GSE Algebra II
GSE Algebra II Pacing Guide Part A and B
Unit 1: Inferences and Conclusions from Data OdysseyWare Unit 1
In this unit students will:
•Describe and compare distributions by using the correct measure of center and spread, and identifying outliers (extreme data points) and their effect on the data set
•Use the mean and standard deviation of the data set to fit it to a normal distribution where appropriate
•Estimate and interpret areas under a normal curve using calculators, spreadsheets or tables
•Design simulations of random sampling: assign digits in appropriate proportions for events, carry out the simulation using random number generators and random number tables and explain the outcomes in context of the population and the known proportions
•Design and evaluate sample surveys, experiments and observational studies with randomization and discuss the importance of randomization in these processes
•Conduct simulations of random sampling to gather sample means and proportions.
Explain what the results mean about variability in a population and use results to calculate margins of error
•Generate data simulating application of two treatments and use the results to evaluate significance of differences
•Read and explain in context data from outside reports
Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
MGSE9-12.S.ID.2 :Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (inter-quartile range, standard deviation) of two or more different data sets.
MGSE9-12.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 procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
MGSE9-12.S.IC.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
MGSE9-12.S.IC.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question themodel?
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
MGSE9-12.S.IC.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
MGSE9-12.S.IC.4:Use data from a sample survey to estimate a population mean or proportion develop a margin of error through the use of simulation models for random sampling.
MGSE9-12.S.IC.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
MGSE9-12.S.IC.6: Evaluate reports based on data.
OdysseyWare Unit 1 should be complete at the end of week 2 of the Minimester.
Unit 2: Polynomial Function OdysseyWare Unit 2
In this unit students will:
•understand the definition of a polynomial
•interpret the structure and parts of a polynomial expression including terms, factors, and coefficients
•simplify polynomial expressions by performing operations, applying the distributive property, and combining like terms
•use the structure of polynomials to identify ways to rewrite them and write polynomials in equivalent forms to solve problems
•perform arithmetic operations on polynomials and understand how closure applies under addition, subtraction, and multiplication
•use Pascal’s Triangle to determine coefficients of binomial expansion
•use polynomial identities to solve problems
•use complex numbers in polynomial identities and equations
•derive the formula for the sum of a finite geometric series and use it to solve problems
•understand and apply the rational Root Theorem
•understand and apply the Remainder Theorem
•understand and apply The Fundamental Theorem of Algebra
•understand the relationship between zeros and factors of polynomials
•represent, analyze, and solve polynomial functions algebraically and graphically
•solve systems consisting of a linear equation and a polynomial equation
Interpret the structure of expressions.
MGSE9‐12.A.SSE.1 :Interpret expressions that represent a quantity in terms of its context.★
MGSE9‐12.A.SSE.1a :Interpret parts of an expression, such as terms, factors, and coefficients.★
MGSE9‐12.A.SSE.1b:Interpret complicated expressions by viewing one or more of their parts as a single entity★
MGSE9‐12.A.SSE.2: Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems.
MGSE9‐12.A.SSE.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.★
Perform arithmetic operations on polynomials.
MGSE9‐12.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.
Use polynomial identities to solve problems.
MGSE9‐12.A.APR.4 : Prove polynomial identities and use them to describe numerical relationships.
MGSE9‐12.A.APR.5 (+): Know and apply that the Binomial Theorem gives the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.
Use complex numbers in polynomial identities and equations.
MGSE9‐12.N.CN.8 (+) Extend polynomial identities to the complex numbers.
MGSE9‐12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Represent and solve equations and inequalities graphically.
MGSE9‐12.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.★
Solve systems of equations.
MGSE9‐12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic polynomial equation in two variables algebraically and graphically.
Understand the relationship between zeros and factors of polynomials.
MGSE9‐12.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x)and a number a, the remainder on division by x –a is p(a), so p(a) = 0 if and only if (x –a) is a factor ofp(x).
MGSE9‐12.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. Analyze functions using different representations.
MGSE9‐12.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.★
MGSE9‐12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.★
OdysseyWare Unit 2 should be complete at the end of Week 5.
Unit 3: Radical and Rational Relationships
In this unit students will:
•Explore Rational and Radical Functions
•Determine rational numbers extend the arithmetic of integers by allowing division by all numbers except zero. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial
•Notice the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers
•Investigate the properties of simple rational and radical functions and then expand their knowledge of the graphical behavior and characteristics of more complex rational functions
•Recall and make use of their knowledge of polynomial functions as well as compositions of functions to investigate the characteristics of these more complex rational functions
•Solve equations and inequalities involving rational and radical functions
•Understand that not all solutions generated algebraically are actually solutions to the equations and extraneous solutions will be explored
•Apply these rational and radical functions with an emphasis on interpretation of real world phenomena as it relates to certain characteristics of the rational expression
Rewrite rational expressions
MGSE9‐12.A.APR.6 : Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
MGSE9‐12.A.APR.7 (+) : Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Create equations that describe numbers or relationships
MGSE9‐12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from simple rational functions.★
MGSE9‐12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★
Understand solving equations as a process of reasoning and explain the reasoning
MGSE9‐12.A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Represent and solve equations and inequalities graphically
MGSE9‐12.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 rational.★
Interpret functions that arise in applications in terms of the context
MGSE9‐12.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; and end behavior.★
MGSE9‐12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Advanced Algebra Pacing Guide Part B
Unit 4: Exponential and Logarithmic Functions OdysseyWare Unit 1
In this unit students will:
•Review exponential functions and their graphs
•Explore exponential growth
•Develop the concept of a logarithm as an exponent along with the inverse relationship with exponents
•Define logarithms and natural logarithms
•Develop the change of base formula
•Develop the concept of logarithmic function
•Solving problems relating to exponential functions and logarithms
Write expressions in equivalent forms to solve problems
MGSE9‐12.A.SSE.3 : Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
MGSE9‐12.A.SSE.3c:Use the properties of exponents to transform expressions for exponential functions. Analyze functions using different representations
MGSE9‐12.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.
