Curriculum Guide - Algebra II Introduction

Transcription

1 Algebra II Curriculum Guide Curriculum Guide - Algebra II Introduction Appropriate Common Core State Standards and Clusters are followed by one of the following symbols. Major Clusters/Standards Supporting Clusters/Standards o Additional Clusters/Standards High school mathematical modeling standards FS Fluency Standard All testable standards (SPIs) from the 'TCAP-EOC Algebra II Framework' have been embedded within this guide. Common Core Mathematical Practice Standards The CCSS for Mathematical Practices are expected to be integrated into every mathematics lesson for all students grades K 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. Last updated on 7/20/2014 Additional resources, CRAs, Instructional Tasks, etc. listed at the bottom of each unit will be found on the sharing server. The file name will be in parentheses after a brief description and will be in: Sharing/Consulting Teachers/Math/Algebra II/Algebra II Resources Questions or comments should be directed to Karl Bittinger, Math Curriculum Consulting Teachers. Page 1 of 57

2 Clarifications, Evidence, and Assessment Curriculum Guide - Algebra II Introduction PBA - Performance Based Assessment (PBA) evidence statements, clarifications, math practice standards, and calculator usage were taken from the following location: EOY - End of Year (EOY) evidence statements, clarifications, math practice standards, and calculator usage were taken from the following location: PLD - Performance Level Descriptors (PLD) Level 5 descriptors were taken from the following location: Calculator - When we begin Common Core assessments in , students will only be permitted to use the online calculator for state assessments, which will be similar to the TI-84. While this calculator policy will not be enacted for the school year, it has been left in this document to help teachers prepare for the upcoming Common Core changes. Yes indicates a calculator will be accessible through the computer for the indicated assessment No indicates a calculator will not be accessible through the computer for the indicated assessment Neutral indicates a calculator will be accessible through the computer for the indicated assessment but may not be needed Item specific indicates the standard will only have a calculator accessible for certain items on the assessment A balanced use of calculators continues to be encouraged. Limitations - Assessment limits for standards assessed on more than one end-of-course (EOC) test: Algebra I, Geometry, and Algebra II. Limitations can be found on pages 56 through 59 of the Model Content Frameworks, Mathematics, Grades 3-11 at: - For the purposes of CMCSS pacing, Learning Targets are written in teacher friendly language. The Learning Targets in our pacing guides should be completely aligned to the content standards and exhaust the meaning of the standards. The Learning Targets may lead to clear targets. Page 2 of 57

3 Unit Schedule 1st Semester Unit Title Dates Days 1 Equations and Inequalities August 6 - August 20, Functions August 21 - September 16, Quadratics September 17 - October 10, Polynomials October 20 - November 13, Rational Equations November 14 - December 10, First Semester Review and Exam December 11 - December 19, Total Days 86 2nd Semester 7 Rational Exponents, Exponentials, and Logarithms January 6 - January 30, Trigonometry February 2 -February 24, Sequences and Series February 25 - March 24, Statistics March 25 - April 29, EOC Review and Assessment April 30 - May 7, Second Semester Review and Exam May 8 - May 21, Total Days 90 Assessments Dates EOC May 7, 2015 Page 3 of 57

4 Unit 1 Unit 1: Equations and Inequalities 10 Days: August 6 - August 20, 2014 Standard Clarifications, Evidence, and Assessment A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. PBA - Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any). - Tasks are limited to simple rational or radical equations. - Simple rational equations are limited to those whose numerators and denominators have degree at most 2. - MP 6 Calculator - Yes Limitations - Tasks are limited to simple rational or radical equations. Suggested use proofs (two column, proof, flow-chart) to prove all types of multi-step equations. Do not include factoring. Example of simple rational: 3/(2x+5)=10 Example of simple radical: sqrt(x) +2=12 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. o - In units 1, 7, and 8 - intro to parent graphs- evaluate shifts on (absolute value, quadratic, logs, exponential) Page 4 of 57

