Algebra Curriculum Map

Transcription

1 Unit Title: Ratios, Rates, and Proportions Unit: 1 Approximate Days: 8 Academic Year: Essential Question: How can we translate quantitative relationships into equations to model situations and use algebraic reasoning to solve? N.Q.1 N.Q.2 N.Q.3 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. Define appropriate quantities for the purpose of descriptive modeling. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Ratios Rates Unit Rate Proportions Dimensional Analysis Ratios, Rates, and Unit Rates Proportions Dimensional Analysis MP: 1 Include word problems where quantities are given in different units, which must be converted to make sense of the problem. For example, a problem might have an object moving 12 foot per second and another at 5 miles per hour. To compare speeds, students convert 12 feet per second to miles per hour.

2 Unit Title: Real Numbers with Exponents Unit: 2 Approximate Days: 10 Academic Year: Essential Question: What is the relationship between exponents and radicals? Standards Content Assessment Notes N.RN.1 N.RN.2 Explains 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. Rewrite expressions involving radicals and rational exponents using the properties of exponents. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Pre-Requisite o Properties of Numbers o Order of Ops. o Scientific Notation Properties and Laws of Integer Exponents Rational Exponents Properties and Order of Ops. Exponents Scientific Notation MP: 1

3 Unit Title: Working with Radicals Unit: 3 Approximate Days: 10 Academic Year: Essential Question: How do you simplify radical numbers? How do you perform operations with radicals? Standards Content Assessment Notes 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. Simplifying Radicals Operations with Radicals Number Systems Simplifying Radicals Operations with Radicals Number Systems MP: 1

4 Unit Title: Pythagorean Theorem Unit: 4 Approximate Days: 6 Academic Year: Essential Question: How can we prove the Pythagorean Theorem? How can we use the Pythagorean Theorem to solve problems? 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. Pythagorean Theorem Distance Formula Pythagorean Theorem Distance Formula 8.G.7 Apply the Pythagorean Theorem to determine unknown sides lengths in right triangles in realworld and mathematical problems in two and three dimensions. MP: 1 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

5 Unit Title: Expressions Unit: 5 Approximate Days: 10 Academic Year: Essential Questions: How can we write and evaluate expressions? A.SSE.1 A.SSE.2 A.APR.1 Interpret expressions that represent a quantity in terms of its context. Use the structure of an expression to identify ways to rewrite it. 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. Identifying Types and Parts of Expressions Simplifying Expressions Distributive Property Add/Subtract Polynomials Laws of Exponents Simplifying Expressions Add/Subtract Polynomials Laws of Exponents MP: 2 Performance Assessment

6 Unit Title: Factoring Polynomials Unit: 6 Approximate Days: 14 Academic Year: Essential Question: How can we rewrite a polynomial expression by factoring it? A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Factor GCF Factor by Grouping Factor with LC = 1 Factor with LC other than 1 Factor Special Cases Factor GCF and Grouping Factor trinomials Factor with special cases MP: 2

7 Unit Title: Solving Equations Unit: 7 Approximate Days: 10 Academic Year: Essential Questions: How can we write and solve equations? Standards Content Assessment Notes A.CED.1 A.CED.4 A.REI.1 Create equations and inequalities in one variable and use them to solve problems. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solve equations. 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. Solving Simple Equations Solving Multi-Step Equations Justifying Steps when Solving Equations Problem Solving with Equations Absolute Value Equations Literal Equations Solving Equations Problem Solving with Equations Absolute Value Equations Literal Equations MP: 2 A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

8 Unit Title: Intro. to Functions Unit: 8 Approximate Days: 10 Academic Year: Essential Question: What is the relationship between a relation and a function? How do we represent functions? Standards Content Assessment Notes F.IF.1 F.IF.2 Understand that a function from one set to another set 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). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Intro. to Relations Domain and Range Intro. to Functions Function Notation Story Graphs Relations, Domain, and Range Functions Story Graphs MP: 2 F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship its describes. 8.F.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.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. N.Q.1 Use units as a way to understand problems and to guide the solutions of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and origin in graphs and displays.

9 Unit Title: Graphing and Writing Equations Unit: 9 Approximate Days: 14 Academic Year: Essential Questions? How do we represent rate of change? How do we graph linear functions? How do we write linear equations? F.IF.4 F.IF.6 A.CED.2 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 Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels/ scales. Slope Rate of Change Graphing Linear Functions Writing Linear Equations Writing Inverse Expressions Slope, Rate of Change Graphing linear functions Writing linear equations MP: 3 A.REI.10 Understand that the graph of an equation is two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve. 8.F.2 Compare properties of two functions each represented in a different way. 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. 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 rate of change. F.BF.4 Find inverse functions 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.

