Algebra I Support Lab CURRICULUM GUIDE AND INSTRUCTIONAL ALIGNMENT

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1 TRENTON PUBLIC SCHOOLS Department of Curriculum and Instruction 108 NORTH CLINTON AVENUE TRENTON, NEW JERSEY Secondary Schools Algebra I Support Lab CURRICULUM GUIDE AND INSTRUCTIONAL ALIGNMENT

2 The Algebra Support Lab is an extension of the curriculum. It is meant to provide support to students who have been identified for intervention based on test scores and progress in previous math classes. All of the concepts presented in Algebra I are progressions of the concepts that were started in grades 6 through 8. In the support lab, the concepts/standards taught in 6-8 are identified and students are presented the material at their level in order to progress to the corresponding standard. This class is meant to work on individual skills and pre-requisites that students are missing. It is a blended approach using MathXL for individual remediation and teacher created group activities to promote a deeper understanding of the math topics using the 8 practices from the common core state standards. 1

3 Grade level: HS /Algebra Lab Algebra Support Lab Scope and Sequence Marking Period 1: Stage 1 Desired Results Enduring Understandings/Goals: Interpreting numbers as quantities with appropriate units, scales, and levels of accuracy allows one to effectively model and make sense of real world problems. Relationship between numbers can be represented by equation, inequalities and systems. Equivalent equations can be found using inverse operations and simplification. Essential Questions: In what ways can the choice of units, quantities and levels of accuracy impact a solution? How can algebra describe the relationship between sets of numbers? Can equations that appear to be different be equivalent? Mathematical Practices: 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. Standard: Student Learning Objectives Prerequisites Necessary for Mastery Suggested Resources N.Q.1 Use units as a way to understand problems and to guide the solution 1. Solve multi-step problems that can be represented 6.RP.A1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Pearson Algebra 1 Common Core Series (in relation of multi-step problems; algebraically with 6.RP.A2 Understand the concept of a unit rate a/b to choose and interpret units accurate and associated with a ratio a:b with b 0, and use rate consistently in formulas; appropriately defined language in the context of a ratio relationship. 2

4 choose and interpret the scale and the origin in graphs and data displays. N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 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. 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.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.CED.4 Rearrange formulas to highlight a units, scales, and models (such as graphs, tables, and data displays). S.C. 2. Interpret terms, factors, coefficients, and expressions (including complex linear and exponential expressions) in terms of context. M.C. 3. Solve linear equations and inequalities in one variable (including literal equations). Justify each step in the process and solution. 6.RP.A3 Use ratio and rate reasoning to solve real-world and problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.A.3 Apply the properties of operations to generate equivalent expressions. 6.EE.A.4 Identify when two expressions are equivalent 6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. activities to develop Pearson Algebra 1 Common Core Series (in relation to activities to develop Pearson Algebra 1 Common Core Series (in relation to 3

5 quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph M.C. 4. Create linear equations and inequalities in one variable and use them to solve problems. Justify each step in the process and the solution. M.C. 5. Create linear equations in two or more variables to represent relationships 6.EE.B.7 Solve real-world and problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 6.EE.B.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. 7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.EE.B.3 Solve multi-step real-life and problems posed with positive and negative rational activities to develop Pearson Algebra 1 Common Core Series (in relation to activities to develop Pearson Algebra 1 Common Core Series (in relation to 4

6 equations on coordinate axes with labels and scales. between quantities; graph equations on coordinate axes with labels and scales. M.C. numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 7.EE.B.4 Use variables to represent quantities in a realworld or problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. activities to develop A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a 6. Model and describe constraints with linear equations and inequalities and systems of equations and/or inequalities to determine if solutions are viable or non-viable. M.C. 8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 8.EE.A.2 Use square root 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 small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 8.EE.A.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 Pearson Algebra 1 Common Core Series (in relation to activities to develop 5

