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Integrated Math III Scope and Sequence Standards Trajectory

Course Name Integrated Math III Grade Level High SchoolIntegrated Math III Common Core State Standards

Conceptual Category Domain Standards Cluster Heading

Number and Quantity The Complex Number System (N-CN) N-CN.C: Use complex numbers in polynomial identities and equations. (Additional) [honors course only]

Algebra

Seeing Structure in Expressions (A-SSE)

A-SSE.A: Interpret the structure of expressions. (Major)A-SSE.B: Write expressions in equivalent forms to solve problems. (Major)

Arithmetic with Polynomials and Rational Expressions (A-APR)

A-APR.A: Perform arithmetic operations on polynomials. (Major)A-APR.B: Understand the relationship between zeros and factors of polynomials. (Major)A-APR.C: Use polynomial identities to solve problems. (Additional)A-APR.D: Rewrite rational expressions. (Additional)

Creating Equations ✯ (A-CED) A-CED.A: Create equations that describe numbers or relationships. (Major)Reasoning with Equations and

Inequalities (A-REI)A-REI.A: Understand solving equations as a process of reasoning and explain the reasoning. (Major)A-REI.D: Represent and solve equations and inequalities graphically. (Major)

Functions

Interpreting Functions (F-IF)F-IF.B: Interpret functions that arise in applications in terms of context. (Major)F-IF.C: Analyze functions using different representations. (Supporting)

Building Functions (F-BF)F-BF.A: Build a function that models a relationship between two quantities. (Supporting)F-BF.B: Build new functions from existing functions. (Additional)

Linear, Quadratic, and Exponential Models ✯ (F-LE) F-LE.A: Construct and compare linear, quadratic, and exponential models and solve problems. (Supporting)

Trigonometric Functions (F-TF)F-TF.A: Extend the domain of trigonometric functions using the unit circle. (Additional)F-TF.B: Model periodic phenomena with trigonometric functions. (Additional)F-TF.C: Prove and apply trigonometric identities. (Additional) [honors course only]

Geometry

Similarity, Right Triangles, and Trigonometry (G-SRT) G-SRT.D: Apply trigonometry to general triangles. (Additional) [honors course only]

Geometric Measurement and Dimension (G-GMD) G-GMD.B: Visualize relationships between two-dimensional and three-dimensional objects. (Additional)

Modeling with Geometry (G-MG) G-MG.A: Apply geometric concepts in modeling situations. (Major)Statistics and Probability

✯Interpreting Categorical and

Quantitative Data ✯ (S-ID) S-ID.A: Summarize, represent, and interpret data on single count or measurement variable. (Supporting)

Making Inferences and Justifying Conclusions ✯ (S-IC)

S-IC.A: Understand and evaluate random processes underlying statistical experiments. (Supporting)S-IC.B: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

(Major)

Denver Public Schools 2018–20191


Use Probability to Make Decisions ✯ (S-MD)

S-MD.B: Use probability to evaluate outcomes of decisions. (Additional) [honors course only]

✯ Modeling Standards

Major clusters require greater emphasis based on depth of ideas, time they take to master, and their importance to future mathematics. An intense focus on these clusters allows in-depth learning carried out through the Standards for Mathematical Practice. Supporting clusters are closely connected to the major clusters and strengthen areas of major emphasis. Additional clusters may not tightly or explicitly connect to the major work of the grade. All standards should be taught.



Competencies Performance Indicators

C1: Write, solve, and interpret solutions to equations, inequalities and systems of equations and inequalities using a variety of tools.

PI1: Write and solve equations in a single variable.

PI2: Write, solve and interpret a system of two equations in two variables or two inequalities in two variables that model real-world situations. Solve a system consisting of a linear and a quadratic equation.

PI3: Solve quadratic equations with real and complex roots using multiple methods.

PI4: Write and solve inequalities in a single variable.

C2: Create functions to model data and fit functions to real-world quantitative relationships. Functions: trigonometric (SLO 2)

PI3: Create a graph and algebraic model to represent real-world data or a situation, to interpret, predict and justify my conclusions.

C3: Analyze the multiple representations of a function in terms of the context. Functions: polynomial and trigonometric (SLO 1)

PI1: Convert between representations (algebraically, graphically, numerically in tables, or by verbal descriptions).

PI3: Determine, interpret, compare and contrast key features of functions.

C4: Apply statistical methods and reasoning to summarize, represent, analyze and interpret quantitative data.

PI1: Create visual displays, describe key characteristics and determine summary statistics to describe data sets.

PI2: Evaluate processes, make inferences and justify conclusions from sample surveys, experiments, and observational studies.

C5: Evaluate and interpret outcomes using a probability model.

PI1: Use the concepts of independence and conditional probability to interpret data and compute probabilities of compound events.

C8: Apply geometric relationships in modeling situations.

PI1: Solve problems using properties of 2- and 3-dimensional figures to analyze, represent and model geometric relationships.

Suggested Student Learning Objective (SLO) Statements1. All students will demonstrate competency in the multiple representations of a function in terms of the context. Functions include quadratic, exponential, logarithmic,



polynomial, and trigonometric. (C3)2. All students will demonstrate competency in equations or functions to model mathematical or real-world situations, use these equations or functions to solve problems

involving the given situation, interpret the solution in terms of the context, and justify reasoning orally and in writing. Functions include linear, quadratic, exponential, and trigonometric. (C2)



Colorado 21st Century Skills

Critical Thinking and Reasoning: Thinking Deeply, Thinking Differently

Information Literacy: Untangling the Web

Collaboration: Working Together, Learning Together

Self-Direction: Own Your Learning

Invention: Creating Solutions

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.

Unit of Study Length of Unit* Time Frame1: Investigations and Functions 14 days (7 lessons) August 20–September 7, 20182: Transformations of Parent Graphs 18 days (8 lessons) September 10–October 3, 20183: Solving and Inequalities 14 days (8 lessons) October 4–October 26, 20184: Normal Distributions and Geometric Modeling 18 days (11 lessons) October 29–November 28, 20185: Inverses and Logarithms 22 days (7 lessons) November 29, 2018–January 11, 20196: Simulating Sampling Variability 13 days (5 lessons)** January 14–January 31, 20197: Logarithms and Triangles 10 days (4 lessons)** February 4–February 15, 20198: Polynomials 13 days (6 lessons)** February 19–March 8, 20199: Trigonometric Functions 20 days (11 lessons) March 11–April 12, 201910: Series 11 days (6 lessons)** April 15–April 30, 201911: Rational Expressions and Three-Variable Systems 8 days (4 lessons)** May 1–May 10, 201912: Analytic Trigonometry 12 days (4 lessons)** May 13–May 30, 2019

*Number of days includes time for review, unit assessment, course assessments, and CMAS.

