Modeling Exponential and Logarithmic Functions

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Modeling Exponential and Logarithmic Functions

Mathematics – Algebra 2 This unit builds on students' understanding of exponential functions and inverses to develop logarithmic functions. Students investigate the characteristics of these functions and solve equations that arise from situations that can be modeled by these functions. Students see interdisciplinary connections to science (pH), social studies (population growth) and finance (stock market scenario). The curriculum embedded performance assessment (CEPA) makes a real-world connection modeling the spread of a flu virus in a school district and the parameters that influence if and when the schools should close. These Model Curriculum Units are designed to exemplify the expectations outlined in the MA Curriculum Frameworks for English Language Arts/Literacy and Mathematics incorporating the Common Core State Standards, as well as all other MA Curriculum Frameworks. These units include lesson plans, Curriculum Embedded Performance Assessments and resources. In using these units, it is important to consider the variability of learners in your class and make adaptations as necessary.

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Table of Contents Stage 1 Desired Results .................................................................................................................................................................................................................... 3

Stage 2 - Evidence ............................................................................................................................................................................................................................ 4

Stage 3 – Learning Plan .................................................................................................................................................................................................................... 5

Lesson 1: Graphing Inverse Exponential Functions ......................................................................................................................................................................... 6

Lesson 2: Defining the Logarithmic Function ................................................................................................................................................................................ 24

Lesson 3: Logarithmic Functions ................................................................................................................................................................................................... 45

Lesson 4: Real World Applications ................................................................................................................................................................................................ 54

Lesson 5: Euler’s number and the natural logarithm ..................................................................................................................................................................... 79

Curriculum Embedded Performance Assessment (CEPA) .............................................................................................................................................................. 94



Stage 1 Desired Results ESTABLISHED GOALS G Analyze functions using different representations (F-IF).

F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, (and trigonometric functions, showing period, midline, and amplitude )

Construct and compare linear, quadratic, and exponential models and solve problems (F-LE). F-LE.4 For exponential models, express as a logarithm the solution to dabct = where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Standards for Mathematical Practice

SMP 3. Construct viable arguments and critique the reasoning of others

SMP 4. Model with mathematics

SMP 7. Look for and make use of structure

Transfer Students will be able to independently use their learning to: T

1. express appropriate mathematical reasoning by constructing viable arguments, critiquing the reasoning of others, and attending to precision when making mathematical statements.

2. apply mathematical knowledge to analyze and model mathematical relationships in the context of a situation in order to make decisions, draw conclusions, and solve problems.

Meaning UNDERSTANDINGS U Students will understand that… U1. real-world phenomenon are often modeled by exponential functions. U2. exponential and logarithmic functions are inverse functions. U3. logarithms can be used to solve exponential equations. U4. mathematical modeling enables us to observe trends and make predictions.

ESSENTIAL QUESTIONS Q Q1. When, why and how would you use the inverse of an exponential function? Q2. What kinds of real-world situations are modeled by exponential functions?

Acquisition Students will know… K K1. the definition of logarithm is bcca a

b =↔= log

K2. the relationship between logarithmic and exponential functions functions K3. the parameters necessary to graph a function including intercepts, end

Students will be skilled at… S S1. graphing exponential functions S2. graphing logarithmic functions S3. using the definition of logarithm to solve equations S4. rewriting exponential function as equivalent logarithmic equation S5. using technology to evaluate logarithms when appropriate



behavior, domain, range, rate of change . K4. the key features of exponential and logarithmic functions such as end behavior. K5. the methods for solving exponential and logarithmic equations. K6. the patterns underlying exponential functions (e.g., multipliers seen in table). K7. a logarithm is a tool to solve exponential equations. K8. academic vocabulary: function, function notation, inverse, logarithm, exponential growth, exponential decay, intercepts, end behavior, natural logarithm, growth rate, decay rate.

S6. identifying key features of exponential and logarithmic functions including intercepts, end behavior, domain, range, rate of change. S7. comparing properties of different exponential and logarithmic functions. S8. determining efficient means to solve exponential problems. S9. recognizing a function’s inverse graphically. S10. analyzing the appropriateness of fit of a graph and adjusting parameters, if necessary to make improvements. S11. translating between different representations of exponential and logarithmic functions. S12. analyzing, justifying, and communicating results.

Stage 2 - Evidence Evaluative Criteria Assessment Evidence CURRICULUM EMBEDED PERFOMANCE ASSESSMENT (PERFORMANCE TASKS)

Superintendent Dilemma: There is a flu epidemic; will we have to close the schools? PT

OTHER EVIDENCE: OE Lesson 1: Teacher check-in during poster graphing exercise. Exit Ticket. Lesson 2: Teacher observations during the discovery work and class consensus (definition of logarithmic function). Lesson summarizer. Lesson 3: Collected fluency practice (option). Numbered heads together. Quiz (formative/summative option). Lesson 4: Exit tickets for each application (Chemistry and Stock Market). Mid-unit assessment.



Lesson 5: Exit ticket. Fluency Building problem set.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

Lesson 1: Graphing Exponential Functions and their Inverse Students explore graphing the inverse of exponential functions. Lesson 2: Defining the Logarithmic Function Students discover the logarithm function as the inverse of an exponential function graphically and algebraically. Students will be introduced to the definition of a log function and use it to rewrite exponential equations in log form and log equations in exponential form. They will begin the practice of solving logarithmic expressions and equations. Lesson 3: Using the Logarithmic Function Students will continue using the definition of the logarithm to develop fluency with rewriting exponential equations in log form and log equations in exponential form and with function notation.

Lesson 4: Real World Applications Students solve real world applications of exponential equations with an unknown exponent using the definition of the logarithm. Students will be introduced to the notation xlog is equivalent to x10log . They will recognize when and how to perform symbolic manipulations to prepare for the use of

the log definition to solve an exponential equation in the form of da ct =⋅10 for t. Lesson 5: Euler’s number e and the natural logarithm ln Students investigate where the irrational number e is found using a population growth real world scenario, define exe lnlog = and solve form equations

in the daect = for t. CEPA



Unit: Modeling Exponential & Logarithmic Functions Content Area/Course: Algebra 2 Lesson 1: Graphing Inverse Exponential Functions Brief Overview of Lesson: Students will explore graphing the inverse of exponential functions. At the conclusion of this lesson, they will be able to graph the inverse of an exponential function. Additionally they will recognize, identify, and determine key features and parameters of the inverse exponential function’s graph including intercepts, end behavior, domain, range and rate of change. As you plan, consider the variability of learners in your class and make adaptations as necessary. Prior Knowledge Required: Students need to know basic properties of exponents, be able to recognize function notation, and the concept of an algebraic inverse (F-BF.4a). In the prior year, students developed a strong understanding of rates of change of linear, quadratic, and exponential functions and compared the rates of change of these three functions. Estimated Time (minutes): 60 minutes Resources for Lesson:

• Graphs (on the board, flip chart or index cards) for the lesson launch.

• Templates (paper, electronic, poster) for graphing xbxf =)( and examining key features and parameters of the graph(s). • Independent practice • Document camera for presenting student work

Time (minutes): 60 minutes (1 class) By the end of this lesson students will be able to: 1. Graph the inverse of an exponential function. 2. Recognize, identify, and determine key features and parameters for exponential function graphs including intercepts, end behavior and domain,

range and rate of change. Essential Question(s) addressed in this lesson: When, why and how would you use the inverse of an exponential function?



Standard(s)/Unit Goal(s) to be addressed in this lesson (type each standard/goal exactly as written in the framework): F-IF 7 e: Graph exponential and logarithmic functions, showing intercepts and end behavior, (and trigonometric functions, showing period, midline, and amplitude) SMP 3. Construct viable arguments and critique the reasoning of others

Instructional Tips/Strategies Engage student curiosity and introduce the concept of population growth visually with a piece of fruit in an enclosed jar (an alternative could be an on-line video clip showing fruits flies). The “Walk this Way” warm-up reinforces key vocabulary for the entire unit in an accessible kinesthetic manner and allows the teacher to formatively assess prior knowledge. Encourage academic vocabulary such as “increasing at an increasing rate…” An activity on graphing stories http://graphingstories.com/ could be done in the days leading up to the unit. The exponential graph and inverse posters should be done separately since we remove the exponential graph in lesson 3. These will be used in subsequent lessons so the teacher may want to encourage students to do a rough draft of their work before creating the poster. Some students may need support in evaluating zero and negative exponents. Some students may need support in graphing fractional outputs. Students should be able to graph the exponential function without further assistance. However, for the example, 3x students may need hints about excluding values that go off the graph, i.e. (3,27). Next, the students will graph the inverse of the exponential function. The teacher should prompt the class to recall how they algebraically found the inverse of a function (switch the x and the y variables in the equation and solve for y). The teacher should now ask the class what procedure should be followed to graph the inverse of a function (switch the x and y values of the ordered pairs and graph the new ordered pairs). NOTE: In lesson 1, the inverse of the exponential function is intentionally not yet defined as a logarithmic function.