7 solution. Construct a viable argument to justify a solution method. 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. A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 1. Solve systems of linear equations in two variables graphically and algebraically. Include solutions that have been found by replacing one equation by the sum of that equation and a multiple of the other. A.C. technology 8.EE.B.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.B.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.C.7 Solve linear equations in one variable. 8.EE.C.8 Analyze and solve pairs of simultaneous linear equations Pearson Algebra 1 Common Core Series (in relation to activities to develop Stage 2 Assessment Evidence Suggested Performance Tasks: Other Evidence: Exemplars Classwork Extended projects Exit Slips Math Webquests Homework Writing in Math/Journal Open-ended questions Portfolio Quizzes Stage 3 Learning Plan Lesson Format 10 min: Do Now/Journaling (whole class) 30 min: Individual computer work - intervention 6

8 30 min: Group work Complex problems used to develop a deeper understanding using practices 10 min: Closure/Assessment/Evaluation UNIT NAME: Linear Relationships Grade level: HS/ALGEBRA LAB Marking Period 2: Stage 1 Desired Results Enduring Understandings/Goals: There is often an optimal method of manipulating equations and inequalities to solve a problem, however, other methods which may not be efficient can still provide insight into the problem. Equations, verbal descriptions, graphs, and tables provide insight into the relationship between quantities. Functions can be represented by using tables, equations and graphs. Real-world situations can be modeled by discrete and continuous functions. Essential Questions: In what ways can the problem be solved and why should one method be chosen over another? How can the relationship between quantities best be represented? How can you represent and describe functions? How can functions describe real-world situations? Mathematical Practices: 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. 7

9 Standard: 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.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 Student Learning Objectives 2. Find approximate solutions of linear equations by making a table of values, using technology to graph and successive approximations. M.C. 3. Graph equations, inequalities, and systems of inequalities in two variables and explain that the solution to an equation is all points along the curve, the solution to a system of linear functions is the point of intersection, and the solution to a system of inequalities is the intersection of the corresponding half- Prerequisites Necessary for Mastery 6.EE 7.EE 8.EE See standards listed in unit 1. 6.EE 7.EE 8.EE See standards listed in unit 1. Suggested Resources Pearson Common Core Series (in relation to activities to develop Pearson Common Core Series (in relation to activities to develop 8

10 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. F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y=f(x). F.IF.2 Use function notation, evaluates functions for inputs in their domains, and interprets statements that use function notation in terms of a context. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f (0) = f (1) = 1, f (n+1) = f (n) + f (n-1) for n 1. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the planes. M. C. 4. Explain and interpret the definition of functions including domain and range and how they are related; correctly use function notation in a context and evaluate functions for inputs and their corresponding outputs. M.C. 5. Write a function for a geometric sequence defined recursively, whose domain is a subset of the integers. M.C. 6. Graph functions by hand (in simple cases) and with technology 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 8.F.A.3 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. 8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the Pearson Common Core Series (in relation to activities to develop Pearson Common Core Series (in relation to activities to develop Pearson Common Core Series (in relation to 9

11 function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear functions. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 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. (in complex cases) to describe linear relationships between two quantities and identify, describe, and compare domain and other key features in one or multiple representations. M.C. 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.B.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. activities to develop F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 7. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). S.C. Pearson Common Core Series (in relation to 10

12 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. 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). 1. Interpret parts of expressions in terms of context including those that represent square and cube roots; use the structure of an expression to identify ways to rewrite it. M. C. 6.EE 7.EE 8.EE See standards listed in unit 1. activities to develop Pearson Common Core Series (in relation to activities to develop 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. 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 2. Manipulate expressions using factoring, completing the square and properties of exponents to produce equivalent forms that highlight particular properties such as the zeros or the maximum or minimum value of the function. S. C. 6.EE 7.EE 8.EE See standards listed in unit 1. Pearson Common Core Series (in relation to activities to develop 11

