ALGEBRA 1 CURRICULUM COMMON CORE BASED

Transcription

1 ALGEBRA 1 CURRICULUM COMMON CORE BASED (Supplemented with 8th grade PSSA anchors ) UPPER MERION AREA SCHOOL DISTRICT 435 CROSSFIELD ROAD KING OF PRUSSIA, PA /20/2012

2 PA COMMON CORE ALIGNED MATHEMATICS CURRICULUM FRAMEWORK (DRAFT VERSION 2012) Mathematics LONG TERM TRANSFER GOALS Transfer goals highlight the effective uses of understanding, knowledge, and skill that we seek in the long run; i.e., what we want students to be able to do when they confront new challenges both in and outside of school. Students will be able to independently use their learning to: 1. Make sense of and persevere in solving complex and novel mathematical problems. 2. Use effective mathematical reasoning to construct viable arguments and critique the reasoning of others. 3. Communicate precisely when making mathematical statements and express answers with a degree of precision appropriate for the context of the problem/situation. 4. Apply mathematical knowledge to analyze and model situations/relationships using multiple representations and appropriate tools in order to make decisions, solve problems, and draw conclusions. 5. Make use of structure and repeated reasoning to gain a mathematical perspective and formulate generalized problem solving strategies. Big Ideas Mathematical relationships can be represented as expressions, equations, and inequalities in mathematical situations. Numerical quantities, calculations, and measurements can be estimated or analyzed by using appropriate strategies and tools. Data can be modeled and used to make inferences. Geometric relationships can be described, analyzed, and classified based on spatial reasoning and/or visualization. Mathematical relations and functions can be modeled through multiple representations and analyzed to raise and answer questions. Mathematical relationships among numbers can be represented, compared, and communicated. Measurement attributes can be quantified and estimated using customary and non-customary units of measure. Patterns exhibit relationships that can be extended described, and generalized. Essential Questions How are relationships represented mathematically? How can expressions, equations, and inequalities be used to quantify, solve, model and/or analyze mathematical situations? What does it mean to estimate or analyze numerical quantities? When is it is appropriate to estimate versus calculate? What makes a tool and/or strategy appropriate for a given task? How does the type of data influence the choice of display? How can probability and data analysis be used to make predictions? How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real situations or solve problems? How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving? How can geometric properties and theorems be used to describe, model, and analyze situations? How can data be organized and represented to provide insight into the relationship between quantities? How is mathematics used to quantify, compare, represent, and model numbers? How can mathematics support effective communication? Why does what we measure influence how we measure? In what ways are the mathematical attributes of objects or processes measured, calculated, and/or interpreted? How precise do measurements and calculations need to be? How can patterns be used to describe relationships in mathematical situations? How can recognizing repetition or regularity assist in solving problems more efficiently? 1

3 ALGEBRA 1 STANDARDS FOR MATHEMATICAL CONTENT 2.1 NUMBERS AND OPERATIONS A) Counting and cardinality B) Number and operations in base ten C) Number and operations - fractions D) Ratios and proportional relationships E) The number system F) Number and quantity 2.2 ALGEBRAIC CONCEPTS A) Operation and algebra thinking B) Expressions and equations C) Functions D) Algebra 2.3 GEOMETRY A) Geometry 2.4 MEASUREMENT, DATA AND PROBABILITY A) Measurement and data B) Statistics and probability Big Ideas and Essential Questions for Algebra 1 1) Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms. 2) Patterns exhibit relationships that can be extended, described, and generalized. How can we show that algebraic properties and processes are extensions of arithmetic properties and processes, and how can we use algebraic properties and processes to solve problems? 3) Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations. 4) There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities. How can we show that algebraic properties and processes are extensions of arithmetic properties and processes, and how can we use algebraic properties and processes to solve problems? How do you decide which functional representation to choose when modeling a real world situation, and how would you explain your solution to the problem? How do you write, solve, graph, and interpret linear equations and inequalities to model relationships between quantities? How do you write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques? 5) Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations. 6) Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations. How do you decide which functional representation to choose when modeling a real world situation, and how would you explain your solution to the problem? How do you write, solve, graph, and interpret linear equations and inequalities to model relationships between quantities? How do you write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques? 7) Bivariate data can be modeled with mathematical functions that approximate the data well and help us make predictions based on the data. 8) Degree and direction of linear association between two variables is measurable How do you write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques? How can we use univariate and bivariate data to analyze relationships and make predictions? 2

