Mathematics Standards for Colorado — Grade 8

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G81.1a

Define irrational numbers.1

G81.1b

Demonstrate informally that every number has a decimal expansion. (CCSS: 8.NS.1) i. For rational numbers show that the decimal expansion repeats eventually. (CCSS: 8.NS.1) ii. Convert a decimal expansion which repeats eventually into a rational number. (CCSS: 8.NS.1)

G81.1b.i

For rational numbers show that the decimal expansion repeats eventually. (CCSS: 8.NS.1)

G81.1b.ii

Convert a decimal expansion which repeats eventually into a rational number. (CCSS: 8.NS.1)

G81.1c

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.2 (CCSS: 8.NS.2)

G81.1d

Apply the properties of integer exponents to generate equivalent numerical expressions.3 (CCSS: 8.EE.1)

G81.1e

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. (CCSS: 8.EE.2)

G81.1f

Evaluate square roots of small perfect squares and cube roots of small perfect cubes.4 (CCSS: 8.EE.2)

G81.1g

Use numbers expressed in the form of a single digit times a whole number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.5 (CCSS: 8.EE.3)

G81.1h

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. (CCSS: 8.EE.4) i. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.6 (CCSS: 8.EE.4) ii. Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.4)

G81.1h.i

Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.6 (CCSS: 8.EE.4)

G81.1h.ii

Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.4)

G82.1a

Describe the connections between proportional relationships, lines, and linear equations. (CCSS: 8.EE)

G82.1b

Graph proportional relationships, interpreting the unit rate as the slope of the graph. (CCSS: 8.EE.5)

G82.1c

Compare two different proportional relationships represented in different ways.1 (CCSS: 8.EE.5)

G82.1d

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. (CCSS: 8.EE.6)

G82.1e

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. (CCSS: 8.EE.6)

G82.2a

Solve linear equations in one variable. (CCSS: 8.EE.7) i. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.2 (CCSS: 8.EE.7a) ii. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.7b)

G82.2a.i

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.2 (CCSS: 8.EE.7a)

G82.2a.ii

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.7b)

G82.2b

Analyze and solve pairs of simultaneous linear equations. (CCSS: 8.EE.8) 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. (CCSS: 8.EE.8a) ii. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.3 (CCSS: 8.EE.8b) iii. Solve real-world and mathematical problems leading to two linear equations in two variables.4 (CCSS: 8.EE.8c)

G82.2b.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. (CCSS: 8.EE.8a)

G82.2b.ii

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.3 (CCSS: 8.EE.8b)

G82.2b.iii

Solve real-world and mathematical problems leading to two linear equations in two variables.4 (CCSS: 8.EE.8c)

G82.3a

Define, evaluate, and compare functions. (CCSS: 8.F) i. Define a function as a rule that assigns to each input exactly one output.5 (CCSS: 8.F.1) ii. Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (CCSS: 8.F.1) iii. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).6 (CCSS: 8.F.2) iv. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. (CCSS: 8.F.3) v. Give examples of functions that are not linear.7

G82.3a.i

Define a function as a rule that assigns to each input exactly one output.5 (CCSS: 8.F.1)

G82.3a.ii

Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (CCSS: 8.F.1)

G82.3a.iii

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).6 (CCSS: 8.F.2)

G82.3a.iv

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. (CCSS: 8.F.3)

G82.3a.v

Give examples of functions that are not linear.7

G82.3b

Use functions to model relationships between quantities. (CCSS: 8.F) i. Construct a function to model a linear relationship between two quantities. (CCSS: 8.F.4) 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. (CCSS: 8.F.4) 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. (CCSS: 8.F.4) iv. Describe qualitatively the functional relationship between two quantities by analyzing a graph.8 (CCSS: 8.F.5) v. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (CCSS: 8.F.5) vi. Analyze how credit and debt impact personal financial goals (PFL)

G82.3b.i

Construct a function to model a linear relationship between two quantities. (CCSS: 8.F.4)

G82.3b.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. (CCSS: 8.F.4)

G82.3b.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. (CCSS: 8.F.4)

G82.3b.iv

Describe qualitatively the functional relationship between two quantities by analyzing a graph.8 (CCSS: 8.F.5)

G82.3b.v

Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (CCSS: 8.F.5)

G82.3b.vi

Analyze how credit and debt impact personal financial goals (PFL)

G83.1a

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. (CCSS: 8.SP.1)

G83.1b

Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (CCSS: 8.SP.1)

G83.1c

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.1 (CCSS: 8.SP.2)

G83.1d

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.2 (CCSS: 8.SP.3)

G83.1e

Explain patterns of association seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. (CCSS: 8.SP.4) i. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. (CCSS: 8.SP.4) ii. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.3 (CCSS: 8.SP.4)

G83.1e.i

Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. (CCSS: 8.SP.4)

G83.1e.ii

Use relative frequencies calculated for rows or columns to describe possible association between the two variables.3 (CCSS: 8.SP.4)

G84.1a

Verify experimentally the properties of rotations, reflections, and translations.1 (CCSS: 8.G.1)

G84.1b

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (CCSS: 8.G.3)

G84.1c

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. (CCSS: 8.G.2)

G84.1d

Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them. (CCSS: 8.G.2)

G84.1e

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. (CCSS: 8.G.4)

G84.1f

Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them. (CCSS: 8.G.4)

G84.1g

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.2 (CCSS: 8.G.5)

G84.2a

Explain a proof of the Pythagorean Theorem and its converse. (CCSS: 8.G.6)

G84.2b

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (CCSS: 8.G.7)

G84.2c

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (CCSS: 8.G.8)

G84.2d

State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (CCSS: 8.G.9)