5 Unit 1 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, polynomial, rational, absolute value, exponential, and logarithmic functions. PBA - MP 1 and 5 Calculator - Item Specific PBA - Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane - MP 3 Calculator - Yes MYA/EOY - solve linear and absolute value equations - use various forms to represent linear and absolute value equations/functions SPI In units 1, 5, and 7 - connect parallel, perpendicular slopes of lines to form possible intersections. (parallel and perpendicular slopes taught in Geometry) A.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. o EOY - Solve algebraically a system of three linear equation in three unknowns - 80% of systems have a unique solution. 20% of systems have no solution or infinitely many solutions. - Coefficients are rational numbers Tasks do not require any specific method to be used. (e.g. prompts do not direct the student to use elimination or any other particular method) - MP 1 and 7 Calculator - Item specific Limitations - Tasks are limited to 3x3 systems SPI , SPI Page 5 of 57

6 Unit 1 A.CED.A.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. F.BF.A.1 Write a function that describes a relationship between two quantities. Limitations - Tasks are limited to exponential equations with rational or real exponents and rational functions - Tasks have a real-world context given a graph you must come up with an equation or an inequality. - In units 1 and 2 - Story/Word Problems 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.B.5 Interpret the parameters in a linear or exponential function in terms of a context. O EOY - Prompts describe a scenario using everyday language. Mathematical language such as "function," "exponential," etc. is not used. - Students autonomously choose and apply appropriate mathematical techniques without prompting. For example, in a situation of doubling, they apply techniques of exponential functions - For some illustrations, see tasks at under F.LE - MP 1, 2, 4, and 6 Calculator - Item specific - In units 1 and 2 - Suggest writing equations given two points or a graph - Identify domain and range of a linear function. - In units 1 and 7 - Suggestion for advancing problems for domain and range (honors): piecewise functions. Page 6 of 57

7 Unit 1 Additional Resources: Common Student Misconceptions 013_14/Math/Alg%20II%20FT%204.pdf 013_14/Math/Alg%20II%20FT%206.pdf module-overview-and-assessments.pdf pdf A.REI.A.1: forget to use reverse Pemdas, students have trouble with inverse operations such as square vs. square root. F.BF.B.3: students mix up the h and k shifts transformations A.REI.D.11: students often forget two cases or extraneous solutions for absolute values A.REI.C.6: students do not know the difference between infinite solutions and no solutions in a system of equations A.CED.A.1: students forget included and excluded points in inequalities F.BF.A.1: forget when writing slope students often put (change x)/(change y) F.LE.A.2: write equations from two points, slope intercept; point slope forms Video Tutorials Solving through substitution, elimination, and graphing Parameters of real-world linear functions lesson Course Resource - sample problems Free student topic practice Page 7 of 57

8 Unit 1 Assessments: Unit Vocabulary: Parent Functions, Transformation, Parameters, Coefficient, Equation vs. Inequality, Solution Set Instructional Tasks: Integration Prerequisite Skills A.REI.A.1: 8.EE.C.7.A- no solution, identity, infinite sol; solving multi-step equations F.LE.A.2: write equations from two points, slope intercept; point slope forms Constructed Response Assessments If the teacher cannot fully complete this unit in the alloted days, then the areas of the Common core which may be demphasized and/or emphasized are: Deemphasize: If you run out of time; minimize the time with radicals and simple rational in this form. We will also be addressing these two topics later. Emphasize: SPI Graph the solution set of two or three linear or quadratic inequalities. SPI Solve quadratic equations and systems, and determine roots of a higher order polynomial. SPI Solve systems of three linear equations in three variables. Page 8 of 57

9 Unit 2 Unit 2: Functions 18 Days: August 21 - September 16, 2014 Standard Clarifications, Evidence, and Assessment F.BF.A.1 Write a function that describes a relationship between two quantities. in units 1 and 2 F.BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. Limitations - Tasks have a real-world context - Tasks may involve linear functions, quadratic functions, and exponential functions - From a real-world scenario create a linear, quadratic or exponential expression/equation to describe the relationship between two quantities Create and apply a step-by-step, repeatable (recursive) process to describe the relationship Create and apply unique steps to solve a given scenario Domain is Real Numbers (Algebra 1 uses only Integer Domains) F.BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. EOY - Represent arithmetic combinations of standard function types algebraically. - Tasks may or may not have context. - For example, given f(x)=e x and g(x)=5, write an expression for h(x)=2f(-3x)+g(x) Calculator - Neutral - Perform arithmetic operations on functions Understand effect of arithmetic operations on functions Use operations to match a real-world model Do NOT teach composition of functions Page 9 of 57