4 Unit 1 EQUATIONS STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of Specific skills the student needs to acquire the to master the learning competencies. 1. Use order of operations to Use order of operations (operations with correctly evaluate algebraic natural, whole, integer, rational, expressions, for mathematical irrational) using paper and pencil. applications, and formulas, for Include powers (positive and negative), real world problems. basic square roots, and absolute value. 2. Solve equations in one variable using a variety of approaches. Use order of operations (operations with natural, whole, integer, rational, irrational) using calculators. Include powers (positive and negative), basic square roots, and absolute value. (**scientific notation) Recognize and use properties (chart) justifying the steps in simplifying equations. PA KEYSTONE A Represent and/or use numbers in equivalent forms (e.g. integers, fractions, decimals, percent, square roots, and exponents). A Use exponents, roots, and/or absolute values to solve problems. A Simplify square roots A Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10). 2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. 2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real world or mathematical problems). 2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. 2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real world or mathematical problems B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions. Solve multi-step equations by combining like terms. Solve multi-step equations by using the distributive property. Solve equations with no solution or a solution which is all real numbers. A Use estimation strategies in problem-solving situations. A Use estimation to solve problems. 2.2.HS.D.9 Use reasoning to solve equations and justify the solution method B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. Solve proportional equations. 3

5 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of Specific skills the student needs to acquire the to master the learning competencies. 3. Rewrite an equation or a Solve equations for a specific variable. formula by solving the equation or the formula for a specific variable. 4. After analyzing a real world problem, write and solve an equation which represents the situation. 5. When presented with a real life problem, determine if a given equation accurately represents the situation and explain how the equation is equivalent to the real life situation it represents. 6. Use absolute value equations to represent and solve real life problems. Write and solve an equation for real world situations. (Including, but not limited to, geometry formulas, temperature, distance, consecutive integer, proportional relationships ). Relate components (variable, constants) of an equation to its real world situation. Solve equations using definition of absolute value. Write absolute value equations to represent real world situations. Interpret equations using absolute value. PA KEYSTONE A Write, solve and/or graph linear equations using various methods. A Write, solve and/or apply a linear equation (including problem situations). A Use and/or identify an algebraic property to justify any step in an equation solving process. Note. Linear equations only. A Interpret solutions to problems in the context of the problem situation. 2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs and data displays. 2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multi-step problems. 2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. 2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. 2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. 2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. 2.2.HS.C.3 Write functions or sequences that model relationships between two quantities B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. 4

6 ADDENDUM TO UNIT 1 - EQUATIONS 8 TH GRADE PSSA REQUIREMENTS PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability What the students will know and be able to do upon completion of the Use order of operations to correctly evaluate algebraic expressions, for mathematical applications, and formulas, for real world problems. Specific skills the student needs to acquire to master the learning competencies. Simplify expressions containing powers, square roots and cube roots. Operate with numbers in scientific notation by applying the power rules. Compare numbers written in scientific notation. 8 th grade PSSA ASSESSMENT M08.B-E.1 Demonstrate an understanding of expressions and equations with radicals and integer exponents. PSSA DESCRIPTOR M08.B-E.1.1 Represent and use expressions and equations to solve problems involving radicals and integer exponents. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.B-E Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents).properties will be provided. Example: 3 12 x2-15 =3-3 =1/(3) 3 M08.B-E Use square and cube root symbols to represent solutions to equations of the form x 2 =p and x 3 = p, where p is a positive rational number. Evaluate square roots of perfect squares (up to and including 12 2 ) and cube roots of perfect cubes (up to and including 5 3 ) without a calculator. Example: If x 2 =25 then x =. PA COMMON CORE STANDARDS B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions. Rewrite a number presented on a calculator as its equivalent number in scientific notation. M08.B-E Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10, and express how many times larger or smaller one number is than another. Example: Estimate the population of the United States as 3x10 8 and the population of the world as 7x10 9, and determine that the world population is more than 20 times larger than the United States population M08.B-E Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Express answers in 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., interpret 4.7EE9 displayed on a calculator as 4.7 x 10 9 ). 5