In this Unit, the )(1 xf − notation is not required prior knowledge and is not in the content of instruction. The inclusion of inverse function notation is at the discretion of the teacher based on student abilities.

Anticipated Student Preconceptions/Misconceptions Students may still not have a clear understanding of domain and range and the graph continuing beyond the “page”. To help students grasp the concept of infinity, use phrases such as “as the value of the input (x) increases, what happens to the value of the output (y)”? And vice versa, “as the value of the input (x) decreases, what happens to the value of the output (y)”?


http://graphingstories.com/


Lesson Sequence Lesson Opening: In this lesson we will be graphing exponential functions and their inverses. Do you remember the relationship that a function has with its inverse? Can you think of an example of a function and its inverse? As you work today, keep the answers to these questions in mind. We’ll use these ideas today and throughout the rest of the unit. Warm up: “Walk this Way”- students discuss with the support of a word bank the rate of change of a function and act it out changing their walking speed to match the rate of change of the function. The activity includes a group or class vocabulary review domain, range, etc. The summarizing graph intentionally has no x intercept foreshadowing the asymptotic behavior of exponential and logarithmic functions.

Present the biologist fruit fly scenario and encourage student curiosity foreshadowing the need for new knowledge and skills to determine when there will be 180 flies. Solicit ideas from the class. It is optional to record (pictures) of the fruit flies in the jar during the course of the unit as a model of population (exponential) growth.

Students accurately graph xxf 2)( = on posters in groups and carousel to compare and self-correct graphs and tables. (SMP3) Once every group has a correct table and graph this cycle is repeated to generate posters for the inverse function. Some groups may need teacher support when graphing the inverse. This should be in the form of probing questions such as, “How are the coordinates of a function and its inverse related?” The third (final) carousel is for groups to reflect and compare the characteristics of both graphs using a tabular graphic organizer.

Students summarize and formalize the lesson content independently creating and recording the characteristics for graphs of xxf 3)( = and its inverse. Teacher will choose one pair of students to put graphs on board or poster paper. Another pair of students will put the answers to the “characteristics” on board or another poster paper. The rest of the class will use these to check their answers.

Formative assessment: Teacher check-in during poster graphing exercise. Students produce accurate graphs for xxf 3)( = and its inverse and correctly identify the characteristics. Lesson Closing: Today we created graphs of exponential functions and their inverses. We’ll use today’s posters in subsequent lessons to define a new function that is the inverse of the exponential. This will help us solve the fruit fly problem.



Written Exit Ticket Done In Pairs – Using today’s vocabulary, write a description of the graph of the inverse of an exponential function and include rate of change. (Teacher can choose class discussion, pair presentations or collect responses.)

Homework: Students should graph the following functions and their inverses: xy 4= , xy 5= .



Lesson 1 Launch – “Walk this Way” (single page handout or on index cards or posted around the room).

Directions: In groups of 3 or 4.

Each member of the group chooses one graph and walks across the room changing his/her speed to model the rate of change illustrated by his/her graph. Using the word bank below, describe the graph of the function and its rate of change. Record your thinking.

INCREASING DECREASING FUNCTION RATE OF CHANGE

INCREASING RATE DECREASING RATE

___________________________ __________________________

__________________________ ___________________________



Discuss with your group and write down the characteristics of the following graph.

Domain

Range

x-intercept

y-intercept

End behavior



Lesson 1 Launch – “Walk this Way” Answer Key

Graph starts at (-3, 1), increases at a decreasing rate, y-intercept (0,6)

Graph starts at (-3,9), decreases at a decreasing rate, y-intercept (0,4)

Continuous graph, increasing at an increasing rate, y-intercept (0,3)

Continuous graph, decreasing at an increasing rate, y-intercept (0,5), x-intercept (2.5,0)



Discuss with your group and write down the characteristics of the following graph.

It’s important to note that all of the following characteristics cannot be fully determined as it’s unclear what’s happening to the graph outside of the window shown and there is no scale on either the horizontal or vertical axis. Please make sure that students understand why this is the case.

Domain: all real numbers greater than -2

Range: all positive real numbers

x-intercept: none shown

y-intercept: approximately (0,3)

End behavior: as x approaches -2, y approaches ∞+ ; as x approaches ∞+ , y approaches 0



Graphing the inverse of the exponential function

Setting the stage: Biologists use fruit flies to study genetics because fruit fly populations grow very quickly. As a result, scientists can follow genetic traits through many generations in a short period of time. If a geneticist starts with 15 fruit flies and the fruit fly population doubles every 10 days, how long will it take for the population to reach 180?

Building a model: Working in groups of 3 or 4: No Calculators; Accuracy is important

On your poster paper graph f(x) = 2x using a scale of 1. You may want to do a draft graph below. When your poster is done check your work with at least one other group. Modify your graph if needed.

x f(x) = 2x

-4

-3

-2

-1

0

1

2

3

4



On your poster paper graph the inverse of xxf 2)( = using a scale of 1.

You may use the table and graph below for a draft.



Graphing the inverse of the exponential function: ANSWER KEY

x f(x) = 2x

-4 1

16

-3 18

-2 14

-1 12

0 1

1 2

2 4

3 8

4 16



On your poster paper graph the inverse of xxf 2)( = using a scale of 1.

𝑥 𝑓−1(𝑥)

116

-4

18

-3

14

-2

12

-1

1 0

2 1

4 2

8 3

16 4



Studying and comparing the characteristics of exponential graphs and their inverses With a partner discuss the characteristics of each graph (on the posters) and record your observations below.

xxf 2)( =

The inverse of xxf 2)( =

Domain: Domain:

Range: Range:

x-intercept: x-intercept:

y-intercept: y-intercept:

Rate of change: Rate of change:

End behavior:

As x approaches ∞+ , y_________

As x approaches ∞− , y________

End behavior:


As x approaches 0, y ________



Studying and comparing the characteristics of exponential graphs and their inverses: ANSWER KEY

xxf 2)( =

The inverse of xxf 2)( =

Domain: all real numbers Domain: all positive numbers

Range: all positive numbers Range: all real numbers

x-intercept: none x-intercept: (1,0)

y-intercept: (0,1) y-intercept: none

Rate of change: increasing at an increasing rate Rate of change: increasing at a decreasing rate

End behavior:

As x approaches ∞+ , y approaches ∞+

As x approaches ∞− , y approaches 0

End behavior:


As x approaches 0, y approaches -∞



Practice No Calculators; Accuracy is important

Graph xxf 3)( = and the inverse function using a scale of 1.

x f(x) = 3x

-3

-2

-1

0

1

2

3



Practice: Answer Key

x f(x) = 3x

-3 1

27

-2 19

-1 13

0 1

1 3

2 9

3 27



Describe characteristics of each graph.

xxf 3)( = The inverse of xxf 3)( =

Domain: Domain:

Range: Range:

x-intercept: x-intercept:

y-intercept: y-intercept:

Rate of change: Rate of change:

End behavior:


As x approaches ∞− , y________

End behavior:

As x approaches + ∞, y_________

As x approaches 0, y ________



Describe characteristics of each graph: Answer Key

xxf 3)( = The inverse of xxf 3)( =

Domain: all real numbers Domain: all positive numbers

Range: all positive numbers Range: all real numbers

x-intercept: none x-intercept: (1,0)

y-intercept: (0,1) y-intercept: none

Rate of change: increasing at an increasing rate Rate of change: increasing at a decreasing rate

End behavior:


As x approaches ∞− , y approaches 0

End behavior:

As x approaches + ∞, y approaches ∞+

As x approaches 0, y approaches −∞



Unit: Modeling Exponential & Logarithmic Functions Content Area/Course: Algebra 2 Lesson 2: Defining the Logarithmic Function Brief Overview of Lesson: Students will discover that the log function is the inverse of an exponential function, both graphically and algebraically. At the conclusion of this lesson they will be able to approximate solutions to exponential equations using graphs and tables and recognize the inverse relationship between exponential and logarithmic functions. Additionally they will be able to rewrite exponential equations in log form and log equations in exponential form, as well as solve logarithmic equations and simplify logarithmic expressions. As you plan, consider the variability of learners in your class and make adaptations as necessary. Prior Knowledge Required: Students need to know the concept of an inverse function (input ↔output). They should also be able to create a table of values of an exponential function using technology. Estimated Time (minutes): 60 minutes Resources for Lesson:

• Graphing calculator or online alternative (desmos.com) • f(x)= 2x and its inverse posters (graphs and tables) from Lesson 1 • Individual student whiteboards or whiteboard space for student work • Lesson 2 discovery activity directions and recording sheets • Logarithmic expression placemat and cut out numbers • Lesson Summarizer • Fluency practice sheet



Time (minutes): 60 minutes (1 class) By the end of this lesson students will be able to: 1. solve equations with the variable in the exponent by approximating solutions to exponential equations using graphs and tables. 2. recognize the inverse relationship between exponential and logarithmic function. 3. rewrite exponential equations in log form and log equations in exponential form. 4. solve logarithmic equations and simplify logarithmic expressions Essential Question(s) addressed in this lesson: When, why and how would you use the inverse of an exponential function? Standard(s)/Unit Goal(s) to be addressed in this lesson : F-LE.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 logarithms using technology.

