Grade 8 Math Colorado standards Standards

Geometry

Statistics & Probability

Functions

Expressions & Equations

The Number System

Standards for Mathematical Practice

Work with radicals and integer exponents.

Expressions & Equations: Work with radicals and integer exponents.

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

Know that an integer exponent of 2 is called “squared” and an integer exponent of 3 is called “cubed” and that these numbers indicate how many times to use a number as a factor.

For example, 42 means 4 times 4 and 43 means 4 times 4 times 4.

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 (up to 100) and cube roots of small perfect cubes (up to 64). Know that √2 is irrational.

Evaluate square roots of small perfect squares (up to 100).

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 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

Use numbers expressed in the form a single digit times a 2nd or 3rd power of 10 to estimate large quantities.

For example, 5000 can be expressed as 5 x 103 .

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.

Perform addition with numbers expressed in scientific notation, including problems where whole numbers are used as a factor times 102 and 103 .

For example, 5 x 102 plus 3 x 102 is 8 x 102 (five ten-squared plus three ten-squared is 8 ten-squared).

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

Expressions & Equations: Understand the connections between proportional relationships, lines, and linear equations.

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Graph simple proportional relationships by connecting the origin to a point representing the ratio in the form y/x.

For example, given a ration of 3 miles per 1 hour, plot the point (1,3) and draw a line through it and the origin in the first quadrant.

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.

Measure the slope of a graph by drawing or using given slope triangles.

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

Expressions & Equations: Analyze and solve linear equations and pairs of simultaneous linear equations.

Solve linear equations in one variable.

Solve two-step linear equations in one variable.

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).

For example, if 6x - 3 = 9, then x = 2.

Solve linear equations with rational number coefficients, including equations with variables on both sides and whose solutions require expanding expressions using the distributive property and collecting like terms.

Analyze and solve pairs of simultaneous linear equations.

Match a pair of simultaneious linear equations to a real-world context.

Explain 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.

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

Solve real-world and mathematical problems leading to two linear equations in two variables.

Define, evaluate, and compare functions.

Functions: Define, evaluate, and compare functions.

Define a function as a rule that assigns to each input exactly one output. Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required for Grade 8.)

Given a function table containing at least 2 ordered pairs, identify a missing number that completes another ordered pair (limited to linear functions). (EE.F.1-3)

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Compare properties of two functions each represented by verbal descriptions.

For example, compare the amount of money received if your allowance of $10 per week instead of $6 per week.

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.

Interpret a function as linear if all of the input/output pairs can be graphed on a single line.

Use functions to model relationships between quantities.

Functions: Use functions to model relationships between quantities.

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.

Determine the values or rule of a function using a graph or a table.

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.

Describe qualitatively how a graph represents a relationship between two quantities (EE.8.F.B.5).

Understand congruence and similarity using physical models, transparencies, or geometry software.

Geometry: Understand congruence and similarity using physical models, transparencies, or geometry software.

Verify experimentally the properties of rotations, reflections, and translations:

Recognize rotations, reflections, and translations (EE.8.G.1).

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Demonstrate 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.

Demonstrate congruence of two-dimensional shapes by rotating, reflecting, and/or translating one shape onto another.

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

Describe the effect of a translation, rotation, or reflection in the coordinate plane by matching a point on the figure with the corresponding point on the image of the future.

Demonstrate 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.

Recognize 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.

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.

Use angle measurements to establish facts about the angle sum of triangles and about the angles created when parallel lines are cut by a transversal

Understand and apply the Pythagorean Theorem.

Geometry: Understand and apply the Pythagorean Theorem.

Explain a proof of the Pythagorean Theorem and its converse.

Explain or otherwise demonstrate how the areas of the squares constructed on the sides of a 3-4-5 right triangle correspond to the parts of the formula for the Pythagorean Theorem.

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

Represent a right triangle on a coordinate plan and measure to find missing side lengths.

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

Measure to find the distance between two points on a coordinate plane.

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

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

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

Use the formulas for the volumes of cylinders to solve real-world and mathematical problems.

Know that there are numbers that are not rational, and approximate them by rational numbers.

The Number System: Know that there are numbers that are not rational, and approximate them by rational numbers.

Demonstrate 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. Define irrational numbers as numbers that are not rational.

Calculate or identify the equivalent decimal values for common fractions less than 1, such as those expressed as halves, thirds, fourths, and fifths.

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).

Compare the size of rational numbers by locating them approximately on a number line diagram.

Investigate patterns of association in bivariate data.

Statistics & Probability: Investigate patterns of association in bivariate data.

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.

Interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.

Describe patterns such as positive or negative association and linear 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.

Informally fit a straight line to a scatterplot and assess the model fit by judging the closeness of the data points to the graph of the line.

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

Use a graph of linear model in the context of bivariate measurement data to interpret the slope and intercept (EE.8.SP.4).

Explain 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.

Construct a graph or table from given categorical data and compare the data categorized in the graph or table.

Algebra and Functions

Data, Statistics, and Probability

Geometry

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.

Number and Quantity