Colorado

$Colorado flag$

Skills available for Colorado eighth-grade math standards

Standards are in black and IXL math skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practice that skill.

Show alignments for:

Colorado Academic Standards Colorado Academic Standards
Colorado Academic Standards (Common Core) Colorado Academic Standards (Common Core)

Actions

Print standards

8.1 Number Sense, Properties, and Operations

8.1.1 In the real number system, rational and irrational numbers are in one to one correspondence to points on the number line
- 8.1.1.a Define irrational numbers.
  - Identify rational and irrational numbers (8-I.2)
- 8.1.1.b Demonstrate informally that every number has a decimal expansion.
  - 8.1.1.b.i For rational numbers show that the decimal expansion repeats eventually.
  - 8.1.1.b.ii Convert a decimal expansion which repeats eventually into a rational number.
    - Convert between decimals and fractions or mixed numbers (8-G.5)
- 8.1.1.c 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.
  - Estimate positive and negative square roots (8-F.4)
  - Estimate cube roots (8-F.10)
- 8.1.1.d Apply the properties of integer exponents to generate equivalent numerical expressions.
- 8.1.1.e 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.
- 8.1.1.f Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
- 8.1.1.g 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.
  - Convert between standard and scientific notation (8-E.1)
- 8.1.1.h Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.
  - 8.1.1.h.i Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
  - 8.1.1.h.ii Interpret scientific notation that has been generated by technology.

8.2 Patterns, Functions, and Algebraic Structures

8.2.1 Linear functions model situations with a constant rate of change and can be represented numerically, algebraically, and graphically
- 8.2.1.a Describe the connections between proportional relationships, lines, and linear equations.
  - Identify proportional relationships by graphing (8-K.3)
- 8.2.1.b Graph proportional relationships, interpreting the unit rate as the slope of the graph.
  - Find the constant of proportionality from a graph (8-K.4)
  - Write equations for proportional relationships from graphs (8-K.5)
- 8.2.1.c Compare two different proportional relationships represented in different ways.
- 8.2.1.d 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.
- 8.2.1.e 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.
8.2.2 Properties of algebra and equality are used to solve linear equations and systems of equations
- 8.2.2.a Solve linear equations in one variable.
  - 8.2.2.a.i Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.
  - 8.2.2.a.ii Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
- 8.2.2.b Analyze and solve pairs of simultaneous linear equations.
  - 8.2.2.b.i 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.
  - 8.2.2.b.ii Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
  - 8.2.2.b.iii Solve real-world and mathematical problems leading to two linear equations in two variables.
8.2.3 Graphs, tables and equations can be used to distinguish between linear and nonlinear functions
- 8.2.3.a Define, evaluate, and compare functions.
  - 8.2.3.a.i Define a function as a rule that assigns to each input exactly one output.
  - 8.2.3.a.ii Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
  - 8.2.3.a.iii Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  - 8.2.3.a.iv Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.
    - Graph a line from an equation in slope-intercept form (8-AA.6)
  - 8.2.3.a.v Give examples of functions that are not linear.
    - Identify linear and nonlinear functions: graphs and equations (8-BB.19)
- 8.2.3.b Use functions to model relationships between quantities.
  - 8.2.3.b.i Construct a function to model a linear relationship between two quantities.
  - 8.2.3.b.ii 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.
  - 8.2.3.b.iii 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.
  - 8.2.3.b.iv Describe qualitatively the functional relationship between two quantities by analyzing a graph.
    - Write linear functions: word problems (8-BB.17)
  - 8.2.3.b.v Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
  - 8.2.3.b.vi Analyze how credit and debt impact personal financial goals.
    - Simple interest (8-M.10)
    - Compound interest (8-M.11)

8.3 Data Analysis, Statistics, and Probability

8.3.1 Visual displays and summary statistics of two-variable data condense the information in data sets into usable knowledge
- 8.3.1.a Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.
  - Identify trends with scatter plots (8-EE.17)
- 8.3.1.b Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
  - Identify trends with scatter plots (8-EE.17)
  - Outliers in scatter plots (8-FF.9)
- 8.3.1.c 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.
- 8.3.1.d Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
- 8.3.1.e Explain patterns of association seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
  - 8.3.1.e.i Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
  - 8.3.1.e.ii Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

