Transcript Algebra II Module 3

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
A Close Look at Grade 11 Module 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Opening Exercise
I would like to assign each person in the room a unique identification
number of equal digits using only the digits 0, 1, and 2. How many
digits does the number need to be to ensure that each person has a
unique ID number?
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NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
•
•
•
•
•
Classroom teacher
Math trainer
Principal or school leader
District representative / leader
Other
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
• Experience and model the instructional approaches to
teaching the content of Grade 11 Module 3 lessons.
• Articulate how the lessons promote mastery of the focus
standards and how the module addresses the major work of
the grade.
• Make connections from the content of previous modules and
grade levels to the content of this module.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Flow of Module 3
• Topic A: Real Numbers
• Topic B: Logarithms
• Topic C: Exponential and Logarithmic Functions and Their
Graphs
• Topic D: Using Logarithms in Modeling Situations
• Topic E: Geometric Series and Finance
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Mathematical Themes of Module 3
• Look for and make use of structure (MP 7).
• Properties of exponents and logarithms are useful for rewriting
expressions, solving equations, and graphing functions.
• Exponential and logarithmic functions can be used to model a
variety of real world situations.
• An understanding of the inverse relationship between
exponential and logarithmic functions is key.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
What prior experiences have students had?
• Evaluating exponential expressions and graphing exponential
functions with a domain of the integers (Algebra I Module 3)
• Identifying arithmetic and geometric sequences and writing
recursive and explicit formulas for the sequences (Algebra I
Module 3)
• Modeling data using exponential functions (Algebra I Module 3
and 5)
• Selecting an appropriate function to model data (Algebra I and
Algebra II)
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic A: Real Numbers
• Students extend their understanding of exponential functions
to a domain of all real numbers.
• Students use their prior knowledge about families of functions
to graph exponential functions.
• The stage is set for logarithms through exploration of base 10
exponentials and a discovery of the number e.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 1: Integer Exponents
• Paper folding and the power of exponential growth:
• http://ed.ted.com/lessons/how-folding-paper-can-get-you-to-themoon
• Properties of exponents are addressed in this lesson.
• Focus on how these properties can be used to rewrite
expressions in a variety of ways rather than on the
vague notion of “simplifying.”
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 1: Integer Exponents
• Consider the expression
is more simplified?
6𝑥 −5
.
−3
𝑥
Which equivalent form
6
• 2
𝑥
•6𝑥
−2
• Why would I prefer the expression
reduced?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
2𝑥 4
4𝑥 2 3
to be
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Base 10 and Scientific Notation
• Students explore multiplying numbers by powers of 10.
• Scientific notation is reviewed and connected to the properties
of exponents covered in lesson 1.
• We are building the foundation for understanding base 10
logarithms in Topic B.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: Rational Exponents – What are
1
2
1
3
2 and 2 ?
Extend the properties of exponents to rational exponents
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 51/3 to be the cube root of 5
because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
N-RN.A.2
Rewrite expressions involving radicals and rational exponents using
the properties of exponents.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: Rational Exponents – What are
1
3
1
2
2 and 2 ?
1
2
• Can you raise 2 to an exponent of ?
• How does the graph of 𝑓 𝑥 = 2𝑥
support the idea that there is a value
1
2
for 2 ?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: Rational Exponents – What are
1
2
1
3
2 and 2 ?
1
2
• Estimate the value of 2 from the
graph.
1
2
• What does (2 )2 equal?
• What does ( 2)2 equal?
1
2
• Therefore, 2 = 2.
1
3
• Estimate the value of 2 from the
graph.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: Rational Exponents – What are
1
2
1
3
2 and 2 ?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Properties of Exponents and
Radicals
• Students continue to make connections between the
properties of exponents, rational exponents, and radicals.
• Students interchange between radicals and rational exponents,
realizing that depending on the situation one or the other may
be more useful.
• Notice the wording of the directions when working the
exercises.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: Irrational Exponents – What Are
2 2 and 2𝜋 ?