** Includes additional honors lessons

End-of-Year Fluency Recommendations



● Students should look at algebraic manipulation as a meaningful enterprise, in which they seek to understand the structure of an expression or equation and use properties to transform it into forms that provide useful information (e.g., features of a function or solutions to an equation). (A/F)

● Seeing mathematics as a tool to model real-world situations should be an underlying perspective in everything students do, including writing algebraic expressions, creating functions, creating geometric models, and understanding statistical relationships. (M)

● In particular, students should recognize that much of mathematics is concerned with understanding quantities and their relationships. They should pick appropriate units for quantities being modeled, using them as a guide to understand a situation, and be attentive to the level of accuracy that is reported in a solution. (N-Q)

● Students should understand the effects of parameter changes and be able to apply them to create a rule modeling the function. (F-BF.B.3)



Integrated 3 Competency Unit Map & Expected Performance Indicator

C1: Solving and Systems of Equations and Inequalities

C2: Math Models (SLO 2)

C3: Interpreting Functions (SLO 1)

C4 : Statistics C5: Probability

C8: Geometric Models

Performance Indicators

PI1 PI2 PI3 PI4 PI1 PI1 PI3 PI1 PI2 PI1 PI1

Unit 1 ✽M

✽A

✽M

✽A

✽A

✽M

✽M

Unit 2 ✽(CP)- M

✽M

✽M

✽A

Unit 3 ✽PM

✽M

✽E

✽A

Unit 4 ✽PM

✽M

✽M

Unit 5 ✽M

✽A

✽A

Unit 6 ✽(CP)- PM

✽(CP)- A

✽M

✽M

Unit 7 ✽M

✽(CP)- A

✽M

✽A

✽A

✽M

Unit 8 ✽M

✽A

✽A

Unit 9 ✽M

✽A

✽A

✽

Unit 10

Unit 11 ✽



(CP)- A

Unit 12 ✽(CP)- M

CP = Checkpoint



Unit of Study 1: Investigations and Functions Length of Unit 14 days (August 20–September 7, 2018)

Unit Learning Trajectory

● The focus of this chapter is to give students the opportunity to share their current mathematical knowledge with their study teams as they work together to solve problems and become comfortable talking about mathematics by investigating mathematical questions and creating logical and convincing arguments to support their findings. In their previous courses, they have established classrooms norms (e.g., working in study teams, explaining thinking, investigating mathematics) and class procedures for graphing organizers, Learning Logs, and homework, and to review vocabulary.

● In Section 1.1, students create multiple representations of a function using a graphing calculator and review describing graphs of functions using precise mathematical language. Students learn how a parent graph and parameters define a family of functions and explore the results of combining linear functions. In Section 1.2, students engage in modeling problems to preview concepts in the course. Mastery of the concepts in Section 1.2 should not be expected yet, as these topics will be revisited, extended, and applied to new situations in upcoming chapters.

● Further work on function families, key features of graphs of functions, and multiple representations of functions will continue in Chapters 2, 5, and 7. In Chapter 2, students will generate families of functions from parent functions starting with the parent function f(x) = x2. They will make generalizations about functions by stretching, compressing, reflecting, and translating parent functions. Students will extend their learning to inverse and logarithmic functions in Chapter 5 and trigonometric functions in Chapter 9.

Standards Content Standards

Arithmetic with Polynomials and Rational Expressions (A-APR)Perform arithmetic operations on polynomials. (Major) [A-APR.A]A-APR.A.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.Interpreting Functions (F-IF)Interpret functions that arise in applications in terms of context. (Major) [F-IF.B]F-IF.B.4: For a function that models a relationships 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. ✯

Analyze functions using different representations. (Supporting) [F-IF.C]F-IF.C.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 and absolute value functions.

Building Functions (F-BF)Build a function that models a relationship between two quantities. (Supporting) [F-BF.A]F-BF.A.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 be modeled. ✯Standards for Mathematical Practice



1. Make sense of problems and persevere in solving them.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.

Fluency Recommendation


Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding

My students Understand that … Factual Conceptual

● Functions model relationships between quantities using a variety of representations (tables, graphs, and equations). (F-IF.B.4)

● How do graphs, equations, and tables show similarities and differences of functions?

● Why is it important to interpret differences and similarities of functions through multiple representations?

● Modeling nonlinear relationships between two quantities requires using appropriate functions. (F-IF.B.4)

● How can we determine, from tables or context, which function models the relationship between two quantities?

● How can we determine key features of graphs of nonlinear functions from equations?

● How do we use functions families’ key features to determine appropriate functions for given situations?

● How can knowing whether or not functions are even or odd be useful?

● Why do we classify functions?

Key Knowledge and Skills (Procedural Skills and Application)

My students will be able to (Do) …● Evaluate f(x) when given x, and solve for x when given a value for f(x). (review of prior course)● Sketch graphs of functions and completely describe the graphs. (F-IF.B.4)● Determine domain and range given a graph. (review of prior course)● Solve linear equations and/or equations with fractions. (review of prior course)● Solve quadratic equations. (review of prior course)● Create multiple representations of a linear relationship. (review of prior course)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.



Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentdomain, equation, expression, function, infinity, input, output, parameter, solution, spread, statistic, variable

asymptote, boxplot, closed set, complete graph, family of functions, histogram, interquartile range (IQR), parent function, point of intersection, polynomial, population standard deviation, quadratic equation, range of data set, range of a function, sample standard deviation

Resources

Core Lessons1.1.1 through 1.1.41.2.1 through 1.2.3Chapter Closure

Connections to Khan Academy Resources for PSAT/SAT

● Heart of Algebra: Solving Linear Equations and Linear Inequalities (Equations in this unit)● Heart of Algebra: Interpret Linear Functions● Heart of Algebra: Graphing Linear Equations● Passport to Advanced Mathematics: Solving Quadratic Equations

Misconceptions ● Students believe it is reasonable for any x-value to be in the domain.● Students believe that each family of functions is independent of the others and do not recognize commonalities among functions.



Unit of Study 2: Transformations of Parent Graphs Length of Unit 18 days (September 10–October 3, 2018)


● An important theme of this chapter is generalizing as students generate families of functions by applying transformations to their parent functions.

