Integrated Algebra with 8th grade Common CoreUnit 1 - Algebraic Expressions & Equations & Real Numbers

1 Evaluating Algebraic Expressions & Order of Operations Vocabulary - variable, constant, power, base, exponent, algebraic expression, evaluate, multiple representation, standard notation, denominator, numerator, fraction, order of operations, hierarchy, parenthesis, brackets, grouping symbols, product, quotient, PEMDAS Define and identify variables, constants, powers, bases, and exponents Translate a word phrase into a power expression Evaluate the power of a number Evaluate algebraic expressions with and without powers Apply PEMDAS Without calculator With calculator (stress parentheses) Include expressions with absolute value Problem solving applicationsA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical conceptA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphicallyA.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written formA.A.2 Write a verbal expression that matches a given mathematical expressionA.N.1 Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s)(no factorials or absolute values)

2 Translating Verbal Phrases into Algebraic Expressions (not equations) Vocabulary verbal model, sum, plus, total, more than, increased by, added to, difference, less than, minus, decreased by, subtracted from, times, twice, product, multiplied by, of, quotient, divided by, divided into, consecutive integers, consecutive even integers, consecutive odd integers, interpretation, quantitative model, rate, unit rate Translate a quantitative verbal phrase into an algebraic expression (no equations) Interpret a verbal expression into its matching mathematical expression (no equations) Problem solving applicationsA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical conceptA.A.1 Translate a quantitative verbal phrase into an algebraic expressionA.A.2 Write a verbal expression that matches a given mathematical expression

3 Translate Word Sentences into Algebraic Equations & Inequalities (Linear in one variable only NO solving, only setup) Vocabulary equation, inequality, open sentence, plus, minus, sum, difference, product, quotient, increased by, decreased by, more than, less than, times, twice, the difference of, subtracted from, added to, consecutive, consecutive even, consecutive odd, break even point, equal to, is less than(), is at least(), is at most(), is not more than(), is not less than(), solution Differentiate between an expression and an equation Translate a quantitative verbal sentence into a mathematical equation Represent a situation with an algebraic equation Translate a quantitative verbal sentence into a mathematical inequality Represent a situation with an algebraic inequalityA.A.1 Translate a quantitative verbal phrase into an algebraic expressionA.A.3 Distinguish the difference between an algebraic expression and an algebraic equationA.A.4 Translate verbal sentences into mathematical equations or inequalitiesA.A.5 Write algebraic equations or inequalities that represent a situationA.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variableA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept

4 Analyze and Solve Verbal Problems (Algebraic solutions of equations/inequalities not applied) Vocabulary review vocabulary from 1.1 through 1.4, formula, strategy, appropriate, interpret, explain, elicit, check, formula, quantity, formulate Analyze a verbal problem whose solution requires solving a linear equation in one variable and formulate a solution plan Analyze a verbal problem whose solution requires solving a linear inequality in one variable and formulate a solution plan Apply Problem solving strategies to determine a problem solving plan & solution Verify the appropriateness of a solution Apply formulas to solve for given quantityA.A.5 Write algebraic equations or inequalities that represent a situationA.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variableA.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) A.PS.6 Use a variety of strategies to extend solution methods to other problemsA.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutionsA.PS.10 Evaluate the relative efficiency of different representations and solution methods of a problemA.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problemA.CM.6 Support or reject arguments or questions raised by others about the correctness of mathematical workA.CM.8 Reflect on strategies of others in relation to ones own strategyA.CM.11 Represent word problems using standard mathematical notation

5 Functions Vocabulary relation, function, domain, range, ordered pair, vertical line test, independent variable, dependent variable Define and identify functions given their input/output tables Determine if an input/output table is a function using its ordered pairs Verbalize based on a graph or table if a function exists (ex: It is a function because there is one y coordinate for each x coordinate) Identify the domain and range of a function Identify the independent and dependent variable Given a scatter plot, determine whether it is a function Apply the vertical line test Given the equation of a linear function, with a limited domain, determine its input/output table Problem solving applicationsA.PS.3 Observe and explain patterns to formulate generalizations and conjecturesA.G.3 Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations

6 Introduction to the Real Numbers Vocabulary natural numbers, whole numbers, opposites, integers, rational numbers, irrational numbers, real numbers, decimals, fractions, set, element, union, intersection, square root, perfect square, absolute value, symbol: | | Define absolute value Determine the absolute value of a number Evaluate absolute value expressions with multiple absolute values. (Ex. |3 7| 2|1 + 9|) Define and identify the following sets and their symbolic representation: Natural Numbers, Whole Numbers, Integers, Rationals, Irrationals, & Real Numbers (include their roster form) Refer to historical development of each number set Define and identify undefined terms Compare & rank order the Real Numbers in different forms (decimals, fractions, integers, & radicals) using a number line Classify a number to which set it is a member of: whole, natural, integer, rational, irrational & real Rank order the Real Numbers in different forms (decimals, fractions, integers, & radicals) Compare & order the Real Numbers in different forms (decimals, fractions, integers, & radicals) using , Determine the exact square root of a rational number Approximate the value of the square root of a irrational number Apply the calculator to find the square root of a number exactly or to a given decimal place Problem solving applicationsA.N.2 Simplify radical terms (no variable in the radicand)A.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written formA.R.2 Recognize, compare, and use an array of representational formsA.CN.8 Develop an appreciation for the historical development of mathematicsA.PS.3 Observe and explain patterns to formulate generalizations and conjecturesA.RP.3 Recognize when an approximation is more appropriate than an exact answer8.NS.1 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.8.NS.2Use 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 expression.8.EE.2Use 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 roots of small perfect cubes. Know is irrational.

7 Addition, Subtraction, Multiplication, Division of Integers, Fractions & Decimals Vocabulary integers, fractions, decimals, additive identity, additive inverse, opposites Define and apply the rules for addition & subtraction of Integers, fractions, and decimals Review operations with mixed numbers & improper fractions Add and subtract positive/negative of integers, fractions, and decimals without the calculator Add and subtract positive/negative of integers, fractions, and decimals with the calculator (stress difference between negative key & subtraction key on calculator) Define and apply the rules for multiplication & division Integers, fractions, and decimals Multiply & divide integers, fractions, and decimals without the calculator Multiply & divide integers, fractions, and decimals with calculator Use calculator to express division in fractional form Verbal applications (expressions not equations, stress difference between an equation and an expression)A.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.CM.11 Represent word problems using standard mathematical notationA.CN.6 Recognize and apply mathematics to situations in the outside worldA.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematicsA.R.4 Select appropriate representations to solve problem situations