10 Unit Title: Inequalities Unit: 10 Approximate Days: 11 Academic Year: Essential Question: How do we write and solve single variable linear inequalities? How do we solve 2 variable inequalities A.CED.1 A.REI.3 A.REI.12 Create equations and inequalities in one variable and use them to solve problems. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Graph the solutions to a linear inequality in two variables as a half-plane, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Solving Single Variable Linear Inequalities Solving Compound Inequalities Solving Absolute Value Inequalities Solving 2-Variable Inequalities Solving Inequalities Compound Inequalities Absolute Value Inequalities 2-Variable Inequalities MP: 3

11 Unit Title: Systems of Equations and Inequalities Unit: 11 Approximate Days: 14 Academic Year: Essential Questions: How do we write and solve systems of equations? How do we write and solve systems of inequalities? Standards Content Assessment Notes A.REI.5 A.REI.6 A.REI.7 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. Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Systems of Equations o Graphing o Substitution o Elimination o Applications Systems of Inequalities Graphing Substitution Elimination Applications Systems of Inequalities MP: 3 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. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. 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.

12 Unit Title: Introduction to Quadratic Functions Unit: 12 Approximate Days: 12 Academic Year: Essential Questions: How do we solve quadratic equations? How do we graph quadratic functions? A.REI.4 F.IF.4 F.IF.7 Solve quadratic equations in one variable. Use the method of completing the square and derive the quadratic formula. 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. Graph functions expressed symbolically to show key features of the graph, by hand in simple cases and using technology for more complicated cases. Solving Quadratic Equations o Factoring o Completing the Square o Quadratic Formula Discriminant Nature of Roots Graphing Quadratic Functions Transforming Functions System of Linear/Quadratic Solving Quadratics Graphing Quadratics Transforming Quadratics MP: 4 F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x+k) for specific values of K; find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

13 Unit Title: Special Functions Unit: 13 Approximate Days: 8 Academic Year: Essential Questions: How do we graph special functions? 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. b. Graph square root, cube root, and piecewise defined functions, including step functions and absolute value functions. Graphing square roots Graphing piece wise functions Graphing Step Functions Graphing Absolute Value Functions Square roots Piecewise Step Absolute Value MP: 4

14 Unit Title: Exponential Functions Unit: 14 Approximate Days: 10 Academic Year: Essential Questions: What is an exponential function? How can we solve problems using exponential function? How do we compare linear, quadratic and exponential functions? F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another Exponential Functions Exponential Growth and Decay Exponential Applications Comparing Linear, Quadratic, and Exponential Exponential functions Exponential growth/decay Applications Comparing Functions MP: 4 F.LE.2 Construct linear and exponential functions given a graph, a description, or two input-output pairs F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or as a polynomial function. F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. F.IF.9 Compare properties of two functions each represented in a different way.

15 Unit Title: Sequences Unit: 15 Approximate Days: 8 Academic Year: Essential Questions: How can we represent arithmetic and geometric sequences? Standards Content Assessment Notes F.IF.3 F.BF.1 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 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 Arithmetic Sequences Geometric Sequences Explicit Formulas Recursive Formulas Arithmetic Sequences Geometric Sequences MP: 4 F.BF.2 Write arithmetic and geometric sequences both recursively and with explicit formula, use them to model situations, and translate between the two forms.

16 Unit Title: Representing Data Unit: 16 Approximate Days: 8 Academic Year: Essential Questions: How can we represent data? S.ID.1 Represent Data with plots on the real number line. Dot Plots Histograms S.ID.6 Represent data on two quantitative variables on a Box Plots scatter plot and describe how the variables are Scatter Plots related. Lines of Best Fit * Fit a function to the data Correlation Coefficient S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. Correlation vs. Causation Dot Plots, Histograms, Box Plots Scatter Plots MP: 4 S.ID.9 8.SP.1 8.SP.2 Distinguish between correlation and causation Construct and interpret scatter plots. Know that straight lines are used to model relationships between two 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.

17 Unit Title: Data Analysis Unit: 17 Approximate Days: 8 Academic Year: Essential Question: How can we analyze data? S.ID.2 S.ID.3 S.ID.7 Use statistics appropriate to the shape of the data distribution to compare center and spread of two ore more data sets Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points Interpret the slope and intercept of a linear model in the context of the data. Median Mean Interquartile Range Standard Deviation Interpreting data displays Mean, median IQR, Standard Deviation Interpreting Data MP: 4 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting slope and intercept.

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