1 PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I
2 Key Ideas and Details Section I: Alignment to the Standards for Mathematical Practice Algebra I Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. The PowerTeaching curriculum consistently encourages students to ask questions, plan for solutions, assess their reasoning and the reasonableness of their answers, and to check their work. The students focus on these good habits as a part of the daily PowerTeaching lesson routine as well as specific strategy lessons throughout the curriculum. Team Huddle During daily Team Huddle activities, students work with their teammates to discuss, plan for, and solve math problems. Within the team, they must work through disagreements, ensure that each teammate understands and can explain the solution, and encourage each other when problems seem difficult. Problem Solving Strategies Students practice the various problem-solving strategies at multiple points. Specific lessons introduce and have students practice the strategies: identify extraneous data, make a model, find a pattern, guess and check, work backwards, and solve a simpler problem. Extended Response Many PowerTeaching learning cycles culminate in an extended response lesson. The math problems in these lessons are complex and combine multiple math topics. The teacher modeling, teamwork activities, and individual practice are all centered on solving these real-world problems in steps: understand the problem, find the parts, make a plan, estimate the answer, find the solution, and assess the reasonableness and correctness of the solution. Standard for Mathematical Practice 2: Reason abstractedly and quantitatively. Throughout PowerTeaching students will routinely approach math concepts using both concrete and abstract tools and methods. Problem Solving Strategies The problem solving strategies that students learn help them break apart word problems and real-world math scenarios into the important information, then represent this information as numeric and algebraic models. Problem Solving Practice In each cycle, students will apply the problem solving strategies they have learned. Many lessons include real-world math problems. The students learn to represent the solutions to these problems concretely and abstractly. Students are also routinely asked to design a math story for a numeric or algebraic model. Project-Based Learning The PowerTeaching curriculum includes quarterly project-based learning opportunities. These activities will be multi-day cycles of learning that include planning, research, modeling, reporting, and presenting. Students will be required to represent their project topic mathematically, use the math to find a solution to the problem they researched or an answer to the question they asked, and then explain how the mathematical model relates back to their original problem or question. Standard for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. Students will support their arguments with sound reasoning as well as critique or support the reasoning of others. They will construct their supports and critiques both in writing as well as verbally. Get the Goof Each lesson includes a Get the Goof activity. Students will discuss a completed problem related to recently studied math topics. They will work with their teams to identify the
3 error in think that led to a mistake in the math work. The students will explain the error and correct the math. Random Reporter A part of the daily PowerTeaching routine includes teamwork and team discussion to solve problems. At various points during each lesson, the teacher will use Random Reporter to have a student from each team share their answer and support that answer with their team s reasoning. Extended Response One type of extended response math problem will have students critique the math reasoning presented in the problem, correctly solve the problem, and construct a viable argument to support their reasoning. These types of extended response situations will represent about one third of the PowerTeaching extended response experiences. Standard for Mathematical Practice 4: Model with mathematics. Students will use tables, graphs, charts and diagrams to represent mathematical information. They will also use number sentences, expressions, and equations to describe a situation. Students will also use the information they gather in tables, graphs, charts, and diagrams to identify patterns, determine relationships, and draw conclusion. Problem Solving Strategy: Modeling Students will receive specific and targeted instruction on modeling as a strategy to solve math problems. Problem Solving Practice The ongoing problem solving experiences, word problems, real-world scenarios, and extended response, often require students to represent the data as a model. Students must determine which model would best help them find the solution or answer the question. Standard for Mathematical Practice 5: Use appropriate tools strategically. Throughout the PowerTeaching curriculum, students will be guided to use various tools to solve math problems and answer math questions. They will also be faced with opportunities to choose which tool would best help them solve more complex math problems or real-world scenarios. The students will more often be faced with choices when completing extended response and project-based learning activities. Standard for Mathematical Practice 