SMP 7 Look for and make use of structure.

Instructional Tips/Strategies Exponential and log graphs from Lesson 1 should be displayed around the classroom in preparation for this lesson. Warm-up A word wall will be interactively created during this warm up. Students would go to the front board (or call out) vocabulary words from Lesson 1 such as, base, exponent, power, input, output, function, inverse, etc. Class Work – Using “What’s a Log?” Handout In this lesson, the students are in small groups. Instructional Tips:



Question 1: Possible guiding questions or sentence starters to use during the discovery might be, using 813 =x as an example, “What is power of 3 that yields 81?” or “The power of 3 that makes 81 is ….” Assign board space to each group or hand out one mini whiteboard to each group. Students will use the Lesson 1 exponential posters to answer #1 of the handout “What’s a Log?” They are to write their answers on the whiteboard(s). The teacher should encourage students to use “think-alouds” in their groups and remind students to talk as they are writing out examples on the board.

Exemplar for Class Consensus – If xxf 2)( = , then x is the exponent of 2 that yields )(xf . Question 3: Students should recognize that they are using the output to locate the input. Question 5: Students should recognize that they are using the input to locate the output. Question 6: Students will need to know how to use the table feature of the graphing calculator or graphing application if using a computer or tablet. Question 8: This question will likely elicit some confusion and is designed to focus on SMP7 (and SMP1). Prompt students to review the steps in prior questions and if needed emphasize the input-output shifts to output-input for the inverse relationship. The teacher may need to guide students to the proper conclusion(s). TEACHER BACKGROUND NOTE: Students are not expected to generate the table given for the inverse of 2x; this was generated using change of base property. Using Log Placemat Manipulative As an option the teacher can prepare laminated Log Form placemats and students could use whiteboard erase markers instead of number cut-outs. For further differentiation inputs and outputs may be color-coded. Using the Fluency Practice Work Problems Handout The fluency practice work problems are designed so that students see one illustration for different conditions (base, exponent, +/-, etc.). Students may struggle with rational exponents – have students use think-alouds to facilitate. Encourage partner work at the board as well. Question 4(g) is a



problem where the teacher can help students develop SMP 7 (look for and make sense of structure) by suggesting they analyze the format of the equation before attempting to solve. Lesson 2 Summarizer This summarizer lays the groundwork for Lesson 4 and student answers may not be correct. Collect student responses and correct during Lesson 4. Additional Notes Advanced students could extend the conversation to domain and range as well as extraneous solutions. For struggling students the teacher might suggest using the log placemat manipulative. Some students may need more time and practice making sense of negative and rational exponents. Supplemental practice can be found at on-line. If needed, this site provides an additional reinforcing example for inverse functions using square roots http://www.oercommons.org/courses/logarithm-concepts-the-logarithm-explained-by-analogy-to-roots/view. Anticipated Student Preconceptions/Misconceptions Some students may confuse inputs/outputs and misread the graphs. Prompt these students to refer to the table used to generate the graph and if needed, label the ordered pairs on the poster. Lesson Sequence Lesson Opening: Yesterday we graphed exponential functions and their inverses. We compared the key characteristics of each graph. Today we will learn the name of the inverse of an exponential function and begin to understand how it behaves, and why it is useful. This will help us get closer to a solution to our fruit fly problem. The teacher collects the students’ homework and does a quick formative check as students start the investigation in groups. The students use the graphs and tables from lesson 1 to define the logarithmic function.

[Optional] Create a word wall of vocabulary from lesson 1.

The students work in pairs to explore input output relationship between an exponential function and its inverse.


http://www.oercommons.org/courses/logarithm-concepts-the-logarithm-explained-by-analogy-to-roots/view


Have a discussion with the whole class to confirm the class consensus (after Q5 and Q7). Then introduce the log function and formally define it in a whole class presentation using Teacher-Student Summary handout. (SMP7) The students’ understanding improves as they complete a placemat/card sorting activity followed by fluency practice (see comments in tips/strategies) and as they apply the log definition rewriting and evaluating expressions and equations. Formative assessment: Student homework Developing the word wall Placemat activity Fluency work (think-alouds) Lesson 2 summarizer Lesson Closing: Today we learned that exponential functions have inverses called logarithmic functions. (This relationship is similar to the one that exists with other functions and their inverses i.e. cubic functions and cube root functions.) How, when and why is the log function useful in solving exponential equations? Write your answer to this question on the Lesson 2 summarizer sheet, and we’ll use it in a think-pair-share. Preview next lesson: Students will apply the skills from lesson 2 to answer the fruit fly problem from lesson 1 and other real world problems.



Lesson 2

What’s a Log?

1) Work in groups or pairs using the posters from Lesson 1. Use the graph of f(x) = 2x and the inverse of f(x)=2x for the following problems.

Using the exponential graph showing f(x) = 2x, estimate the value of x and complete the statements below. Use words like exponent, power, base, yields, makes, etc. The first one is structured for you as an example.

a) If 3 = 2x then x ≈ _____ x is the _________of ________that _______ _______

b) If 10 = 2x then x ≈ ____ x is the

c) If 5.5 = 2x then x ≈ ___ x is the

CLASS CONSENSUS

If f(x) = 2x then x is the

2) Find and circle the points you used to answer question 1 parts a) b) and c) on the exponential graph.

3) Describe the process you used for finding the x values using the graph.

4) Find and circle the points you used to answer question 1 parts a) b) and c) on the inverse graph.

5) Describe the process you used for finding the x values using the graph. For example where did you look first?



CLASS CONJECTURE: Compare your answers to questions 3 and 5.

Using Our Tables to Find Solutions 6) Using the table feature of a graphing calculator with the table step set to 0.01 and the function f(x) =2x , find solutions to the equations from question 1.

a) If x23 = , then x ≈ _____

b) If x210 = , then x ≈ ____

c) If x25.5 = , then x ≈ ___

7) Describe the process you used for finding the x values from the table. 8) The input-output table below gives some values for the inverse of 2x. Where can you locate your answers for question 6?

Input Output

3

10

5.5



9) Describe the process you used for locating the answers.


10) a) Whether using a graph or table, write a statement in words regarding the input output of f(x) = 2x and the input output of the corresponding inverse function.

b) Complete the table summarizing the investigation using function notation.

Exponential Function Equation f(x)=2x

Function Notation

3=21.5850 f (____ ) = __________

5.5=22.4594

10=23.3219



Lesson 2 What’s a Log? ANSWER KEY

1) Work in groups or pairs using the posters from Lesson 1. Use the graph of f(x) = 2x and the inverse of f(x)=2x for the following problems.

Using the exponential graph showing f(x) = 2x, estimate the value of x and complete the statements below. Use words like exponent, power, base, yields, makes, etc. The first one is structured for you as an example.

d) If 3 = 2x then x ≈ _____ x is the power that you raise 2 to which yields 3

e) If 10 = 2x then x ≈ ____ x is the power that you raise 2 to which yields 10

f) If 5.5 = 2x then x ≈ ___ x is the power that you raise 2 to which yields 5.5

CLASS CONSENSUS

If f(x) = 2x then x is the power that you raise 2 to which yields f(x)

2) Find and circle the points you used to answer question 1 parts a) b) and c) on the exponential graph.

Circled points (1.58, 3); (3.32, 10); (2.46, 5.5)

3) Describe the process you used for finding the x values using the graph.

Answers will vary. Most students will identify the points using the y-coordinate then estimate the x-coordinate.

4) Find and circle the points you used to answer question 1 parts a) b) and c) on the inverse graph.

Circled points (3, 1.58); (10, 3.32); (5.5, 2.46)

5) Describe the process you used for finding the x values using the graph. For example where did you look first?



Answers will vary. Most students will identify the points using the x-coordinate then estimate the y-coordinate.


Answers will vary. Students should notice the process that they used in question 5 is the reverse of the process they used in question 3.