8 Properties of the Real Numbers & Distributive Property Vocabulary closure, commutative, associative, distributive, identity, opposite, additive inverse, multiplicative inverse, logical argument, commutative property of addition, commutative property of multiplication, associative property of addition, associative property of multiplication, distributive property of multiplication over addition, identity element of addition, identity element of multiplication, equivalent expressions, term, coefficient, constant, like terms, variable, combining like terms Identify and apply the properties of Real Numbers (closure, commutative property of addition, commutative property of multiplication, associative property of addition, associative property of multiplication, distributive property of multiplication over addition, identity element of addition, identity element of multiplication, additive inverse, multiplicative inverse) Note: Students do not need to identify groups & fields, but students should be engaged in the ideas. Apply properties to simplify calculations (i.e. (25)(37)(4) = (25)(4)(37) = (254)(37) = (100)(37) = 3700) Define and apply the distributive property to numerical expressions (no variables) Define and apply the distributive property to algebraic expressions (with degree 2 or lower) Applications using the distributive property Ex. 13(12) = 13( 10 + 2) = 13(10) + 13(2) = 130 + 26 = 156 Ex. 13(95) = 13(100 5) = 13(100) 13(5) = 1300 65 = 1235A.N.1Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverseA.RP.4 Develop, verify, and explain an argument, using appropriate mathematical ideas and languageA.RP.7 Evaluate written arguments for validityA.RP.9 Devise ways to verify results or use counterexamples to refute incorrect statements.A.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept

Unit 2 Solutions of Linear Equations

1 Solution of 1-step, 2-step, & multi-step equations applying addition, subtraction, multiplication, or division Vocabulary inverse operations, equivalent equations, additive inverse, opposite, reciprocal, multiplicative inverse, like terms, solutions, check, simplify, distributive property Distinguish the difference between an algebraic expression and an algebraic equation (i.e. Can you solve 3x + 9?) Given algebraic expressions and algebraic equations determine which can be solved and which cannot be solved Solve a one-step equation with addition, subtraction, multiplication or division Solve two-step & multi-step linear equations containing variable one side Solve multi-step equations by applying the distributive property Justify the solution (check) Problem solving applicationsA.A.3 Distinguish the difference between an algebraic expression and an algebraic equation A.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variableA.A.22 Solve all types of linear equations in one variableA.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problemA.RP.2 Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjectureA.RP.9 Devise ways to verify results or use counterexamples to refute incorrect statementsA.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

2 Solving Equations with variables on both sides Solve multi-step equations with variables on both sides of equation Solve multi-step equations that apply the distributive property Solve multi-step equations that apply the distributive property where there is a negative outside the parentheses Solve equations that lead to x = 0 Justify the solution (check) Problem solving applications (include consecutive integers, consecutive even & consecutive odd)A.A.5 Determine and construct an algebraic equations or inequalities that represent a situationA.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variableA.A.22 Solve all types of linear equations in one variableA.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problemA.RP.2 Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjectureA.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

3 Solving Literal Equations Vocabulary - literal equation, formula Define & identify literal equations Solve for a given variable in a literal equation Apply formulas to solve for a certain variable Problem solving applicationsA.A.23 Solve literal equations for a given variableA.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

4 Solving Proportions Vocabulary ratio, proportion, simplest form, cross product, cross product property Find the ratio of two quantities in simplest form Solve proportions that lead to linear equations Applications of ratios Applications of proportions Problem solving applicationsA.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variationA.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variableA.A.22 Solve all types of linear equations in one variableA.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equationsA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

5 Conversions Vocabulary metric system, customary system, inches, feet, etc. Given a relationship between two units, convert within the measurement system Calculate rates Solve verbal problems involving conversions within measurement systems given the relationship between the units (i.e. Ellen traveled six times as far as Beth. If Ellen traveled 150 km, how many meters did Beth travel? 1 km = 1,000 m or A dress on sale was reduced to $25. If that was 1/6 of the original price, find the original price in Euros. The current conversion is 1 Euro = $1.50) Include examples where students need to do multiple conversions to arrive at an answer. A.A.5 Write algebraic equations or inequalities that represent a situationA.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variableA.M.1 Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail)A.M.2 Solve problems involving conversions within measurement systems, given the relationship between the unitsA.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) A.PS.6 Use a variety of strategies to extend solution methods to other problemsA.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solvingA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

6 Solving Word Problems in Percents, leading to linear equations Vocabulary percent, proportion, percent of change, percent of increase, percent of decrease, discount Determine the percent of a number (ex. What is 9% of 12?) Determine the percentage (ex. What % of 20 is 17?) Given a percentage, determine the number (ex. 19 is 50% of what number?) Determine the percent of increase Determine the percent of decrease Determine the percent of change Problem solving applicationsA.A.5 Write algebraic equations or inequalities that represent a situationA.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variationA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

7 Applications Apply knowledge of linear equations to the solution of consecutive integers word problems Apply knowledge of linear equations to the solution of coin word problems Apply knowledge of linear equations to the solution of break even word problems Stress the use of let statements and give examples where they are appropriate. A.A.5 Write algebraic equations or inequalities that represent a situationA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

Unit #3 Geometry

8.G.1Understand congruence and similarity using physical models, transparencies, or geometry software.1. Verify experimentally the properties of rotations, reflections, and translations:a. Lines are taken to lines, and lime segments to lime segments of the same length.b. Angles are taken to angles of the same measure.c. 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 the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

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

4. Understand 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; 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 the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Unit 4 Graphing Two Variable Linear Equations

1 Introduction to graphing linear equations Vocabulary coordinate plane, abscissa, ordinate, ordered pair, x-coordinate, y-coordinate, x-axis, y-axis, axes, quadrants, origin, linear function, vertical, horizontal, slant, oblique Define & identify coordinate plane, abscissa, ordinate, ordered pair, quadrants, axes Plot points (include those that lie on the x & y axis) Explore & generalize how to determine the quadrant in which a point lies Identify the quadrant in which a point lies (stress quadrantals, and the Roman numeral names of quadrants) Introduce general form for equation of a line (Equation of a line can have only x1, y1, or x1 and y1. General form is Ax + By = C) Discuss the concept that a line is a linear function Given a linear function, with a limited domain, set up a table & graph the points (not a line) Given an ordered pair, check whether it is a solution to an equation Given a linear equation in two variables, using the table method, graph the line (this is to be done by hand, not the calculator, the students are not putting the equation in y = mx + b form) Given a graph, determine the ordinate when given abscissa Given a graph, determine the abscissa when given ordinate Introduce horizontal lines Graph horizontal lines Introduce vertical lines Graph horizontal linesA.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical conceptsA.A.4 Translate verbal sentences into mathematical equations or inequalitiesA.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variableA.A.36 Write the equation of a line parallel to the x- or y-axisA.A.39 Determine whether a given point is on a line, given the equation of the lineA.PS.3 Observe and explain patterns to formulate generalizations and conjecturesA.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CM.4 Explain relationships among different representations of a problem

