Algebra II Pacing Guides, Quarter 3 SY 2017-18

Updated 12/20/17

Q1

SY 17-18

Q2

SY 17-18

Q3

SY 17-18

Q4

SY 17-18

Overview

Module 3 Exponential and Logarithmic Functions

Topic A: Real Numbers

Students prepare to generalize what they know about various function families by examining the behavior of exponential functions. In Topic B, students extend their work with exponential functions to include solving exponential equations numerically and developing an understanding of the relationship between logarithms and exponentials.

Mid-Module Assessment Task After Topic B

Topic C: Exponential and Logarithmic Functions and their Graphs

Topic C reintroduces exponential functions, introduces logarithmic functions, explains their inverse relationship, and explores the features of their graphs and how they can be used to model data.

Topic D: Using Logarithms in Modeling Situations

Topic D opens with a simulation and modeling activity where students start with one bean, roll it out of a cup onto the table, and add more beans each time the marked side is up. The lesson unfolds by having students discover an exponential relationship by examining patterns when the data is presented numerically and graphically. Topic E is a culminating series of lessons driven by MP.4, Modeling with Mathematics. Students apply what they have learned about mathematical models and exponential growth to financial literacy, while developing and practicing the formula for the sum of a finite geometric series.

End-of-Module Assessment Task After Topic E

  • Week 1: February 5

    Module

    3

    Topic

    B

    Lesson

    11

    12

    13

    Standards Addressed in Week's Lessons

    N-Q.A.2

    A-CED.A.1

    F-BF.A.1a

    F-LE.A.4

    Notes:

    Lesson 11: The Most Important Property of Logarithms (P)

    In Lessons 10 and 11, students discover the logarithmic properties by completing carefully structured logarithmic tables and answering sets of directed questions. Throughout these two lessons, students look for structure in the table and use that structure to extract logarithmic properties (MP.7).

     

    Lesson 12: Properties of Logarithms (P)

    Lesson 12 continues the consideration of properties of the logarithm function, while remaining focused solely on base-10 logarithms. Its centerpiece is the demonstration of basic properties of logarithms such as the power, product, and quotient  properties, which allows students to practice MP.3 and A-SSE.A.2, providing justification in terms of the definition of logarithm and the properties already developed. In this lesson, students begin to learn how to solve exponential equations, beginning with base-10 exponential equations that can be solved by taking the common logarithm of both sides of the equation.

     

    Lesson 13: Changing the Base (P)

    Lesson 13 again focuses on the structure of expressions (A-SSE.A.2), as students change logarithms from one base to another.

  • Week 2: February 12

    Module

    3

    Topic

    B

    Lesson

    14

    15

    Assessment

    Mid-Module Assessment (Topics A - B)

    Standards Addressed in Week's Lessons

    N-Q.A.2

    A-CED.A.1

    F-BF.A.1a

    F-LE.A.4

    Standards Addressed in Assessment

    N-RN.A.1

    N-RN.A.2

    N-Q.A.2,

    A.CED.A.1

    F-IF.B.6,

    F-BF.A.1a

    F-LE.A.4

    Notes:

    Lesson 14: Solving Logarithmic Equations (P)

    In this lesson, students apply the definition of the logarithm to rewrite logarithmic equations in exponential form, so the equations must first be rewritten in the form log๐‘(๐‘‹) = ๐‘, for an algebraic expression ๐‘‹ and some constant c.

     

    Lesson 15: Why Were Logarithms Developed? (P)

    Students learn a bit of the history of how and why logarithms first appeared. The materials for this lesson contain a base-10 logarithm table. Although modern technology has made logarithm tables functionally obsolete, there is still value in understanding the historical development of logarithms.

  • Week 3: February 19

    Standards Addressed in Week's Lessons

    F-IF.B.4

    F-IF.B.5

    F-IF.C.7e

    F-BF.A.1a

    Module

    3

    Topic

    C

    Lesson

    16

    17

    18

    F-BF.B.3

    F-BF.B.4a

    F-LE.A.2

    F-LE.A.4

    Notes:

    Lesson 16: Rational and Irrational Numbers (S)

    Lesson 16 ties back to work in Topic A by helping students to further extend their understanding of the properties of real numbers, both rational and irrational (N-RN.B.3). This Algebra I standard is revisited in Algebra II so that students know and understand that the exponential functions are defined for all real numbers, and, thus, the graphs of the exponential functions can be represented by a smooth curve.

