Mendham Borough Public Schools Mendham, New Jersey. Curriculum and Instruction. Course of Study

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1 Mendham Borough Public Schools Mendham, New Jersey Curriculum and Instruction Course of Study Mathematics 5-8 June 2013 I. RATIONAL, DESCRIPTION AND PURPOSE Mendham Borough Public Schools strive to provide outstanding learning experiences by utilizing best practices for instruction. As the need to incorporate mathematics in the workplace and everyday life increases, providing our students with the tools to navigate a mathematically inundated society is ever so important. In designing our program, we look to the Principles and Standards for Mathematics, as championed by the National Council of Teachers of Mathematics. Program design will revolve around six basic principles. Starting with the establishment of high expectations and support for all students, we will in turn provide any necessary accommodations in addition to the challenges they must endure. Links across grade levels will both connect with and build upon knowledge in a way which will promote students ability to solve mathematical problems. As teachers work to understand their students as mathematicians, they will be provided with ample professional learning opportunities to refine their practice. Student learning will focus primarily on conceptual knowledge and necessitate resolve as they are asked to work through intricate tasks. In assessing our students, we provide both teachers and students alike with the feedback necessary to adjust the ways in which learning takes place. This level of rigor, in conjunction with the acceleration of more ready students, represents best practice in mathematics for all of our students. II. DISTRICT OBJECTIVES The district adopts the Standards for Mathematical Practice as outlined in the Common Core State Standards. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one s own efficacy). (Common Core State Standards, 2010)

2 The standards are presented in the Common Core State Standards as follows: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. In addition, the mathematics curriculum will reflect the National Council of Teachers of Mathematics Curriculum Focal Points for Mathematics in Prekindergarten through Grade 8 Mathematics through the incorporation of the following from the Focal Points document: the use of mathematics to solve problems an application of logical reasoning to justify procedures and solutions; and an involvement in the design and analysis of multiple representations to learn, make connections among, and communicate about the ideas within and outside of mathematics III. ALIGNMENT TO STANDARDS Mathematics instruction aligns with the following Common Core State Standards for Mathematics: Grade 5: Operations and Algebraic Thinking (5.OA) Write and interpret numerical expressions. Analyze patterns and relationships. Number and Operations in Base Ten (5.NBT) Understand the place value system. Perform operations with multi-digit whole numbers and with decimals to hundredths. Number and Operations Fractions (5.NF)

3 Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Measurement and Data (5.MD) Convert like measurement units within a given measurement system. Represent and interpret data. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Geometry (5.G) Graph points on the coordinate plane to solve real-world and mathematical problems. Classify two-dimensional figures into categories based on their properties. Grade 6: Ratios and Proportional Relationships (6.RP) Understand ratio concepts and use ratio reasoning to solve problems. The Number System (6.NS) Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and multiples. Apply and extend previous understandings of numbers to the system of rational numbers. Expressions and Equations (6.EE) Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables. Geometry (6.G) Solve real-world and mathematical problems involving area, surface area, and volume. Statistics and Probability (6.SP)

4 Grade 7: Develop understanding of statistical variability. Summarize and describe distributions. Ratios and Proportional Relationships (7.RP) Analyze proportional relationships and use them to solve real-world and mathematical problems. The Number System (7.NS) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Expressions and Equations (7.EE) Use properties of operations to generate equivalent expressions. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Geometry (7.G, 8.G) Draw, construct and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. Understand congruence and similarity using physical models, transparencies, or geometry software. Understand and apply the Pythagorean Theorem. Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. Statistics and Probability (7.SP, 8.SP) Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models. Investigate patterns of association in bivariate data. Seeing Structure in Expressions (A-SSE) Interpret the structure of expressions