8.F.5Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

2 Graphing Linear Equations using x & y intercepts Vocabulary linear equation, x-intercept, y-intercept, general form of a line Given the graph of a line determine the x and y intercepts Develop the concepts of intercepts and their connections to the equation of a line Explore, conjecture & generalize about the x & y-intercepts of horizontal & vertical lines Given the equation of line in Ax + By = C form, find the x-intercept (A or B or C can equal zero) Given the equation of line in Ax + By = C form, find the y-intercept (A or B or C can equal zero) Given the equation of a line use the x and y intercepts to graph the line Conjecture & rationalize the appropriateness of this method and when other methods are more appropriate (intercept is a fraction)A.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical conceptsA.A.4 Translate verbal sentences into mathematical equations or inequalitiesA.A.36 Write the equation of a line parallel to the x- or y-axisA.A.39 Determine whether a given point is on a line, given the equation of the lineA.PS.3 Observe and explain patterns to formulate generalizations and conjecturesA.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CM.4 Explain relationships among different representations of a problem

3 Slope of a Line Vocabulary slope, rate of change, positive slope, negative slope, zero slope, undefined slope Define and introduce slope as a rate of change (do not simply use x and y as the variables) Discover by looking at a graph, if the slope of a line is positive, negative, zero, or no slope (stress difference between zero & no slope) Determine the exact slope of a line by looking at its graph Determine the slope of a line given in any form Discover the formula for slope of a line by investigating graphs of lines Apply the slope formula to find the slope of a line given two points on the line Apply the slope formula to find the rate of change Given a point on a line & the slope of a line, find the coordinates of another point on that line Given the slope of a line & a point on the line, find the missing x or y-coordinate of the point (i.e. A line with a slope of goes through the points (2, 7) and (6, y). What is the value of y?)A.A.32 Explain slope as a rate of change between dependent and independent variablesA.A.33 Determine the slope of a line, given the coordinates of two points on the lineA.A.36 Write the equation of a line parallel to the x- or y-axisA.A.37 Determine the slope of a line, given its equation in any formA.CM.7 Read and listen for logical understanding of mathematical thinking shared by other studentsA.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students conjectures

8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships in different ways. For example, compare a distance-time graph to a distance-tme equation to determine which of two moving objects has greater speed.8.EE.6Use 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 the equation y=mx+b for a line intercepting the vertical axis at b.

4 Graphing Linear Equations using Slope-Intercept form (y = mx + b) Vocabulary slope, y-intercept, slope intercept form, parameter changes, general form of a line, m, b Given an equation in y = mx + b form, identify the slope and y-intercept Given an equation of a line, write as an equivalent equation in slope-intercept form Graph a linear equation given in slope-intercept form by using the slope & y-intercept Given the line in Ax + By = C form, apply the slope-intercept form to graph Given the equation of a line in any form, graph by hand Given the equation of a line in any form, graph by using the graphing calculator Discuss direct variation (i.e. what happens when y-intercept = 0) Conjecture & generalize the effects of changing the slope or y-intercept to a family of lines Determine if lines are parallel, given the equation in any form Problem solving applicationsA.A.37 Determine the slope of a line, given its equation in any formA.A.38 Determine if two lines are parallel, given their equations in any formA.G.4 Identify and graph linear, quadratic (parabolic), absolute value,and exponential functionsA.CN.2 Understand the corresponding procedures for similar problems or mathematical conceptsA.RP.10 Extend specific results to more general cases

Unit 5 Equation of Lines

1 Slope-Intercept form (without calculator) & Families of Lines Vocabulary slope, y-intercept, slope-intercept form, parallel Conjecture & generalize the effects of changing the slope or y-intercept to a family of lines Draw conclusions of the effects of m & b on a line Given the slope & y-intercept of a line, determine the equation Given the slope & y-intercept of a line, determine if a point lies on that line Determine the equation of a line, given its slope and y-intercept Determine the equation of a line, given its slope and the coordinates of a point on the line Determine the equation of a line, given the graph of a line that includes the y-intercept Determine the equation of a line, given two points Determine the equation of a line, given its graph Determine the equation of a line, parallel to the x- or y-axis Problem solving applicationsA.A.5 Write algebraic equations or inequalities that represent a situationA.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line A.A.35 Write the equation of a line, given the coordinates of two points on the lineA.A.36 Write the equation of a line parallel to the x- or y-axisA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical conceptA.PS.3 Observe and explain patterns to formulate generalizations and conjecturesA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written formA.CN.2 Understand the corresponding procedures for similar problems or mathematical conceptsA.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

2 Equations of lines continued Vocabulary y-intercept, slope, slope-intercept form, parallel Determine the equation of a line in slope-intercept form, given its slope and a point on the line Determine the equation of a line in slope-intercept form, given any two points on the line Determine the equation of a line given its graph Problem solving applicationsA.A.33 Determine the slope of a line, given the coordinates of two points on the lineA.A.35 Write the equation of a line, given the coordinates of two points on the lineA.A.36 Write the equation of a line parallel to the x- or y-axisA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical conceptA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CN.2 Understand the corresponding procedures for similar problems or mathematical conceptsA.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

3 Point-Slope form of a line Vocabulary point-slope form, two-point form, parallel Define & apply the point-slope form Determine the equation of a line given the slope & a point Graph a line given in point-slope form Define & apply two-point form Graph a line given in two-point form Problem solving applicationsA.A.34 Write the equation of a line, given its slope and the coordinates of a point on the lineA.A.35 Write the equation of a line, given the coordinates of two points on the lineA.A.36 Write the equation of a line parallel to the x- or y-axisA.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts

4 Standard Form Equation of a Line Vocabulary standard form, Ax + By = C, parallel Define and identify a line in standard form Determine the standard form equation of a line, parallel to the x- or y- axis Discover equivalent standard form equations (Ax + By = C and its multiple) Reason about A, B & C and their values in parallel lines Determine the equation in standard form, given the slope & y-intercept Determine the equation in standard form, given the slope & a point on the line Determine the equation in standard form, given two points on the line Determine the slope of a line, given the equation in any form Problem solving applicationsA.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line A.A.35 Write the equation of a line, given the coordinates of two points on the lineA.A.37 Determine the slope of a line, given its equation in any formA.A.38 Determine if two lines are parallel, given their equations in any formA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept

