Gizmos for Colorado - Mathematics: 8th Grade (Academic Standards adopted 2010)

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.

1: : Number Sense, Properties, and Operations

1.1: : Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent real-world quantities

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

1.1.a: : Students can: Define irrational numbers.

1.1.b: : Students can: Demonstrate informally that every number has a decimal expansion.

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

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

1.1.c: : Students can: 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.

1.1.d: : Students can: Apply the properties of integer exponents to generate equivalent numerical expressions.

1.1.e: : Students can: 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.

1.1.f: : Students can: Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

1.1.g: : Students can: 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.

1.1.h: : Students can: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

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

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

2: : Patterns, Functions, and Algebraic Structures

2.1: : Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

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

2.1.b: : Students can: Graph proportional relationships, interpreting the unit rate as the slope of the graph.

2.1.c: : Students can: Compare two different proportional relationships represented in different ways.

2.1.d: : Students can: 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.

2.1.e: : Students can: 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.

2.2: : Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

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

2.2.a: : Students can: Solve linear equations in one variable.

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

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.

2.2.b: : Students can: Analyze and solve pairs of simultaneous linear equations.

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.

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.

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

2.3: : Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

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

2.3.a: : Students can: Define, evaluate, and compare functions.

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

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.

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

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

2.3.a.v: : Give examples of functions that are not linear.

2.3.b: : Students can: Use functions to model relationships between quantities.

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

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.

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.

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

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

3: : Data Analysis, Statistics, and Probability

3.1: : Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

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

3.1.a: : Students can: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.

3.1.b: : Students can: Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

3.1.c: : Students can: 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.

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

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

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

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

4: : Shape, Dimension, and Geometric Relationships

4.1: : Apply transformation to numbers, shapes, functional representations, and data

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

4.1.a: : Students can: Verify experimentally the properties of rotations, reflections, and translations:

4.1.b: : Students can: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

4.1.c: : Students can: 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.

4.1.d: : Students can: Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them.

4.1.e: : Students can: 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.

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

4.1.g: : Students can: 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.

4.2: : Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

4.2: : Direct and indirect measurement can be used to describe and make comparisons

4.2.a: : Students can: Explain a proof of the Pythagorean Theorem and its converse.

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

4.2.c: : Students can: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

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

Correlation last revised: 9/22/2020

Colorado - Mathematics: 8th Grade

Academic Standards | Adopted: 2010

1: : Number Sense, Properties, and Operations

2: : Patterns, Functions, and Algebraic Structures

3: : Data Analysis, Statistics, and Probability

4: : Shape, Dimension, and Geometric Relationships