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### 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.

- 8.2.2.a Solve linear equations in one variable.
- 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.

- 8.2.3.a Define, evaluate, and compare functions.

### 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.