5 Equations of Parallel Lines & Perpendicular Lines Vocabulary parallel Define & discuss parallel & perpendicular lines and their slopes Determine if two lines are parallel or perpendicular given their graphs Given the equation of two lines in any form, determine if they are parallel & perpendicular Determine the equation of a line parallel to a given line and passing through a given point Problem solving applicationsA.A.38 Determine if two lines are parallel, given their equations in any formA.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept

6 Best Fit Line Vocabulary scatter plot, univariate data, bivariate data, positive correlation, negative correlation, no correlation Define and recognize the difference between univariate and bivariate data Given a set of data, determine whether it is univariate or bivariate Create and investigate a scatter plot of bivariate data (minimal calculator use) Construct manually a reasonable line of best fit for a scatter plot Determine the equation of the reasonable line of best fit MANUALLY Determine predictions from line of best fit using linear interpolation and linear extrapolation Justify predictions found from line of best fit Verbal applicationsA.S.2 Determine whether the data to be analyzed is univariate or bivariateA.S.7 Create a scatter plot of bivariate dataA.S.8 Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that lineA.S.12 Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)A.S.17 Use a reasonable line of best fit to make a prediction involving interpolation or extrapolationA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)A.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions A.PS.9 Interpret solutions within the given constraints of a problem

8.SP.4Understand that patterns of association can also 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.

7 Applications of Scatter Plots Vocabulary scatter plot, best-fit line, linear regression, interpolation, extrapolation, zero of a function Create and investigate a scatter plot of bivariate data Define linear interpolation and linear extrapolation Apply linear interpolation & linear extrapolation to predictions Problem solving applicationsA.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutionsA.PS.9 Interpret solutions within the given constraints of a problemA.RP.2 Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjectureA.RP.3 Recognize when an approximation is more appropriate than an exact answerA.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CM.8 Reflect on strategies of others in relation to ones own strategyA.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situationsA.R.4 Select appropriate representations to solve problem situationsA.R.5 Investigate relationships between different representations and their impact on a given problemA.S.7 Create a scatter plot of bivariate dataA.S.8 Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that lineA.S.17 Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation

Unit 6 Solving & Graphing Linear Inequalities

1 Solving Linear Inequalities in one variable Vocabulary - , , , , inequality, solution set, braces {}, number line Define linear inequalities Graph linear inequalities on a number line (stress open & closed circle) Determine the values that make the inequality true or false Determine the linear inequality represented by a graph on the number line Define and apply the Addition/Subtraction property of inequality Define and apply the Multiplication/Division property of inequality Explain & prove the effect of multiply or dividing by a negative in a given inequality Solve linear inequalities in one variable Graph the solution to a linear inequality on a number lineA.A.5 Write algebraic equations or inequalities that represent a situationA.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variableA.A.24 Solve linear inequalities in one variable A.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.RP.12 Apply inductive reasoning in making and supporting mathematical conjecturesA.CN.1 Understand and make connections among multiple representations of the same mathematical idea

2 Applications of Linear Inequalities Translate verbal sentences into mathematical inequalities and solve. Translate real life situations into mathematical inequalities and solve. Determine acceptable solutions to applications of linear inequalities.A.A.4 Translate verbal sentences into mathematical equations or inequalitiesA.A.5 Write algebraic equations or inequalities that represent a situationA.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variableA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.PS.8 Determine information required to solve a problem, chose methods for obtaining the information, and define parameters for acceptable solutions. A.CM.11 Represent word problems using standard mathematical notationA.CN.6 Recognize & apply mathematics to situations in the outside world.

3 Compound Inequalities in one Variable Vocabulary compound inequality, intersection, union, and, or Define and graph on the number line compound inequalities with no solution needed (ex. -2 x < 5) Solve compound inequalities (and ex. -15 < 15p 10 25) Solve compound inequalities with and (ex. -2 < 3x + 1 and 7x 3 < 19) Solve compound inequalities with or Graph the solution set to a compound inequality (stress open & closed circle) Write the solution to a compound inequality as a solution set with braces (ex. {x| -2 < x < 5} Solve and graph compound inequalities on a number line using the graphing calculator (stress inability to see open/closed circle on calculator) Problem solving applicationsA.A.5 Write algebraic equations or inequalities that represent a situationA.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variableA.A.24 Solve linear inequalities in one variable A.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.RP.12 Apply inductive reasoning in making and supporting mathematical conjecturesA.CM.11 Represent word problems using standard mathematical notationA.CN.1 Understand and make connections among multiple representations of the same mathematical idea

4 Solving Absolute Value Equations Vocabulary absolute value, symbol: | |, absolute value equation Solve absolute linear equationA.A.22 Solve all types of linear equations in one variable

5 Graphing Absolute Value Functions Vocabulary absolute value, function Properties of the parent function y = |x| Graph absolute value linear functions with the calculator Compare & contrast the graphs of linear absolute value functions with the parent functionA.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s)A.A.22 Solve all types of linear equations in one variable

6 Graphing Linear Inequalities in Two Variables On the Coordinate Plane Vocabulary coordinate plane, linear inequality, solution Given an ordered pair, determine if it is a solution of a two variable inequality Graphing a linear inequality in one variable Graphing a linear inequality in two variables given in any form Graph linear inequalities using the graphing calculator (with shading) Determine a point in the solution set & verify your solution Determine a point not in the solution set & prove it is not in the solution setA.G.6 Graph linear inequalitiesA.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable

Unit 7 Systems of Equations & Inequalities

1 Solving Linear Systems Graphically Vocabulary linear system, solution, consistent system, inconsistent system, consistent dependent Given a graph of a linear system, find the solution and check algebraically Review graphing lines in any form Review graphing lines in the calculator Graph a linear system to find the solution and check algebraically (include lines that are parallel to x- & y- axis) Discuss and identify a system with no solution (inconsistent) Discuss and identify a system with an infinite number of solutions (consistent dependent) Find and label the solution Problem solving applicationsA.A.10 Solve systems of two linear equations in two variables algebraicallyA.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variableA.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variablesA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.CM.11 Represent word problems using standard mathematical notationA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

2 Solving Linear Systems by Substitution Vocabulary - linear system, solution, consistent system, inconsistent system Discuss the steps in the problem solving process Solve a linear system by substitution and check Setup and solve a system of equations from verbal applications Verbal problem applicationsA.A.7 Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variablesA.A.10 Solve systems of two linear equations in two variables algebraicallyA.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variablesA.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problemA.CN.6 Recognize and apply mathematics to situations in the outside worldA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