     

    Lesson 17: Graphing the Logarithm Function (P) and Lesson 18: Graphs of Exponential Functions and Logarithmic Functions (P)

    Lessons 17 and 18 introduce the graphs of logarithmic functions and exponential functions. Students compare the properties of graphs of logarithm functions for different bases and identify common features, which align with standards F-IF.B.4, F-IF.B.5, and F-IF.C.7.

  • Week 4: February 26

    Standards Addressed in Week's Lessons

    F-IF.B.4

    F-IF.B.5

    F-IF.C.7e

    F-BF.A.1a

    Module

    3

    Topic

    C

    Lesson

    19

    20

    21

    F-BF.B.3

    F-BF.B.4a

    F-LE.A.2

    F-LE.A.4

    Notes:

    Lesson 19: The Inverse Relationship between Logarithmic and Exponential Functions (P)

    Lesson 19 addresses standards F-BF.B.4a and F-LE.A.4 while continuing the ideas introduced graphically in esson 18 to help students make the connection that the logarithmic function base ๐‘ and the exponential function base ๐‘ are inverses of each other.

     

    Lesson 20: Transformations of the Graphs of Logarithmic and Exponential Functions (E) and Lesson 21: The Graph of the Natural Logarithm Function (E)

    The relationship between graphs of these functions, the process of sketching a graph by transforming a parent function, and the properties associated with these functions are linked in Lessons 20 and 21, showcasing standards F-IF.C.7e and F-BF.B.3.

     

    Lesson 21: The Graph of the Natural Logarithm Function (E)

    Lesson 21 revisits the natural logarithm function, and students see how the change of base property of logarithms implies that we can write a logarithm function of any base ๐‘ as a vertical scaling of the natural logarithm function (or any other base logarithm function we choose).

  • Week 5: March 5

    Module

    3

    Topic

    C

    D

    Lesson

    22

    23

    24

    25

    Standards Addressed in Week's Lessons

    A-SSE.B.3c

    A-CED.A.1

    A-REI.D.11

    F-IF.B.3

    F-IF.B.6

    F-IF.C.8b

    F-IF.C.9

    F-BF.A.1

    F-BF.A.2

    F-BF.B.4a

    F-LE.A.4

    F-LE.B.5

    Notes:

    Lesson 22: Choosing a Model (P)

    In Lesson 22, students must synthesize knowledge across both Algebra I and Algebra II to decide whether a linear, quadratic, sinusoidal, or exponential function will best model a real-world scenario by analyzing the way in which we expect the quantity in question to change.

     

    Lesson 23: Bean Counting (M)

    This topic opens with a simulation and modeling activity where students start with one bean, roll it out of a cup onto the table, and add more beans each time the marked side is up. The lesson unfolds by having students discover an exponential relationship by examining patterns when the data is presented numerically and graphically.

     

    Lesson 24: Solving Exponential Equations (P)

    Lesson 24 shows students how to use logarithms to solve these types of equations analytically and makes the connections between numeric, graphical, and analytical approaches explicit, invoking the related standards F-LE.A.4, F-BF.B.4a, and A-REI.D.11.

     

    Lesson 25: Geometric Sequences and Exponential Growth and Decay (P)

    A general growth/decay rate formula is presented to students to help construct models from data and descriptions of situations. Students must use properties of exponents to rewrite exponential expressions in order to interpret the properties of the function (F-IF.C.8b).