6: Attend to precision. Students will use symbols, math vocabulary, and clear explanations in their team discussions and written and oral explanations. Students will also make choices to best represent their solution and reasoning clearly and efficiently. Rubrics Students will use rubrics to assess the completeness and clarity of their oral and written explanations. They will also use the rubrics to critique the explanations of their peers. Complete explanations include the correct answer stated as a complete sentence that identifies the question and a clear explanation in words, as a diagram, using symbols. Vocabulary PowerTeaching key vocabulary is highlighted in each lesson. The definition is built into the lesson instead of only existing in a separate glossary. Students will see the vocabulary used correctly within the teacher modeling and be expected to use key vocabulary to support their mathematical thinking. Standard for Mathematical Practice 7: Look for and make use of structure. Specific targeted skills in the PowerTeaching curriculum address the topics of structure and patterns. Problem Solving Strategy: Look for a Pattern Students will learn to identify problems that can be solved by finding and describing a pattern. They will learn how to represent the data to most efficiently identify the pattern. In later lessons, students will apply this strategy to new and more
4 complex problem solving situations. Expressions and Equations Within the Expressions and Equations content area students will consistently work to make sense of data by defining any patterns they notice and translating those patterns into expressions, equations, and graphs. Formulas and Mathematical Rules In the PowerTeaching curriculum, students will be guided through instruction, modeling, teamwork, and individual practice, to define rules and formulas based on work with multiple examples. Instead of being given the rule, they will have to write the rule, and then prove it by applying it to new situations. Standard for Mathematical Practice 8: Look for and express regularity in repeated reasoning. Specific targeted skills in the PowerTeaching curriculum address the topic of repeated reasoning to find shortcuts, processes, and formulas. Expressions and equations Within the Expressions and Equations content area, students will prove expressions equivalent, prove or disprove solutions to equations and inequalities, and use the properties of addition and multiplication. Geometry Within the Geometry content area of PowerTeaching, students will apply their knowledge of expressions and equations to geometry and derive formulas for area, volume, and surface area.
5 Key Ideas and Details Section II: Alignment to the Standards for Mathematical Content Algebra I Standard for Mathematical Content N.Q 1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Unit 2 Cycle 1 Lesson 1 Using Appropriate Units Objective: Select and use appropriate units of measure to solve multiple-step problems Unit 2 Cycle 1 Lesson 2 Units in Formulas Objective: Choose and interpret units correctly in formulas Unit 2 Cycle 1 Lesson 3 Units and Graphing Objective: Determine the scale and origin to graph data and use the graph to answer questions Unit 2 Cycle 2 Lesson 5 Scientific Notation Objective: Write numbers in scientific notation and perform operations on numbers written in scientific notation Standard for Mathematical Content N.Q 2: Define appropriate quantities for the purpose of descriptive modeling. Unit 2 Cycle 2 Lesson 4 Defining Quantities Objective: Define appropriate quantities and calculate multiple quantities to describe real world situations Standard for Mathematical Content N.Q 3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities Unit 2 Cycle 2 Lesson 6 Significant Digits Objective: Write quantities using the correct number of significant digits Unit 2 Cycle 2 Lesson 7 Accuracy with Measurement Instruments Objective: Select and use an appropriate level of accuracy based on measurement instruments Unit 2 Cycle 2 Lesson 8 Quantities in Complex Problems Objective: Solve real-world, complex problems involving quantities, accurate measures, and significant digits Standard for Mathematical Content 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. Unit 2 Cycle 3 Lesson 9 Parts of Expressions Objective: Identify and name the parts of an algebraic expression Unit 10 Cycle 1 Lesson 1 Identify and Interpret Parts of an Expression Objective: Interpret complex expressions in different ways, focusing on quadratic expressions b. Interpret complicated expressions by viewing one or more of their parts as a single entity. Unit 2 Cycle 3 Lesson 10 Describing Complicated Algebraic Expressions Objective: Write math statements to represent complicated algebraic expressions Unit 2 Cycle 3 Lesson 11 Expressions in Context Objective: Interpret and describe the parts of an expression in context of a real-world situation