● In Section 2.1, students apply transformations to the parent function f(x) = x2 to develop a general equation in the form f(x) = a(x - h)2 + k for the family of quadratic functions. Students review how to sketch a parabola by identifying its orientation, vertical stretch (or compression), and vertex. In Section 2.2, students continue to develop families of functions by applying transformations to other parent functions, including f(x) = 1/x, f(x) = √ x, f(x) = bx, f(x) =|x|, and f(x) = x3 and create a Function Family Graphic Organizer describing the role of (h, k) for each. Students create functions to model relationships between quantities and explore different ways to compare the growth of their models over time. In Section 2.3, students review completing the square for parabolas and circles, which they learned in Core Connections Integrated II.

● Further work on families of functions will continue in Chapters 5 and 9. In Chapter 5, students will be introduced to logarithmic functions, and in Chapter 9, students will be introduced to trigonometric functions. Students will be asked to graph members of these new families of functions and write a general equation for the function family. Throughout the course, they will learn to choose the appropriate function to model a particular relationship.

Standards Content StandardsSeeing Structure in Expressions (A-SSE)Interpret structures of expressions. (Major) [A-SSE.A]A-SSE.A.1: Interpret expressions that represent a quantity in terms of its context. ✯

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Creating Equations (A-CED)✯Create equations that describe numbers or relationships. (Major) [A-CED.A]A-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales.Interpreting Functions (F-IF)Interpret functions that arise in applications in terms of the context. (Major) [F-IF.B]F-IF.B.4: For a function that models a relationships between two quantities, interpret key features of graphs and tables in terms of the quantities,


F-IF.B.5: Relate the domain of a function to its graph and, where applicable, to 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.B.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. ✯


complicated cases.✯b. Graph square root, cube root, and piecewise-defined functions, including step and absolute value functions.



e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period midline, and amplitude.



by adding a constant function to a decaying exponential, and relate these functions to be modeled. ✯Build new functions from existing functions. (Additional) [F-BF.B]F-BF.B.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.

Standards for Mathematical Practice

5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure



● Students should understand the effects of parameter changes and be able to apply them to create a rule modeling the function. (F-BF.B.3)



● New classes of functions emerge by performing operations on functions with constants and/or other functions. (F-BF.B.3)

● How can tables, graphs, and function notations be used to explain how one function family is different or similar to another?

● What are function families? Why are they important or useful?

● How can you operate on linear functions to create other classes of functions?

● Modeling nonlinear relationships between two quantities requires using appropriate functions. (F-IF.B.4)

● How can we determine, from tables or context, which function models the relationship between two quantities?

● How can we determine key features of graphs of nonlinear functions from equations?

● How do we use functions families’ key features to determine appropriate functions for given situations?

● How can knowing whether or not functions are even or odd be useful?





My students will be able to (Do) …● Graph parabolas by identifying the intercepts and vertex. (F-IF.B.4)● Model and solve everyday problems using quadratic functions. (A-CED.A.2, F-BF.A.1a)● Write equations of transformed functions/equations and/or sketch graphs, focusing on quadratic, square root, absolute value, and cubic functions, and equations of

circles. (F-BF.B.3)● Solve quadratic equations, including ones with complex solutions. (Checkpoint 2)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentcompress, parameter, reflect, stretch, transformation, translate absolute value function, average rate of change, circle (equation of), completing the

square, complex number, cubic function, even function, exponential function, family of functions, graphing form, (h, k), horizontal translation, imaginary number, linear function, locator point, odd function, parabola, parent graph, piecewise-defined function, quadratic equation, quadratic function, standard form, vertical translation

Resources

Core Lessons

2.1.1 through 2.1.22.2.1 through 2.2.52.3.1Chapter Closure

Connections to Khan Academy Resources

● Passport to Advanced Mathematics: Solving Quadratic Equations● Passport to Advanced Mathematics: Polynomial Factors and Graphs



for PSAT/SAT ● Passport to Advanced Mathematics: Quadratic and Exponential Word Problems (Quadratic only in this unit)● Additional Topics in Math: Circle Equations

Misconceptions● Students often think that a “+” next to x is a shift to the right due to the addition sign, which indicates positive change. Students should explore

examples analytically and graphically to overcome this notion.● Students confuse the shift of a function with the stretch of a function.



Unit of Study 3: Solving and Inequalities Length of Unit 14 days (October 4–October 26, 2018)


● The focus of this chapter is to gain a stronger understanding of the meaning of solutions and to apply prior knowledge of equations, inequalities, and systems to solve problems.

● In Section 3.1, students review strategies for solving single-variable equations and use graphing as a tool for solving equations and systems as well as visualizing the solutions. Students gain a deeper understanding of the meaning of solutions. They learn multiple ways to determine and represent solutions. In Section 3.2, students extend their understanding of solving and solutions to inequalities and systems of inequalities. Students also explore linear programming as a way to determine solutions that meet a set of constraints.

● Further work on solving and solutions will continue in Chapter 11 as students understanding of linear systems in two-variables to solving systems of linear equations in three variables. At the end of Chapter 11, they will solve a system of three equations in three variables to determine the equation of a parabola, y = ax2 + bx + c, that passes through three unknown points. They will continue to use graphing as tool to solve polynomial equations in Chapters 8 and trigonometric equations in Chapter 9.


Seeing Structure in Expressions (A-SSE)Interpret the structure of expressions. (Major) [A-SSE.A]A-SSE.A.1: Interpret expressions that represent a quantity in terms of its context. ✯

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

A-SSE.A.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).Arithmetic with Polynomials and Rational Expressions (A-APR)Use polynomial identities to solve problems. (Additional) [A-APR.C]A-APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 +

(2xy)2 can be used to generate Pythagorean triples.Creating Equations (A-CED)✯Create equations that describe numbers or relationships. (Major) [A-CED.A]A-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales.A-CED.A.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 foods.✯

Reasoning with Equations and Inequalities (A-REI)Understand solving equations as a process of reasoning and explain the reasoning. (Major) [A-REI.A]A-REI.A.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. (Major) [A-REI.D]A-REI.D.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 functions. ✯Interpreting Functions (F-IF)Interpret functions that arise in applications in terms of the context. (Major) [F-IF.B]F-IF.B.4: For a function that models a relationships between two quantities, interpret key features of graphs and tables in terms of the quantities,




by adding a constant function to a decaying exponential, and relate these functions to be modeled. ✯

Standards for Mathematical Practice1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.