5 Write expressions in equivalent forms to solve problems Creating Equations (A-CED) Create equations that describe numbers or relationships Reasoning with Equations and Inequalities (A-REI) Grade 8: Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically The Number System (8.NS) Know that there are numbers that are not rational, and approximate them by rational numbers. Expressions and Equations (8.EE) Work with radicals and integer exponents. Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations. Functions (8.F) Define, evaluate, and compare functions. Use functions to model relationships between quantities. Geometry (8.G) Understand congruence and similarity using physical models, transparencies, or geometry software. Understand and apply the Pythagorean Theorem. Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. Statistics and Probability (8.SP) Investigate patterns of association in bivariate data. Seeing Structure in Expressions (A-SSE)

6 Interpret the structure of expressions Write expressions in equivalent forms to solve problems Arithmetic with Polynomials and Rational Expressions (A-APR) Perform arithmetic operations on polynomials Understand the relationship between zeros and factors of polynomials Use polynomial identities to solve problems Rewrite rational expressions Creating Equations (A-CED) Create equations that describe numbers or relationships Reasoning with Equations and Inequalities (A-REI) Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically

7 IV. INSTRUCTIONAL TECHNIQUES Mathematics education relies on a variety of instructional techniques to meet the continuum of learners interests, learning profiles, and readiness levels. Individual differences are supported by differentiation. Differentiation is the framework through which the developmental needs of a range of readiness levels are met. Differentiated instruction is accomplished through pre-assessment and development of appropriate learning experiences. Differentiation in content, product and/or process addresses a complete range of students; providing support from the exceptionally talented and scaffolding of varying degree to support less ready students in meeting appropriately rigorous learning outcomes. Instructional objectives, strategies and materials emphasize relevance, authenticity, and student-centered learning. In addition, cross curricular learning experiences and global communication skills are an important part of the transfer of knowledge and skills and will be incorporated when possible. Instructional techniques in Mathematics Education include the following: Whole class instruction and modeling Small group instruction and modeling Individual instruction and modeling Inquiry based laboratory experiences Communication and collaboration V. ASSESSMENT Assessment in Mathematics is designed to provide evidence which shows the extent to which students have achieved desired results. Best practices in the Mathematics classroom are extensive and varied and may include: Pre-assessment of skills, knowledge, and understandings Periodic student-teacher conferences to assess development of skills, knowledge, and understandings Teachers observation of students independent inquiry skills Teachers observation of students independent problem solving skills Performance assessments requiring students to apply concepts, skills, and problem-solving strategies to real-world scenarios Written responses to essential questions Summative assessments compiled of computer adaptive quizzes, projects, oral reports, and studio created portfolios Ongoing formative assessment

8 VI. CONTENT, SCOPE, AND SEQUENCE, LEARNING OUTCOMES Building upon knowledge and skills gained in preceding grades, the following desired results will guide staff in the selection of appropriate assessment evidence, and ultimately in the creation and implementation of a learning plan which uses that evidence to determine attainment of the desired results.

9 COURSE: Mathematics LEVEL: Grade 8 UNIT/FOCUS: Linear Systems Desired Results Related standard(s): Students will be able to independently use their learning to Transfer 8.EE.5 8.EE.6 8.EE.7a 8.EE.7b 8.EE.8a 8.EE.8b 8.EE.8c 8.F.4 8.F.5 A-SSE.1a A-CED.2 A-CED.3 A-REI.4a A-REI.4b A-REI.5 A-REI.6 A-REI.10 A-REI.11 A-REI.12 Create models using the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations. Use functions to model relationships between quantities. Interpret the structure of expressions. Create equations that describe numbers or relationships. Solve equations and inequalities in one variable. Solve systems of equations. Represent and solve equations and inequalities graphically. Enduring Understandings (EUs) Students will understand that The use and manipulation of symbols and expressions provide a variety or representations for solving problems and expressing mathematical concepts, relationships, and reasoning. Understandings of number how many or how much and number types extend applications of arithmetic properties, operations, and number systems and guide the use of computational strategies and algorithms. Measurement attributes, processes, and tools, help us to quantify, compare, and solve problems involving objects, situations, and events. Patterns, relations, and functions are used to represent and analyze change in various contexts, make predictions and generalizations, and provide models and explanations for real-world problems. Visualizations, spatial reasoning, and properties of two- and three- dimensional figures can be used to analyze, represent, and model geometric concepts and relationships. Questions are posed and investigated by collecting data or retrieving existing data, and representing, analyzing, and interpreting data. Investigations, inferences, and predictions are used to make critical and informed decisions. Meaning Essential Questions (EQs) Students will keep considering How are symbols and expressions utilized as representations in mathematics? In what ways can a quantity be represented? In what ways is measurement useful? How can I use patterns, relations, and functions? How are geometric concepts and relations conceptualized? What roles can data play in the decision making process?