Using Our Tables to Find Solutions 6) Using the table feature of a graphing calculator with the table step set to 0.01 and the function f(x) =2x , find solutions to the equations from question 1.

d) If x23 = , then x ≈ 1.58

e) If x210 = , then x ≈ 3.32

f) If x25.5 = , then x ≈ 2.46

7) Describe the process you used for finding the x values from the table. Answers will vary. Students might scroll down until they find a y-value close to their desired output value, and then choose the corresponding x-value. 8) The input-output table below gives some values for the inverse of 2x. Where can you locate your answers for question 6?

Input Output

1.58 3

3.32 10

2.46 5.5



Students will circle points (3, 1.585); (5.5, 2.4594); (10, 3.3219).

9) Describe the process you used for locating the answers. Answers will vary. Students might look for the x-values which match the outputs from question 6 and circle those points.


Answers will vary. Students should notice the process that they used in question 9 is the reverse of the process they used in question 7.

10) a) Whether using a graph or table, write a statement in words regarding the input output of f(x) = 2x and the input output of the corresponding inverse function.

For any point (input, output) on the function f(x) = 2x , the point (output, input) will be on the corresponding inverse function.

b) Complete the table summarizing the investigation using function notation.

Exponential Function Equation f(x)=2x

Function Notation

3=21.5850 f(1.5850)=3

5.5=22.4594 f(2.4594)=5.5

10=23.3219 f(3.3219)=10



Teacher-Student Summary

Solving for x when x is an exponent requires us to define a new function, the log function.

For the given exponential equation 322 =x when we want to find “x”, we are thinking:

“ x is the exponent on base 2 that yields 32 ”

This is written symbolically as:

x = log2 (32)

x is the exponent on base 2 that yields 32

Since 25 = 32, the equation can be written and log 2 (32) = 5 so if we are solving for

x = log2 (32) this means x = 5

The “log” function helps us solve for a variable when it is an exponent.

Generally, if a b = c , then b = loga (c) Examples:

24 =16 can be rewritten as log2 (16) = 4 log2 (8) =3 can be rewritten as 23 = 8

TEACHER NOTE: Using the cards and placemats provided each student creates at least two accurate statements in the form.

if a b = c, then loga (c)=b



Log Placemat Manipulative Use your number cutouts and the log “placemat” to express the following exponential equations as logarithmic equations.

EXPONENTIAL FORM ↔ LOGARITHMIC FORM

23 = 8 _____________

525 21

= _____________

3921

= _____________

32 = 9 _____________

rt = p _____________



Log Placemat Manipulative ANSWER KEY Use your number cutouts and the log “placemat” to express the following exponential equations as logarithmic equations.

EXPONENTIAL FORM ↔ LOGARITHMIC FORM

23 = 8 log28=3

525 21

= log255=12

3921

= log93=12

32 = 9 log39=2

rt = p logrp=t



Cut out each number.

2 2 5 21

9 8 3 2

1−

5 r p 31

t 1− 25 51



Log =



Lesson Closing Summarizer: Today we learned that the log function is the inverse of the exponential function. How, when, and why is the log function useful in solving exponential equations? Answers will vary. Students should realize that the log function is useful when you encounter an exponential equation with an unknown variable in the exponent. The log function (for base b) takes as its input a number and returns as its output the power that you must raise b to, to get that input.



Fluency Practice Work Problems

Please do without a calculator.

1. Write each equation in logarithmic form.

a) 12553 = b) 170 = c) 8134 =

d) 8113 4 =−

e) 641

41 3

=

f) 67776 51

=

2. Write each equation in exponential form.

a) 3216log6 = b) 01log10 = c) 4

811log3 −=

d) 500001.0log10 −= e) 2

15log25 = f) 2

3216log36 =

Generally, if a b = c then b = loga (c) Examples: 24 =16 can be rewritten as log2 (16) = 4 log2 (8) =3 can be rewritten as 23 = 8



3. Evaluate each expression.

a) 81log3 b) 001.0log10 c) 491log7

d) 27log31 e) 1log2 f) 8log4

4. Solve each equation. Verify your solutions.

a) 3log10 −=n b) 23log4 =x

c) 416log −=m d) 0log4 =n

e) 2log

51 −=t

f) 381log −=p

g) )64(log)8(log 22 += xx



ANSWER SHEET

1. Write each equation in logarithmic form.

a) log5125=3 b) log71=0 c) log381=4

d) log3181

=4 e) log14 164

=3 f) log77766=15

2. Write each equation in exponential form.

a) 63=216 b) 100=1 c) 3-4= 181

d) 10-5=0.00001 e) 2512=5 f) 36

32=216



3. Evaluate each expression.

a) 4 b) -3 c) 491log7 = -2

d) -3 e) 0 f) 32

4. Solve each equation. Verify your solutions.

a) n=0.001 b) x=8

c) m=12 d) n=1

e) t=25 f) p=2

g) x=1.5



Unit: Modeling Exponential & Logarithmic Functions Content Area/Course: Algebra 2 Lesson 3: Logarithmic Functions

Brief Overview: Students will estimate the solution to an exponential equation graphically, solve logarithmic equations, evaluate log expressions using the definition of the logarithm, and deepen their understanding of function notation. At the conclusion of this lesson, they will be able to estimate graphically the solution to a real world problem involving exponential growth. Additionally they will be able to translate between exponential and logarithmic functions correctly using function notation. As you plan, consider the variability of learners in your class and make adaptations as necessary.

Prior Knowledge Required: Properties of exponents, function notation, definition of the logarithm, and symbolic manipulation of equations.

Estimated Time: 60 minute periods

Resources for Lesson 3: • Graphing calculator or online alternative (desmos.com) • Log poster ONLY (graphs and tables) from Lesson 1 • Cards/Placemats from lesson 2 as needed • Individual student whiteboards or whiteboard space for student work • Concept building problem set • Quiz

Time (minutes): 60 minutes (1 class)

By the end of this lesson students will be able to: 1. estimate graphically, the solution to a real world problem involving exponential growth. 2. translate between exponential and logarithmic functions correctly using function notation 3. solve logarithmic equations and evaluate log expressions using the definition of the logarithm.

Essential Question(s) addressed in this lesson: When, why, and how would you use the inverse of an exponential function?



What kinds of real-world situations are modeled by exponential functions? Standard(s)/Unit Goal(s) to be addressed in this lesson: F-LE.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.


Instructional Tips/ Strategies Removing the exponential graph and table and relabeling the inverse graph/table as y = log2(x) highlights the newly defined logarithmic function and emphasizes using it. Students would have been able to do the “warm up” mid way through lesson 2 prior to the definition of a logarithmic function since it is a graphical and not an algebraic approach. It is used here instead to reinforce the “new” function in graphical and tabular form. Engage the whole class using “number heads together (http://www.teachervision.fen.com/group-work/cooperative-learning/48538.html) or a similar strategy to building on the prior lesson [and the word wall from lesson 2]. Alternatively this work could continue with the teacher circulating among student pairs or small groups using “think alouds”. Anticipated Student Preconceptions/Misconceptions Students may either see the exponential and logarithmic functions as either non-related or at the other extreme as the same function. Students may still have misconceptions regarding the language and concepts of inverse function and f-1(x) notation. This site provides an additional example for inverse functions using square roots http://www.oercommons.org/courses/logarithm-concepts-the-logarithm-explained-by-analogy-to-roots/view. Lesson Sequence Lesson Opening: Yesterday we defined a new function, the logarithmic function, which is the inverse of the exponential function. Today we will use that function to solve our fruit fly problem. Additionally, we will practice working with the logarithmic function and evaluating logarithmic expressions. Warm Up:


http://www.teachervision.fen.com/group-work/cooperative-learning/48538.html

http://www.oercommons.org/courses/logarithm-concepts-the-logarithm-explained-by-analogy-to-roots/view


Students revisit the poster graph for the inverse of 2x and re-label it y = log2(x). Students then use the logarithmic function graph to solve the fruit fly problem. NOTE: The exponential f(x)=2x poster should be hidden/removed so that students are required to work with the inverse (logarithmic) function. Using “numbered heads together” (see instructional tips/strategies) groups of students work through the concept building problems to reinforce the definition of logarithmic functions using proper function notation. (SMP7) As needed, additional fluency practice sheets could be generated by the teacher. Formative assessment: Collected fluency practice (option) Numbered heads together Quiz (formative/summative option) Lesson Closing: Today we used the logarithmic function to solve our fruit fly problem. We practiced solving logarithmic equations and saw one reason why their useful mathematically. Tomorrow, we’ll take a short quiz, then explore more real world problems that require logarithms to solve them. Preview outcomes for the next lesson: Real world examples



Lesson 3 Warm Up

Working in groups, using your y = log2 (x) graph, and without a calculator please answer the question.