Fluency Recommendations


● In particular, students should recognize that much of mathematics is concerned with understanding quantities and their relationships. They should pick appropriate units for quantities being modeled, using them as a guide to understand a situation, and be attentive to the level of accuracy that is reported in a solution. (N-Q)



● Mathematicians evaluate mathematical solutions for their relevance to models; not all solutions to systems are viable in context. (A-CED.A.3)

● What are characteristics of non-viable solutions?● How do we know when solutions will be viable?

● Why would we model a context with an inequality rather than an equation?

● Why is it important to evaluate all solutions within the original context?



● The (x, y) coordinate that satisfies two or more linear equations is a solution to that system. (A-REI.D.11)

● How do you know if a given point is a solution of a given system of equations?

● What is the real-world meaning/context of the solution?

● Why are systems of equations used to model situations?

● What situations could be modeled by systems of equations?

● Solving radical equations can result in extraneous solutions. (A-REI.A.2)

● How do we check for extraneous solutions?● When do extraneous solutions arise?● How can we determine if solutions are not viable?

● Why do extraneous solutions occur?


My students will be able to (Do) …● Solve a wide range of equations including equations that have an extraneous solution. (A-REI.A.2)● Solve systems of equations algebraically and/or graphically, or use a graph of a system to solve a one-variable equation. (A-REI.D.11)● Write equations or systems of equations to solve application problems. (A-CED.A.3)● Solve one-variable inequalities. (review of prior course)● Apply function notation and describe functions. (Checkpoint 3)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentboundary point, equation, graph, inequality, intersection point, solution, substitution, variable

boundary curve or line, extraneous solution, linear programming, Looking Inside, Rewriting, system of equations, system of inequalities, Undoing

Resources





● Heart of Algebra: Solving Linear Equations and Linear Inequalities● Heart of Algebra: Linear Inequality Word Problems● Heart of Algebra: Solving Systems of Linear Equations● Heart of Algebra: Systems of Linear Equations Word Problems● Heart of Algebra: Systems of Linear Inequalities Word Problems

Misconceptions

● Many mistakes students make when solving systems are careless, rather than conceptual. Teach students to check their answers by substituting answers back into all equations.

● Ensure students understand that a graph represents all solutions to an equation; it is not simply a line or curve “connecting the dots.”● Students believe that systems of inequalities have no application in the real world, which is why Lessons 3.2.2 and 3.2.3 are important.



Unit of Study 4: Normal Distributions Length of Unit 18 days (October 29–November 28, 2018)


● The focus of this chapter is to introduce students to the fundamentals of experimental design and studies and the limitations of each. This unit emphasizes the importance of random sampling for studies and random assignment for experiments and develops the concept of using a normal probability density model to calculate percentiles.

● In Section 4.1, students learn about surveys as a type of observational study and gain experience with two common sources of bias in surveys. In Section 4.2, students compare and contrast experiments and observational studies through an experiment and learn the importance of randomness for experiments. In Section 4.3, students construct relative frequency histograms and model them with normal distributions using the calculator to compute proportions and percentiles. In Section 4.4, students explore geometric modeling. Students visualize the cross-sections of different objects by exploring where two-dimensional shapes occur within three-dimensional figures and visualize solids of revolution. This section ends with a design problem that students need to optimize.

● Further work on statistics will continue in Chapter 6, as students explore inferential statistics, using samples to make predictions about populations.

StandardsGeometric Measurement and Dimension (G-GMD)Visualize relationships between two-dimensional and three-dimensional objects. (Additional) [G-GMD.B]G-GMD.B.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.Modeling with Geometry (G-MG)Apply geometric concepts in modeling situations. ✯(Major) [G-MG.A]G-MG.A.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). ✯G-MG.A.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). ✯Interpreting Categorical and Quantitative Data ✯ (S-ID)Summarize, represent, and interpret data on a single count or measurement variable. ✯ (Additional) [S-ID.A]S-ID.A.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 procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Making Inferences and Justifying Conclusions (S-IC)✯Understand and evaluate random processes underlying statistical experiments. (Supporting) [S-IC.A]✯S-IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. ✯S-IC.A.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 the model? ✯Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (Major) [S-IC.B]✯S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization

relates to each. ✯S-IC.B.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. ✯S-IC.B.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are

significant. ✯S-IC.B.6: Evaluate reports based on data. ✯


3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.





● Random population samples allow statisticians to make inferences about population parameters. (S-IC.A.1)

● How can we reduce margins of error in population predictions?

● How can we use mean and standard deviations of data sets to draw normal distributions?

● What happens to sample-to-sample variability when sample size is increased?

● Why is the normal distribution commonly used to model populations, and when is it not appropriate?

● How can statistical investigation results be used to support arguments?

● Why is margin of error in studies important?● How do we know study results are not simply due

to chance?

● Validity in sampling, surveys, experiments, observational studies, and interpretations of statistical results depends on randomization. (S-IC.B.3)

● In what ways can surveys be biased?● How does randomization factor into experiment

design?

● Why is randomization an important component of sampling?

● When should sampling be used? When is sampling better than a census?


My students will be able to (Do) …● Distinguish between a census and a sample, and identify which is appropriate for a given question or situation. (S-IC.B.3)● Evaluate and create survey questions. (S-IC.B.6)● Identify sampling techniques. (S-IC.B.3)● Design experiments. (S-IC.B.3, S-IC.B.6)● Create histograms, boxplots, and relative frequency histograms; describe center, shape, spread, and outliers; calculate sample standard deviation. (review of prior



course)● Simplify expressions with integer and rational exponents. (Checkpoint 4)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentbias, cause and effect, census, cross-section, experiment, mean, parameter, percentile, placebo, population, preface, proportion, sample, solid, statistic, survey, treatment

closed question, cluster sample, convenience sample, desire to please, geometric model, histogram, lurking variable, normal distribution, observational study, open question, question order, random sample, relative frequency, solid of revolution, standard deviation, two questions in one

Resources

Core Lessons

4.1.1 through 4.1.34.2.1 through 4.2.24.3.1 through 4.3.34.4.1 through 4.4.3Chapter Closure


● Problem Solving and Data Analysis: Data Inferences● Problem Solving and Data Analysis: Center, Spread, and Shape of Distributions● Problem Solving and Data Analysis: Data Collection and Conclusions

Misconceptions

● Students believe inferences from samples to populations can be done only in experiments. They need to realize that inferences can be made in samplings and observational studies if data are collected through random processes.

● Students believe population parameters and sample statistics are the same. For example, they may think there is no difference between population mean, which is a constant, and sample mean, which is a variable.