10 Students will know Knowledge Grade Level Benchmarks Students will be able to Skills A solution of a linear system is an ordered pair that must satisfy each equation in a system. Consistent independent describes a system with one solution. Consistent dependent describes a system with infinitely many solutions. Inconsistent/coincident lines describe a system with no solution (parallel lines). The solutions for systems of inequalities are where shading overlaps. There are only two types of solutions for systems of inequalities: infinitely many or none. Find solutions of systems through graphing. Find solutions of systems through substitution. Find solutions of systems through multiplication-addition algorithm. Determine if a system has one, none, or infinitely many solutions. Graph and find solutions for systems of inequalities. Use graphing calculator to graph systems of inequalities.

11 TASKS from CCSS for Linear Systems 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. 8.EE.7. Solve linear equations in one variable. a. Give examples of 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). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.EE.8. Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. c. Solve real-world and mathematical problems leading to two linear equations in two variables. For 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. 8.F.4. 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. 8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. A-SSE.1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A-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. b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a± bi for real numbers a and b. A-REI.5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

12 A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

13 COURSE: Mathematics LEVEL: Grade 8 UNIT/FOCUS: Exponents Desired Results Related standard(s): Students will be able to independently use their learning to Transfer 8.NS.1 8.NS.2 8.EE.1 8.EE.2 8.EE.3 8.EE.4 A-SSE.1b A-SSE.2 A-SSE.3c A-SSE.4 A-REI.11 Approximate irrational numbers using rational numbers. Work with radicals and integer exponents. Interpret the structure of expressions. Write expressions in equivalent forms to solve problems. Represent and solve equations and inequalities graphically. Enduring Understandings (EUs) Students will understand that The use and manipulation of symbols and expressions provide a variety or representations for solving problems and expressing mathematical concepts, relationships, and reasoning. Understandings of number how many or how much and number types extend applications of arithmetic properties, operations, and number systems and guide the use of computational strategies and algorithms. Measurement attributes, processes, and tools, help us to quantify, compare, and solve problems involving objects, situations, and events. Patterns, relations, and functions are used to represent and analyze change in various contexts, make predictions and generalizations, and provide models and explanations for real-world problems. Visualizations, spatial reasoning, and properties of two- and three- dimensional figures can be used to analyze, represent, and model geometric concepts and relationships. Questions are posed and investigated by collecting data or retrieving existing data, and representing, analyzing, and interpreting data. Investigations, inferences, and predictions are used to make critical and informed decisions. Meaning Essential Questions (EQs) Students will keep considering How are symbols and expressions utilized as representations in mathematics? In what ways can a quantity be represented? In what ways is measurement useful? How can I use patterns, relations, and functions? How are geometric concepts and relations conceptualized? What roles can data play in the decision making process?

14 Students will know Knowledge Grade Level Benchmarks Students will be able to Skills Compound interest is interest that earns interest. Simple interest is a linear function; compound interest is an exponential function. Constant increase is in the form of y = mx + b; where b is your initial value; and m is your slope/constant increase x number of times. Exponential growth is modeled by the equation y = b * g x ; where b is your initial value; g is the growth rate factor/percent change greater than 1; and x is the number of time periods. Exponential decay is modeled by the equation y = b * g x ; where b is your initial value; g is the growth rate factor/percent change greater than 0, but less than 1; and x is the number of time periods. The seven properties of powers: zero exponent; negative exponent; product of powers; quotient of powers; power of a power; power of a product; and power of a quotient. Evaluate integer powers of real numbers. Simplify products, quotients, and powers of power. Apply properties of exponents to simplify expressions. Solve problems involving exponential growth and decay. Calculate compound interest. Write and graph exponential functions. Graph and compare simple and compound interests by hand and graphing calculator.