• In lesson 3, we extended the domain of exponentials by considering
rational exponents.
• In order to extend the domain to the set of all real numbers, we need to
consider irrational exponents.
• Students approximate 2 2 in a similar fashion to how students in Geometry
first defined the area of a circle and how students in Calculus approximate
a limit.
• This review of irrational numbers is a nice lead in to the number 𝑒 which is
studied in lesson 6.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: Irrational Exponents – What Are
2 2 and 2𝜋 ?
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 5: Irrational Exponents – What Are
2 2 and 2𝜋 ?
1< 2<2
1.4 < 2 < 1.5
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21 < 2
2
< 22
2< 2
2
<4
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Euler’s Number, 𝑒
•
•
•
Students discover Euler’s number, 𝑒, which is an irrational number and
therefore cannot be expressed exactly in numeric form and is denoted by
the letter 𝑒.
𝑒 ≈ 2.7182818284
Water tank demo
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic A
• The domain of an exponential function is the set of all real
numbers.
• Therefore, exponential expressions can be evaluated for any
rational or irrational value.
• It is useful to be able to manipulate expressions containing
exponents and/or radicals into forms that are convenient to
use.
• The number 𝑒 is an irrational number that is important in a
variety of mathematical applications.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic B: Logarithms
• Students develop an understanding of the relationship
between exponentials and logarithms.
• Students explore a variety of methods for solving exponential
equations.
• Ultimately, students solve equations in the form abct = d by
expressing the exponential equation as a logarithm (F-LE.A.4).
• Students discover basic properties of logarithms.
• Why do logarithms even exist?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Bacteria and Exponential Growth
• Some exponential functions can be solved by
equating exponential terms of similar bases:
•2𝑥 = 64
•2𝑥 = 26
•𝑥 = 6
• But what if the equation is 2𝑥 = 46 instead of
2𝑥 = 64?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Bacteria and Exponential Growth
• 2𝑥 = 46
•
•
•
•
25 = 32 and 26 = 64
25.5 = 45.255 and 25.6 = 48.503
25.52 = 45.887 and 25.53 = 46.206
25.523 = 45.982 and 25.524 = 46.014
• Since both 5.523 and 5.524 round to 5.52, we can say
𝑥 ≈ 5.52.
• What if we wanted the solution to be accurate to 4
decimal places?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: The “WhatPower” Function
• Work the opening exercise.
• Students are making sense of the definition of a logarithm
through a more intuitive function.
• Students continue investigating the “WhatPower” function
coming to the realization that bases of 0 and 1 do not make
sense and negative bases are only valid sometimes.
• Students also make sense of the domain of the “WhatPower”
function, realizing that only inputs greater than 0 have a real
output.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: The “WhatPower” Function
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Rapid White Board Exchange
Evaluate each logarithm
𝑙𝑜𝑔4 (16)
1
𝑙𝑜𝑔16 ( )
4
1
𝑙𝑜𝑔4 ( )
16
𝑙𝑜𝑔4 ( 4)
𝑙𝑜𝑔16 (4)
𝑙𝑜𝑔4 (412 )
© 2012 Common Core, Inc. All rights reserved. commoncore.org
1
𝑙𝑜𝑔4 ( 12 )
4
4
𝑙𝑜𝑔4 (8)
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Opening Exercise
I would like to assign each person in the room an identification number
using only the digits 0, 1, and 2. How many digits does the number
need to be to ensure that each person has a unique ID number?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 9: Logarithms – How Many Digits
Do You Need?
• Why are social security numbers 9 digits?
• How do we know that 9 digits are enough for each person in
the U.S. to have a unique number?
• Would 8 digits be enough to ensure each person had a unique
number?
• At what point would we need to increase to 10 digits?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: Building Logarithmic Tables
• Students explore base 10 logs first through a “squeeze”
process and then using the log button.
• Emphasize the meaning of the output on the calculator.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: The Most Important Property
of Logarithms
• Students build on the properties discovered in lesson 10.