Biologists use fruit flies to study genetics because fruit fly populations grow very quickly. As a result, scientists can follow genetic traits through many generations in a short period of time.

If a geneticist starts with 15 fruit flies and the fruit fly population doubles every 10 days, approximately how long it will take for the population to reach 180?

Explain your process and justify the reasonableness of your answer.

Lesson 3 Warm Up Answer Key

Working in groups, using your y = log2 (x) graph, and without a calculator please answer the question.

Biologists use fruit flies to study genetics because fruit fly populations grow very quickly. As a result, scientists can follow genetic traits through many generations in a short period of time.

If a geneticist starts with 15 fruit flies and the fruit fly population doubles every 10 days, approximately how long it will take for the population to reach 180?

Explain your process and justify the reasonableness of your answer.

It takes about 35 days for the fruit fly population to reach 180. Student justifications will vary.



Concept Building

Exponential and Logarithmic functions using function notation

1) Fill in the boxes given: f(x) = 2x then: a) =)5(f

b) =)3(f

c) =)6(f

2) Fill in the boxes given: xxf 2log)( =

a) =)64(f

b) =)32(f

c) =)8(f

3) Fill in the boxes with values to correctly complete the equality. log ) =



Concept Building Answer Key

Exponential and Logarithmic functions using function notation

2) Fill in the boxes given: f(x) = 2x then: c) =)5(f 32

d) =)3(f 8

c) =)6(f 64

2) Fill in the boxes given: xxf 2log)( =

a) =)64(f 6

d) =)32(f 5

e) =)8(f 3

4) Fill in the boxes with values to correctly complete the equality. Answers will vary. As long as the equation created by students is mathematically sound, then the answer should be marked correct.



Summarize:

If xxf 2)( = , then =)4(f ____________

If g(x) = log2(x) then =)16(g ____________

How does )4(f relate to )16(g ? Summarize:

If xxf 2)( = , then =)4(f 16

If g(x) = log2(x) then =)16(g 4

How does )4(f relate to )16(g ? Functions f and g are inverses of each other so f(4)=g(16).



QUIZ Find the value of each

• log8(2)

• log51

125

Solve for x

• logx(81) = 4

• log9(x) = 32

• log4(2x + 6) = log4(-10x + 8)

• 272x = 81x + 4 Hint: This can be solved without using logarithms.

• log8 (-x + 13) = 0

Complete the following statements: If g(x) = log2(x), then g(2) = _______________

If h(x) = x2 , then h(2) = ___________________



QUIZ ANSWER KEY

Find the value of each

• log8(2)=13

• log51

125

=-3

Solve for x

• logx(81) = 4; x=3

• log9(x) = 32

; x=27

• log4(2x + 6) = log4(-10x + 8); x=16

• 272x = 81x + 4 ; x=8

• log8 (-x + 13) = 0; x=12

Complete the following statements: If g(x) = log2(x), then g(2) = 1

If h(x) = x2 , then h(2) = 4



Unit: Modeling Exponential & Logarithmic Functions Content Area/Course: Algebra 2 Lesson 4: Real World Applications Brief Overview: Students will solve real world applications of exponential equations with an unknown exponent using the definition of the logarithm. Students will realize that the notation log(x) is equivalent to log10(x). At the conclusion of this lesson they will be able to understand how to use logarithms to solve real world problems and solve problems involving unknown exponents in base 10.Additionally they will recognize when and how to perform symbolic manipulations to prepare for the use of the log definition to solve an exponential equation in the form of da ct =⋅10 for t. Finally, students will use the definition of a logarithm solve for the unknown exponent and use technology to evaluate a common logarithm, log10.

As you plan, consider the variability of learners in your class and make adaptations as necessary. Prior Knowledge Required: Students need to know how to create an exponential equation from a real world problem, the definition of the logarithm, and how to perform symbolic manipulation of equations. Estimated Time: Two 60 minute periods (see teacher tips/strategies) Resources for Lesson 4:

• Graphing calculator or online alternative (desmos.com) • Common solutions and pH cards for lesson launch • Guided student handouts for Chemistry (pH) application; Developing log10 • Did they write this right? Opener for Stock marker scenario • Guided student handouts for the Stock market scenario. Optional enlarged stock market graphs. • Exit ticket



Time: 2 days one day each application. By the end of this lesson students will be able to: 1. understand how to use logarithms to solve real world problems. 2. solve problems involving unknown exponents in base 10. 3. recognize when and how to perform symbolic manipulations to prepare for the use of the log definition. 4. use the definition of a logarithm solve for the unknown exponent. 5. use technology to evaluate a common logarithm, log10. Essential Question addressed in this lesson:

When, why and how would you use the inverse of an exponential function? What kinds of real-world situations are modeled by exponential functions? Standard(s)/Unit Goal(s) to be addressed in this lesson):

F-LE.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.

SMP 3. Construct viable arguments and critique the reasoning of others.

SMP 4. Model with mathematics

SMP 7. Look for and make use of structure.

Instructional Tips/Strategies/Teacher Notes:

The teacher may opt to divide the class into groups and assign either the pH or Stock market application/scenario. On the second day the teacher will facilitate groups “teaching each other”. Other appropriate base 10 applications might be Decibel or Richter scales. Each application has an opener/launch. The guiding questions for the exit tickets are based on Polya’s work as stated in this blog http://grantwiggins.wordpress.com/2013/02/08/on-genuine-vs-bogus-inquiry-using-eqs-properly/


http://grantwiggins.wordpress.com/2013/02/08/on-genuine-vs-bogus-inquiry-using-eqs-properly/

http://grantwiggins.wordpress.com/2013/02/08/on-genuine-vs-bogus-inquiry-using-eqs-properly/


Chemistry pH As students are working on the class opener, the teacher circulates around the room remediating, prompting, and assisting student and/or groups to help them move forward. The intent is to hook the students and not focus on getting all the solutions/pH matches correct. Guide the activity with probing questions such as “In what other class/course have you seen/used pH? Are there pH/solutions your group is certain of? “

As students work through the guided handout the teacher could suggest a placemat model for struggling students with boxes such as: = and guiding question such as “What is the input? What is the output?” Student may want to use the logarithm placemat model from Lesson 2. Student groups should report out on the board their answers to (d) – (g) using the pH=-log [H+] showing their work and there could be a class carousel to check answers. Alternatively, this could be structured as another “numbered heads together” activity. Stock Market Only two coordinates on the graph are given so that students study the shape of the graph, conclude the exponential model fits the best, and realize the common ratio is 10. Permit students to answer part c using any means. For example they may estimate an answer or guess and check. They may use tables in their calculator looking in the y column for 2000. Emphasize that they are looking at the output to find the input.

You may need to support students in extending the graph to include current data. This part of the exploration is included to encourage discussions about the effectiveness of the model and why models are used. The “how to plot the model” has been intentionally not given (e.g. table of values, calculator) so that students have the freedom to choose a method. As a class, discuss any differences. This extension could be included as an exit ticket where students reflect why we create function models and if they believe their model is reasonable and effective to make predictions. (SMP3 and SMP4)

Students /groups should be able to work thought the Building Fluency without much teacher assistance. This would be a good opportunity for formative assessment and/or interventions as needed.

Anticipated Student Preconceptions/Misconceptions

Students may want to find the value of the unknown exponent by putting the exponential equation in their calculator and use tables and/or graphs to estimate the value, instead of reasoning symbolically.

Students may have trouble recognizing the difference between using the definition of log to solve problems vs. using the log as an inverse operation.



Chemistry pH

Students may incorrectly use the log definition by putting parameters in the wrong place, i.e., thinking x107 = is equivalent to 10log7 =x

The teacher may need to explicitly make sure students write the log formula as “ =pH “ and not leave it as “ =− pH “. If students do not, then it can be clarified when they record the differences in their answer and the standard formula for calculating the pH. The key difference is base 10 is assumed, 10log is written as log. (Question 2)

Students may not know the units “moles per liter”. In this case, a definition should be looked up or provided.

Stock Market Students may not recall writing exponential equations from word problems. The lesson launch is designed to be a formative refresher. For simplification, a time interval of 35 years is used. Students may need to be reminded of this.

Lesson Sequence

Lesson Opening: So far in this unit we’ve defined the logarithmic function and used it in one (limited) real world application. Our work in the rest of this unit will focus specifically on real world applications of logarithms. Over the next two days we’ll see how logarithms are useful in chemistry and finance.