10 Unit 2 F.BF.B.4 Find inverse functions. o PBA - Solve multi-step contextual problems with degree of difficulty appropriate to the course that require writing an expression for an inverse function. Calculator - Yes SPI F.BF.B.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. For example, f(x) =2 x3 or f(x) = (x+1)/(x 1) for x 1. o EOY - For example, see - As another example, given a function C(L)=750L 2 f or the cost C(L) of planting seeds in a square field of edge length L, write a function for the edge length L, of a square field that can be planted for a given amount of money C; graph the function, labeling the axes. - MP 1, 6, and 8 Calculator - Item specific SPI Additional Learning Target - ONLY focus on findings inverse functions algebraically (graphical representations of inverses can be used for understanding/teaching purposes but should not be included as part of assessment for this standard) Page 10 of 57

11 Unit 2 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. 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). EOY - Tasks have context. - Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. - MP 1, 4, 5, and 7 Calculator - Item specific F.IF.B.6 is in units 3, 4, 7, and 8 PLD (PBA and EOY) - Calculate and interpret the average rate of change of polynomial, exponential, logarithmic, or trigonometric functions (presented symbolically or as a table) over specified interval, and estimates the rate of change from a graph. - Compare rates of change associated with different intervals - Understand the relationship between arithmetic sequences and linear functions Understand the relationship between geometric sequences and exponential functions Create linear or exponential functions when given o Table of values (from arithmetic or geometric sequence) o Written description of relationship Graph of Relationship SPI Interpret graphs that depict real-world phenomenon. Additional Resources: Integration _14/Math/Alg%20II%20FT%202.pdf 013_14/Math/Alg%20II%20FT%203.pdf cal_appliedsciences_math/_assoc/958aca7292b8492ab9f2e4cb242aa8c6/geo _FT_7.pdf Page 11 of 57

12 Unit 2 cal_appliedsciences_math/_assoc/f1b50b120f624e98bebd6c67ed3f9496/alg_ II_FT_8.pdf pdf Assessments: Common Student Misconceptions F.BF.A.1a - Function, Recursive Process, Explicit Expression F.BF.B.4a - Inverse F.LE.A.2 - Sequence, Arithmetic, Geometric, Input-Output Pair, Common Ratio, Common Difference F.BF.A.1a Students have difficulty identifying dependent and independent variables Students tend to always identify x as independent and y as dependent variable F.BF.B.4(a) Students may take inverse to mean simply reversing sign (+ to -) within an expression, switching variables without solving after switch or simply reversing the order of the expression/equation F.LE.A.2 Improper determination of common ratio in Geometric Sequences (often divide term i by i+1 instead of dividing i+1 by i Unit Vocabulary: Prerequisite Skills F.BF.A.1 students should have completed F.BF.A.1a during Algebra 1 with Domain limited to Integers. F.BF.A.1b understanding of combining like terms operations with polynomials Page 12 of 57

13 Unit 2 Instructional Tasks: Combining functions (A2 exploring polynomials.pdf) TNCore Instructional Task Arc 1 Task 1: Polynomial functions can be subtracted. The sum or difference of two linear functions will be a linear function. F.BF.B.4(a) solving multi-step equations basic algebraic manipulation F.LE.A.2 Familiarity with Arithmetic and Geometric Sequences Constructed Response Assessments A2 Amusement SG A2 Honeybees SG Task 2: Polynomial functions can be added and subtracted. The sum or difference of a linear function and a quadratic function is quadratic. Functions can be added or subtracted using graphs and tables or using the algebraic representations. Task 3: Solidify understanding that polynomial functions can be added and subtracted using tables, graphs, and equations and that the degree of the sum function is dependent on the degree of the addends developed in Tasks 1-2. Task 4: the product of two non-constant linear functions is always a quadratic function. The x-intercepts of the linear function are the x- intercepts of the quadratic. Functions can be multiplied using graphs and tables or using their algebraic representations. Task 5: The product of two non-constant linear functions is always a quadratic function. The graph of the quadratic function has predictable characteristics determined by whether the linear factors are parallel, perpendicular, intersecting/no perpendicular, or represent the same line. Regardless of the relationship between the linear factors, however, the x-intercepts of the linear factors are the x-intercepts of the quadratic function. Page 13 of 57