• Emphasis on Math Practice 8:
• Look for and express regularity in repeated reasoning.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 12: Properties of Logarithms
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 13: Changing the Base
• Why does the calculator only have the base 10 and
base e logarithms?
• How would one evaluate a base 2 logarithm on the
𝑙𝑜𝑔2 (7)
calculator?
• Students develop and then use the change of base
formula for logarithms.
𝑙𝑜𝑔10 (7)
𝑙𝑜𝑔2 7 =
𝑙𝑜𝑔10 (2)
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 14: Solving Logarithmic Equations
• Emphasize:
• looking for and making use of structure (MP 7).
• checking for extraneous solutions.
• Share different approaches and discuss advantages
and disadvantages. (MP 3)
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 15: Why Were Logarithms
Developed?
Student Outcomes
• Students use logarithm tables to calculate products and
quotients of multi-digit numbers without technology.
• Students understand that logarithms were developed to speed
up arithmetic calculations by reducing multiplication and
division to the simpler operations of addition and subtraction.
• Students solve logarithmic equations of the form log 𝑋 =
log 𝑌 by equating 𝑋 = 𝑌.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic B
• Logarithmic functions have an important historical context.
• Logarithms help us make sense of very large and very small
numbers.
• Logarithms give us a way of solving exponential equations.
• Properties of logarithms are essential for evaluating logarithms
and solving exponential or logarithmic equations.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment
Work with a partner on this assessment
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Scoring the Assessment
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic C: Exponential and Logarithmic
Functions and Their Graphs
• Students recognize that exponential functions are defined for
all real numbers and logarithmic functions are defined for all
positive numbers.
• Students explore the idea that exponential and logarithmic
functions are inverses of each other.
• Students connect the properties of logarithms from Topic B to
transformations of the graphs of logarithmic functions.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Rational and Irrational
Numbers
• On a number line extending infinitely in both
directions, are there more:
• integers or natural numbers?
• rational numbers or integers?
• irrational numbers or rational numbers?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Rational and Irrational
Numbers
• In topic A, we expanded the domain of exponential functions
to include all real numbers.
• Since we know that we can evaluate an exponential function
for both rational and irrational numbers, it is logical that
logarithms can have an output that is either rational or
irrational.
• Most logarithms (even of rational numbers) are irrational.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Graphing the Logarithm
Function
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Graphing the Logarithm
Function
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NYS COMMON CORE MATHEMATICS CURRICULUM
𝑙𝑜𝑔𝑏 𝑥
𝑙𝑜𝑔𝑏 𝑥
𝑙𝑜𝑔1 𝑥 =
=
= −𝑙𝑜𝑔𝑏 𝑥
1
−1
𝑏
𝑙𝑜𝑔𝑏
𝑏
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 18: Graphs of Exponential
Functions and Logarithmic Functions
• Students realize that if the graph of an exponential function is
reflected across the line y = x the graph produced is a
logarithmic function.
• Students explore the relationship between ordered pairs on
the exponential curve and the logarithmic curve.
• Students use this relationship to make sense of the domain
and range of each function as well as the end behavior.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: The Inverse Relationship
Between Logarithmic and Exponential
Functions
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) =2 x3 or
f(x) = (x+1)/(x-1) for x ≠ 1.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that
the function has an inverse.
d. (+) Produce an invertible function from a non-invertible function by
restricting the domain.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: The Inverse Relationship
Between Logarithmic and Exponential
Functions
F-BF.B.5
(+) Understand the inverse relationship between exponents and logarithms
and use this relationship to solve problems involving logarithms and
exponents.
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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 20: Transformations of the Graphs
of Logarithmic and Exponential Functions
• Student Outcomes
• Students study transformations of the graphs of logarithmic functions.
• Students use the properties of logarithms and exponents to produce
equivalent forms of exponential and logarithmic expressions. In
particular, they notice that different types of transformations can produce
the same graph due to these properties.
• Revisits transformations to produce exponential and
logarithmic graphs.