● Students believe that making decisions is simply comparing the value of one observation of a sample statistic to the value of a population parameter, not realizing the distribution of the sample statistic needs to be created.

● Students believe all bell-shaped curves are normal distributions. A bell-shaped curve is normal when 68% of the distribution is within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.



Unit of Study 5: Inverses and Logarithms Length of Unit 22 days (November 29, 2018–January 11, 2019)


● The focus of this chapter is to learn about inverse functions and investigate the relationship between functions and their inverses. Students explore multiple representations of functions and their inverses, and recognize that many functions have inverses that are not functions.

● In Section 5.1, students examine relationships called inverses and investigate multiple representations of inverse functions. In Section 5.2, students investigate a new family of functions called logarithms, which are the inverse of exponential functions. Students convert equations in exponential form to logarithmic form and vice versa. They also apply transformations to the graph of the parent function, f(x) = log(x).

● Further work on transformations of parent functions will continue in Chapter 8 with polynomial functions and Chapter 9 with trigonometric functions.



Standards

Content Standards

Interpreting Functions (F-IF)Analyze functions using different representations. (Supporting) [F-IF.C]F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

complicated cases.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period midline,

and amplitude. ✯Building Functions (F-BF)Build new functions from existing functions. (Additional) [F-BF.B]F-BF.B.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


F-BF.B.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. For example, f(x) =

2x3 or f(x) = (x + 1)/(x – 1) for x ≠ 1.Linear, Quadratic, and Exponential Models ✯ (F-LE)Construct and compare linear, quadratic, and exponential models and solve problems. ✯ (Supporting) [F-LE.A]F-LE.A.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).F-LE.A.4: For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate

the logarithm using technology. ✯Standards for Mathematical Practice

3. Construct viable arguments and critique the reasoning of others.5. Use appropriate tools strategically.7. Look for and make use of structure.


● Students should understand the effects of parameter changes and be able to apply them to create a rule modeling the function. (F-BF.B.3)● Students should look at algebraic manipulation as a meaningful enterprise, in which they seek to understand the structure of an expression or

equation and use properties to transform it into forms that provide useful information (e.g., features of a function or solutions to an equation). (A/F)



● Inverse functions facilitate the efficient ● What is the relationship of the graph of a function ● How do inverse functions expand our



computation of inputs of the original function. and the graph of its inverse?● When is it necessary to limit the domain of an

inverse function?

understanding of an original function?● Why are inverses important in mathematical

modeling?

● Logarithms, the inverse of exponential functions, provide a mechanism to transform and solve exponential functions. (F-LE.A.4)

● What is the relationship of graphs of exponential functions and their inverse?

● How can we use properties of exponents to represent exponential functions as logarithms?

● How are logarithms used to solve exponential functions?

● Why are logarithms inverses of exponential functions?


My students will be able to (Do) …● Write the equations of inverses for simple functions by “undoing”. (F-BF.B.4a)● Create the inverse graph or table and determine the domain and range of f(x) and f-1(x). (F-IF.C.7)● Rewrite logarithmic equations in exponential form and exponential equations in logarithmic form. (F-LE.A.4)● Solve simple logarithmic equations. (F-LE.A.4)● Describe transformations of a function relative to its parent function. (Checkpoint 5)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentdomain, line of symmetry, range, transformation, undo asymptote, base (of a logarithm), exponent, exponential equation, inverse function,

logarithm, logarithmic function, restricted domain, y=x

Resources





● Passport to Advanced Mathematics: Manipulating Quadratic and Exponential Expressions (Exponential only in this unit)

Misconceptions ● Students believe that the inverse of all functions are functions themselves and need to see counterexamples.● Students may confuse f -1(x) with 1/f(x).



Unit of Study 6: Simulating Sampling Variability Length of Unit 13 days (January 14–January 31, 2019)


● The focus of this chapter is to determine complex probabilities through simulations and introduce inferential statistics and statistical hypothesis testing. Students use area models or tree diagrams to understand counterintuitive probability problems.

● In Section 6.1, students perform simulations to estimate complex probabilities. Students determine sample-to-sample variability which they use to put a margin of error on characteristics they predict about a population. In Section 6.2, students extend their learning to conduct a statistical hypothesis test and observe the effect of sample size on sample-to-sample variability. Students learn how statistics are used in manufacturing for both quality control and process control. Section 6.3, students evaluate decisions and strategies based on area models of probability and use probability to solve problems whose solutions are counterintuitive.

● Further work on inferential statistics will continue in a formal statistics course.

Standards

Content StandardsMaking Inferences and Justifying Conclusions (S-IC)✯Understand and evaluate random processes underlying statistical experiments. (Supporting) [S-IC.A]✯S-IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. ✯S-IC.A.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 the model? ✯Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (Major) [S-IC.B]✯S-IC.B.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. ✯S-IC.B.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are

significant. ✯S-IC.B.6: Evaluate reports based on data. ✯Using Probability to Make Decisions (+S-MD)✯Use probability to evaluate outcomes of decisions. (Additional) [+S-MD.B]✯S-MD.B.6: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). ✯ [honors course only]S-MD.B.7: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of

a game). ✯ [honors course only]


4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.







● Random population samples allow statisticians to make inferences about population parameters. (S-IC.A.1)

● How can we reduce margins of error in population predictions?

● How can we use mean and standard deviations of data sets to draw normal distributions?

● What happens to sample-to-sample variability when sample size is increased?

● Why is the normal distribution commonly used to model populations, and when is it not appropriate?

● How can statistical investigation results be used to support arguments?

● Why is margin of error in studies important?● How do we know study results are not simply due

to chance?

● Simulation provides a means to indirectly collect data. (S-IC.B.4)

● How do we design simulations to model the collection of data not easily obtained?

● How has technology enhanced our ability to study difficult-to-measure phenomena?


My students will be able to (Do) …● Use simulations to describe probabilities. (S-IC.A.2)● Determine sample-to-sample variability and/or a margin of error for the population. (S-IC.B.4, S-IC.B.5)● Solve complicated equations and systems. (Checkpoint 6)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentclaim, margin of error, population, simulation, upper/lower control limit area model (probability), bounds (lower/upper), conditional probability, control limit,

false positive, hypothesis test, in/out of control, Law of Large Numbers, process control, quality control, random number, sample-to-sample variability, sampling distribution, statistical test, x-bar process control chart



Resources

Core Lessons

6.1.1 through 6.1.36.2.1 through 6.2.2Chapter Closure

Honors course also includes: 6.2.3, 6.2.4, and 6.3.1


● Problem Solving and Data Analysis: Data Collection and Conclusions

Misconceptions ● Students believe that making decisions is simply comparing the value of one observation of a sample statistic to the value of a population parameter, not realizing the distribution of the sample statistic needs to be created.