14 Task 6: All polynomial functions can be written as a product of linear factors. The value of the polynomial function is equal to zero when one or more of the linear actors is equal to zero. Polynomials can be multiplied algebraically by applying the distributive property and collecting like terms. Task 7: The product of two polynomial functions is a polynomial function. The x-intercepts of the original functions are the x intercepts of the product function. Polynomial functions can be multiplied using graphs and tables or using the algebraic representations. Task 8: Solidify understanding of the relationships between factors of a polynomial function and its zeros developed in Tasks 4 7. (A1 Task Arc 1 Functions) Curriculum Guide - Algebra II Unit 2 If the teacher cannot fully complete this unit in the alloted days, then the areas of the Common core which may be demphasized and/or emphasized are: Deemphasize: IF.BF.A.1(a & b) focus on creation and evaluation of various types of functions with focus on linear, quadratic and exponential functions; manipulation of functions should be emphasized and real-world applications can afford to be deemphasized Emphasize: F.BF.B.4(a) Be sure to include graphical understanding of inverse functions as reflections about a line (y=0, x=0 and y=x) SPI Interpret graphs that depict real-world phenomenon When needed, first year teachers should collaborate with mentor teachers to aid in understand which EOC specific standards should be emphasized Page 14 of 57

15 Unit 3 Unit 3: Quadratics 18 Days: September 17 - October 10, 2014 Standard Clarifications, Evidence, and Assessment A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). FS PBA and EOY - Examples: In the equation x 2 + 2x y 2 = 9, see an opportunity to rewrite the first three terms as (x+1) 2, thus recognizing the equation of a circle with radius 3 and center (-1,0). See (x 2 +4)/(x 2 +3) as ((x 2 +3)+1)/(x 2 +3), thus recognizing an opportunity to write it as 1+[(1)/(x 2 +3)]. EOY - Example:Factor completely: 6cx-3cy-2dx+dy. (A first iteration might give 3c(2x-y)+d(-2x+y), which could be recognized as 3c(2xy)-d(2x-y) on the way to factoring completely as (3c-d)(2x-y).) - Tasks do not have a context. Calculator - Neutral Limitations - Tasks are limited to polynomial, rational, or exponential expressions. Additional Learning Targerts - In the example of x 4 -y 4 they are emphasizing that the possibility exists that an even degree polynomial can be written as a quadratic. - -Students have learned to write the equation of a circle by completing the square, you could tie this into the recognizing the equation of a circle with a radius and center. - Also the example of (x 2 +4)/(x 2 +3) is and honors class example. You will not need to know how to simplify it in this fashion until you work with intervals in calculus. Page 15 of 57

16 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. Curriculum Guide - Algebra II Unit 3 PBA - Base explanations/reasoning on the relationship between zeros and factors of polynomials. - MP 3 Calculator - Yes Limitations - Tasks include quadratic, cubic, and quartic polynomials and polynomials for which factors are not provided. For example, find the zeros of (x 2-1)(x 2 +1). - -Also when you mention the relationship between zeros and factors of polynomials mention the vocabulary: roots, x-intercepts, and zeros and have the solution discussion. -make sure the focus is factoring 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. A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. In units 3, 4, 6, and 7 - Given a graph or a table a student can state the average rate of change within a certain interval, focus only on quadratic. Average rate of change for linear and piece-wise functions is taught in Algebra I. In units 3, 4, 7, and 8 - This is tied into numerous standards. They are asking you to change one form to the next. Example writing from standard form to vertex form and completing the square and factoring. Page 16 of 57

17 Unit 3 N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Limitations - This standard will be assess in Algebra II by ensuring that some modeling tasks (involving Algebra II content or securely held content from previous grades and courses) require the student to create a quantity of interest in the situation being described (i.e., this is not provided in the task). For example, in a situation involving periodic phenomena, the student might autonomously decide that the amplitude is a key variable in a situation, and then choose to work with peak amplitude. - The standard will be used for real world applications limited to quadratics. EOY - Tasks may or may not have context. - MP 1, 3, 5, 6, and 8 Calculator - Item specific Limitations - Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. - We are limiting this to quadratics only. Compare two quadratic functions representing in different ways. - This is tied into numerous standards. They are asking you to change one form to the next. Example writing from standard form to vertex form and completing the square and factoring. N.CN.A.1 Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. o PBA and EOY - MP 7 Calculator - Item Specific - Being able to simplify complex numbers and operations. Page 17 of 57