6 Unit 2 Cycle 3 Lesson 12 Write Complicated Expressions Objective: Given a contextual situation, write multiple term algebraic expressions Standard for Mathematical Content A.SSE 2: Use the structure of an expression to identify ways to rewrite it. Unit 10 Cycle 1 Lesson 2 Factoring Quadratic Expressions I Objective: Factor quadratic expressions using common rules such as the difference of two squares Unit 10 Cycle 1 Lesson 3 Rewrite Expressions Objective: Rewrite complex expressions in different ways, focusing on quadratic expressions Standard for Mathematical Content 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. Unit 10 Cycle 2 Lesson 4 Factoring Quadratic Expressions II Objective: Use additional methods to factor quadratics; find the zeros, Zero-Product Property, connect factors, etc. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Unit 10 Cycle 2 Lesson 5 Completing the Square Objective: Complete the square in a quadratic expression to find the maximum and minimum value of the function c. Use the properties of exponents to transform expressions for exponential functions Unit 10 Cycle 2 Lesson 6 Rewriting Exponential Expressions Objective: Use the properties of exponents to transform exponential functions to make equivalent expressions, answer a question, or find a solution Standard for Mathematical Content A.CED 1: Create equations and inequalities in one variable and use them to solve problems. Unit 3 Cycle 1 Lesson 1 Creating Equation and Inequalities in One Variable Objective: Create linear and exponential equations and linear inequalities in one variable to solve problems Unit 11 Cycle 1 Lesson 1 Creating Equations and Inequality in One Variable Objective: Write equations for situation including linear, exponential, and quadratic equation in one variable Standard for Mathematical Content A.CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Unit 3 Cycle 1 Lesson 2 Creating Equation and Inequalities in Two Variables Objective: Create linear and exponential equations and linear inequalities in two variables to solve problems Unit 11 Cycle 1 Lesson 2 Creating Equations and Inequalities in Two Variables Objective: Write equations for situations including linear, exponential, and quadratic equations in two variables Standard for Mathematical Content 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.
7 Unit 3 Cycle 1 Lesson 3 Representing Constraints Objective: Represent constraints by equations and/or inequalities; determine if a solution in a viable solution to a contextual problem Standard for Mathematical Content A.CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Unit 3 Cycle 1 Lesson 4 Rearranging Formulas Objective: Rearrange formulas for a particular variable of interest Unit 11 Cycle 1 Lesson 3 Solve Formulas for a Given Variable Objective: Solve a given equation or formula for a particular variable, including linear, exponential, and quadratic equations Standard for Mathematical Content 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. Unit 3 Cycle 2 Lesson 5 Equation Proofs Objective: Use the properties of multiplication and addition to explain each step in solving an equation Standard for Mathematical Content A.REI 3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Also aligns to 8.EE.8 Unit 3 Cycle 2 Lesson 6 Solving Linear Equations Objective: Solve linear equations in one variable Unit 3 Cycle 2 Lesson 7 Solving Linear Inequalities Objective: Solve linear inequalities in one variable Standard for Mathematical Content 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. Unit 11 Cycle 2 Lesson 4 Solve Quadratic Equations I Objective: Solve quadratic equations by factoring the equation b. Solve quadratic equations by inspection (e.g., for x 2 = 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. Unit 11 Cycle 2 Lesson 5 Derive the Quadratic Formula Objective: Solve a quadratic equation for a given variable to derive the quadratic formula Unit 11 Cycle 2 Lesson 6 Complex Numbers Objective: Explore what complex numbers are, what they represent, and how to use them Unit 11 Cycle 2 Lesson 7 Solve Quadratic Equations II Objective Use the quadratic formula to solve quadratic equations Standard for Mathematical Content 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. Also aligns to 8.EE.8
8 Unit 5 Cycle 2 Lesson 5 Solve Systems of Equations Algebraically Objective: Use the process of substitution and elimination to solve a system of two-variable linear equations Unit 5 Cycle 2 Lesson 6 Problem Solving with Systems of Equations Objective: Solve a pair of equations graphically and/or algebraically to solve a real-world math situation Unit 11 Project-Based Cycle Solve Systems of Equations with Quadratic Equations Objective: During a three-day cycle, solve systems of equations that include quadratic equations by solving the equations algebraically and graphically Standard for Mathematical Content A.REI 6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Also aligns to 8.EE.8 Unit 5 Cycle 2 Lesson 4 Solve Systems of Equation by Graphing Objective: Solve a system of two-variable linear equations by graphing both equations Unit 5 Cycle 2 Lesson 6 Problem Solving with Systems of Equations Objective: Solve a pair of equations graphically and/or algebraically to solve a real-world math situation Standard for Mathematical Content 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). Unit 5 Cycle 1 Lesson 1 Graphs of Two-Variable Equations Objective: Discover that the graph of a two-variable equation represents all possible solutions for the equation Standard for Mathematical Content 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. Unit 5 Cycle 1 Lesson 2 Graphs of Multiple Equations Objective: Discover that any points of intersection when two equations are graphed represent the solution(s) to both equations and test that knowledge Unit 5 Cycle 1 Lesson 3 Approximate Solutions to Systems of Equations