7 PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability What the students will know and be able to do upon completion of the (First 2 in eligible content are covered in unit one and the remaining are covered in unit 5.) Specific skills the student needs to acquire to master the learning competencies. 8 th grade PSSA ASSESSMENT M08.B-E.3 Analyze and solve linear equations and pairs of simultaneous linear equations. PSSA DESCRIPTOR M08.B -E.3.1 Write, solve, graph, and interpret linear equations in one or two variables, using various methods. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.B-E Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). PA COMMON CORE STANDARDS B.3 Analyze and solve linear equations and pairs of simultaneous linear equations M08.B-E Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. M08.B-E Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. M08.B-E Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. M08.B-E Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. 6

8 ALGEBRA 1 Unit 2 INEQUALITIES STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the Specific skills the student needs to acquire to master the learning competencies. 1. Understand the relationship Locate and place any real number on between different number sets and the number line. use those relationships to order real numbers and represent them Order elements of the number systems on a number line. from least to greatest. Graph inequalities in one variable. PA KEYSTONE A Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents). A Compare and/or order any real numbers (rational and irrational may be mixed). A Identify or graph the solution set to a linear inequality on a number line. 2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. 2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real world or mathematical problems E.1 Distinguish between rational and irrational numbers using their properties E.4 Estimate irrational numbers by comparing them to rational numbers. 2. Solve inequalities in their many forms and represent the results graphically. Solve inequalities by using the same methods learned for equations, with the exception of multiplying or dividing by a negative number. Solve compound inequalities which use and or or. Solve absolute value inequalities. Graph the solution set for all of the above inequalities. A Write, solve, and/or graph linear inequalities using various methods. A Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities). A Identify or graph the solution set to a linear inequality on a number line. 2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. 2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. 2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. 7

9 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the Specific skills the student needs to acquire to master the learning competencies. 3. After analyzing a real world Write and solve an inequality for real problem, write and solve an world situations, including absolute inequality which represents the value relationships. situation. PA KEYSTONE A Interpret solutions to problems in the context of the problem situation. Note: Limit to linear inequalities. 2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 4. When presented with a real life problem, determine if a given inequality accurately represents the situation and explain how the inequality is equivalent to the real life situation it represent Relate components (variable, constants) of an inequality to its real world situation. 8

10 ADDENDUM TO UNIT 2 - INEQUALITIES 8 TH GRADE PSSA REQUIREMENTS PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 What the students will know and be able to do upon completion of the Use properties of rational and irrational numbers to estimate values and convert between systems. CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability Specific skills the student needs to acquire to master the learning competencies. Estimate the value of an irrational number by placing it between two adjacent integers. Rewrite a rational number in any one of its equivalent forms. 8 th grade PSSA ASSESSMENT M08.A-N.1 Demonstrate an understanding of rational and irrational numbers. PSSA DESCRIPTOR M08.A-N.1.1 Apply concepts of rational and irrational numbers. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.A-N Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). M08.A-N Convert a terminating or repeating decimal into a rational number (limit repeating decimals to thousandths). M08.A-N Estimate the value of irrational numbers without a calculator (limit whole number radicands to less than 144. Example: is between 2 and 3 but closer to 2. PA COMMON CORE STANDARDS E.1 Distinguish between rational and irrational numbers using their properties E.4 Estimate irrational numbers by comparing them to rational numbers. M08.A-N Use rational approximations of irrational numbers to compare and order irrational numbers. M08.A-N Locate/identify rational and irrational numbers at their approximate locations on a number line. 9