18 Unit 3 N.CN.A.2 Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. o PBA and EOY - MP 6 and 7 Calculator - No SPI Being able to simplify complex numbers and operations. N.CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. o PBA and EOY - Tasks are limited to equations with non-real solutions. - MP 5 Calculator - Item Specific SPI , SPI Being able to simplify complex numbers and operations. A.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. G.GPE.A.2 Derive the equation of a parabola given a focus and directrix. o A.REI.B.4 Solve quadratic equations in one variable. EOY - Tasks have thin context or no context - MP 1 Calculator - Item specific SPI To include intersections of linear and quadratic functions PBA - Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. - MP 3 Calculator - Yes - Solving quadratics using methods previously taught Page 18 of 57

19 Unit 3 A.REI.B.4b Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. PBA - Tasks involve recognizing an equation with complex solutions, e.g., Which of the following equations has no real solutions? with one of the options being a quadratic equation with non-real solutions. - Writing solutions in the form a ± bi is not assessed here. (N.CN.C.7) - MP 5 and 7 Calculator - Neutral EOY - Solve quadratic equations in one variable. - Recognize when the quadratic formula gives complex solutions. - Tasks involve recognizing an equation with complex solutions, e.g., "Which of the following equations has no real solution?" with one of the options being a quadratic equation with non-real solutions. MP 5 and 7 Calculator - Neutral Limitations - In the case of equations that have roots with nonzero imaginary parts, students write the solution as a ± bi for real numbers a and b. SPI , SPI , SPI Solving quadratics using methods previously taught Page 19 of 57

20 Unit 3 Additional Resources: Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_AdvancedAlgebra_Unit2SE.pdf Integration _14/Math/Alg%20II%20FT%202.pdf cal_appliedsciences_math/_assoc/f1b50b120f624e98bebd6c67ed3f9496/alg_ II_FT_8.pdf cal_appliedsciences_math/_assoc/1e70c6f8ceea4ac08c556fe2407f1979/alg_ II_FT_9.pdf pdf Pages Assessments: A.APR.B.3 Roots, zeros, x-intercepts, solutions A.SSE.B.3 Completing the square is the vertex form A.REI.C.7 Quadratic equation with two variables Common Student Misconceptions A.SSE.A.2 They may not understand that the possibility exists that a higher degree even polynomials can be written as a quadratic. A.APR.B.3 The solutions to quadratics are not just the zeros or roots. Vocabulary will confuse them F.IF.B.6 One misconception could be that it only applies to linear function Page 20 of 57

21 Unit 3 G.GPE.A.2 Students will have to know the new vocabulary directix,focus, A.SSE.B.3 Understanding the name of the different forms and being able to identify. N.Q.A.2 Too broad depends on the questions F.IF.C.9 Students may misread graphs, and the interval amounts. Unit Vocabulary: Instructional Tasks: A.REI.C.7 They will solve for x and not for the y They wont put in the positive and the negative in their calculator Prerequisite Skills A.APR.B.3 This is taught in Algebra 1 but limited to quadratic and cubic s F.IF.B.6 They do this in Algebra 1 with all the graphs A.SSE.B.3 Same as A.SSE.A.2 N.Q.A.2 They do this in algebra 1 but not with functions using a data set only F.IF.C.9 This is taught in Algebra 1 A.REI.C.7 They have done this in geometry by graphing a circle A.SSE.A.2 - This is also an Algebra 1 standard that focuses on just a degree of 2 and numerical difference of squares. - You would have to know how to factor a trinomial - Geometry also uses completing the square to write an equation of a circle Constructed Response Assessments Page 21 of 57