     

  • Week 6: March 12

    Module

    3

    Topic

    D

    Lesson

    26

    27

    28

    Standards Addressed in Week's Lessons

    A-SSE.B.3c

    A-CED.A.1

    A-REI.D.11

    F-IF.B.3

    F-IF.B.6

    F-IF.C.8b

    F-IF.C.9

    F-BF.A.1

    F-BF.A.2

    F-BF.B.4a

    F-LE.A.4

    F-LE.B.5

    Notes:

    Lesson 26: Percent Rate of Change (P) and Lesson 27: Modeling with Exponential Functions (M)

    In Lessons 26 and 27, a general growth/decay rate formula is presented to students to help construct models from data and descriptions of situations. Students must use properties of exponents to rewrite exponential expressions in order to interpret the properties of the function (F-IF.C.8b).

     

    Lesson 28: Newtonโ€™s Law of Cooling, Revisited (M)

    Lesson 28 closes this topic and addresses F-BF.A.1b by revisiting Newtonโ€™s law of cooling, a formula that involves the sum of an exponential function and a constant function. Students first learned about this formula in Algebra I, but now that they are armed with logarithms and have more experience understanding how transformations affect the graph of a function, they can find the precise value of the decay constant using logarithms and, thus, can solve problems related to this formula more precisely and with greater depth of understanding.

  • Week 7: March 19

    Module

    3

    Topic

    E

    Lesson

    29

    30

    31

    Standards Addressed in Week's Lessons

    A-SSE.B.4

    F-IF.C.7e

    F-IF.C.8b

    F-IF.C.9

    F-BF.A.1b

    F-BF.A.2

    F-LE.B.5

    Notes:

    Lesson 29: The Mathematics Behind a Structured Savings Plan (M)

    Lesson 29 develops the future value formula for a structured savings plan and, in the process, develops the formula for the sum of a finite

    geometric series (A-SSE.B.4).

     

    Lesson 30: Buying a Car (M)

    Lesson 30 introduces loans through the context of purchasing a car. To develop the formula for the present value of an annuity, students combine two formulas for the future value of the annuity (F-BF.A.1b) and apply the sum of a finite geometric series formula.

     

    Lesson 31: Credit Cards (M)

    Lesson 31 addresses the issue of revolving credit such as credit cards, for which the borrower can choose how much of the debt to pay each cycle. Students again sum a geometric series to develop a formula for this scenario, and it turns out to be equivalent to the formula used for car loans.

  • Week 8: April 2

    Module

    3

    Topic

    E

    Lesson

    32

    33

    Assessment

    End of Module Assessment

    Standards Addressed in Week's Lessons

    A-SSE.B.4

    F-IF.C.7e

    F-IF.C.8b

    F-IF.C.9

    F-BF.A.1b

    F-BF.A.2

    F-LE.B.5

    Standards Addressed in Assessment

    A-SSE.B.3c

    A-SSE.B.4

    A-CED.A.1

    A-REI.D.11

    F-IF.A.3

    F-IF.B.4

    F-IF.B.5

    F-IF.B.6

    F-IF.C.7e

    F-IF.C.8b

    F-IF.C.9

    F-BF-A.1a

    F-BF.A.1b

    F-BF.A.2,

    F-BF.B.3,

    F-BF.B.4a

    F-LE.A.2,

    F-LE.A.4

    F-LE.B.5

    Notes:

    Lesson 32: Buying a House (M)

    Lesson 32 may be extended to an open-ended project in which students research buying a home and justify its affordability.

     

    Lesson 33: The Million Dollar Problem (M)

    Lesson 33, the final lesson of the module, is primarily a summative lesson in which students formulate a plan to have $1,000,000 in assets within a fixed time frame,using the formulas developed in the prior lessons in the topic. Students graph the present value function and compare that with an amortization table, in accordance with F-IF.C.9.

  • Week 9: April 9

    Module

    3

    Topic

    E

    Assessment

    End of Module Assessment

    Standards Addressed in Assessment

    A-SSE.B.3c

    A-SSE.B.4

    A-CED.A.1

    A-REI.D.11

    F-IF.A.3

    F-IF.B.4

    F-IF.B.5

    F-IF.B.6

    F-IF.C.7e

    F-IF.C.8b

    F-IF.C.9

    F-BF-A.1a

    F-BF.A.1b

    F-BF.A.2,

    F-BF.B.3,

    F-BF.B.4a

    F-LE.A.2,

    F-LE.A.4

    F-LE.B.5