3 Solving Linear Systems by Elimination Vocabulary - linear system, solution, consistent system, inconsistent system, elimination, least common multiple Convert a system as an equivalent system with both equations in standard form (Ax + By = C) Solve a linear system by elimination when addition can be used to eliminate a variable Solve a linear system by elimination when multiplying one of the equations by -1 can be used to eliminate a variable Solve a linear system by elimination when one equation must be multiplied by a constant to eliminate a variable Solve a linear system by elimination when both equations must be multiplied by a constant to eliminate a variable Verify solution by substitution into each of the original equationsA.A.10 Solve systems of two linear equations in two variables algebraicallyA.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

4 Solving Systems of Linear Equations Algebraically Vocabulary - linear system, solution, consistent system, inconsistent system, elimination, least common multiple Determine and apply the most appropriate method for solving a system of linear equations Problem solving applications (include all past Regents problems from parts II, III, and IV) Solve coin problems that involve systems of linear equation Include sum and difference problems (Ex: The sum of two numbers is 47, and their difference is 15. What is the larger number?) Include examples where the students must find only the x coordinate or y coordinate. A.A.5 Write algebraic equations or inequalities that represent a situationA.A.7 Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variablesA.A.10 Solve systems of two linear equations in two variables algebraically (See A.G.7)A.R.4 Select appropriate representations to solve problem situationsA.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)A.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problemA.CN.6 Recognize and apply mathematics to situations in the outside worldA.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies.A.PS.1 Use a variety of Problem solving applications strategies to understand new mathematical contentA.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)A.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)A.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions A.PS.9 Interpret solutions within the given constraints of a problemA.PS.10 Evaluate the relative efficiency of different representations and solution methods of a problem

5 Solve Systems of Linear Inequalities Graphically Vocabulary linear inequalities in two variables, systems of linear inequalities Review graphing a linear inequality on a coordinate plane Graph a system of linear inequalities on a coordinate plane Graph a system of linear inequalities on a coordinate plane using the graphing calculator Determine a point in the solution set of the linear inequality system Label the solution set on the graph Determine whether a given point is in the solution set of the linear inequality system from the graph Determine whether a given point is in the solution set of the linear inequality system algebraicallyA.A.40 Determine whether a given point is in the solution set of a system of linear inequalitiesA.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables

Unit 8 Exponents & Exponential Functions

1 Laws of Exponents Involving Products Vocabulary power, base, exponent, reciprocalApply positive, negative, and zero exponents to all of the below (integral exponents only) Final answers may not be left with negative exponents Define x0, xa, x a Evaluate expressions Multiply two powers of the same base (do examples with the like bases as numbers and/or variables) Multiply monomials by monomials Power of a Power Property Power of a Product Property Product of a Power Property Define and apply zero and negative exponents Compare and rank different bases raised to powers Applications with verbal problems (i.e. area of a square, area of a rectangle)A.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents (Note: Use integral exponents only.)A.PS.3 Observe and explain patterns to formulate generalizations and conjectures

2 Laws of Exponents involving Quotients Vocabulary - power, base, exponent, quotient, reciprocalApply positive, negative, and zero exponents to all of the below (integral exponents only) Power of a Quotient Property Quotient of a Power Property Define and apply zero and negative exponentsA.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents (Note: Use integral exponents only.)A.PS.3 Observe and explain patterns to formulate generalizations and conjectures

3 Laws of Exponents Mixed Review

4 Scientific Notation Vocabulary scientific notation, standard form of a number Define scientific notation Given a number in standard form, convert to scientific notation Given a number in scientific notation, convert to standard form Determine the product/quotient of numbers in scientific notation Apply the graphing calculator to convert from one form to another Apply the graphing calculator to add, subtract, multiply and divide in scientific notation Include examples where students find the product or quotient of two numbers where one is in scientific notation and the other is not. A.N.4 Understand and use scientific notation to compute products and quotients of numbers

8.EE4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.

5 Applications of Exponential Functions Vocabulary exponential function, exponential growth, exponential decay, initial amount Generalize y = ax, where a > 1, as exponential growth Generalize y = ax, where 0 < a < 1, or y = a -x where a > 1as exponential decay Compare and contrast exponential growth and decay Stress the difference in the exponential growth, y = a(1 + r)x, and exponential decay, y = a(1 r)x , STUDENTS MUST MEMORIZE THESE!! Identify a as the initial amount, r as the rate expressed as a decimal, and x as the time. Problem solving applicationsA.A.9 Analyze and solve verbal problems that involve exponential growth and decayA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)A.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)A.CN.6 Recognize and apply mathematics to situations in the outside worldA.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematicsA.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students conjectures

6 Graphs of Exponential Functions Vocabulary exponential function, exponential growth, exponential decay, initial amount, asymptote Define & graph y = ax, where a > 1, with a graphing calculator Define & graph y = ax, where 0 < a < 1, with a graphing calculator Compare and contrast the graphs where a > 1 with the graphs where 0 < a < 1 Generalize y = ax, where a > 1, as exponential growth Generalize y = ax, where 0 < a < 1, as exponential decay Compare and contrast exponential growth and decay Identify a graph as exponential growth or exponential decay Identify and define the asymptote of the exponential function Define & graph y = a bx, where a is an integer and b is a natural number Compare graphs of exponential functions Problem solving applicationsA.A.9 Analyze and solve verbal problems that involve exponential growth and decayA.G.4Identify and graph linear, quadratic (parabolic), absolute value, and exponential functionsA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)A.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

A.R.8 Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations and )A.CN.6 Recognize and apply mathematics to situations in the outside worldA.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematicsA.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students conjectures

Unit 9 Polynomials

1 Adding and Subtracting Polynomials Vocabulary variable, constant, degree, exponent, degree of polynomial, monomial, binomial, trinomial, polynomial, leading coefficient, simplify Define and identify polynomial, monomial, binomial and trinomial Define and identify the degree of a polynomial Identify a polynomial in standard form Put a polynomial in standard form Determine the leading coefficient of a polynomial Add polynomials Subtract polynomials (include subtract from problems)A.A.13 Add, subtract, and multiply monomials and polynomials

2 Multiplying Polynomials & Special Products Vocabulary variable, constant, exponent, monomial, binomial, trinomial, polynomial, standard form, simplify Multiply a monomial by a monomial Multiply a polynomial by a monomial using the distributive property Multiply two binomials using distributive property Multiply two binomials using FOIL method Multiply a polynomial by a binomial Find the product of two polynomials Identify the product of the sum and difference of two binomials ex. (2x 1)(2x + 1) Determine the product of the sum & difference of two binomials ex. (2x 1)(2x + 1) Determine the square of a binomials ex. (3x + 5)2 Stress that final answer must be in standard form Problem solving applications Ex. Geometric problems, area shaded and word problemsA.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents (Note: Use integral exponents only.)A.A.13 Add, subtract, and multiply monomials and polynomialsA.PS.3 Observe and explain patterns to formulate generalizations and conjectures