8.4 Shape, Dimension, and Geometric Relationships

8.4.1 Transformations of objects can be used to define the concepts of congruence and similarity
- 8.4.1.a Verify experimentally the properties of rotations, reflections, and translations:
- 8.4.1.b Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
- 8.4.1.c 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.
- 8.4.1.d Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them.
- 8.4.1.e 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.
  - Similar and congruent figures (8-S.1)
- 8.4.1.f Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them.
- 8.4.1.g 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.
8.4.2 Direct and indirect measurement can be used to describe and make comparisons
- 8.4.2.a Explain a proof of the Pythagorean Theorem and its converse.
  - Converse of the Pythagorean theorem: is it a right triangle? (8-T.6)
- 8.4.2.b Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
- 8.4.2.c Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
  - Find the distance between two points (8-P.4)
- 8.4.2.d State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Colorado

Show alignments for:

Actions

8.1 Number Sense, Properties, and Operations

8.1.1 In the real number system, rational and irrational numbers are in one to one correspondence to points on the number line

8.1.1.a Define irrational numbers.

8.1.1.b Demonstrate informally that every number has a decimal expansion.

8.1.1.b.i For rational numbers show that the decimal expansion repeats eventually.

8.1.1.b.ii Convert a decimal expansion which repeats eventually into a rational number.

8.1.1.c 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.

8.1.1.d Apply the properties of integer exponents to generate equivalent numerical expressions.

8.1.1.e 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.

8.1.1.f Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

8.1.1.g 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.

8.1.1.h Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

8.1.1.h.i Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.

8.1.1.h.ii Interpret scientific notation that has been generated by technology.

8.2 Patterns, Functions, and Algebraic Structures

8.2.1 Linear functions model situations with a constant rate of change and can be represented numerically, algebraically, and graphically

8.2.1.a Describe the connections between proportional relationships, lines, and linear equations.

8.2.1.b Graph proportional relationships, interpreting the unit rate as the slope of the graph.

8.2.1.c Compare two different proportional relationships represented in different ways.

8.2.1.d 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.

8.2.1.e 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.

8.2.2 Properties of algebra and equality are used to solve linear equations and systems of equations

8.2.2.a Solve linear equations in one variable.

8.2.2.a.i Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.

8.2.2.a.ii Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

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

8.2.2.b.i 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.

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

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

8.2.3 Graphs, tables and equations can be used to distinguish between linear and nonlinear functions

8.2.3.a Define, evaluate, and compare functions.

8.2.3.a.i Define a function as a rule that assigns to each input exactly one output.

8.2.3.a.ii Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

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

8.2.3.a.iv Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.

8.2.3.a.v Give examples of functions that are not linear.

8.2.3.b Use functions to model relationships between quantities.

8.2.3.b.i Construct a function to model a linear relationship between two quantities.

8.2.3.b.ii 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.

8.2.3.b.iii 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.

8.2.3.b.iv Describe qualitatively the functional relationship between two quantities by analyzing a graph.

8.2.3.b.v Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

8.2.3.b.vi Analyze how credit and debt impact personal financial goals.

8.3 Data Analysis, Statistics, and Probability

8.3.1 Visual displays and summary statistics of two-variable data condense the information in data sets into usable knowledge

8.3.1.a Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.

8.3.1.b Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.3.1.c 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.

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

8.3.1.e Explain patterns of association seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.

8.3.1.e.i Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.

8.3.1.e.ii Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

8.4 Shape, Dimension, and Geometric Relationships

8.4.1 Transformations of objects can be used to define the concepts of congruence and similarity

8.4.1.a Verify experimentally the properties of rotations, reflections, and translations:

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

8.4.1.c 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.

8.4.1.d Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them.

8.4.1.e 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.

8.4.1.f Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them.

8.4.1.g 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.

8.4.2 Direct and indirect measurement can be used to describe and make comparisons

8.4.2.a Explain a proof of the Pythagorean Theorem and its converse.

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

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

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