22 Unit 3 Functions can be represented in multiple ways (A2 3rd degree polynomial.pdf) Functions can be represented in multiple ways (A2 4th degree polynomial.pdf) TNCore Instructional Task Arc 1 A2 Boxed SG A2 Rocket SG A2 Writing SG Task 4: the product of two non-constant linear functions is always a quadratic function. The x-intercepts of the linear function are the x- intercepts of the quadratic. Functions can be multiplied using graphs and tables or using their algebraic representations. Task 5: The product of two non-constant linear functions is always a quadratic function. The graph of the quadratic function has predictable characteristics determined by whether the linear factors are parallel, perpendicular, intersecting/nonperpendicular, or represent the same line. Regardless of the relationship between the linear factors, however, the x-intercepts of the linear factors are the x-intercepts of the quadratic function. Task 6: All polynomial functions can be written as a product of linear factors. The value of the polynomial function is equal to zero when one or more of the linear actors is equal to zero. Polynomials can be multiplied algebraically by applying the distributive property and collecting like terms. Task 7: The product of two polynomial functions is a polynomial function. The x-intercepts of the original functions are the x intercepts of the product function. Polynomial functions can be multiplied using graphs and tables or using the algebraic representations. Task 8: Solidify understanding of the relationships between factors of a polynomial function and its zeros developed in Tasks 4 7. (A1 Task Arc 1 Functions) Page 22 of 57

23 Unit 3 If the teacher cannot fully complete this unit in the alloted days, then the areas of the Common core which may be demphasized and/or emphasized are: Deemphasize: - G.GPE.A.2- It involves them using the focus and directrix of a parabola - Rationalizing the denominators including complex conjugates. Emphasize: Try to combine several standards in a lesson, keep from reteaching for example A.APR.B.3 (falls with polynomials) Page 23 of 57

24 Unit 4 Unit 4: Polynomials 17 Days: October 20 - November 13, 2014 Standard Clarifications, Evidence, and Assessment 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). EOY - MP 6 Calculator - No PBA - Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about polynomials, rational expressions, or rational exponents. - Factor polynomials using a variety of methods to include the Factor Theorem, synthetic division, long division, sums and differences of cubes, and grouping. 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. - Taught in unit 3 and 4 - Identify zeros of Cubic s and higher order polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 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 - Identify the effect on a graph by manipulating a graph with a negative); find the value of k given the graphs. Experiment with constant, which shows a transformation. Include recognizing cases and illustrate an explanation of the effects on the graph difference between even and off functions, which re-integrates end using technology. Include recognizing even and odd functions from behavior of a function. their graphs and algebraic expressions for them. o Page 24 of 57

25 Unit 4 F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.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. F.IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. EOY - See illustrations for F.IF.B.4 at e.g., MP 4 and 6 Calculator - Yes Limitations - Tasks have a real-world context. - Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. - Compare second limitation with F.IF.C.7. - The function types listed here are the same as those listed in the Algebra II limitation for standards F.IF.B.6 and F.IF.C.9. - Identify key features of linear functions and move to more advanced functions. Include domain, range and end behavior as key features. - Compare, contrast and label parts of graphs to include linear, absolute value, quadratic, square root, rational, exponential, logarithmic, trigonometric. Teach concepts of graphing by hand and then introduce using the calculator EOY - MP 1, 5, and 6 Calculator - Item specific Page 25 of 57

26 Unit 4 A.APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. o PBA - Construct, autonomously, chains of reasoning that will justify or refute algebraic propositions or conjectures. - MP 3 Calculator - Yes PBA - Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about polynomials, rational expressions, or rational exponents. - In units 3, 4, and 6 - Derive polynomial identities to expedite problem solving. A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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. SPI Add, subtract, multiply polynomials; divide a polynomial by a lower degree polynomial. - In units 3, 4, and 6 - Convert between different forms of equations ( i.e. factor, vertex, standard form, translation form, graphical representation) - In units 3, 4, 7, and 8 - Use graphing calculator to calculate and interpret rate of change of any function through use of regression line of best fit presented symbolically or as a table. Multiplying and dividing is addressed within A.APR.B.2 / A.APR.B.3, but adding and subtracting will need to be covered for EOC compliance. Page 26 of 57