Chemistry pH Students are arranged in groups of 3 or 4 for collaborative learning. Class opener: Students complete the common solution pH card matching activity. (source) http://www.emc.maricopa.edu/faculty/farabee/biobk/acidbase.gif Students are given an envelope with cut out slips and work cooperatively to match the solution with its pH. Teacher posts or distributes the answer key after approximately 5 minutes.

What’s the pH? Students work cooperatively on the application problem. If needed, a placemat model could be used to scaffold for struggling students.


http://www.emc.maricopa.edu/faculty/farabee/biobk/acidbase.gif


The Building Fluency, Section 2 work is designed to guide students in identifying the structure in algebraic expressions (SMP7). By noting what is different in each section they recognize implications as the equations build in complexity with changes in the coefficients of both the base and the exponent.

The assessment provided can be either formative or summative.

NOTE: Students may question the fact that the problem statement says doubling and the equation in base 10 (because 100. 30103= 2) is used because students at this point don’t know how to solve using base 2.

Exit ticket: Enhanced Essential Question1: When, why and how would you use the inverse of an exponential function?

Lesson Sequence

Stock Market Lesson launch use handout “Did They Write These Right?” refresher. Hand to students and have informal teacher facilitated/remediated conversations as needed. Students work in pairs on guided stock market scenario handout “Let’s Make Some Money!” The assessment provided can be either formative or summative. Exit ticket: Essential Question 2: What kinds of real-world situations are modeled by exponential functions? Assessment Both Chemistry pH and Stock Market The assessment provided is to be used whether the class followed the Chemistry scenario or the Stock Market scenario, or both. It can be either formative or summative at the teacher’s discretion. NOTE: Problem #1a is meant to assist problem #2. Problem #2 is actually a base 2 scenario as the students should discover. The given equation has already converted base 2 to base 10, since change of base is beyond the scope of this unit.



Formative Assessment: Ticket to leave Lesson Closing: Over the last two days we’ve explored the use of the logarithmic function in real world applications. We’ve seen how the logarithm is crucial to understanding pH and the stock market. Tomorrow we’ll work with a unique type of logarithm that’s used in population growth models. This will prepare us for a final unit project in the area of epidemiology. Preview next lesson: Students will build on their knowledge and work through a novel discovery of the natural log, e, ln using population growth.



Real World Application: Chemistry, pH

Lesson launch – card sorting/matching pH activity: The pH measures the acidity or alkalinity of a substance. A pH of 7 is neutral, a pH less than 7 is acidic, and a pH greater than 7 is alkaline.

Using the slips provided by your teacher, match the solution with its pH.

NOTE: this is the solution page and is NOT provided to students immediately. Adapted from http://www.emc.maricopa.edu/faculty/farabee/biobk/acidbase.gif

BATTERY ACID pH = 0

STOMACH ACID pH = 1

SODA pH = 3

TOMATO JUICE pH = 4

SALIVA pH = 7

SEA WATER pH = 8

BAKING SODA pH = 9

HOUSEHOLD AMMONIA pH = 11

BLEACH pH = 13

LIQUID DRAIN CLEANER pH = 14


http://www.emc.maricopa.edu/faculty/farabee/biobk/acidbase.gif


What’s the pH?

1) The pH measures the acidity or alkalinity of a substance. A pH of 7 is neutral, a pH less than 7 is acidic, and a pH greater than 7 is alkaline. The pH depends on the hydrogen ion concentration in the substance measured in moles per liter.

a) Examine the table above and determine the exponential equation for the Hydrogen Ion

concentration as a function of pH.

b) Using the definition of log, write an equation for pH rating as a function of the hydrogen ion

concentration.

c) If the hydrogen ion concentration in a substance is 4.5 x 10-5 moles per liter, find the pH rating

using the log equation in part b) and technology.

pH rating 0 1 2 3 4 5 6 7 8 9 10 Hydrogen Ion Concentration in moles per liter

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10



2) The formula (function) used to calculate the pH of a substance given its concentration of hydrogen ions, in moles per liter, is:

𝑝𝐻 = −log [𝐻+]

Compare this formula to your answer to 1(b). What is the same? What is different?

3) Locate the LOG button on your calculator and using the formula 𝑝𝐻 = −log [𝐻+] find the pH of each substance below. Check your answers with at least two other groups. Milk [H+] = 2.51 x 10-7 mole per liter Acid Rain [H+] = 2.51 x 10-6 mole per liter Black coffee [H+] = 2.51 x 10-5 mole per liter Milk of Magnesia [H+] = 3.16 x 10-11 mole per liter



What’s the pH? Answer Key

4) The pH measures the acidity or alkalinity of a substance. A pH of 7 is neutral, a pH less than 7 is acidic, and a pH greater than 7 is alkaline. The pH depends on the hydrogen ion concentration in the substance measured in moles per liter.

d) Examine the table above and determine the exponential equation for the Hydrogen Ion

concentration as a function of pH.

Hydrogen Ion concentration = 10-pH

e) Using the definition of log, write an equation for pH rating as a function of the hydrogen ion concentration.

pH=-log(Hydrogen Ion concentration)

f) If the hydrogen ion concentration in a substance is 4.5 x 10-5 moles per liter, find the pH rating

using the log equation in part b) and technology.

pH=4.3

pH rating 0 1 2 3 4 5 6 7 8 9 10 Hydrogen Ion Concentration in moles per liter

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10



5) The formula (function) used to calculate the pH of a substance given its concentration of hydrogen ions, in moles per liter, is:

𝑝𝐻 = −log [𝐻+]

Compare this formula to your answer to 1(b). What is the same? What is different?

Answers will vary. The formula is the same as the one from 1(b).

6) Locate the LOG button on your calculator and using the formula 𝑝𝐻 = −log [𝐻+] find the pH of each substance below. Check your answers with at least two other groups. Milk [H+] = 2.51 x 10-7 mole per liter; pH=6.6 Acid Rain [H+] = 2.51 x 10-6 mole per liter; pH=5.6 Black coffee [H+] = 2.51 x 10-5 mole per liter; pH=4.6 Milk of Magnesia [H+] = 3.16 x 10-11 mole per liter; pH=10.5



BUILDING FLUENCY Using the definition of log and technology, solve the following equations for t. Discuss your methods and compare your answers with your partner as you progress through each section. Section 1: a) 10t = 2 b) 10t = 20 c) 10t = 55 d) 10t = d What do the equalities above tell you about how any number can be written? Answer Check: Section 2:

a) Is 3•10t the same as 30t ? Why or why not? Explain your reasoning.

b) Compared to section 1 what would be the first step in solving the following exponential equations?

3•10t = 900 5•10t = 400 21 •10t = 44 a•10t = d

With your group members, review your work in section 1 and then solve the exponential equations and record any similarities and differences in the structure of the equations.



Equation WORK Similarities and Differences between equations, the relationships they represent, and their solutions

3•10t = 900

5•10t = 400

21 •10t = 44

a•10t = d

Section 3: With your group members, build on your skills from sections 1 and 2 to solve the following equations.

a) 11•102t = 2560 b) 74 •102.56t = 641

Answer Check:

c) 150107 32

=•t

d) a•10ct = d SUMMARIZE YOUR WORK: What is the order of the algebraic process required to solve for the exponent?



BUILDING FLUENCY Answer Key Using the definition of log and technology, solve the following equations for t. Discuss your methods and compare your answers with your partner as you progress through each section. Section 1: a) 10t = 2; t=log(2) b) 10t = 20; t=log(20) c) 10t = 55; t=log(55) d) 10t = d; t=log(d) What do the equalities above tell you about how any number can be written? Answers will vary. Any exponential equation can be rewritten as a logarithmic equation and vice-versa. Answer Check: Section 2:

a) Is 3•10t the same as 30t ? No, they are not the same expression (order of operations.) Have students verify this fact by choosing values of t to substitute into the expression. This will show that the values of the two expressions are different.

b) Compared to section 1 what would be the first step in solving the following exponential equations?

3•10t = 900

divide both sides of the equation by 3

5•10t = 400

divide both sides of the equation by 5

21 •10t = 44

multiply both sides of the equation by 2

a•10t = d

divide both sides of the equation by a

With your group members, review your work in section 1 and then solve the exponential equations and record any similarities and differences in the structure of the equations.



Equation WORK Similarities and Differences between equations, the relationships they represent, and their solutions

3•10t = 900

t=log(300) t is approx. 2.48

Answers will vary. Solving an exponential equation with a variable in the exponent yields a logarithmic equation. Students should understand that every exponential equation can be written as a logarithmic equation and vice-versa.

5•10t = 400



21 •10t = 44



a•10t = d

t=log(𝑑𝑎) Answers will vary. Solving an exponential equation with a variable in the exponent yields a logarithmic equation. Students should understand that every exponential equation can be written as a logarithmic equation and vice-versa.