3 Factoring GCF Vocabulary equation, zero product property, GCF Find the GCF of a pair of numbers, by hand and with the calculator Find the GCF of a pair of monomials Introduce factoring of polynomials Find the greatest common monomial factor of a polynomial Factor a polynomial by factoring out a GCFA.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF)

4 Factoring ax2 + bx + c Vocabulary trinomial, FOIL backwards, factoring, factors Use same method when a = 1 and a 1 Factor trinomials by FOIL backwards (this is only one method of factoring there are many other perfectly acceptable methods (i.e. possible factors method (see below), area rugs) i.e. x2 + 7x + 10 (FOIL Backwards) step 1: multiply ax2 by c x2 * 10 = 10x2 step 2: find two factors whose sum is bx and product is acx2 2x, 5x step 3: write factors from step 2 between ax2 and c x2 + 2x + 5x + 10 step 4: group and factor G.C.F. from first two terms and last two terms: x(x + 2) + 5(x +2) step 5: factor out common binomial factor (x + 5)(x + 2) Stress prime polynomials Stress the difference between polynomials and equations ( no solutions to polynomials)A.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF)A.PS.1 Use a variety of problem solving strategies to understand new mathematical contentA.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)A.CM.4 Explain relationships among different representations of a problemA.CN.1 Understand and make connections among multiple representations of the same mathematical ideaA.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

5 Factoring Special Products Vocabulary binomial, trinomial, FOIL backwards, factoring, factors Stress that the difference of two perfect squares is actually ax2 + 0x c Identify quadratic binomial Write in factored form as sum and difference Verbal problem applicationsA.A.19 Identify and factor the difference of two perfect squares

6 Factor Completely Vocabulary - binomial, trinomial, FOIL backwards, factoring, factors, completely Three steps to factoring completely 1st Factor out GCF 2nd Factor trinomials 3rd Difference of two perfect squares Identify and factor out GCF in a polynomial (numerical only) and then factor the remaining polynomial into two binomials (remaining polynomial will have a leading coefficient of 1)A.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF)A.PS.1 Use a variety of problem solving strategies to understand new mathematical contentA.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)A.CM.4 Explain relationships among different representations of a problemA.CN.1 Understand and make connections among multiple representations of the same mathematical ideaA.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

7 Solving Quadratic Equations by FactoringVocabulary algebraic expression, algebraic equation, multiplication property of zero, roots Distinguish the difference between an algebraic expression and an algebraic equation Determine the standard form of the quadratic equation Understand and apply multiplication property of zero Solve quadratic equations by factoring (integral coefficients and integral roots) Explain the difference between factors and rootsA.A.8 Analyze and solve verbal problems that involve quadratic equationsA.A.27 Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral rootsA.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression

1 day on equations

1 day on applications

Unit 10 Quadratic Equations

1 Parent Quadratic Function Vocabulary quadratic function, parabola, parent quadratic function, vertex, axis of symmetry Define quadratic functions and parabola (stress that a parabola is a function) Define parabola in standard form Examine the graph of y = x2, looking at the curve, the vertex and the axis of symmetry Graph y = ax2, by the table method, for both positive & negative values of a Graph y = ax2 + c, by the table method for both positive & negative values of c Generalize the effects of different values for a & c Graph all the above using the graphing calculatorA.G.3 Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relationsA.G.4 Identify and graph linear, quadratic (parabolic), absolute value, and exponential functionsA.G.5 Investigate and generalize how changing the coefficients of a function affects its graph

2 Graphing Parabolas & Solving Quadratic Equations by Graphing Vocabulary - quadratic function, parabola, standard form, vertex, axis of symmetry, zeros of a function, roots, solution set(Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.) Define & identify the standard form of a parabola: y = ax2 + bx + c Identify a parabola given its equation or graph Determine axis of symmetry (use x = -b/2a) given an equation or a graph Define and determine vertex given an equation or a graph Define and identify maximum or minimum point by inspection of leading coefficient Apply axis of symmetry to set up a table of values Utilize table to graph parabola Determine the roots of the quadratic from the graph of the parabola (roots must be integral values) Emphasize the difference and connection between the roots of a quadratic equation and the factor of a quadratic expression Graph parabola on calculator Determine window of best fit Utilize calculator to find maximum or minimum point Copy table from calculator Copy graph on graph paper Answers must be written as a solution set with braces { } Define roots of an equation and identify on the graph Show the connections between the factors of a quadratic equation and the roots of a parabola Solve a quadratic equation by placing it in standard form & graphing Solving a quadratic equation having no solutions, one solution, and two solutions Find the zeros of a quadratic function Problem solving applicationsA.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expressionA.A.41 Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 )A.G.8 Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.A.G.10 Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.

A.R.8 Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations and )A.RP.10 Extend specific results to more general solutions.

3 Solving Quadratic Equations ax2 + b = c Vocabulary perfect square, square root, irrational number, simplifying square roots Solve quadratic equations of the form x2 = c, by taking the square root of each side (stress that you will have two answers: one positive & one negative) Solving quadratic equations of the form ax2 = c by dividing by a, then taking the square root of each side Solve quadratic equations in the form ax2 + b = c Find solutions exactly, in radical form, or to the given decimal place Problem solving applicationsA.A.8 Analyze and solve verbal problems that involve quadratic equationsA.REI.4 Solve quadratic equations in one variable.a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from this form. *Note this is a prerequisite skill for High School Geometry.

4 Compare Linear, Exponential, Absolute Value & Quadratic Models Vocabulary linear function, exponential function, absolute value function, quadratic function Given a graph, identify the function as: linear, exponential, absolute value or quadratic Given an equation, identify the function as: linear, exponential, absolute value or quadratic Given points, draw a scatter plot & identify as: linear, exponential, absolute value or quadraticA.G.4Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in table, or by verbal descriptions)

Unit 11 Systems of Quadratic Linear Equations

1 Solving a Quadratic-Linear System Graphically Graph line and parabola and find intersection(s) Utilize calculator to graph line and parabola and find intersection(s) Check and label the solution set in both equations Stress that solutions may not appear on the table and that graphs sometimes need to be extended. Problem solving applicationsA.G.9 Solve systems of linear and quadratic equations graphically (Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.)

2 Solving a QuadraticLinear system algebraically Review solving systems of linear equations using substitution Solve a system of a line and parabola by substitution (The quadratic equation should represent a parabola and the solution should be integers.) Identify all solutions Check all solutions in both equations Problem solving applicationsA.A.11 Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.