• Connects transformations of graphs of functions to properties
of logarithms.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 20: Transformations of the Graphs
of Logarithmic and Exponential Functions
Lesson Summary
• GENERAL FORM OF A LOGARITHMIC FUNCTION: 𝑓 𝑥 = 𝑘 + 𝑎 𝑙𝑜𝑔𝑏 (𝑥 − ℎ) such
that 𝑎, ℎ, and 𝑘 are real numbers, 𝑏 is any positive number not equal to 1,
and 𝑥 − ℎ > 0.
• GENERAL FORM OF AN EXPONENTIAL FUNCTION: 𝑓 𝑥 = 𝑎 ∙ 𝑏 𝑥 + 𝑘 such that 𝑎
and 𝑘 are real numbers, and 𝑏 is any positive number not equal to 1.
• The properties of logarithms and exponents can be used to rewrite
expressions for functions in equivalent forms that can then be graphed by
applying transformations.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: The Graph of the Natural
Logarithm Function
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: The Graph of the Natural
Logarithm Function
•
•
The change of base formula gives us a way to write any logarithmic
function in the form 𝑓 𝑥 = 𝑘 + 𝑎𝑙𝑛 𝑥 − ℎ .
𝑔 𝑥 = 𝑙𝑜𝑔6 𝑥 − 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 22: Choosing a Model
•
•
At his point, students have modeled data using linear, quadratic,
exponential, and sinusoidal function.
How do we know which function is an appropriate model based on a
context?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic C
• A logarithmic function is the inverse of an exponential
function.
• Properties of logarithms and exponents are useful for graphing
transformations of logarithmic and exponential functions.
• Using the change of base formula, any logarithmic function can
be rewritten as a vertical scaling of the natural logarithm
function (or any other base logarithm).
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic D: Using Logarithms in Modeling
Situations
• Students create models for contexts that involve exponential
growth and decay.
• Problems similar to those studied in Algebra I Module 3 are reexamined now that students have the logarithm as a tool for
solving the exponential equations that arise.
• Students connect geometric sequences to exponential growth
and decay which serves as a lead-in to the study of geometric
series and finance in Topic E.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 23: Bean Counting
• Read the student outcomes and then look at the exit ticket.
Identify the key takeaways of the lesson.
• Read the lesson notes and opening.
• Work through Mathematical Modeling Exercise 1 and then
read the discussion that follows.
• Work through Mathematical Modeling Exercise 2 and then
read the discussion that follows.
• Read through the closing and scan the problem set.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 24: Solving Exponential Equations
Opening Exercise
In Lesson 7, we modeled a population of bacteria that doubled
every day by the function 𝑃 𝑡 = 2𝑡 , where 𝑡 was the time in
days. We wanted to know the value of 𝑡 when there were 10
bacteria. Since we did not know about logarithms at the time,
we approximated the value of 𝑡 numerically, and we found
that 𝑃 𝑡 = 10 at approximately 𝑡 ≈ 3.32 days.
Use your knowledge of logarithms to find an exact value for 𝑡
when 𝑃 𝑡 = 10, and then use your calculator to approximate
that value to 4 decimal places.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 25: Geometric Sequences and
Exponential Growth and Decay
• Algebra I Module 3 Lesson 2:
• For the sequence below, an explicit formula is given. Write the
first five terms of each sequence. Then, write a recursive
formula for the sequence.
• 𝑎𝑛 =
1 𝑛−1
for
2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝑛≥1
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 25: Geometric Sequences and
Exponential Growth and Decay
• Algebra I Module 3 Lesson 3:
ARITHMETIC SEQUENCE - described as follows: A sequence is called
arithmetic if there is a real number 𝑑 such that each term in the
sequence is the sum of the previous term and 𝑑.
GEOMETRIC SEQUENCE - described as follows: A sequence is called
geometric if there is a real number 𝑟 such that each term in the
sequence is a product of the previous term and 𝑟.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 25: Geometric Sequences and
Exponential Growth and Decay
• Algebra I Module 3 Lesson 6:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 25: Geometric Sequences and
Exponential Growth and Decay
• Algebra I Module 3 Lesson 14:
A California Population Projection Engineer in 1920 was tasked with finding
a model that predicts the state’s population growth. He modeled the
population growth as a function of time, 𝑡 years since 1900. Census data
shows that the population in 1900, in thousands, was 1,490. In 1920, the
population of the state of California was 3,554 thousand. He decided to
explore both a linear and an exponential model.