Section 3: With your group members, build on your skills from sections 1 and 2 to solve the following equations. a) 11•102t = 2560

t=12log(256011 ); t≈ 1.18



b) 74 •102.56t = 641

t=log (1121.75)2.56

; t≈ 1.19 Answer Check:

c) 150107 32

=•t

d) a•10ct = d t=3

2log (1507 ) t=1𝑐log(𝑑𝑎)

SUMMARIZE YOUR WORK: What is the order of the algebraic process required to solve for the exponent? Answers will vary. Check to see that students have written an algorithm that is mathematically sound and reflects the order of operations.



Assessment: Solving equations using the definition of a logarithm

1) Solve for x using algebra and logarithms. Show your work and check your answer.

a) 210 =x

b) 3•10x = 50

c) 7•105x = 120000

3) Bacteria populations can grow very quickly. One bacteria cell will split and create 2 new cells every hour. If we start with 5 cells, the equation for the bacteria populations as a function of t hours is

ttP 30103.0105)( ⋅=

How long it would take for the population to reach 350 cells? Show your work and check your answer.



Assessment: Solving equations using the definition of a logarithm- Answer Key

1) Solve for x using algebra and logarithms. Show your work and check your answer.

d) 210 =x ; x=log2

e) 3•10x = 50; x=log(503 )

f) 7•105x = 120,000; x=15log(120,000

7 )

4) Bacteria populations can grow very quickly. One bacteria cell will split and create 2 new cells every hour. If we start with 5 cells, the equation for the bacteria populations as a function of t hours is

ttP 30103.0105)( ⋅=

How long it would take for the population to reach 350 cells? Show your work and check your answer. t= 𝑙𝑜𝑔70

0.30103 ; t≈ 6.13 ℎ𝑜𝑢𝑟𝑠



Chemistry Application Exit Ticket

Summarize solving for t in da ct =⋅10 and explain in words, when, why and how you use the inverse of an exponential function.

To help you answer the exit ticket question you should be thinking:

• WHAT exactly does this problem ask me to find? • What STRATEGY or operation will you use to solve this problem? • Do you know a RELATED problem? • Have you SEEN the problem before? • Could you RESTATE the problem? • Can you CHECK the result? • Can you derive the answer DIFFERENTLY? (Is there more than one way to do this?)



Real World Application: Stock Market Trends

Lesson Launch: DID THEY WRITE THESE RIGHT?

In a laboratory, a culture of 5000 organisms increases at a rate of 3% an hour. Which exponential equation models the population growth of the culture? Justify your reasoning.

a) P(t) = 5000(0.03)t b) P(t) = 5000(1.03)t c) P(t) = 5000(0.97)t

With inflation, an item purchased now costs more than the same item purchased 5 years ago. If the rate of inflation is 4% and a pair of sneakers cost $120 today, which equation models the cost of the sneaker? Justify your reasoning.

a) C(t) = 120(0.96)t b) C(t) = 120(0.04)t c) C(t) = 120(1.04)t

Real World Application: Stock Market Trends – Answer Key

DID THEY WRITE THESE RIGHT?

In a laboratory, a culture of 5000 organisms increases at a rate of 3% an hour. Which exponential equation models the population growth of the culture? Justify your reasoning.

P(t) = 5000(1.03)t is the correct equation. The base is 1.03 because in exponential growth equations, the base is represented by 1+r, where r is the growth rate. In this case, the growth rate is 3% or 0.03 and 1+0.03=1.03.

With inflation, an item purchased now costs more than the same item purchased 5 years ago. If the rate of inflation is 4% and a pair of sneakers cost $120 today, which equation models the cost of the sneaker? Justify your reasoning.

C(t) = 120(1.04)t is the correct equation. The base is 1.04 because in exponential growth equations, the base is represented by 1+r, where r is the growth rate. In this case, the growth rate is 4% or 0.04 and 1+0.04=1.04.



Let’s Make Some Money!

1) The S&P 500 is a weighted average of the share value of 500 leading companies. The value of the S&P 500 stock market index in 1950 was 17 and 1985 was 170. The graph of the market is shown below. NOTE: The graph is labeled such that 35 years = 1 time interval in years since 1950.

Working in groups, answer the following questions. a) Assuming the market continues to grow as shown, discuss when you think it will reach 1700. Record two

estimates, one yours and one from your group, when you think it will reach 1700. Justify and give supporting evidence.

YOUR ESTIMATE: __________________________ REASONING: _____________________________ GROUP ESTIMATE: _________________________REASONING: _____________________________ b) Based on the two data points given and the shape of the S&P graph, determine an exponential equation

using f(x) = abx for market value as a function of time that will best fit the graph. Use the points as labeled where time is given in 35-year intervals.



c) Using the equation from part b, predict when the market value will reach 2000. Compare your answer with two other groups, note any similarities and differences.

PREDICTION: _____________________ OTHER GROUP COMPARISON: __________________ OTHER GROUP COMPARISON:__________________ d) How can you be sure that your function accurately represents the data? Plot your model on the graph of

the S&P and discuss the effectiveness of your model and why models are used.



Let’s Make Some Money! ANSWER KEY

1) The S&P 500 is a weighted average of the share value of 500 leading companies. The value of the S&P 500 stock market index in 1950 was 17 and 1985 was 170. The graph of the market is shown below. NOTE: The graph is labeled such that 35 years = 1 time interval in years since 1950. Working in groups, answer the following questions. a) Assuming the market continues to grow as shown, discuss when you think it will reach 1700. Record two

estimates, one yours and one from your group, when you think it will reach 1700. Justify and give supporting evidence.

Estimates will vary. If the groups are assuming that the graph continues to follow an exponential pattern, then the market should reach 1700 in 2020.

b) Based on the two data points given and the shape of the S&P graph, determine an exponential equation using f(x) = abx for market value as a function of time that will best fit the graph. Use the points as labeled where time is given in 35-year intervals. 𝑓(𝑥) = 17 ∙ 10𝑥

c) Using the equation from part b, predict when the market value will reach 2000. Compare your answer with two other groups, note any similarities and differences.

2000 = 17 ∙ 10𝑥

𝑥 = 𝑙𝑜𝑔 (200017 )

𝑥 ≈ 2.07



According to this model the market will reach 2000 about halfway through the year 2022.

d) How can you be sure that your function accurately represents the data? Plot your model on the graph of the S&P and discuss the effectiveness of your model and why models are used.

After they graph the model, the students will realize the limitations of their model. The accuracy of the model breaks down because the stock market doesn’t grow exactly as an exponential function does. Have a discussion with your students about why they think the market dropped so dramatically when it did.



Stock Market Application Exit Ticket

What kinds of real-world situations are modeled by exponential functions? What kinds of information can you decipher about log functions from the graphical, tabular, symbolic, and verbal representations? How can you tell when the graph of an exponential function appears reasonable? To help you answer the exit ticket question you should be thinking:

• WHAT exactly does this question ask me to find?

• What STRATEGY or operation will you use to answer this question?

• Have you SEEN the problem before? • Could you RESTATE the problem?



Unit: Modeling Exponential & Logarithmic Functions Content Area/Course: Algebra 2 Lesson 5: Euler’s number and the natural logarithm

Brief Overview: In this lesson students investigate where the irrational number e is found, define xxe lnlog = and solve daect =

for t. At the conclusion of this lesson students will be able to know why the number e is useful and one way to define it. The students will then solve real world problems involving unknown exponents with a base e and the natural log. They recognize when and how to perform symbolic manipulation in preparation for the ln definition. Additionally they will apply the ln definition in order to be able to solve for the unknown exponent. Finally, the students will use technology to solve for the unknown exponent. As you plan, consider the variability of learners in your class and make adaptations as necessary.

Prior Knowledge Required: Students should have prior knowledge of problems involving interest compounded continuously and population growth. Students must be able to solve for an unknown exponent using the definition of a logarithm.

Estimated Time: 60 minutes

Resources for Lesson: • On-line access for lesson launch and population growth video clip. • Graphing calculator or online alternative (desmos.com) • Guided student handouts for Lesson 5 investigation

Time (minutes): 60 minutes

By the end of this lesson students will know and be able to:

1. know why e is useful and one way to define it. 2. solve real world problems involving unknown exponents with a base e and the natural log. 3. recognize when and how to perform symbolic manipulation in preparation for the ln definition. 4. apply the ln definition in order to be able to solve for the unknown exponent. 5. use technology to solve for the unknown exponent.



Essential Question addressed in this lesson:

When, why and how would you use the inverse of an exponential function? What kinds of real-world situations are modeled by exponential functions?

Guiding Question: How can I use the definition of the natural logarithm to solve for unknown exponents of base e and find out how long it will take for a population to grow?

