The Number System

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

1. Know that numbers that are not rational are called irrational. Understand informally that every numberhas 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.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locatethem approximately on a number line diagram, and estimate the value of expressions (e.g., π2). Forexample, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between1.4 and 1.5, and explain how to continue on to get better approximations.

Expressions and Equations

Work with radicals and integer exponents.

1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. Forexample, 32 × 3–5 = 3–3 = 1/33 = 1/27.

2. 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 and cube rootsof small perfect cubes. Know that √2 is irrational.

3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very largeor 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 × 108 and the population of the world as 7 × 109, anddetermine that the world population is more than 20 times larger.

4. Perform operations with numbers expressed in scientific notation, including problems where bothdecimal and scientific notation are used. Use scientific notation and choose units of appropriate size formeasurements of very large or very small quantities (e.g., use millimeters per year for seafloorspreading). Interpret scientific notation that has been generated by technology.


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

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare twodifferent proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

6. 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 theequation y = mx + b for a line intercepting the vertical axis at b.

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

Solve linear equations in one variable.

1. Give examples of linear equations in one variable with one solution, infinitely manysolutions, or no solutions. Show which of these possibilities is the case by successivelytransforming the given equation into simpler forms, until an equivalent equation of the formx = a, a = a, or a = b results (where a and b are different numbers).

2. Solve linear equations with rational number coefficients, including equations whosesolutions require expanding expressions using the distributive property and collecting liketerms.

Analyze and solve pairs of simultaneous linear equations.

1. Understand that solutions to a system of two linear equations in two variables correspond topoints of intersection of their graphs, because points of intersection satisfy both equationssimultaneously.

2. Solve systems of two linear equations in two variables algebraically, and estimate solutionsby graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

3. Solve real-world and mathematical problems leading to two linear equations in twovariables. For example, given coordinates for two pairs of points, determine whether theline through the first pair of points intersects the line through the second pair.

Functions

Define, evaluate, and compare functions.

1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function isthe set of ordered pairs consisting of an input and the corresponding output.1

2. Compare properties of two functions each represented in a different way (algebraically, graphically,numerically in tables, or by verbal descriptions). For example, given a linear function represented by a tableof values and a linear function represented by an algebraic expression, determine which function has thegreater rate of change.

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examplesof functions that are not linear. For example, the function A = s2 giving the area of a square as a function ofits side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on astraight line.




Use functions to model relationships between quantities.

4. Construct a function to model a linear relationship between two quantities. Determine the rate of changeand initial value of the function from a description of a relationship or from two (x, y) values, includingreading these from a table or from a graph. Interpret the rate of change and initial value of a linear functionin terms of the situation it models, and in terms of its graph or a table of values.

5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., wherethe function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitativefeatures of a function that has been described verbally.

Geometry 

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

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

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

   2. Angles are taken to angles of the same measure.

   3. Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from thefirst by a sequence of rotations, reflections, and translations; given two congruent figures, describe asequence that exhibits the congruence between them.

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

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first bya sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures,describe a sequence that exhibits the similarity between them.

5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about theangles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity oftriangles. For example, arrange three copies of the same triangle so that the sum of the three anglesappears to form a line, and give an argument in terms of transversals why this is so.

Understand and apply the Pythagorean Theorem.

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

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

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

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

9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world andmathematical problems.

Function notation is not required in Grade 8.