13 the annual rate is 15% Stage 2 Assessment Evidence Suggested Performance Tasks: Other Evidence: Exemplars Classwork Extended projects Exit Slips Math Webquests Homework Writing in Math/Journal Open-ended questions Portfolio Quizzes Stage 3 Learning Plan Lesson Format 10 min: Do Now/Journaling (whole class) 30 min: Individual computer work - intervention 30 min: Group work Complex problems used to develop a deeper understanding using practices 10 min: Closure/Assessment/Evaluation 12

14 Grade level: HS/ALGEBRA LAB Marking Period 3: UNIT NAME: Expressions and Equations Stage 1 Desired Results Enduring Understandings/Goals: Expressions can be written in multiple ways using the rules of algebra: each version of the expression tells something about the problem it represents. The relationships between numbers can be represented by equations, inequalities and systems. Quadratic equations can be solved by graphing, factoring, completing the square, and using the quadratic formula. Essential Questions: Why structure expressions in different ways? How can algebra describe the relationship between sets of numbers? How can you solve a quadratic equation? Mathematical Practices: 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. Standard: Student Learning Objectives Prerequisites for Mastery Suggested Resources 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, 3. Perform addition, subtraction and multiplication with polynomials and relate it to arithmetic operations with integers. M. C. 6.EE 7.EE 8.EE See standards listed in MP1 Pearson Common Core Series (in relation to 13

15 and multiply polynomials. F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula,, use them to model situations, and translate between the two forms. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales Write linear and exponential functions (e.g. growth/decay and arithmetic and geometric sequences) from graphs, tables, or a description of the relationship, recursively and with an explicit formula, and describe how quantities increase linearly and exponentially over equal intervals. M. C. 5. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, simple rational and exponential functions and highlighting a quantity of interest in a formula. M. C. 6. Create linear and quadratic equations that represent a relationship between two or more variables. Graph equations on the coordinate axes with labels and scale. M. C. 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 8.F.B.5 See standards listed in MP 2 6.EE 7.EE 8.EE See standards listed in MP 1 6.EE 7.EE 8.EE See standards listed in MP1 Pearson MathXL learning. (group activities to develop

16 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.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. N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of 7. Derive the quadratic formula by completing the square and recognize when there are no real solutions. M. C. 8. Solve quadratic equations in one variable using a variety of methods [including inspection (e.g. x 2 = 81), factoring, completing the square, and the quadratic formula]. M. C. 1. 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. S.C. 2. Use properties of integer exponents to explain and convert between expressions involving radicals and rational exponents, using 6.EE 7.EE 8.EE See standards listed in MP1 6.EE 7.EE 8.EE See standards listed in MP1 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, 6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities 15

17 rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5( 1/3 ) 3 to hold, so (5 1/3 ) 3 must equal 5. N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational. correct notation. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5( 1/3 ) 3 to hold, so (5 1/3 ) 3 must equal 5. A.C. 3. Use the properties of rational and irrational numbers to explain why the sum or product of two rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational. A.C. having opposite directions or values use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.C.7 Understand ordering and absolute value of rational numbers. 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.A.3 Solve real-world and problems involving the four operations with rational numbers. 8.NS.A.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 16

18 F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for Sketch the graph of a function that models a relationship between two quantities (expressed symbolically or from a verbal description) showing key features ( including intercepts, minimums/maximums, domain, and rate of change) by hand in simple cases and using technology in more complicated cases and relate the domain of the function to its graph. M.C. eventually into a rational number. 8.NS.A.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., π 2 ). 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 8.F.B.5 See standards listed in MP 2 Pearson Common Core Series (in relation to Pearson MathXL learning. (group activities to develop

19 more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Exponential, growth or decay. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude Stage 2 Assessment Evidence Suggested Performance Tasks: Other Evidence: Exemplars Classwork Extended projects Exit Slips Math Webquests Homework Writing in Math/Journal Open-ended questions Portfolio Quizzes Stage 3 Learning Plan Lesson Format 10 min: Do Now/Journaling (whole class) 30 min: Individual computer work - intervention 30 min: Group work Complex problems used to develop a deeper understanding using practices 10 min: Closure/Assessment/Evaluation UNIT NAME: Functions and Descriptive Statistics 18