MGSE9‐12.F.IF.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior.
MGSE9‐12.F.IF.8 :Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
MGSE9‐12.F.IF.8b Use the properties of exponents to interpret expressions for exponential functions.
Build new functions from existing functions
MGSE9‐12.F.BF.5 (+) : Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Construct and compare linear, quadratic, and exponential models and solve problems
MGSE9‐12.F.LE.4: For exponential models, express as a logarithm the solution to ab(ct)= d where a, c,and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology
OdysseyWare Unit 1 should be complete by week 2 of the Minimester.
Unit 5: Trigonometric Functions OdysseyWare Unit 2
In this unit students will:
•Expand their understanding of angle with the concept of a rotation angle
•Explore the definition of radian
•Define angles in standard position and consider them in relationship to the unit circle
•Make connections to see how a real number is connected to the radian measure of an angle in standard position which is connected to an intercepted arc on the unit circle which is connected to a terminal point of this arc whose coordinates are connected to the sine and cosine functions
•Gain a better understanding of the unit circle and its connection to trigonometric functions. Develop an understanding of the graphs of the sine and cosine functions and learn to recognize the basic characteristics of their graphs
•Realize transformations of y = sin (x) and y = cos (x) behave just as transformations of other parent functions
•Learn that the concepts of amplitude, midline, frequency, and period are related to the transformations of trigonometric functions
•Learn how to look at a graph of a transformed sine or cosine function and to write a function to represent that graph explore several real-world settings and represent the situation with a trigonometric function that can be used to answer questions about the situation.
•Develop and use the Pythagorean identity (sin^2 + cos^2 = 1) is developed and used to solve problems
Analyze functions using different representations
MGSE9‐12.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.★
MGSE9‐12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.★
Extend the domain of trigonometric functions using the unit circle
MGSE9‐12.F.TF.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
MGSE9‐12.F.TF.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Model periodic phenomena with trigonometric functions
MGSE9‐12.F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
Prove and apply trigonometric identities
MGSE9‐12.F.TF.8: Prove the Pythagorean identity (sin A)2+ (cos A)2= 1 and use it to find sin A, cos A, or tan A, given sin A, cosA, or tan A, and the quadrant of the angle.
OdysseyWare Unit 2 must be complete by week 4 of the Minimester.
Unit 6: Mathematical Modeling OdysseyWare Unit 3
In this unit students will:
•Synthesize and generalize what they have learned about a variety of function families
•Explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying functions
•Identify appropriate types of functions to model a situation,
•Adjust parameters to improve the model,
•Compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit
•Determine whether it is best to model with multiple functions creating a piecewise function
Create equations that describe numbers or relationships
MGSE9‐12.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.
MGSE9‐12.A.CED.2 ;Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MGSE9‐12.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.
MGSE9‐12.A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Interpret functions that arise in applications in terms of the context
MGSE9‐12.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.
MGSE9‐12.F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes
MGSE9‐12.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
Analyze functions using different representations
MGSE9‐12.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.
MGSE9‐12.F.IF.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
MGSE9‐12.F.IF.7b: Graph square root, cube root, and piecewise defined functions, including step functions and absolute value functions.
MGSE9‐12.F.IF.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
MGSE9‐12.F.IF.7d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
MGSE9‐12.F.IF.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
MGSE9‐12.F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
MGSE9‐12.F.IF.8a: 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.
MGSE9‐12.F.IF.8b: Use the properties of exponents to interpret expressions for exponential functions.
MGSE9‐12.F.IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Build a function that models a relationship between two quantities
MGSE9‐12.F.BF.1 Write a function that describes a relationship between two quantities.
MGSE9‐12.F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.
MCC9‐12.F.BF.1b Combine standard function types using arithmetic operations.
MCC9‐12.F.BF.1c Compose functions. Build new functions from existing functions
MCC9‐12.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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
MCC9‐12.F.BF.4 Find inverse functions.
MCC9‐12.F.BF.4a 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.
MCC9‐12.F.BF.4b Verify by composition that one function is the inverse of another.
MCC9‐12.F.BF.4c Read values of an inverse function from a graph or a table, given that the function has an inverse. Visualize relationships between two‐dimensional and three dimensional objects
MCC9‐12.G.GMD.4 Identify the shapes of two‐dimensional cross‐sections of three‐dimensional objects, and identify three‐
dimensional objects generated by rotations of two dimensional objects.
Apply geometric concepts in modeling situations
MCC9‐12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
MCC9‐12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
MCC9‐12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios
OdysseyWare Unit 3 must be complete by Week 8 of the Minimester.
Review for the Final and Final exam will be held Week 9.
Apply geometric concepts in modeling situations
MCC9‐12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
MCC9‐12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
MCC9‐12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios
OdysseyWare Unit 3 must be complete by Week 8 of the Minimester.
Review for the Final and Final exam will be held Week 9.