27 Unit 4 Additional Resources: Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_AdvancedAlgebra_Unit3SE.pdf 013_14/Math/Alg%20II%20FT%201.pdf 6.pdf pdf Integration Assessments: Common Student Misconceptions A.APR.B.2: Students may not understand that using various methods produce the same results. FIF.B.4: Students may believe that each family of functions is independent of the other functions and not recognize common features. Students may believe skills, such as factoring, is not useful in understanding characteristics of functions. Students may not see the different features exhibited by manipulating equations Page 27 of 57

28 Unit 4 F.BF.B.3: Students confuse the shift of the function with the stretch of the function. Students confuse the constant in the parentheses with the shift of the graph. FIF.C.7 Students may believe that each family of functions is independent of the other functions and not recognize common features. Students may believe skills, such as factoring, is not useful in understanding characteristics of functions. Students may not see the different features exhibited by manipulating equations Unit Vocabulary: End Behavior, Zeros, Roots, Remainder Theorem F.IF.B.6 Students may believe that each family of functions is independent of the other functions and not recognize common features. Students may believe skills, such as factoring, is not useful in understanding characteristics of functions. Students may not see the different features exhibited by manipulating equations Prerequisite Skills A.APR.B.2: Perform mathematical operations on numbers (i.e. distributive prop, factor binomials) A.APR.B.3 Interpret attributes of the coordinate plane to include intersecting lines. Page 28 of 57

29 Unit 4 Instructional Tasks: FIF.B.4: Interpret attributes of the coordinate plane to include intersecting lines. Include domain (x), range (y) and end behavior as key features. F.BF.B.3: Interpret attributes of the coordinate plane as functions relate to a graph. FIF.C.7 Interpret attributes of the coordinate plane as functions relate to a graph. A.APR.C.4 How to construct a proof; factoring A.SSE.B.3 Manipulate an equation (solve for a specific variable) F.IF.B.6 Use graphing calculator to find line of best fit Constructed Response Assessments A2 Root SG There are no CCSS within this unit which may be deemphasized for the purposes of the EOC if the teacher begins to fall behind in pacing. Page 29 of 57

30 Unit 5 Unit 5: Rational Equations 16 Days: November 14 - December 10, 2014 Standard Clarifications, Evidence, and Assessment 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, polynomial, rational, absolute value, exponential, and logarithmic functions. PBA - MP 1 and 5 Calculator - Item Specific PBA - Given an equation or system of equations, reason about the number or nature of the solutions. Content scope: A.REI.D.11, involving any of the functions types measured in the standards. - MP 3 Calculator - Yes PLD - The explain part of standard A.REI.D.11 is not assessed here. For example, student might be asked how many positive solutions there are to the equation e x = x + 2 or the equation e x = x + 1, explaining how they know. The student might use technology strategically to plot both sides of the equation without prompting. - in units 1, 5, and 7 Limitations - Tasks may involve any of the function types mentioned in the standard. SPI , SPI Linear is taught in unit 1, new cases include polynomial, rational, and absolute value. Page 30 of 57

31 Unit 5 A.REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. PBA - Simple rational equations are limited to those whose numerators and denominators have degree at most 2. - MP 3 and 6 Calculator- Item Specific EOY - Simple rational equations are limited to those whose numerator and denominators have a degree at most 2. - MP 3 and 6 Calculator - No PLD (PBA and EOY) - Rewrites exponential expressions to reveal quantities of interest that may be useful. SPI , SPI , SPI radical are given in radical form and not rational exponent (unit 7) - radical expressions can be up to cube root. Page 31 of 57

32 Unit 5 Additional Resources: Integration Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_AdvancedAlgebra_Unit3SE.pdf Core/Common%20Core%20Frameworks/CCGPS_Math_9-12_AdvancedAlgebra_Unit4SE.pdf _14/Math/Alg%20II%20FT%201.pdf cal_appliedsciences_math/_assoc/73fd978d6cdb4a03a3c8e5bc1717eb51/alg _II_FT_10.pdf pdf Assessments: Common Student Misconceptions Page 32 of 57