20 Grade level: HS/ALGEBRA LAB Marking Period 4: Stage 1 Desired Results Enduring Understandings/Goals: Statisticians summarize, represent and interpret categorical and quantitative data in multiple ways. Data can be collected and analyzed to help you make decisions and predictions Essential Questions: How can the properties of data be communicated to eliminate its important features? How can data be collected and analyzed to help you make and predictions? Mathematical Practices: 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. Standard: Student Learning Objectives Prerequisites Needed for Mastery F.IF.9 Compare properties 5. Compare 8.F.A.1 of two functions each properties of two 8.F.A.2 represented in a different functions each 8.F.A.3 way (algebraically, represented in a 8.F.B.4. graphically, numerically in different way 8.F.B.5 tables, or by verbal (algebraically, descriptions). For example, graphically, See standards listed in MP 2 given a graph of one numerically in tables, quadratic function and an or by verbal algebraic expression for descriptions). For another, say which has the example, given a Suggested Resources Pearson Common Core Series (in relation to Pearson MathXL 19

21 larger maximum. F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. c. Use the properties of exponents to interpret expressions for exponential graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. S.C. 6. Calculate (over a specified period if presented symbolically or as a table) or estimate (if presented graphically) and interpret the average rate of change of a function. M.C. 7. Write functions in different but equivalent forms by manipulating quadratic expressions using methods such as factoring and completing the square, or exponential expressions using the properties of exponents, to reveal and explain properties of the function. S.C. 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 8.F.B.5 See standards listed in MP 2 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 8.F.B.5 See standards listed in MP 2 learning. (group activities to develop 20

22 functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01)12 t, y = (1.2) t /10, and classify them as representing exponential growth or decay. F.BF.1 Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for 8. Write a function that describes a linear or quadratic relationship between two quantities given in context using an explicit expression, a recursive process, or steps for calculation (include contexts that require a combination of various function types). For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. S.C. 9. Identify the effects of translations [ f(x) + k, k f(x), f(kx), and f(x + k)] on a 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 8.F.B.5 See standards listed in MP 2 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 21

23 specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a 22 function and find the value of k given the graphs. A.C. 10. Compare (using graphs and tables) linear, quadratic, and exponential models to determine that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function, include interpretation of parameters in terms of a context. S.C. 3. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets 8.F.B.5 See standards listed in MP 2 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4. 8.F.B.5 See standards listed in MP 2 6.SP.A.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. 6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described

24 procedure is not appropriate. Use calculators, spreadsheets, and tables estimate areas under the normal curve. S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables estimate areas under the normal curve. A.C. 4. Summarize and interpret categorical data for two categories in two-way frequency tables; recognize associations and trends in the data. S.C. 5. Represent and describe data for two variables on a scatter plot, fit a function to the data, analyze residuals (in order to informally assess fit), and use the function to solve problems. Uses a given function or choose a function suggested by the context. Emphasize linear and exponential models. S.C. by its center, spread, and overall shape. 6.SP.A.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. 7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variability, measuring the difference between the centers by expressing it as a multiple of a measure of variability. 7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. 8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, 23

25 c. Fit a linear function for a scatter plot that suggests a linear association. S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. S.ID.9 Distinguish between correlation and causation. 6. Interpret the slope, intercept and correlation coefficient (compute using technology) of a linear model. M.C. 7. Distinguish between correlation and causation in a data context. M.C. and nonlinear association. 8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Suggested Performance Tasks: Exemplars Extended projects Math Webquests Writing in Math/Journal Stage 2 Assessment Evidence Other Evidence: Classwork Exit Slips Homework Open-ended questions Portfolio Quizzes Stage 3 Learning Plan Lesson Format 10 min: Do Now/Journaling (whole class) 30 min: Individual computer work - intervention 24

26 30 min: Group work Complex problems used to develop a deeper understanding using practices 10 min: Closure/Assessment/Evaluation 25

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