15 TASKS from CCSS for Exponents 8.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.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations 8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² 3 ⁵ = 3 ³= 1/3² = 1/27. 8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x² =p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. 8.EE.4. Perform 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. A-SSE.1. Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ y⁴ as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

16 COURSE: Mathematics LEVEL: Grade 8 UNIT/FOCUS: Polynomials Desired Results Related standard(s): Students will be able to independently use their learning to Transfer 8.G.9 A-SSE.2 A-APR.1 A-APR.4 A-APR.5 Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. Interpret the structure of expressions. Perform arithmetic operations on polynomials. Understand the relationship between zeros and factors of polynomials. Use polynomial identities to solve problems. Enduring Understandings (EUs) Students will understand that Meaning Essential Questions (EQs) Students will keep considering The use and manipulation of symbols and expressions provide a variety or representations for solving problems and expressing mathematical concepts, relationships, and reasoning. Understandings of number how many or how much and number types extend applications of arithmetic properties, operations, and number systems and guide the use of computational strategies and algorithms. Measurement attributes, processes, and tools, help us to quantify, compare, and solve problems involving objects, situations, and events. Patterns, relations, and functions are used to represent and analyze change in various contexts, make predictions and generalizations, and provide models and explanations for real-world problems. Visualizations, spatial reasoning, and properties of two- and three- dimensional figures can be used to analyze, represent, and model geometric concepts and relationships. Questions are posed and investigated by collecting data or retrieving existing data, and representing, analyzing, and interpreting data. Investigations, inferences, and predictions are used to make critical and informed decisions. How are symbols and expressions utilized as representations in mathematics? In what ways can a quantity be represented? In what ways is measurement useful? How can I use patterns, relations, and functions? How are geometric concepts and relations conceptualized? What roles can data play in the decision making process?

17 Students will know Knowledge Grade Level Benchmarks Students will be able to Skills A polynomial is an expression of many terms. There a variety of ways to multiply polynomials. FOIL is the distributive property using two terms (First, Outside, Inside, Last). Binomial has two terms; and a trinomial has three terms. Degree of a monomial is the sum of the exponents of the variables in the expression. Standard form of a polynomial is when the exponents are in descending order. Perfect square patterns and difference of two squares patterns. Add and subtract polynomials. Translate investment situations into polynomials. Multiply polynomials. Classify polynomials by degree and term. Multiply binomials. Model polynomial multiplication using area. Square a binomial. Multiply a binomial by its opposite; multiply the sum of (a + b) by its difference (a b).

18 TASKS from CCSS for Polynomials 8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ y⁴ as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² y²)² + (2xy)² can be used to generate Pythagorean triples. A-APR.5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle.1

19 COURSE: Mathematics LEVEL: Grade 8 UNIT/FOCUS: Factoring Desired Results Related standard(s): A-SSE.2 A-APR.3 A-APR.4 A-APR.5 A-APR.6 A-APR.7 A-REI.2 Students will be able to independently use their learning to Interpret the structure of expressions. Perform arithmetic operations on polynomials. Relate zeros and factors of polynomials. Use polynomial identities to solve problems. Rewrite rational expressions. Explain the reasoning used to solve equations. Enduring Understandings (EUs) Students will understand that Transfer Meaning Essential Questions (EQs) Students will keep considering The use and manipulation of symbols and expressions provide a variety or representations for solving problems and expressing mathematical concepts, relationships, and reasoning. Understandings of number how many or how much and number types extend applications of arithmetic properties, operations, and number systems and guide the use of computational strategies and algorithms. Measurement attributes, processes, and tools, help us to quantify, compare, and solve problems involving objects, situations, and events. Patterns, relations, and functions are used to represent and analyze change in various contexts, make predictions and generalizations, and provide models and explanations for real-world problems. Visualizations, spatial reasoning, and properties of two- and three- dimensional figures can be used to analyze, represent, and model geometric concepts and relationships. Questions are posed and investigated by collecting data or retrieving existing data, and representing, analyzing, and interpreting data. Investigations, inferences, and predictions are used to make critical and informed decisions. How are symbols and expressions utilized as representations in mathematics? In what ways can a quantity be represented? In what ways is measurement useful? How can I use patterns, relations, and functions? How are geometric concepts and relations conceptualized? What roles can data play in the decision making process?