11 ALGEBRA 1 Unit 3 RELATIONSHIPS AND FUNCTIONS STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the 1. Identify a relationship, its parts and the multiple ways of representing the relationship. Specific skills the student needs to acquire to master the learning competencies. Recognize, compare and contrast, and convert amongst the different ways to represent a relationship: table, graph, verbal statement, or an equation. Identify domain range and inverse of any relationship. Given an equation in two variables represent the equation as a table and a graph. (Look at different graphs including discrete and continuous e.g., linear, exponential, absolute value, step, rational ) PA KEYSTONE A Analyze and/or use patterns or relations. A Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically. A Determine whether a relation is a function, given a set of points or a graph. A Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table) 2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations. 2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. 2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. 2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables.). 2. Determine if a relationship, given in any of its multiple forms, is a function (one-to-one) relationship. Determine if equations, graphs, tables, verbal examples represent functions. (vertical line test for graphs) A Interpret and/or use linear functions and their equations, graphs, or tables. A Create, interpret, and/or use the equation, graph, or table of a linear function. A Translate from one representation of a linear function to another (i.e., graph, table, and equation). 2.2.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. 2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs and data displays. 2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multi-step problems. 10

12 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the 3.. Recognize and evaluate functional notation Specific skills the student needs to acquire to master the learning competencies. Evaluate functional equations for specific values. PA KEYSTONE 2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations. 2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. 2.2.HS.C.4 Interpret the effects transformations have on functions and find the inverses of functions. 2.2.HS.C.6 Interpret functions in terms of the situation they model B.2 Understand the connections between proportional relationships, lines, and linear equations. 11

13 ADDENDUM TO UNIT 3- RELATIONSHIPS AND FUNCTIONS 8 TH GRADE PSSA REQUIREMENTS PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 What the students will know and be able to do upon completion of the Identify a proportional relationship as a form of a linear relationship which has a y-intercept of 0 and a slope which can vary. CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability Specific skills the student needs to acquire to master the learning competencies. Determine if a linear relationship (given in any form) is proportional. Use similar right triangles to show that the non-vertical lines will have the same slope. 8 th grade PSSA ASSESSMENT M08.B-E.2 Understand the connections between proportional relationships, lines, and linear equations. PSSA DESCRIPTOR M08.B-E.2.1 Analyze and describe linear relationships between two variables, using slope. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.B-E Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. M08.B-E Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. PA COMMON CORE STANDARDS B.2 Understand the connections between proportional relationships, lines, and linear equations. M08.B-E Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. 12

14 ALGEBRA 1 Unit 4 LINEAR FUNCTIONS STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the 1. Given a relationship in the form of a graph, table, equation or a verbal statement, determine if the relationship is linear. 2. Identify the parts of a linear relationship: x- intercept, y-intercept, and slope. The relationship can be presented as a graph, table, equation or a verbal statement. 3. Identify and use the three forms of a linear relationship. Specific skills the student needs to acquire to master the learning competencies. Determine if a graph, table, equation or expression represents a linear relationship. Using the concept of initial values and rate of change, write linear equations for real life problems. (include examples with unit rateproportional relationships) Given a graph or an equation find the x- and y- intercepts. On a statistical graph of real life data find the rate of change (slope) for selected parts of the data displayed. Given a graph on the coordinate plane find the slope (rate of change) and the y-intercept (start value). Given an equation in slope-intercept form graph the equation using the y-intercept and the rate of change. Determine the slope for two ordered pairs using the slope formula (include examples of undefined and zero slopes). PA KEYSTONE A Interpret and/or use linear functions and their equations, graphs, or tables. A Create, interpret, and/or use the equation, graph, or table of a linear function. A Translate from one representation of a linear function to another (i.e., graph, table, and equation). 2.4.HS.B.2 Summarize, represent and interpret data on two categorical and quantitative variables HS.F.3. Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs and data displays. 2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multi-step problems. 2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations. 2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. 2.2.HS.C.4 Interpret the effects transformations have on functions and find the inverses of functions. 2.2.HS.C.6 Interpret functions in terms of the situation they model B.2 Understand the connections between proportional relationships, lines, and linear equations. 13

