Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
Unit Overview
Common Core State Standards
Content Standards
Major Focus: Students will understand the concept of a series and will connect series to real-world
problems. Students will also understand the major elements of exponential and logarithmic functions and
will apply them to modeling situations.
Tasks:
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Create exponential equations to solve problems.
Rewrite exponential expressions with different exponential values using exponent rules.
Derive the formula for a finite geometric series.
Define arithmetic and geometric sequences both recursively and with an explicit formula.
Find the logarithmic inverse of an exponential function and vice versa.
Identify the rate of growth/decay in an exponential function.
Evaluate logarithms of base 2, 10, and e using the exponential equivalent and using technology.
Textbook Resources
Mathematics Formative Assessment System Tasks
Pearson Prentice Hall Algebra 2 copyright 2011
Sections: 9-2, 9-3, 9-4, 9-5, 7-1, 7-2, CC8, 7-3, 7-4, 7-5, 7-6
The system includes tasks or problems that teachers
can implement with their students, and rubrics that
help the teacher interpret students' responses.
Teachers using MFAS ask students to perform
mathematical tasks, explain their reasoning, and
justify their solutions. Rubrics for interpreting and
evaluating student responses are included so that
teachers can differentiate instruction based on
students' strategies instead of relying solely on correct
or incorrect answers. The objective is to understand
student thinking so that teaching can be adapted to
improve student achievement of mathematical goals
related to the standards. Like all formative
assessment, MFAS is a process rather than a test.
Research
suggests
that
well-designed
and
implemented formative assessment is an effective
strategy for enhancing student learning.
http://www.cpalms.org/resource/mfas.aspx
MUST ADD: Algebra 2 Common Core Additional Lessons:
CC-8: Transforming Exponential Functions
Available under Teacher Resources of the Algebra 2
content at www.pearsonsuccessnet.com.
This a working document that will continue to be revised and improved taking your feedback into consideration.
MAFS.912.A-CED.1.1
MAFS.912.A-REI.4.11
MAFS.912.A-SSE.2.3c
MAFS.912.A-SSE.2.4
MAFS.912.F-BF.1.2
MAFS.912.F-BF.2.5
MAFS.912.F-IF.3.7e
MAFS.912.F-IF.3.8b
MAFS.912.F-LE.1.4
MAFS.912.F-LE.2.5
Standards for
Mathematical Practice
MAFS.K12.MP.1.1
MAFS.K12.MP.2.1
MAFS.K12.MP.4.1
MAFS.K12.MP.8.1
Other Resources
Kuta Software
Purple Math
Algebra Nation
Online Graphing Calculator
National Library of Virtual Manipulatives
Geogebra
Virtual Nerd
YouTube
Khan Academy—Math
Engage NY
TI Nspired Resource Center for Educators
Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:


3.0
Students can derive a formula for an infinite geometric series and can use the formula to evaluate a problem involving an infinite series.
Students can make a connection between exponential and logarithmic models and series by inspecting the graphs of each.
The Student will:
 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and
simple rational, absolute, and exponential functions. (MAFS.912.A-CED.1.1)
 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. (MAFS.912.AREI.4.11)
 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
o Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten
as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. (MAFS.912.A-SSE.2.3c)
 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. (MAFS.912.A-SSE.2.4)
 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two
forms. (MAFS.912.F-BF.1.2)
 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
(MAFS.912.F-BF.2.5)
 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
o Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and
amplitude, and using phase shift. (MAFS.912.F-IF.3.7e)
 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
o Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such
as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. (MAFS.912.F-IF.3.8b)
 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. (MAFS.912.F-LE.1.4)
 Interpret the parameters in a linear or exponential function in terms of a context. (MAFS.912.F-LE.2.5)
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Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
2.0
The student will recognize or recall specific vocabulary, such as:
 Exponential function, base, logarithmic function, arithmetic and geometric sequence, arithmetic and geometric series, the natural logarithm
The student will perform basic processes, such as:
 Evaluate exponential and logarithmic expressions.
 Find the nth term in an arithmetic or geometric sequence.
 Identify a given function as having exponential growth or decay.
 Solve basic exponential and logarithmic equations.
1.0
With help, partial success at 2.0 content but not at score 3.0 content
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Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Functions – Building Functions
Cluster: Build a function that models a relationship between two quantities. (Major)
Standard: MAFS.912.F-BF.1.2: (Write) arithmetic and geometric sequences both recursively and with an explicit formula, (use) them to model situations, and (translate)
between the two forms.
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Students will understand that sequential phenomena occur throughout life that can be modeled using recursive and explicit formulas.
Know
Declarative knowledge: Facts, vocab., information
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Use an explicit arithmetic sequence formula.
Sequence, term, explicit formula, recursive
formula, arithmetic sequence, common
difference, arithmetic mean, geometric
sequence, common ratio, geometric mean,
finite and infinite sequence, explicit and
recursive sequence
Know that formulas are used to find the nth
term in recursive or explicit sequences
Use an explicit geometric sequence formula.
Find the next term of an arithmetic and geometric sequence, given the explicit formula
Find an explicit arithmetic sequence formula.
Find an explicit geometric sequence formula.
Find the next term of a recursively-defined sequence.
Apply the arithmetic and geometric sequence to real-life.
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Identify patterns, simplify expression, writing function rules
Learning Goals: To generate an arithmetic or geometric sequence.
Moving Beyond: In Precalculus, to fully understand binomial expansion using the nCr definition, students must have a thorough understanding of sequences.
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Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
Common Core State Standard:
Learning Progression
Score
Create a real-world problem and write the corresponding arithmetic or
geometric sequence
I can…
4.0
3.5

Sample Tasks
Create a real-world problem and write the corresponding arithmetic
or geometric sequence
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
I can…
3.0

Generate an arithmetic or geometric sequence.
**Glencoe McGraw-Hill, PreCalculus, pg. 596 #11.
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Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
2.5
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
Find the next four terms of the sequence 2, 7, 12, 17, …
I can…
2.0


Find the next term of a sequence
Use an explicit formula
Find the first four terms of the sequence given by an=2n(-1)n.
*Glencoe McGraw-Hill, PreCalculus, pg. 590 Ex. 1 a & c
1.0
I need prompting and/or support to complete 2.0 tasks.
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Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
CCSS Mathematical Content Standard(s)
Design Question 1; Element 1
CCSS Mathematical Practice(s)
Design Question 1; Element 1
Marzano’s Taxonomy
MAFS.912.F-BF.1.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to
model situations, and translate between the two forms.
MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them.
MAFS.K12.MP.2.1: Reason abstractly.
MAFS.K12.MP.4.1: Model with mathematics.
MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning.
Level 4—Knowledge Utilization: “Experimenting”
Teacher Notes
Reference: Prentice Hall Algebra 2 Honors 2010 textbook page 576 problem #53
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Pasco County Schools, 2014-2015
Algebra 2 | Unit E: Series, Exponential and Logarithmic Functions
This task would serve as a compelling why one would find a formula helpful to determine the nth term. Also the task
would be useful to introduce the standard: MACC.912.A-SSE.2.4 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.
Task
Design Question 4; Element 23
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015