20 Students will know Knowledge Grade Level Benchmarks Students will be able to Skills Factoring reverses polynomial multiplication. Solutions to a quadratic equation through factoring are the roots or x-intercepts of the graph. All quadratic equations cannot be factored. Signs in the quadratic equations dictate the signs in the factored binomials. If the product of two factors is zero, then one factor must be zero. Factoring a quadratic equation is done in a variety of ways. If a positive integer is not a perfect square, then its square root is an irrational number. Factor greatest common factor. Factor positive integers into primes by factorization. Factor quadratic equations into the product of two binomials. Solve quadratic equations through factoring. Solve using the zero-product property. Determine if a quadratic equation can be factored. Determine if roots of real numbers are rational or irrational.

21 TASKS from CCSS for Factoring A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ y⁴ as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x² + y²)² = (x² y²)² + (2xy)² can be used to generate Pythagorean triples. A-APR.5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle.1 A-APR.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-APR.7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

22 COURSE: Mathematics LEVEL: Grade 8 UNIT/FOCUS: Quadratics Desired Results Related standard(s): 8.NS.1 8.NS.2 8.EE.1 8.EE.2 8.EE.3 8.EE.4 A-SSE.3a A-SSE.3b A-APR.3 A-REI.11 Students will be able to independently use their learning to Approximate irrational numbers using rational numbers. Work with radicals and integer exponents. Write expressions in equivalent forms to solve problems. Relate zeros and factors of polynomials. Represent and solve equations and inequalities graphically. Enduring Understandings (EUs) Students will understand that Transfer Meaning Essential Questions (EQs) Students will keep considering The use and manipulation of symbols and expressions provide a variety or representations for solving problems and expressing mathematical concepts, relationships, and reasoning. Understandings of number how many or how much and number types extend applications of arithmetic properties, operations, and number systems and guide the use of computational strategies and algorithms. Measurement attributes, processes, and tools, help us to quantify, compare, and solve problems involving objects, situations, and events. Patterns, relations, and functions are used to represent and analyze change in various contexts, make predictions and generalizations, and provide models and explanations for real-world problems. Visualizations, spatial reasoning, and properties of two- and three- dimensional figures can be used to analyze, represent, and model geometric concepts and relationships. Questions are posed and investigated by collecting data or retrieving existing data, and representing, analyzing, and interpreting data. Investigations, inferences, and predictions are used to make critical and informed decisions. How are symbols and expressions utilized as representations in mathematics? In what ways can a quantity be represented? In what ways is measurement useful? How can I use patterns, relations, and functions? How are geometric concepts and relations conceptualized? What roles can data play in the decision making process?

23 Students will know Knowledge Grade Level Benchmarks Students will be able to Skills A quadratic equation is in the form of: y = ax 2 + bx + c; where a determines if the graph opens up or down, and if it is wide or narrow. A quadratic equation has a degree equal to two. The solution(s) to a quadratic equation is also called your roots. The roots are the x-intercepts of the graph. Pattern is determined by a * 1, a * 3, and a * 5. When graphing y = ax 2, the vertex is always at the origin. To find the axis of symmetry, calculate the opposite of b/2a, which is also the x- value of the vertex. Vertex of a parabola is the point of direction change in the graph. Projectile motion describes the path of an object in space. A quadratic equation has one, two, or no solutions. The discriminant is the value inside the radical of the quadratic formula; if the discriminant is: < 0, there is no solution; = 0, there is one solution; or > 0, there are two solutions. The quadratic formula is: b± b 2 4ac 2a (where a, b, and c are from the given quadratic equation). Identify and use properties of quadratic equations. Graph quadratic equations. Solve quadratic equations through graphing. Determine direction and size of a parabola. Using a, determine pattern of the graph from the vertex. Find the vertex of a graph. Simplify square roots. Evaluate the quadratic formula to solve a quadratic equation. Analyze equation to determine placement of the graph in a coordinate plane. Calculate the vertex. Write and solve quadratic equations for paths of projectiles. Calculate the discriminant to determine number of solutions to a quadratic equation. Use graphing calculator to evaluate quadratic formula and graph quadratic equations.