(a) Use the data provided to determine the equation of the linear function that
models the population growth from 1900–1920.
(b) Use the data provided and your calculator to determine the equation of
the exponential function that models the population growth.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 26: Percent of Change
• Students develop and use the following formulas:
•𝐹 𝑡 =𝑃 1+𝑟 𝑡
•𝐹 𝑡 =𝑃 1+
𝑟 𝑛𝑡
𝑛
• 𝐹 𝑡 = 𝑃𝑒 𝑟𝑡
Where 𝐹 is the future value (or ending amount)
𝑃 is the principal value (or initial amount)
r is the unit rate
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 27: Modeling with Exponential
Functions
• Students revisit a problem they explored in Algebra I Module 3
Lesson 6 – creating a function to model the U.S. population.
• Students create a model using a variety of techniques.
• Logarithms are used to solve exponential equations where in
Algebra I we estimated either graphically or numerically to
solve.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 28: Newton’s Law of Cooling,
Revisited
• Algebra I Module 3 Lesson 23, we explored Newton’s Law of
Cooling with a focus on transformations of the exponential
function 𝑓 𝑥 = 𝑎 ∙ 𝑏 𝑥 .
• We used an approximate value, 2.718, as the base of the
exponential.
𝑇 𝑡 = 𝑇𝑎 + 𝑇0 – 𝑇𝑎 ∙ 2.718−𝑘𝑡
𝑓 𝑡 = 2.718𝑡
• Students explored the effect of each parameter in Newton’s
Law of Cooling by using a demonstration on Wolfram Alpha.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 28: Newton’s Law of Cooling,
Revisited
𝑇 𝑡 = 𝑇𝑎 + 𝑇0 – 𝑇𝑎 ∙ 𝑒 −𝑘𝑡
𝑇(𝑡) is the temperature of the object after a time of t hours has
elapsed,
𝑇𝑎 is the ambient temperature (the temperature of the
surroundings), assumed to be constant, not impacted by the
cooling process,
𝑇0 is the initial temperature of the object, and
𝑘 is the decay constant
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic D
• When modeling with exponential functions, logarithms are
useful for solving the exponential equations that arise.
• The number e appears in many applications of exponential
functions.
• There is a powerful connection between geometric sequences,
exponential functions, and logarithms.
• We will continue to explore applications of exponentials and
logarithms in Topic E.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic E: Geometric Series and Finance
• Students learn about series and use sigma notation for the first
time.
• Students apply their knowledge of geometric sequences to
various financial contexts.
• The emphasis in this topic is on MP4 – Modeling with
mathematics.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 29: The Mathematics Behind a
Structured Savings Plan
• SERIES: Let 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 , … be a sequence of numbers. A sum of the form
𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
for some positive integer 𝑛 is called a series (or finite series) and is denoted 𝑆𝑛 .
The 𝑎𝑖 ’s are called the terms of the series. The number 𝑆𝑛 that the series adds
to is called the sum of the series.
• GEOMETRIC SERIES: A geometric series is a series whose terms form a geometric
sequence.
Since a geometric sequence is of the form 𝑎, 𝑎𝑟, 𝑎𝑟 2 , 𝑎𝑟 3 , …, the general form
of a finite geometric sequence is of the form
𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + ⋯ + 𝑎𝑟 𝑛−1 .
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 29: The Mathematics Behind a
Structured Savings Plan
• SUM OF A FINITE GEOMETRIC SERIES: The sum 𝑺𝒏 of the first 𝒏 terms of the
geometric series 𝑺𝒏 = 𝒂 + 𝒂𝒓 + ⋯ + 𝒂𝒓𝒏−𝟏 (when 𝒓 ≠ 𝟏) is given by
𝟏 − 𝒓𝒏
𝑺𝒏 = 𝒂
.