15 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the 4. When presented with a real life problem, determine if a given linear equation accurately represents the situation and explain how the equation is equivalent to the real life situation it represents. 5. Use the properties of arithmetic sequences(identify, extend, find the nth term, and find the specific rule for a sequence using the general formula-linear relationship) Specific skills the student needs to acquire to master the learning competencies. Given different pieces of data (slope and intercept, slope and point, two points) which are linearly related, write the equation of their line in any of the three forms. Find the equations for parallel and perpendicular lines using the special rules for their slopes. Write linear equations for real world situations. Relate components (variable, constants) of a linear equation to its real world situation. After identifying an arithmetic pattern, extend the pattern. Using the 1 st term and the common difference find the nth term of an arithmetic sequence. Find the linear equation for a given sequence. PA KEYSTONE A Describe, compute, and/or use the rate of change (slope) of a line. A Identify, describe, and/or use constant rates of change. A Apply the concept of linear rate of change (slope) to solve problems. A Write or identify a linear equation when given the graph of the line, or two points on the line, or The slope and a point on the line. Note: Linear equation may be in point-slope, standard, and/or slope-intercept form. A Determine the slope and /or the y-intercept represented by a linear equation or graph. 2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. 2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. 2.2.HS.C.5 Construct and compare linear, quadratic and/or exponential models to solve problems C.1 Define, evaluate, and compare functions C.2 Use concepts of functions to model relationships between quantities. 2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. 6. Realize that the line of best fit on a scatter plot is an application of linear equations and be able to use that line to further extend the data. Analyze data presented in a scatter plot to determine if a relationship exists between the two sets of data and if it does exist, what type of relationship it represents, positive or negative. Use the line of best fit to extend the data. Identify a linear equation for the line of best fit A Analyze and/or interpret data on a scatter plot. A Draw, identify, and/or write an equation for a line of best fit for a scatter plot. 2.4.HS.B.3 Analyze linear models to make interpretations based on the data. 2.2.HS.C.6 Interpret functions in terms of the situation they model. 14

16 ADDENDUM TO UNIT 4 LINEAR FUNCTIONS 8 TH GRADE PSSA REQUIREMENTS PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 What the students will know and be able to do upon completion of the Covered in this unit and previous units. CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability Specific skills the student needs to acquire to master the learning competencies. 8 th grade PSSA ASSESSMENT M08.B-E.2 Understand the connections between proportional relationships, lines, and linear equations. PSSA DESCRIPTOR M08.B-E.2.1 Analyze and describe linear relationships between two variables, using slope. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.B-E Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. M08.B-E Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. M08.B-E Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. PA COMMON CORE STANDARDS B.2 Understand the connections between proportional relationships, lines, and linear Compare various forms of relations and functions. Compare 2 linear relations (functions) to determine greater rate of change (may be presented graphically, algebraically, as a table or in a verbal description. M08.B-F.1 Analyze and interpret functions. M08.B-F.1.1 Define, evaluate, and compare functions displayed algebraically, graphically, numerically in tables, or by verbal descriptions.. M08.B-F Determine whether a relation is a function. M08.B-F Compare properties of two functions each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change C.1 Define, evaluate, and compare functions. M08.B-F 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. 15

17 PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 What the students will know and be able to do upon completion of the Covered in the unit and previous units. CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability Specific skills the student needs to acquire to master the learning competencies. 8 th grade PSSA ASSESSMENT. M08.B-F.2 Use functions to model relationships between quantities PSSA DESCRIPTOR M08.B-F.2.1 Represent or interpret functional relationships between quantities using tables, graphs, and descriptions. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.B-F 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. M08.B-F 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 or determine a graph that exhibits the qualitative features of a function that has been described verbally. PA COMMON CORE STANDARDS C.2 Use concepts of functions to model relationships between quantities. Covered in analyzing scatter plots. M08.D-S..1 Investigate patterns of association in bivariate data. M08.D-S.1.1 Analyze and interpret bivariate data displayed in multiple representations. M08.D-S 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 correlation, linear association, and nonlinear association B.1 Analyze and/or interpret bivariate data displayed in multiple representations. M08.D-S For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line. M08.D-S Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Example: 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. 16