24 TASKS from CCSS for Quadratics 8.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.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations 8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² 3 ⁵ = 3 ³= 1/3² = 1/27. 8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x² =p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. 8.EE.4. Perform 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

25 COURSE: Mathematics LEVEL: Grade 8 UNIT/FOCUS: Non-Linear Functions Desired Results Related standard(s): Students will be able to independently use their learning to Transfer 8.NS.1 8.NS.2 8.EE.1 8.EE.2 8.EE.3 8.EE.4 8.G.6 8.G.7 8.G.8 A-SSE.4 A-APR.6 A-APR.7 A-REI.2 A-REI.11 Approximate irrational numbers using rational numbers. Work with radicals and integer exponents. Write expressions in equivalent forms to solve problems. Apply the Pythagorean Theorem. Write expressions in equivalent forms to solve problems. Rewrite rational expressions. Explain the reasoning used to solve equations. Represent and solve equations and inequalities graphically. Enduring Understandings (EUs) Students will understand that The use and manipulation of symbols and expressions provide a variety or representations for solving problems and expressing mathematical concepts, relationships, and reasoning. Understandings of number how many or how much and number types extend applications of arithmetic properties, operations, and number systems and guide the use of computational strategies and algorithms. Measurement attributes, processes, and tools, help us to quantify, compare, and solve problems involving objects, situations, and events. Patterns, relations, and functions are used to represent and analyze change in various contexts, make predictions and generalizations, and provide models and explanations for real-world problems. Visualizations, spatial reasoning, and properties of two- and three- dimensional figures can be used to analyze, represent, and model geometric concepts and relationships. Questions are posed and investigated by collecting data or retrieving existing data, and representing, analyzing, and interpreting data. Investigations, inferences, and predictions are used to make critical and informed decisions. Meaning Essential Questions (EQs) Students will keep considering How are symbols and expressions utilized as representations in mathematics? In what ways can a quantity be represented? In what ways is measurement useful? How can I use patterns, relations, and functions? How are geometric concepts and relations conceptualized? What roles can data play in the decision making process?

26 Students will know Knowledge Grade Level Benchmarks Students will be able to Skills A function is a set of ordered pairs in which each first coordinate appears with exactly one second coordinate. F(x) is read the function of x (do not mean multiplication). An absolute value graph is in the shape of a v. Coefficients in an absolute value equation mimic actions of the coefficients of a quadratic equation and the graphs. Domain is the input value/x-value of a function. Range is the output value/y-value. Functions can have restricted domains and ranges. Non-linear functions do not have a constant rate of change. A solution to a rational equation may not always work if it sets the denominator equal to zero. Evaluate functions. Solve equations involving function notation f(x). Graph functions. Identify a function. Identify domain and range of functions. Predict size and shape of graphs of functions. Simplify radical expressions. Identify restricted values of rational functions. Simplify rational expressions. Add, subtract, multiply, and divide rational expressions. Solve rational equations and identify extraneous solutions. Use function keys on a calculator. Graph non-linear functions on a graphing calculator.

27 TASKS from CCSS for Non-Linear Functions 8.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.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations 8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1/33 = 1/27. 8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x² =p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. 8.EE.4. Perform 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. 8.G.6. Explain a proof of the Pythagorean Theorem and its converse. 8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. A-SSE.4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. A-APR.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-APR.7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.