Objective: Approximate the solution to various systems of equations using graphing software Standard for Mathematical Content A.REI 12: Graph the solutions to a linear inequality in two variables as a halfplane (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. Unit 5 Cycle 3 Lesson 7 Graph an Inequality Objective: Represent a linear inequality on a graph Unit 5 Cycle 3 Lesson 8 Inequalities in Context Objective: Graph linear inequalities to describe solutions in context and determine if given values are solutions to contextual linear inequalities by graphing them Unit 5 Cycle 3 Lesson 9 Solving Paris of Linear Inequalities Objective: Solve a pair of linear inequalities by graphing and identifying the overlap of solutions to their graphs Standard for Mathematical Content 8.EE 1: Know and apply the properties of integer exponents to generate equivalent numerical expressions
9 Unit 4 Cycle 1 Lesson 1 Understanding Negative Exponents Objective: Explain what a negative exponent represent in a numeric or algebraic expression and rewrite expressions with negative exponents Standard for Mathematical Content 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 rational exponents. Unit 4 Cycle 1 Lesson 2 Operations with Exponents Objective: Add, subtract, multiply, and divide numbers and algebraic terms with exponents Unit 4 Cycle 1 Lesson 4 Simplify and Rewrite Expressions with Radicals Objective: Simplify and rewrite numeric and algebraic steps with multiple steps of exponents and radicals Standard for Mathematical Content N.RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. Unit 4 Cycle 1 Lesson 3 Fractional Exponents and Radicals Objective: Rewrite numeric and algebraic expression with fractional exponents and radicals Unit 4 Cycle 1 Lesson 4 Simplify and Rewrite Expressions with Radicals Objective: Simplify and rewrite numeric and algebraic steps with multiple steps of exponents and radicals Standard for Mathematical Content 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 nonzero rational number and an irrational number is irrational. Unit 2 Cycle 1 Lesson 2 Units in Formulas Objective: Choose and interpret units correctly in formulas Standard for Mathematical Content 8.F 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. Unit 6 Cycle 2 Lesson 4 Linear Functions Objective: Discover that y = mx + b is a function that describes a straight line Unit 6 Cycle 2 Lesson 5 Exponential Functions Objective: Compare linear and exponential functions using graphs, data sets, and equations Standard for Mathematical Content 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). Also aligns to 8.IF.1 Unit 6 Cycle 1 Lesson 1 Defining Functions Objective: Define and identify function in words, in tables, and graphically by definition Unit 6 Cycle 1 Lesson 2 Domain and Range Objective: Determine the domain and range of functions given different representations Standard for Mathematical Content F.IF 2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Also aligns to 8.IF.2 Unit 6 Cycle 1 Lesson 3 Evaluate Functions Objective: Evaluate functions for given values and use function notation Standard for Mathematical Content F.IF 3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers Also aligns to 8.IF.2
10 Unit 6 Cycle 2 Lesson 6 Sequences as Functions Objective: Recognize that sequences represent function that are sometimes represented recursively Standard for Mathematical Content 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. Also aligns to 8.F.5 Unit 6 Cycle 3 Lesson 8 Functions and Graphs I Objective: Analyze the graph of a function and use the graph to describe different aspects of the function and sketch a function given a description Standard for Mathematical Content F.IF 5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Unit 6 Cycle 3 Lesson 9 Functions and Graphs II Objective: Given a graph or other information about a function, discuss the domain in detail Standard for Mathematical Content 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. Unit 6 Cycle 3 Lesson 7 Construct and Analyze Linear Functions Objective: Given different information about a function, determine the rate of change Standard for Mathematical Content F.IF 7: 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. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Also aligns to 8.IF.3 Unit 7 Cycle 1 Lesson 1 Linear and Quadratic Functions Objective: Graph and analyze linear and quadratic functions to show intercepts, maxima, and minima Unit 12 Cycle 1 Lesson 1 Graph Linear and Quadratic Functions Objective: Graph linear and quadratic functions Unit 12 Cycle 1 Lesson 2 Graph Absolute Value, Step, and Piece-Wise Functions Objective: Graph absolute value, step, and piece-wise functions b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Unit 12 Cycle 1 Lesson 1 Graph Linear and Quadratic Functions Objective: Graph linear and quadratic functions Unit 12 Cycle 1 Lesson 2 Graph Absolute Value, Step, and Piece-Wise Functions Objective: Graph absolute value, step, and piece-wise functions e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Unit 7 Cycle 1 Lesson 2 Exponential, Logarithmic, and Trigonometric Functions Objective: Graph and analyze exponential, logarithmic, and trigonometric functions to show various descriptors such as intercepts, end behavior, period, midline, and amplitude Standard for Mathematical Content 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.