18 ALGEBRA 1 Unit 5 SYSTEMS OF EQUATIONS STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the 1. Solve a system of equations by selecting the most appropriate method from a variety of approaches. Specific skills the student needs to acquire to master the learning competencies. Solve a system of equations by graphing the equations on the coordinate plane (include those with no solution and an infinite solution). Solve a system of equations by using substitution. (include those with no solution and an infinite solution). PA KEYSTONE A Write, solve, and/or graph systems of linear equations using various methods. A Write and/or solve a system of linear equations (including problem situations) using graphing, substitution, and/or elimination. Note: Limit systems to two linear equations. 2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. 2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. Solve a system of equations using linear combination (elimination) method. (include those with no solution or an infinite solution). Prove that a given point is a solution to a system of equations. A Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear equations. 2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. 2. After analyzing a real world problem, write and solve a system of equations which represents the situation. Write a system to represent a real life problem. (e.g., distance, mixture, current ) 3. When presented with a real life problem, determine if a given system accurately represents the situation and explain how the system is equivalent to the real life situation it represents. Interpret how the solution to a system relates to the real life situation. Include the break-even point. Relate components (variables, constants) of a system to its real world situation. 17

19 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the Specific skills the student needs to acquire to master the learning competencies. PA KEYSTONE 4. When presented with a real life problem, determine if a given linear inequality accurately represents the situation and explain how the inequality is equivalent to the real life situation it represents. 5. Use the graph of linear inequalities to identify possible solutions to real world problems Write linear inequalities in two variables. Graph linear inequalities in two variables. Find integral solutions for the graph of a linear inequality. Graphically solve a system of inequalities. A Write, solve, and/or graph systems of linear inequalities using various methods. A Write and/or solve a system of linear inequalities using graphing. Note: Limit systems to two linear inequalities. A Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear inequalities. 2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. 2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 6. Solve a system of inequalities. Match the graph of a system of equations/inequalities to its algebraic representation. 7. Analyze the graph of a system of equations or inequalities to determine the correct equations/inequalities for that system. 18

20 ADDENDUM TO UNIT 5 SYSTEM OF EQUATIONS 8 TH GRADE PSSA REQUIREMENTS PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 What the students will know and be able to do upon completion of the First 2 were covered parts of the eligible content were covered in previous The remaining concepts are covered in the CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability CONCEPT LIST: Specific skills the student needs to acquire to master the learning competencies. 8 th grade PSSA ASSESSMENT M08.B-E.3 Analyze and solve linear equations and pairs of simultaneous linear equations. PSSA DESCRIPTOR M08.B-E.3.1 Write, solve, graph, and interpret linear equations in one or two variables, using various methods. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.B-E Write and identify linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). M08.B-E Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. M08.B-E Interpret solutions to a system of two linea equations in two variables as points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. M08.B-E Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. M08.B-E Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. PA COMMON CORE STANDARDS B.3 Analyze and solve linear equations and pairs of simultaneous linear equations 19

21 ALGEBRA 1 Unit 6 DATA ANALYSIS AND PROBABILITY STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will Specific skills the student needs to know and be able to do acquire to master the learning upon completion of the competencies. PA KEYSTONE 1) Use appropriate numeric forms to represent the probability of compound events. 2. Analyze data, presented in graphical form, to solve problems and/or make predictions. Find the probability of an event which uses and (as a fraction, decimal or percent). Find the probability of an event which uses or (as a fraction, decimal or percent). Recognize the correct data display for a given set of data. Use data displays to make predictions and/or solve real world problems. Use measures of central tendency. Collect data, median and mode from a stem and leaf plot. Interpret a box-and-whisker plot by identifying median, quartiles and interquartile ranges. A Apply probability to practical situations. A Use measures of dispersion to describe a set of data. A Use data displays in problem-solving settings and/or to make predictions. A Find probabilities for compound events (e.g., find probability of red and blue, find probability of red or blue) and represent as a fraction, decimal, or percent. A Calculate and/or interpret the range, quartiles, and interquartile range of data. A Estimate or calculate to make predictions based on a circle, line, bar graph, measures of central tendency, or other representations. A Analyze data, make predictions, and/or answer questions based on displayed data (box-and-whisker plots, stem-andleaf plots, scatter plots, measures of central tendency, or other representations). A Make predictions using the equations or graphs of best-fit lines of scatter plots. 2.4.HS.B.4 Recognize and evaluate random processes underlying statistical experiments. 2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model. 2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. 2.4.HS.B.3 Analyze linear models to make interpretations based on the data. 2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. 2.4.HS.B.3 Analyze linear models to make interpretations based on the data. 2.4.HS.B.5 Make inferences and justify conclusions based on sample surveys, experiments, and observational studies. 20