Unit 12 Sets

1 Sets (use set symbolic notation) Define sets and subsets Define and identify the difference between a ( ) and [ ] and { } Apply set-builder notation and/or interval notation to illustrate the elements of a set (given the elements in roster form) Apply a number line to express the elements of a set (stress open & closed circles) Determine an inequality that represents a set Determine the subsets of a set (improper & proper subsets) Find the complement of a subset of a given set (within a given universe)A.A.29 Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster formA.A.30 Find the complement of a subset of a given set, within a given universe

2 - Union and Intersection of Sets (No more than 3 sets) Use proper set notation! Define and determine the union of 2 or 3 sets Define and determine the intersection of 2 or 3 sets Apply set notation! Problem solving applicationsA.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets)

3 - Venn Diagrams (No more than 3 sets) Define and give examples of Venn diagrams Construct Venn diagrams Construct a Venn diagram to support a logical argument Problem solving applicationsA.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets)A.RP.11 Use a Venn diagram to support a logical argument

Unit 13 Radicals

1 Simplifying Radical Expressions (no variables in the radicand) Vocabulary square root, radical, radicand, simplest form Define square roots Simplify numeral square roots Find the product of two square roots and simply Simplify the square root of a quotient Find quotient of two square roots and simplify Find the sum of square roots and simplify Find the difference of square roots and simplify Rationalize the denominatorA.N.2 Simplify radical terms (no variable in the radicand)A.N.3 Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest formA.N.4 Understand and use scientific notation to compute products and quotients of numbers

2 Pythagorean Theorem Vocabulary right triangle, right angle, legs, hypotenuse, Pythagorean Theorem, a2 + b2 = c2 Define a right triangle and its parts Define Pythagorean Theorem as leg2 + leg2 = hyp2 Given two sides of a right triangle in numerical form, find the third side Given the sides of a triangle in variable form, solve for the variable & find the sides (this will lead to quadratic equations only where a = 1 after taking out the GCF) Given the length of three sides of a triangle, determine if the triangle is a right triangle Problem solving applicationsA.A.5 Write algebraic equations or inequalities that represent a situationA.A.45 Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides8.G.6Explain a proof of the Pythagorean Theorem and its converse.

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

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

Unit 14 Rational Expressions & Equations

1 Simplify Rational Expressions Vocabulary rational expression, excluded value, undefined, fractional expression Define & identify rational expressions Find the value for which a rational expression is undefined (NYS term is undefined, Textbook term is excluded value) Simplify a rational expression by dividing out the GCF Simplify a rational expression by dividing out binomials & monomials Geometric problem solving applicationsA.A.14 Divide a polynomial by a monomial or binomial, where the quotient has no remainderA.A.15 Find values of a variable for which an algebraic fraction is undefinedA.A.16 Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms

2 Multiplying & Dividing Rational Expressions Vocabulary multiplicative inverse, reciprocal Only problems that factor (no long division) Multiply rational expressions involving monomials and simplify the answer Multiply rational expressions involving polynomials and simplify the answer Multiply a rational expression by a polynomial and simplify the answer Divide rational expressions involving polynomials and simplify the answer Divide a rational expression by a polynomialA.A.18 Multiply and divide algebraic fractions and express the product or quotient in simplest form

3 Adding & Subtracting Rational Expressions (only with monomial or LIKE binomial denominators) Vocabulary Least Common Denominator (LCD) Adding and subtracting rational expressions with common monomial denominators Adding and subtracting rational expressions with UNLIKE monomial denominators Adding and subtracting rational expressions with LIKE binomial denominatorsA.A.17 Add or subtract fractional expressions with monomial or like binomial denominators

4 Solving Rational Equations Vocabulary rational equation, cross product, extraneous roots, check Stress difference between expression and equations Solve algebraic proportions that result in linear or quadratic equations that can be solved by factoring Solve by LCD Solve by product of the means = product of the extremesA.A.25 Solve equations involving fractional expressions (Note: Expressions which result in linear equations in one variable)A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equations

Unit 15 Geometric Figures

1 Polygons & Circles Vocabulary - triangle, quadrilaterals, parallelograms, rectangles, squares, rhombi, trapezoids, isosceles trapezoids, pentagons, hexagons, heptagons, octagons, nonagons, decagons, dodecagons, radius, center, diameter, circle, circumference, area Define and identify triangle, quadrilaterals, parallelograms, rectangles, squares, rhombi, trapezoids, isosceles trapezoids, pentagons, hexagons, heptagons, octagons, nonagons, decagons, dodecagons Given the name of the polygon, identify number of sides and number of angles Given the number of sides or number of angles in a polygon, determine the name of the polygon Define and identify regular polygons Define a circle Define radius and diameter Given the radius, find the diameter (vice-versa) Determine the circumference of a circle Determine the circumference of semi-circles & quarter-circles Determine the area of a circle Determine the area of semi-circles and quarter-circles Given the area, find the circumference of the circle (vice-versa) Verbal applicationsA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)

2 Perimeter Vocabulary perimeter, triangle, square, rectangle, polygon, rhombus, trapezoid, & regular polygon, complex figure Determine the perimeter of a triangle, square, rectangle, polygon, rhombus, trapezoid, & regular polygon Determine the perimeter of a complex figure (i.e. Figure has more than two shapes attached) Problem solving applicationsA.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms ,rhombuses, trapezoids, circles, semi-circles, quarter-circles ,and regular polygons (perimeter only).A.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)

1

3 Area Vocabulary area, triangle, square, rectangle, polygon, rhombus, trapezoid, & regular polygon, complex figure, regular polygon Develop and apply the formulas for area of a triangle, square, rectangle, polygon, rhombus, trapezoid, & regular polygon Determine the area of a complex figure (i.e. Figure that separates into more than one shape or shaded area) Calculate the relative error in measuring square and/or cubic units, when there is an error in the linear measure Verbal applicationsA.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).A.M.3 Calculate the relative error in measuring square and/or cubic units, when there is an error in the linear measureA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)

4 Volume Vocabulary - volume, cube, rectangular solid, cylinder Define the volume ONLY for: a cube, a rectangular solid, and a cylinder Apply volume formulas to find the volume of a cube, rectangular solid, and cylinder Given the volume, find the length of a missing dimension in a cube, rectangular solid, and cylinder Calculate the relative error in measuring square and/or cubic units, when there is an error in the linear measure Problem solving applicationsA.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylinders A.M.3 Calculate the relative error in measuring square and/or cubic units, when there is an error in the linear measureA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

5 Total Surface Area Define total surface area of a cube, rectangular solid, and cylinder Apply total surface area formulas to find the volume of a cube, rectangular solid, and cylinder Given the total surface area, find the length of a missing dimension in a cube, rectangular solid, and cylinder Calculate the relative error in measuring square and/or cubic units, when there is an error in the linear measure Verbal applicationsA.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylindersA.M.3 Calculate the relative error in measuring square and/or cubic units, when there is an error in the linear measureA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)