Unit of Study 7: Logarithms and Triangles Length of Unit 10 days (February 4–February 15, 2019)


● The focus of this chapter is to further explore logarithms and expand knowledge of triangles to include solving for the sides and angles of non-right triangles.

● In Section 7.1, students further develop their understanding of the properties of logarithms that will enable them to solve application problems that can be modeled by exponential functions. In Section 7.2, students investigate how to solve for side lengths and angle measures given different types of information about a triangle and then develop tools to solve for missing side lengths and angles in any triangle. [Honors only]

● Further work on logarithms will continue in future math courses, such as pre-calculus or a mathematics analysis course. The work with Law of Sines and Cosines and reviewing the use of right triangle trigonometry and special right triangles will prepare students for work with trigonometric functions in Chapter 9.


Seeing Structure in Expressions (A-SSE)Interpret the structure of expressions. (Major) [A-SSE.A]A-SSE.A.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).Interpreting Functions (F-IF)Analyze functions using different representations. (Supporting) [F-IF.C]F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

complicated cases.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period midline,

and amplitude. ✯Building Functions (F-BF)Build a function that models a relationship between two quantities. (Supporting) [F-BF.A]F-BF.A.1: Write a function that describes a relationship between two quantities. ✯


by adding a constant function to a decaying exponential, and relate these functions to be modeled. ✯Linear, Quadratic, and Exponential Models ✯ (F-LE)Construct and compare linear, quadratic, and exponential models and solve problems. ✯ (Supporting) [F-LE.A]F-LE.A.4: For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate

the logarithm using technology. ✯Similarity, Right Triangles, and Trigonometry (G-SRT)Apply trigonometry to general triangles. (Additional) [+G-SRT.D]G-SRT.D.9: (+) Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite

side. [honors course only]G-SRT.D.10: (+) Prove the Law of Sines and Cosines and use them to solve problems. [honors course only]G-SRT.D.11: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g.,



surveying problems, resultant forces). [honors course only]


1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.6. Attend to precision.8. Look for and express regularity in repeated reasoning.





● ● ●

● ● ●


My students will be able to (Do) …● Use logarithms to solve equations, including exponential equations. (F-LE.A.4)● Write equations of exponential functions from two points and/or use them to solve application problems. (F-BF.A.1a)● Calculate missing side lengths or areas of non-right triangles. (G-SRT.D.11+) [honors course only]● Solve and graph inequalities. (Checkpoint 7)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Content



area (of a non-right triangle), inverse ambiguous case, change of base, curve fitting, exponential equation, Law of Cosines [honors course only], Law of Sines [honors course only], logarithm, Power Property of Logarithms, Product Property of Logarithms, Quotient Property of Logarithms

Resources

Core Lessons

7.1.1 through 7.1.4Chapter Closure

Honors course also includes: 7.2.1 through 7.2.5


● Passport to Advanced Mathematics: Quadratic and Exponential Word Problems (Exponential only in this unit)

Misconceptions ●



Unit of Study 8: Polynomials Length of Unit 13 days (February 19–March 8, 2019)


● The focus of this chapter is to apply knowledge of families of functions from Chapters 2, 5, and 7 to include polynomial functions.● In Section 8.1, students investigate the connection between the graph and equation of a polynomial function and understand the relationship

between the factors of a polynomial equation and the x-intercepts of the graph of a polynomial. Students sketch the graphs of polynomials without using a graphing calculator and write equations from graphs. In Section 8.2, students further develop their understanding of imaginary and complex numbers (introduced in Core Connections Integrated II) and recognize that polynomial functions can have complex roots. In Section 8.3, students divide polynomials by a known factor to find other factors, allowing them to determine complex and irrational roots of some cubic and quartic functions.

● Further work on function families will continue in Chapter 9. In Chapter 9, students study trigonometric functions and investigate the effects of transformations on the graphs of sine and cosine functions. Students will further develop their algebra skills in proving formulas for sums of series in Chapter 10, studying rational functions in Chapter 11, and solving trigonometric identities in Chapter 12.


The Complex Number System (N-CN)Use complex numbers in polynomial identities and equations. (Additional) [N-CN.C]N-CN.C.8: (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x+2i)(x-2i). [honors course only]N-CN.C.9: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [honors course only]Seeing Structure in Expressions (A-SSE)Interpret the structure of expressions. (Major) [A-SSE.A]A-SSE.A.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).Arithmetic with Polynomials and Rational Expressions (A-APR)Perform arithmetic operations on polynomials. (Major) [A-APR.A]A-APR.A.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.Understand the relationship between zeros and factors of polynomials. (Major) [A-APR.B]A-APR.B.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 of p(x).A-APR.B.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.Use polynomial identities to solve problems. (Additional) [A-APR.C]A-APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 +

(2xy)2 can be used to generate Pythagorean triples.Rewrite rational expressions. (Additional) [ELG.MA.HS.A.6]A-APR.D.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 more complicated examples, a computer algebra system.



Creating Equations (A-CED)✯Create equations that describe numbers or relationships. (Major) [A-CED.A]A-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales.Interpreting Functions (F-IF)Interpret functions that arise in applications in terms of the context. (Major) [F-IF.B]F-IF.B.4: For a function that models a relationships between two quantities, interpret key features of graphs and tables in terms of the quantities,



complicated cases. ✯c. Graph polynomial functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. ✯



by adding a constant function to a decaying exponential, and relate these functions to be modeled. ✯Standards for Mathematical Practice

3. Construct viable arguments and critique the reasoning of others.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.





● Modeling nonlinear relationships between two quantities requires using appropriate functions. (F-BF.A.1a)

● How can we determine key features of polynomial function graphs from their equations?

● What phenomena can be modeled with polynomial functions?

● How do we use function families’ key features to determine appropriate functions for given situations?


● Transforming polynomial expressions and ● What are the different ways to solve quadratic ● How can polynomial identities be used to describe



equations can reveal underlying structures and solutions. (A-APR.B.2, A-APR.B.3)

equations?● How is factoring used to solve polynomials with

degrees greater than two?

numerical relationships?● Why is the Remainder Theorem useful?

● Properties of operations transform rational expressions with the intention of creating more efficient forms of expressions. (A-SSE.A.2)

● How can inspection, long division, and computer algebra systems be used to rewrite rational expressions?