11 Unit 12 Cycle 1 Lesson 3 Rewriting Functions Objective: Use factoring, completing the square, and the properties of exponents to rewrite functions to highlight a property of the function b. Use the properties of exponents to interpret expressions for exponential functions. Unit 12 Cycle 1 Lesson 3 Rewriting Functions Objective: Use factoring, completing the square, and the properties of exponents to rewrite functions to highlight a property of the function Standard for Mathematical Content F.IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Unit 7 Cycle 1 Lesson 3 Compare Functional Representations Objective: Compare the graphs of different functions and compare the properties of functions given in different ways Unit 12 Cycle 1 Lesson 4 Compare Functions Objective: Compare the properties of two functions including quadratic functions Standard for Mathematical Content 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. Also aligns to 8.F.4. Unit 7 Cycle 2 Lesson 4 Functions and Context Objective: Determine an explicit expression, a recursive process, or steps for a calculation from a given context for a function Unit 12 Cycle 2 Lesson 5 Write a Function from a Context Objective: Determine the elicit expression, a recursive process, or the steps for a calculation for a contextual situation, including Quadratic functions b. Combine standard function types using arithmetic operations. Unit 7 Cycle 2 Lesson 5 Combine Function Objective: Add, subtract, multiply, and divide separate functions Unit 10 Cycle 2 Lesson 6 Combine Functions Objective: Add, subtract, multiply, and divide functions to create new functions including quadratic functions Standard for Mathematical Content 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. Unit 7 Cycle 2 Lesson 6 Model Situations with Geometric and Arithmetic Sequences Objective: Write arithmetic and geometric sequences both recursively and with an explicit formula to model situations Standard for Mathematical Content 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 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. Unit 7 Cycle 3 Lesson 7 Even and Odd Functions Objective: Recognize even and odd functions from their graph and the algebraic expressions for them Unit 7 Cycle 3 Lesson 8 Changes to Function Graphs Objective: Record the changes to the output of a function when the same value is combined with the function in different ways
12 Unit 7 Cycle 3 Lesson 9 Functions of Functions Objective: Given two functions, find a function of the other function; find f(gx) Unit 10 Cycle 2 Lesson 7 Building New Functions Objective: Record the changes to the output of a function when the same value is combining with the quadratic function in different ways Standard for Mathematical Content F.BF 4: Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Unit 10 Cycle 2 Lesson 8 Find the Inverse of a Function Objective: Understand the inverse of a function and find it for linear and simple exponential functions Standard for Mathematical Content F.LE 1: Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Unit 8 Cycle 1 Lesson 1 Growth of Functions Objective: Establish a pattern of growth for linear and exponential functions b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Unit 8 Cycle 1 Lesson 2 Recognize Linear and Exponential Situations Objective: Given mathematical situation, identify whether they are linear of exponential c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Unit 8 Cycle 1 Lesson 2 Recognize Linear and Exponential Situations Objective: Given mathematical situations, identify whether they are linear or exponential Standard for Mathematical Content F.LE 2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Unit 8 Cycle 1 Lesson 3 Construct Linear and Exponential Functions Objective: Given information like a graph or input/output pairs, construct linear and exponential functions Standard for Mathematical Content 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. Unit 8 Cycle 1 Lesson 4 Compare Linear and Exponential Models Objective: Compare linear and exponential models of functions by observing that the quantities starting at the same point behave differently Unit 10 Project-Based Cycle Linear vs. Quadratic Functions Objective: During the three-day cycle, use graphs, tables, and functions written symbolically to compare linear and quadratic functions and solve real-world math problems Standard for Mathematical Content F.LE 5: Interpret the parameters in a linear or exponential function in terms of a context. Unit 8 Cycle 1 Lesson 5 Functions and Parameters Objective: Interpret the parameters of linear and exponential functions in context Unit 11 Project-Based Cycle Functions in Context Objective: During a three-day cycle, apply all function work to real-world situations