Unit 16 Right Triangle Trigonometry

1 - Introduction to Right Triangle Trigonometry Define sine, cosine and tangent functions Determine the opposite and adjacent legs to an angle Express the ratios for sine, cosine, & tangent, given a right triangle and the lengths of its sidesA.A.42 Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides

2 Right Triangle Trigonometry sides Determine a missing side given an angle & the length of a sideA.A.44 Find the measure of a side of a right triangle, given an acute angle and the length of another side

3 Right Triangle Trigonometry angles Determine a missing angle given two sidesA.A.43 Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle

4 Right Triangle Trigonometry applications Determine and apply the correct trigonometry ratio to solve a verbal right triangle problem Apply right triangle trigonometry to verbal problem with angle of elevation and angle of depression Include examples where the students must use the same triangle to answer multiple questions. A.RP.3 Recognize when an approximation is more appropriate than an exact answerA.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)A.CN.6 Recognize and apply mathematics to situations in the outside world

Unit 17 Probability

1 Empirical vs. Theoretical & Probability and complementVocabulary probability, empirical probability, theoretical probability, outcome, finite sample space, event, favorable event, independent event, dependent event, complement, certainty, impossibility, Define empirical probability Define theoretical probability Distinguish between empirical vs. theoretical Define outcome, finite sample space, event, & favorable event Distinguish between dependent and independent events Determine empirical probabilities based on specific sample data (simple one stage) Define probability and complement Calculate the probability of an event and its complement Define and apply certainty and impossibility Solve for simple probabilities in a finite sample space Determine, based on calculated probability of a set of events, if some or all are equally likely to occur Determine, based on calculated probability of a set of events, if one is more likely to occur than another Determine, based on calculated probability of a set of events, if whether or not an event is certain to happen or not to happenA.S.20 Calculate the probability of an event and its complementA.S.21 Determine empirical probabilities based on specific sample dataA.S.22 Determine, based on calculated probability of a set of events, if: some or all are equally likely to occur one is more likely to occur than another whether or not an event is certain to happen or not to happen

2 Multi-Stage ExperimentsVocabulary compound event, sample space, tree-diagram, mutually exclusive events, independent events, dependent events, counting principle, with & without replacement Determine the total number of outcome in a multi-stage experiment List the sample space of an experiment Determine the number of elements in a sample space Determine the number of favorable events in a sample space Construct a tree diagram Define and apply the concept of mutually exclusive events and not mutually exclusive events Calculate the probability of a series of independent events Calculate the probability of a series of dependent events (with & without replacement) Calculate the probability of two mutually exclusive events (with & without replacement) Calculate the probability of two events that are not mutually exclusive (with & without replacement)A.N.7 Determine the number of possible events, using counting techniques or the Fundamental Principle of CountingA.S.18 Know the definition of conditional probability and use it to solve for probabilities in finite sample spacesA.S.19 Determine the number of elements in a sample space and the number of favorable events2

3 Permutations & Applications Vocabulary factorial, x!, permutation, nPr Define factorial Evaluate expressions involving factorials Define permutations Evaluate permutations using the graphing calculator and by hand Determine the number of possible arrangements of a list of items with no repeats Determine the number of possible arrangements of a list of items with repeats Apply permutations to probability Problem solving applicationsA.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s)A.N.8 Determine the number of possible arrangements (permutations) of a list of items

4 Probability mixed review

Unit 18 Statistics

1 - Introduction to Statistics Vocabulary survey, sample, random sample, bias, correlation, causation, biased sampler, qualitative, quantitative Categorize data as qualitative versus quantitative Determine when collected data or display of data may be biased Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions Define and apply the difference between correlation and causation Identify variables that might have a correlation but not a causal relationship Identify and describe sources of bias and its effect on drawing conclusions from dataA.S.1 Categorize data as qualitative or quantitativeA.S.3 Determine when collected data or display of data may be biasedA.S.10 Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusionsA.S.13 Understand the difference between correlation and causationA.S.14 Identify variables that might have a correlation but not a causal relationshipA.S.15 Identify and describe sources of bias and its effect, drawing conclusions from dataA.CM.6 Support or reject arguments or questions raised by others about the correctness of mathematical workA.CM.9 Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjecturesof othersA.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situationsA.CN.5 Understand how quantitative models connect to various physical models and representations

2 Measures of Central Tendency Vocabulary mean, median, mode, range, measures of central tendency By hand & by calculator Define range, mean, median and mode Given a set of data find the range Given a set of data find the mean Given a set of data find the median Given a set of data find the mode Apply linear transformation to mean, median, mode and range and describe its effect Given a set of data, determine whether it is univariate or bivariate Compare & contrast the appropriateness of different measures of central tendency for a given data set Recognize how linear transformations of one-variable (univariate) data affect the datas mean, median, mode, and rangeA.S.4 Compare and contrast the appropriateness of different measures of central tendency for a given data setA.S.16 Recognize how linear transformations of one-variable data affect the datas mean, median, mode, and rangeA.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations

3 Frequency Distribution Tables Vocabulary frequency, frequency distribution tables, cumulative frequency distribution tables, histograms, cumulative histograms Define and interpret frequency distribution tables Define and interpret cumulative frequency distribution tables Construct frequency distribution tables Construct cumulative frequency distribution tables Analyze frequency distribution tables Analyze cumulative frequency distribution tables Verbal applicationsA.S.5 Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of dataA.S.9 Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plotA.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CN.5 Understand how quantitative models connect to various physical models and representationsA.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations

4 HistogramsVocabulary histogram, cumulative histogram By hand only Define and interpret histograms Define and interpret cumulative frequency histograms Construct histograms Construct cumulative frequency histograms Analyze histograms Analyze cumulative frequency histogramsA.S.5 Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of dataA.S.9 Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plotA.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CN.5 Understand how quantitative models connect to various physical models and representationsA.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations

5 Box & Whisker Plots Vocabulary box & whisker plot, quartiles, Q1, Q2, Q3, lower quartile, median quartile, upper quartile, inter quartile range, outlier By hand & by calculator Define box-and-whisker plots Given a box-and-whisker plot, identify the five statistical summary (minimum, maximum, and three quartiles) Apply the five statistical summary construct a box-and-whisker plot Identify point values in the 1st, 2nd, and 3rd quartiles (or lower, middle, and upper quartiles) Define and determine percentile ranks Analyze and interpret percentile ranks Problem solving applicationsA.S.5 Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of dataA.S.6 Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box-and-whisker plotA.S.9 Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plotA.S.11 Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartilesA.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagramsA.CN.5 Understand how quantitative models connect to various physical models and representationsA.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations

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