● How do we use factoring to rewrite rational expressions?

● What are applications of rational equations?● Why do we rewrite rational expressions in different

forms?


My students will be able to (Do) …● Given the x-intercepts and a point on a graph, write the equation of a polynomial function that represents the graph. (A-CED.A.2)● Sketch a polynomial that has a given number of real and complex roots, or determine the minimum number of real and complex roots from the graph or vice versa. (F-

IF.C.7c, F-IF.B.4)● Given the equation of a polynomial function of degree three or less with only real roots, determine the roots or factors, and/or sketch a graph. (A-APR.B.3)● Given the roots of a polynomial, write a possible equation. (A-CED.A.2)● Determine the equation for the inverse of a function. (Checkpoint 8)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentfactor, remainder, root coefficient, complex conjugate [honors course only], complex number [honors

course only], degree (of a polynomial), difference of cubes, difference of squares, double root, Fundamental Theorem of Algebra, polynomial, polynomial division, quotient, real number, sum of cubes, x-intercept, zero (of a function)

Resources



Core Lessons

8.1.1 through 8.1.38.3.1 through 8.3.3Chapter ClosureHonors course also includes: 8.2.1, 8.2.2, and 8.3.4


● Passport to Advanced Mathematics: Polynomial Factors and Graphs

Misconceptions● Students also believe expressions cannot be factored because they do not fit recognizable forms. They need help manipulating terms until

structures become evident.● Students try to combine terms that are not like or change the degree of variables when combining terms.



Unit of Study 9: Trigonometric Functions Length of Unit 20 days (March 11–April 12, 2019)


● The focus of this chapter is to extend understanding of trigonometric ratios in right triangles to trigonometric functions. This chapter gives students the tools they need to model situations involving periodic phenomena.

● In Section 9.1, students investigate periodic relationships and explore the connections between the unit circle and graphs of trigonometric functions. Students create multiple representations for two new functions, y = sin(θ) and y = cos(θ). Also, students use radians to describe angle measures. In Section 9.2, students apply transformations to the parent function f(x) = sin(x) to develop general equations for the family of trigonometric functions.

● Further work on trigonometry will continue in Chapter 12. In Chapter 12, students will solve trigonometric equations, learn reciprocal trigonometric functions, develop trigonometric identities, and further explore the connections between sine, cosine, and tangent.


Interpreting Functions (F-IF)Interpret functions that arise in applications in terms of the context. (Major) [F-IF.B]F-IF.B.4: For a function that models a relationships between two quantities, interpret key features of graphs and tables in terms of the quantities,



complicated cases.✯e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period midline, and amplitude. ✯



by adding a constant function to a decaying exponential, and relate these functions to be modeled. ✯Build new functions from existing functions. (Additional) [F-BF.B]F-BF.B.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


Trigonometric Functions (F-TF)Extend the domain of trigonometric functions using the unit circle. (Additional) [F-TF.A]F-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.F-TF.A.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. (Additional) [F-TF.B]



F-TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. ✯Standards for Mathematical Practice

4. Model with mathematics.5. Use appropriate tools strategically.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.


● Students should understand the effects of parameter changes and be able to apply them to create a rule modeling the function. (F-BF.B.3)● Seeing mathematics as a tool to model real-world situations should be an underlying perspective in everything students do, including writing

algebraic expressions, creating functions, creating geometric models, and understanding statistical relationships. (M)● In particular, students should recognize that much of mathematics is concerned with understanding quantities and their relationships. They

should pick appropriate units for quantities being modeled, using them as a guide to understand a situation, and be attentive to the level of accuracy that is reported in a solution. (N-Q)



● Modeling nonlinear relationships between two quantities requires using appropriate functions. (F-TF.B.5, F-IF.C.7e)

● How can we determine key features of graphs of trigonometric functions from their equations?

● What phenomena can be modeled with trigonometric functions?

● How do we use function families’ key features to determine appropriate functions for given situations?


● The unit circle in the coordinate plane represents the trigonometric functions for any angle. (F-TF.A.2)

● How is the circumference of a unit circle used to determine radian measures of angles?

● Given an angle, how is the unit circle used to determine each trigonometric function?

● How are the relationships of right triangles used to determine trigonometric functions of angles?

● How does the periodicity in the unit circle correspond to the periodicity in graphs of models of periodic phenomena?

● How does the Pythagorean identity illustrate the inverse nature of the relationship between sine and cosine?

● Trigonometric functions model periodic phenomena. (F-TF.B.5)

● In which situations would it be appropriate to model with trigonometry?

● How are period, midline, and amplitude reflected in trigonometric function equations?

● Why would the parent trigonometric function change in period, midline, and amplitude for a given situation?

● Why can the same class of functions model diverse types of situations (for example, manufacturing, sales, temperature, amusement park rides)?




My students will be able to (Do) …

● Identify side measures of special right triangles. (review of prior course)● Use a circle, the unit circle, or sin2(ɵ) + cos2(ɵ) = 1 to determine missing information. (F-TF.A.2)● Calculate sine, cosine, and tangent ratios for benchmark angles in the unit circle without a calculator. (F-TF.A.2)● Convert between degrees and radians and vice versa. (F-TF.A.1)● Solve equations with exponents. (Checkpoint 9A)● Rewrite expressions and solve equations with logarithms. (Checkpoint 9B)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentamplitude, angle, cycle, degree, midline, parameter, period cosine function, horizontal shift, periodic function, Pythagorean Identity, radian,

reference angle, sine function, special right triangles, tangent function, theta ( ), 𝜽trigonometric function, unit circle, vertical shift

Resources



● Additional Topics in Math: Right Triangle Word Problems● Additional Topics in Math: Right Triangle Trigonometry

Misconceptions



Unit of Study 10: Series Length of Unit 11 days (April 15–April 30, 2019)


● The focus of this chapter is to apply and extend knowledge about arithmetic and geometric sequences to the development of formulas for sums of geometric and arithmetic sequences while further developing their algebraic skills.

● In Section 10.1, students devise methods for determining sums of arithmetic sequences and explore proof by induction. In Section 10.2, students devise methods for determining sums of geometric sequences where the last term is defined. Students also learn to represent and calculate sums of some infinite series. In Section 10.3, students develop the Binomial Theorem and use it to expand binomials such as (x + 3)4 and solve probability problems.

● Further work on summation notation and series will be developed further in precalculus and calculus. Also, using a graphical approach to develop a formula for the sum of an arithmetic series is a precursor for using the area of rectangles to calculate the area under curves in calculus.