22 ADDENDUM TO UNIT 6 DATA ANALYSIS AND PROBABILITY 8 TH GRADE PSSA REQUIREMENTS (NOT COVERED IN ALGEBRA 1 TOPICS) PENNSYLVANIA STATE SYSTEM OF ASSESSMENT- GRADE 8 What the students will know and be able to do upon completion of the Create, read and interpret data using frequency tables. CLASSIFICATIONS A = Numbers and operations B = Algebraic concepts C= Geometry D = Data analysis and probability Specific skills the student needs to acquire to master the learning competencies. Create a frequency table to collect information on two topics. Analyze data presented in a frequency table to determine if there is a relationship between the topics represented by the data. 8 th grade PSSA ASSESSMENT M08.D-S.1 Investigate patterns of association in bivariate data. PSSA DESCRIPTOR M08.D-S.1.2 Understand that patterns of association can be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. REPORTING CATEGORIES Classification-domain A-N = The number system B-E = Expressions and equations B-F = Functions C-G = Geometry D-S = Statistics and probability M08.D-S 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 associations between the two variables. Example: Given data on whether students have a curfew on school nights and whether they have assigned chores at home, is there evidence that those who have a curfew also tend to have chores? PA COMMON CORE STANDARDS B.2 Understand that patterns of association can be seen in bivariate data utilizing frequencies. 21

23 ALGEBRA 1 Unit 7 POLYNOMIALS AND FACTORING STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Use appropriate tools strategically. Reason abstractly and quantitatively. Attend to precision. What the students will know and be able to do upon completion of the 1. Understand the difference between terms and factors, the difference between factors and multiples, and the meaning of degree of a polynomial. Specific skills the student needs to acquire to master the learning competencies. Identify the degree of a polynomial and the number of terms in the polynomial. Find the Greatest Common Factor of given monomials. PA KEYSTONE A Apply number theory concepts to show relationships between real numbers in problemsolving settings. A Find the Greatest Common Factor (GCF) and/or the Least Common Multiple (LCM) for sets of monomials E.3 Develop and/or apply number theory concepts to find common factors and multiples. Find the Least Common Multiple of given monomials. 2. Use addition, subtraction, multiplication and division of polynomials to simplify expressions and equations. 3. Given a polynomial of two or more terms find the prime factorization of the polynomial. Add and subtract polynomials. Use rules of exponents to multiply and divide monomials. Multiply polynomials using the distributive property. Multiply polynomials using FOIL. Recognize that some polynomials are special products and that they can be multiplied using special patterns. Use GCF to factor polynomials. Factor trinomials (using guess and check, factoring by grouping, patterns, table method or any valid approach). A Simplify expressions involving polynomials. A Add, subtract, and/or multiply polynomial expressions (express answers in simplest form). A Factor algebraic expressions, including difference of squares and trinomials. Note: trinomials are limited to the form ax 2 +bx+c where a is equal to 1 after factoring out all monomial factors. A Simplify/reduce a rational algebraic expression. 2.2.HS.D.1 Interpret the structure of expressions to represent a quantity in terms of its context. 2.2.HS.D.2 Write expressions in equivalent forms to solve problems. 2.2.HS.D.5 Extend the knowledge of arithmetic operations and apply to polynomials. 2.2.HS.D.6 Extend the knowledge of rational functions to rewrite in equivalent forms. 22