments of the same length, angles are taken to angles of the same measure, parallel lines are taken to parallel lines.

Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that demonstrates congruence. 

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Understand that a two-dimensional figure is similar to another if, and only if, one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that demonstrates similarity.

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.

Understand the Pythagorean Theorem and its converse.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world context and mathematical problems in two and three dimensions. 

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Understand and use formulas for volumes of cones, cylinders and spheres and use them to solve real-world context and mathematical problems.

Construct and interpret scatter plots for bivariate measurement data to investigate and describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 

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.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

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. 

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that 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 using organized lists, tables, tree diagrams and other methods. Identify the outcomes in the sample space which compose the event. c. Design and use a simulation to generate frequencies for compound events.

Identify π as an irrational number.

Round irrational numbers to the hundredths place. 

Use approximations of irrational numbers to locate them on a number line.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

Convert a number expressed in scientific notation up to 10,000. 

No CCC developed for this standard.

Represent proportional relationships on a line graph. 

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

No CCC developed for this standard.

Given two graphs, describe the function as linear and not linear.

 Identify the rate of change (slope) and initial value (y-intercept) from graphs. 

Given two graphs, describe the function as linear and not linear.

Given a verbal description of a situation, create or identify a graph to model the situation.

Given a graph of a situation, generate a description of the situation.

describe or select the relationship between the two quantities given a line graph of a situation.

Analyze provided information (e.g., a graph) to describe the relationship between two quantities.

Recognize a rotation, reflection, or translation of a figure.

Use the reflections, rotations, or translations in the coordinate plane to solve problems with right angles.

No CCC developed for this standard.

Identify a rotation, reflection, or translation of a plane figure when given coordinates.

Recognize congruent and similar figures. 

Describe the changes in surface area, area, and volume when the figure is changed in some (e.g., scale drawings).

Compare area and volume of similar figures.

Use angle relationships to find the value of a missing angle.

No CCC developed for this standard.

Apply the Pythagorean theorem to determine lengths/distances in real-world situations.

Find the hypotenuse of a two-dimensional right triangle (Pythagorean Theorem).

Find the missing side lengths of a two-dimensional right triangle (Pythagorean Theorem).

Find the hypotenuse of a two-dimensional right triangle (Pythagorean Theorem).

Find the missing side lengths of a two-dimensional right triangle (Pythagorean Theorem).

No CCC developed for this standard. 

Apply the formula to find the volume of 3 dimensional shapes (i.e., cubes, spheres, and cylinders).

Recognize a pattern of association using existing data.

Graph data using line graphs, histograms, or box plots.

Graph bivariate data using scatter plots and identify possible associations between the variables. 

using box plots and scatter plots, identify data points that appear to be outliers.

Distinguish between a linear and non-linear association when analyzing bivariate data on a scatter plot

Interpret the slope and the y-intercept of a line in the context of a problem.

Analyze displays of bivariate data to develop or select appropriate claims about those data.

Construct a two-way table summarizing data on two categorical variables collected from the same subjects; identify possible association between the two variables.

No CCC developed for this standard.