𝟏−𝒓
• The sum of a finite geometric series can be written in summation
notation as
𝒏−𝟏
𝒂𝒓𝒌
𝒌=𝟎
𝟏 − 𝒓𝒏
= 𝒂⋅
.
𝟏−𝒓
• The generic formula for calculating the future value of an annuity 𝑨𝒇 in
terms of the recurring payment 𝑹, interest rate 𝒊, and number of periods
𝒏 is given by
𝟏+𝒊 𝒏−𝟏
𝑨𝒇 = 𝑹 ⋅
.
𝒊
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 29: The Mathematics Behind a
Structured Savings Plan
A $𝟏𝟎𝟎 deposit is made at the end of every month for 𝟏𝟐 months in an
account that earns interest at an annual interest rate of 𝟑% compounded
monthly. How much will be in the account immediately after the last
payment?
100 + 100 1.025 +
100 1.025 2 + ⋯ + 100 1.025
© 2012 Common Core, Inc. All rights reserved. commoncore.org
11
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 30: Buying a Car
• Students derive a formula for the monthly payment on a car
loan using the formula for the sum of a finite geometric series.
• Students complete an open-ended modeling task where they
research the price of a potential car purchase and from there
calculate their loan.
• Students realize there is more to consider when purchasing a
car than just the selling price.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 31: Credit Cards
You have charged $𝟏, 𝟓𝟎𝟎 for the down
payment on your car to a credit card that
charges 𝟏𝟗. 𝟗𝟗% annual interest, and you
plan to pay a fixed amount toward this debt
each month until it is paid off. We will
denote the balance owed after the 𝒏th
payment has been made as 𝒃𝒏 .
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 32: Buying a House
Can you buy the house you have chosen on the salary of the career you have
chosen? You need to adhere to the following constraints:
• Mortgages are loans that are usually offered with 𝟑𝟎-, 𝟐𝟎-, or 𝟏𝟓-year
repayment options. You will start with a 𝟑𝟎-year mortgage.
• The annual interest rate for your mortgage will be 𝟓%.
• Your payment includes the payment of the loan for the house and
payments into an account called an escrow account, which is used to pay
for taxes and insurance on your home. We will approximate the annual
payment to escrow as 𝟏. 𝟐% of the home’s selling price.
• The bank will only approve a mortgage if the total monthly payment for
your house, including the payment to the escrow account, does not
exceed 𝟑𝟎% of your monthly salary.
• You have saved up enough money to put a 𝟏𝟎% down payment on this
house.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 33: The Million Dollar Problem
• How can I accumulate $1,000,000 in assets?
• Students apply knowledge from the previous lessons to
explore this question.
• Students consider savings, property which generally
appreciates in value, and vehicles which depreciate in value.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – Topic E
• Geometric sequences and series are prevalent in finance
calculations.
• Good modeling problems are often open-ended and
sometimes messy!
• Technology is useful for exploring a problem graphically or
numerically.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Key Points – Module 3 Lessons
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
•
•
•
•
•
•
•
•
Overview of Module 3
Topic A
Topic B
Mid-Module Assessment
Topic C
Topic D
Topic E
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment
Work with a partner on this assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Scoring the Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Points – End-of-Module Assessment
• End of Module assessment are designed to assess all standards
of the module (at least at the cluster level) with an emphasis
on assessing thoroughly those presented in the second half of
the module.
• Recall, as much as possible, assessment items are designed to
asses the standards while emulating PARCC Type 2 and Type 3
tasks.
• Recall, rubrics are designed to inform each district / school /
teacher as they make decisions about the use of assessments
in the assignment of grades.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Biggest Takeaway
What are your biggest takeaways from the study of
Module 3?
How can you support successful implementation of
these materials at your schools given your role as a
teacher, trainer, school or district leader,
administrator or other representative?
© 2012 Common Core, Inc. All rights reserved. commoncore.org