33 Unit 5 Unit Vocabulary: Know the difference between monomial and binomial so students can simplify rational expressions. - Strong vocabulary to know the different between solutions to the equation, roots, zeros, solutions to the system. - Also extraneous solutions are a new concept. Instructional Tasks: A.REI.A.2 Students want to simplify rational expressions with x in the numerator and x+2 in the denominator by marking out the x s. A.REI.A.2 students want to add or subtract without a common denominators Prerequisite Skills A.REI.11 This standard is taught in Algebra I and review in Algebra II unit 1. A.REI.A.2 This standard is new. Students need to be able to add, subtract, multiply and divide fractions. Also students need to be able to factor polynomials ( unit 3) Constructed Response Assessments If the teacher cannot fully complete this unit in the alloted days, then the areas of the Common core which may be emphasized are: Emphasize: - SPI Move flexibly between multiple representations (contextual, physical, written, verbal, iconic/pictorial, graphical, tabular, and symbolic) of non-linear and transcendental functions to solve problems, to model mathematical ideas, and to communicate solution strategies. - SPI Add, subtract, multiply, divide and simplify rational expressions including those with rational and negative exponents. - SPI Use the number system, from real to complex, to solve equations and contextual problems. - SPI Solve contextual problems using quadratic, rational, radical and exponential equations, finite geometric series or systems of equations. (do not include rational exponential till unit 7 ) Page 33 of 57

34 Unit 6 Unit 6: First Semester Review and Exam 7 Days: December 11 - December 19, 2014 Additional Resources: Assessments: Page 34 of 57

35 Unit 7 Unit 7: Rational Exponents, Exponentials, and Logarithms 18 Days: January 6 - January 30, 2014 Standard N.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. N.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Clarifications, Evidence, and Assessment PBA and EOY - MP 7 Calculator - Item Specific PLD (PBA and EOY) - Uses mathematical properties and structure of polynomial, exponential, rational, and radical expressions to create equivalent expressions that aid in solving mathematical and contextual problems with three or more steps SPI In units 3, 4, and 6 Page 35 of 57

36 Unit 7 A.SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. PBA and EOY - Tasks have a context. As described in the standard, there is an interplay between the mathematical structure of the expression and the context of the situation, such that choosing and producing an equivalent form of the expression reveals something about the situation. - MP 1, 2, 4, and 7 Calculator - Neutral PLD (PBA and EOY) - Uses mathematical properties and structure of polynomial, exponential, rational, and radical expressions to create equivalent expressions that aid in solving mathematical and contextual problems with three or more steps. Limitations - Tasks have a real-world context. As described in the standard, there is an interplay between mathematical structure of the expression and the structure of the situation such that choosing and producing and equivalent from of the expression reveals something about the situation. - Tasks are limited to exponential expressions with rational or real exponents. 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, polynomial, rational, absolute value, exponential, and logarithmic functions. PBA and EOY - MP 1 and 5 - The "explain" part of standard A.REI.D.11 is assessed on the PBA but not on the EOY Calculator - Item Specific Limitations - Tasks may involve any of the function types mentioned in the standard. SPI In Units 1&5. The only new content is exponential/logarithms Page 36 of 57

37 Unit 7 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. o PBA - Express reasoning about transformations of functions. Content scope: F.BF.B.3, which may involve polynomial, exponential, logarithmic or trigonometric functions. - Tasks also may involve even and odd functions. - MP 3 Calculator - Yes F.BF.B.3 is in units 1, 6, and 7 EOY - Experimenting with cases and illustrating an explanation are not assessed here. - Include recognizing even and odd functions from their graphs and algebraic expression for them. - MP 3, 5, 7, and 8 Calculator - Item Specific PLD (PBA and EOY) - Given a context that infers particular transformations, identifies the effects on graphs of polynomial, exponential, logarithmic, and trigonometric functions, and determines if the resulting function is even or odd. - ONLY transformations. Even/Odd is in unit 4 Page 37 of 57

38 Unit 7 F.IF.B.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. Calculator - Yes Limitations - Tasks have a real-world context. - Tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. - Compare second limitation with F.IF.C.7. - The function types listed here are the same as those listed in the Algebra II limitation for standards F.IF.B.6 and F.IF.C.9. - Identify key features of linear functions and move to more advanced functions. Include domain, range and end behavior as key features. - In units 1, 2, 3, 5, 8 - Only doing exponents and logs F.IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. EOY - MP 1, 5, and 6 - About half of tasks involve logarithmic functions, while the other half involve trigonometric functions. Calculator - Item specific Page 38 of 57