13 Standard for Mathematical Content S.ID 1: Represent data with plots on the real number line (dot plots, histograms, and box plots). Unit 9 Cycle 1 Lesson 1 Numeric Data Displays Objective: Represent data on various numeric data displays, choose the best display and interpret the data display Standard for Mathematical Content S.ID 2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Unit 9 Cycle 1 Lesson 2 Measure of Center and Spread Objective: Compare the center and spread of various date given on dot numeric data displays using measure like mean and standard deviation Standard for Mathematical Content S.ID 3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Unit 9 Cycle 1 Lesson 3 Data Displays and Problem Solving Objective: Interpret the difference in shape, center, spread, trend, outliers, and clusters of data on different graphs in context Standard for Mathematical Content S.ID 5: 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 procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Unit 9 Cycle 2 Lesson 4 Frequency Tables Objective: Organize, summarize, and interpret categorical data for two categories into frequency tables and discuss the characteristics of each data display Standard for Mathematical Content 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. Also aligns to 8.SP.1. Unit 9 Cycle 2 Lesson 5 Scatter Plots Objective: Construct scatter plots for bivariate data and interpret the shape of the plot by discussing clusters, outliers, and general association b. Informally assess the fit of a function by plotting and analyzing residuals. Also aligns to 8.SP.2 Unit 9 Cycle 2 Lesson 6 Functions and Scatter Plots Objective: Assess how well a function fits the data of a scatter plot by determining the residuals c. Fit a linear function for a scatter plot that suggests a linear association. Also aligns to 8.SP.2 Unit 9 Cycle 2 Lesson 7 Lines of Best Fit Objective: Create a line of best fit for scatter plots with linear associations Standard for Mathematical Content S.ID 7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Also aligns to 8.SP.3 Unit 9 Cycle 3 Lesson 8 Slope and Intercept Objective: Interpret the slope and intercept of a linear association in context Standard for Mathematical Content S.ID 8: Compute (using technology) and interpret the correlation coefficient of a linear fit. Also aligns to 8.SP.4
14 Unit 9 Cycle 3 Lesson 9 Correlation Coefficient Objective: Find and analyze the correlation coefficient of a linear line of best fit Standard for Mathematical Content S.ID 9: Distinguish between correlation and causation. Also aligns to 8.SP.4 Unit 9 Cycle 3 Lesson 10 Correlation vs. Causation Objective: Understand the different between correlation and causation and recognize when there is a strong correlation but no causation Standard for Mathematical Content 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. Unit 10 Cycle 3 Lesson 7 Adding and Subtracting Polynomials Objective: Add and subtract polynomials to rewrite or simplify the polynomials Unit 10 Cycle 3 Lesson 8 Multiplying and Dividing Polynomials Objective: Multiply and divide polynomials to rewrite or simplify the polynomials Unit 10 Cycle 3 Lesson 9 Special Polynomial Combinations Objective: Use special polynomial combinations to rewrite or simplify polynomials Standard for Mathematical Content 8.G 6: Explain a proof of the Pythagorean Theorem and its converse. Unit 11 Cycle 1 Lesson 1 Explain the Pythagorean Theorem Objective: Use algebra and functions to explain the Pythagorean Theorem and its converse Standard for Mathematical Content 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. Unit 11 Cycle 1 Lesson 2 Apple the Pythagorean Theorem Objective: Use the Pythagorean Theorem to find missing side length on triangles and explore special right triangles Unit 11 Cycle 1 Lesson 3 Distance Formula Objective: Use the Pythagorean Theorem to derive the distance formula