Standards

Content Standards

Seeing Structure in Expressions (A-SSE)Interpret the structure of expressions. (Major) [A-SSE.A]A-SSE.A.1: Interpret expressions that represent a quantity in terms of its context. ✯

a. Interpret parts of an expression, such as terms, factors, and coefficients.A-SSE.A.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).Write expressions in equivalent forms to solve problems. (Major) [A-SSE.B]A-SSE.B.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For

example, calculate mortgage payments. ✯Use polynomial identities to solve problems. (Additional) [A-APR.C]A-APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 +

(2xy)2 can be used to generate Pythagorean triples.A-APR.C.5: (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are

any numbers, with coefficients determined for example by Pascal’s Triangle. [honors course only]


1. Make sense of problems and persevere in solving them.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.


● N/A





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My students will be able to (Do) …● Given a finite arithmetic series or a situation, represent it with summation notation and/or calculate the sum. (A-SSE.A.1a)● Given a finite geometric series or a situation, represent it with summation notation and/or calculate the sum. (A-SSE.B.4)● Given an infinite geometric series or a situation, represent it with summation notation and/or calculate the sum. (extension of A-SSE.B.4) [Honors only]● Solve triangles for missing sides and angles. (Checkpoint 10) [some honors only problems within this checkpoint]

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentargument, combinations, geometric, index, infinite, sequence, series arithmetic, Binomial Theorem [honors course only], common difference, common

multiplier, compound continuously, compound interest, e, Pascal’s Triangle [honors course only], proof by induction, sigma, summation notation

ResourcesCore Lessons 10.1.1 through 10.1.5




Honors course also includes: 10.2.2 and 10.3.1 through 10.3.3


● N/A

Misconceptions

● Students do not understand what it means to find sums of series. For example, if asked to find the sum of the first 17 terms of a series, they only find the 17th term.

● Students should recognize they have multiple ways to find sums of series. It is not always necessary to use formulas; students can also use conceptual methods.



Unit of Study 11: Rational Expressions and Three-Variable Systems Length of Unit 8 days (May 1–May 10, 2019)


● The focus of this chapter is to deepen algebraic skills through simplifying rational expressions and solving three-variable systems of equations.● In Section 11.1, students rewrite and simplify rational expressions using properties of the number 1 and apply understanding of operations on

fractions to add, subtract, multiply, and divide rational expressions [honors course only]. In Section 11.2, students extend their understanding of two-variable systems of equations to solve systems of equations in three variables graphically in the three-dimensional Cartesian coordinate system and algebraically. Students apply this new learning to solve problems, including writing the equation of a parabola passing through three points.

● Further work on algebraic skills will continue in Chapter 12. In Chapter 12, students use their algebra skills to solve trigonometric identities. Three-dimensional visualization in this chapter prepares students for visualizing intersections of planes with cones to form conic sections in future courses. Also, in future courses, students may use matrices to solve systems of equations.

Standards

Content Standards

Arithmetic with Polynomials and Rational Expressions (A-APR)Rewrite rational expressions. (Additional) [A-APR.D]A-APR.D.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. [honors course only]Creating Equations (A-CED)✯Create equations that describe numbers or relationships. (Major) [A-CED.A]A-CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels

and scales.Standards for Mathematical Practice

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.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.





● Properties of operations transform rational expressions with the intention of creating more

● How can inspection, long division, and computer algebra systems be used to rewrite rational

● What are applications of rational equations?● Why do we rewrite rational expressions in different



efficient forms of expressions. (+A-APR.D.7) expressions?● How do we use factoring to rewrite rational

expressions?

forms?


My students will be able to (Do) …● Simplify rational expressions. (A-APR.D.7+) [honors course only]● Multiply and divide rational expressions. (A-APR.D.7+) [honors course only]● Determine the location of or graph points and planes in three dimensions. (preparation for A-CED.A.2)● Given the equation, determine the roots of a polynomial function and given the graph, write a possible equation for a polynomial function. (Checkpoint 11)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentplane closed set, coordinate system, excluded value, ordered triple, rational expression

[honors course only], rational number [honors course only], system of equations, three-variable equation

Resources

Core Lessons




● Passport to Advanced Mathematics: Radical and Rational Equations● Passport to Advanced Mathematics: Operations with Rational Expressions



Misconceptions ●

Unit of Study 12: Analytic Trigonometry Length of Unit 12 days (May 13–May 30, 2019)


● The focus of this chapter is to solve trigonometric equations and ask students to think about the circumstances under which an equation is true.

● In Section 12.1, students learn about inverse and reciprocal trigonometric functions. Students understand solutions to trigonometric functions in multiple representations: algebraically, graphically, and based on the unit circles. They also learn how to determine the number of solutions for a trigonometric equation. In Section 12.2, students use trigonometric identities to rewrite trigonometric equations [honors course only].

● Further work on trigonometry will continue in future courses such as precalculus or calculus.

Standards

Content Standards

Interpreting Functions (F-IF)Analyze functions using different representations. (Supporting) [F-IF.C]F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

complicated cases. ✯e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline,

and amplitude. ✯Trigonometric Functions (T-FT)Prove and apply trigonometric identities. (Additional) [F-TF.C]F-TF.C.9: (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. [honors course only]


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.







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My students will be able to (Do) …● Solve trigonometric equations. (F-TF.C.9+) [honors course only]● For periodic functions, given an equation, determine key features and sketch the graph and given a graph, write a possible equation. (Checkpoint 12)

WIDA English Language Development (ELD) Standards1: English language learners communicate for social and instructional purposes within the school setting.3: English language learners communicate information, ideas, and concepts necessary for academic success in the content area of mathematics.Use WIDA Can-Do Descriptors and WIDA Performance Definitions to determine appropriate sensory, graphic, and interactional supports and scaffolds and differentiate appropriate outputs based on English proficiency levels.Critical language development includes:● Academic vocabulary to use orally and in writing at the word/phrase and sentence levels to understand and engage in communication of academic content.● Academic language not specific to any particular content area, but necessary to communicate complex ideas and critical thinking.Goal is for students to understand and use cross-content academic and technical words. This vocabulary may help in creating CLOs.Cross-Content Academic Words Technical Words Specific to Contentidentity, inverse, solution cosecant, cotangent, inverse cosine, inverse sine, inverse tangent, reciprocal, secant,

trigonometric function, unit circle

Resources

Core Lessons




● N/A

Misconceptions ●


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