8th Grade Algebra

Equations analyze and solve linear equations and pairs of simultaneous linear equations. (2-3 weeks)

Transformations

Exponents 

Radical Expressions & Equations 

Pythagorean Theorem & Volume 

Introduction to Functions 

Linear Functions

Linear Models & Tables 

Systems of Equations

Exponential & Quadratic

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

 Understand the angle relationships 

Understand and describe transformations in the coordinate plane 

Evaluate and solve expressions using integer exponents 

Evaluate expressions in scientific notation

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

Simplify expressions involving radicals (limited to square roots) 

 Understand and apply the Pythagorean Theorem

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

 Define, evaluate, and compare functions

Understand the connections between proportional relationships, lines, and slope

Graphing and analyzing linear functions

Writing Linear Equations

Comparing linear functions in multiple representations

 Investigate patterns of association in bi-variate data.

 Analyze and solve pairs of simultaneous linear equations algebraically in context

Analyze and solve pairs of simultaneous linear equations both graphically and algebraically 

Graphing exponential functions and interpreting key features

Graphing quadratic functions and interpreting key features

Solving quadratic equations

solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms

solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms

justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties

rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (e.g., rearrange Ohm’s law V=IR to highlight resistance R)

use informal arguments to establish facts about the angle sum and exterior angles of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles (e.g. arrange three copies of the same triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so)

verify experimentally the congruence properties of rotations, reflections, and translations: a. lines taken to lines, and line segments taken to line segments of the same length; b. angles are taken to angles of the same measure; c. parallel lines are taken to parallel lines

understand 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; given two congruent figures, describe a sequence that exhibits the congruence between them

describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates

recognize that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations and dilations; describe a sequence of transformations, that when given, proves similarity between two figures

apply and know the properties of integer exponents to generate equivalent numerical expressions

add, subtract, and multiply polynomials

express and use numbers in scientific notation to estimate very large or very small quantities; express how many times as much ones is than the other

add, subtract, multiply, and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are used; interpret and use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g. use millimeters per year for seafloor spreading); interpret scientific notation that has been generated by technology (e.g. calculators)

know that numbers that are not rational are called irrational; understand informally that every number has 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

use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line and estimate the value of expressions

use square root and cube root symbols to represent solutions to equations. Recognize that x2 = p (where p is a positive rational number) has two solutions and x3 = p (where p is a negative or positive rational number) has one solution. Evaluate square roots of perfect squares up to and including 625 and cube roots of perfect cubes up to and included 1000 and – 1000

simplify, add, subtract, multiply, and divide radical expression to include rationalizing the denominator

explain why the sum or product of rational numbers is rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational

explain a proof of the Pythagorean Theorem and its converse

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

apply the Pythagorean Theorem to find the distance between two points in a coordinate system

solve real world and mathematical problems involving the volume of cylinders, cones, and spheres

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

evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context

relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 < x ≤ 10})

construct a function to model a linear relationship between two quantities. 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. 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

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

graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways (e.g. compare a distance-time graph to a distance-time equation to determine which of the two moving objects has greater speed).

determine the meaning of slope by using similar right 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 and graph linear equations in slope-intercept form y = mx + b

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

graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)

create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales

compare properties of two functions each represented among verbal, tabular, graphic and algebraic representations of functions (e.g. given a linear function represented by a table of values and a linear function represented by an algebraic equation, determine which function has a greater rate of change)

construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

know that straight lines are widely used to model relationships between two quantitative variables; for scatter plots that suggest linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line

apply the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting slope and intercept (e.g. in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height)

recognize that patterns of association can be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table; construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects; use relative frequencies calculated for rows or columns to describe possible association between the two variables (e.g. collect data from your students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home; is there evidence that those who have a curfew also tend to have chores?)

solve systems of equations algebraically and estimate solutions by graphing the equations

solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations; solve simple cases by inspection (e.g. 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)

solve real world mathematical problems leading to two linear equations in two variables (e.g. given coordinates for two pairs of points, determine whether the lines through the first pair of points intersects the line through the second pair) show and explain why the elimination method works to solve a system of two-variable equations

show and explain why the elimination method works to solve a system of two-variable equations

solve systems of equations algebraically and estimate solutions by graphing the equations

correspond points of intersection of graphs to solutions to a system of two linear equations in two variables because points of intersection satisfy both equations simultaneously

graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior

interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship

graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)

interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship

factor any quadratic expression to reveal the zeros of the function it defines

complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines

use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (e.g., compare and contrast quadratic functions in standard, vertex, and intercept forms)