Standard(s)/Unit Goal(s) to be addressed in this lesson


SMP 4. Model with mathematics.

SMP 7. Look for and make use of structure.

Instructional Tips/Strategies/Suggestions:

Of all the lessons in the unit, this will need the most teacher guided whole class work. The Niger population model used to discover “e” is dense and authentic and, as such, may be complicated for all students to grasp initially. The title of the handout should be fruit for discussion. At the end of the activity, students should be able to connect the fact that π and e are both irrational numbers that are so special they have special names. (SMP7)

In the exploration, the rationale behind question 2 is to lead students to the conclusions that the compounding (rate) for population

growth is continuous. Question 4 relies on the property of exponents mnnm aa =)( to rewrite the population equation. This is necessary as

a rewrite since the purpose is to derive e which is the limit (end behavior) of rn

nr

+1 .

Question 5 reinforces vocabulary from lesson 1 and leads to the model for Niger population growth as p(t) =16.3* e0.0368t (SMP4)



Question 7 will have students wondering how to use their calculators since the log (base 10) button will not work. This question is intended to elicit questions from students and lead to the conversation that loge is written as ln. It will be at the teacher’s discretion whether or not to include the “spoiler alert”.

Anticipated Student Preconceptions/Misconceptions

Students may take some time to connect that xxe lnlog = and will not realize the letter e represents a number.

Students having just successfully used 10log to solve equations may resist using ln.

Students may want to find the value of the unknown exponent by putting the exponential equation in their calculator and using tables and/or graphs to estimate the value, instead of reasoning symbolically.

Students may forget to divide both sides by “a” in abct = d before using the definition of ln.

Lesson Sequence

Lesson Opening: So far in this unit we’ve learned about logarithmic functions and how crucial they are to solving real world problems in science and finance. Today we’ll explore a special type of logarithm, the natural logarithm, and discover the specific applications of this logarithm. We’ll use the natural logarithm today, and in our curriculum embedded performance assessment.

Lesson Launch 100 people video http://100people.org/wp/2012/08/the-100-people-project-an-introduction/ (first 1 minute and 40 seconds only).

Students discover e using “π Has Some Company” problem. This handout also reinforces their knowledge of continuous compounding, end behavior, academic language and compound interest.

Students use e to solve the Niger population problem algebraically using the definition of log to rewrite the exponential equation. In the process, they define xxe lnlog = .


http://100people.org/wp/2012/08/the-100-people-project-an-introduction/


Students will work with a partner to practice solving exponential equations in the form daect = with natural logs on the “Building Fluency” handout.

Lesson Closing: In today’s lesson we used our “log base e” or natural logarithm to model population growth. Scientists are concerned with population growth in a variety of contexts, so this logarithm is of particular importance to them. For today’s final activity, you will graph the natural logarithm function with its inverse, and making comparisons between the graphs of both functions. Tomorrow, we’ll begin our curriculum embedded performance assessment, where you’ll use the skills that you learned in this unit to solve an epidemiology problem.

Formative Assessment

Exit ticket



π Has Some Company

According to the CIA World Factbook (https://www.cia.gov/library/publications/the-world-factbook/fields/2002.html) Niger is the country with one of the fastest growing populations. Niger’s population growth rate is approximately 3.68% annually. In comparison, U.S. population growth rate is approximately 0.93% and world population growth approximately 1.165%. PROBLEM STATEMENT: If Niger’s population is currently 16.3 million, what will its population be one year from now? We will be able to answer this accurately using algebra the end of the lesson. 1) Write an equation to model Niger’s population at any year t.

2) In your group answer the following: a) How would you describe the relationship/trend?

b) Is this model reasonable? Justify your thinking. For example discuss in your groups how you could adjust your model to make it more realistic. Does the population of a country only change at the end of a year?

3) If t is in years, how would you rewrite your population equation to determine the population if it grew only


https://www.cia.gov/library/publications/the-world-factbook/fields/2002.html



a) at the end of each month?

b) at the end of each second?

4) Recall that for compound interest

nt

nrPtA

+= 1)(

�

A = Amount at any time

P = Principal (starting amount at t = 0)

r = interest rate as a number not a percent

n= number of compoundings annually

t = time in years

Compare the compound interest model to the model for Niger’s population growth

𝑝(𝑡) = 16.3 �1 + 0.0368𝑛

�𝑛𝑡

p = the population at any time 16.3 = the starting population for this problem 0.0368 = is the growth rate as a number n = number of times in a year you will calculate population, t = time in years. What is the same? What is different? Considering exponent properties, explain why it is equivalent to rewrite the population model as

𝑝(𝑡) = 16.3 ��1 + 0.0368𝑛

�𝑛

0.0368�

0.0368𝑡



5) Looking at just the base of the exponential expression, evaluate �1 + 0.0368𝑛

�𝑛

0.0368 for values of n in the table below. Record your answers to 4 decimal places.

n 1 3 10 100 1000 10000 50000

�1 + 0.0368𝑛

�𝑛

0.0368

Evaluate the expression for 000,100=n and 000,200=n until you reach a conclusion. Write your conclusion on the board and have one person volunteer to summarize the conclusion to the class.



6) Rewrite the model for Niger’s population growth 𝑝(𝑡) = 16.3 ��1 + 0.0368𝑛

�𝑛

0.0368�0.0368𝑡

with the base

replaced with e

𝑝(𝑡) = This is the formula for any growth rate compounded continuously. It is also used in the financial business to calculate continuous compounding of money. PROBLEM STATEMENT: If Niger’s population is currently 16.3 million, what will its population be one year from now? 7) Use your model to try to solve for when Niger’s population will reach 25 million. SPOILER ALERT: You need to know that loge(x) = ln(x) and is called the natural log. Find the calculator button and then you can finish solving the problem.



π Has Some Company – Answer Key

According to the CIA World Factbook (https://www.cia.gov/library/publications/the-world-factbook/fields/2002.html) Niger is the country with one of the fastest growing populations. Niger’s population growth rate is approximately 3.68% annually. In comparison, U.S. population growth rate is approximately 0.93% and world population growth approximately 1.165%. PROBLEM STATEMENT: If Niger’s population is currently 16.3 million, what will its population be one year from now? We will be able to answer this accurately using algebra the end of the lesson. 1) Write an equation to model Niger’s population at any year t.

𝑃(𝑡) = 16.3(1.0368)𝑡

2) In your group answer the following: a) How would you describe the relationship/trend?

The population model shows uncontrolled exponential growth.

b) Is this model reasonable? Justify your thinking. For example discuss in your groups how you could adjust your model to make it more realistic. Does the population of a country only change at the end of a year?

Students should realize the limitations of this model. It may be accurate for a short time, but it does not make sense to assume that a population will grow that quickly indefinitely. Additionally, populations are constantly changing. The change in a country’s population does not always occur at the end of a year.





3) If t is in years, how would you rewrite your population equation to determine the population if it grew only a) at the end of each month?

𝑝(𝑡) = 16.3 �1 + 0.0368

12�12𝑡

b) at the end of each second?

𝑝(𝑡) = 16.3 �1 + 0.0368

31,536,000�31,536,000𝑡

4) Recall that for compound interest

nt

nrPtA

+= 1)(

�

A = Amount at any time

P = Principal (starting amount at t = 0)

r = interest rate as a number not a percent

n= number of compoundings annually

t = time in years

Compare the compound interest model to the model for Niger’s population growth

𝑝(𝑡) = 16.3 �1 + 0.0368𝑛

�𝑛𝑡

p = the population at any time 16.3 = the starting population for this problem 0.0368 = is the growth rate as a number n = number of times in a year you will calculate population, t = time in years. What is the same? What is different? The equation in problem 3 is the same as this equation.



Considering exponent properties, explain why it is equivalent to rewrite the population model as:

𝑝(𝑡) = 16.3 ��1 + 0.0368𝑛

�𝑛

0.0368�

0.0368𝑡

Students can verify that the two equations are the same using exponent rules.

5) Looking at just the base of the exponential expression, evaluate �1 + 0.0368𝑛

�𝑛

0.0368 for values of n in the table below. Record your answers to 4 decimal places.

n 1 3 10 100 1000 10000 50000

�1 + 0.0368𝑛

�𝑛

0.0368

2.6699

2.7018

2.7133

2.7178

2.7182

2.7183

2.7183

Evaluate the expression for 000,100=n and 000,200=n until you reach a conclusion. Write your conclusion on the board and have one person volunteer to summarize the conclusion to the class. As n approaches infinity, the value of the expression approaches e.

6) Rewrite the model for Niger’s population growth 𝑝(𝑡) = 16.3 ��1 + 0.0368𝑛

�𝑛

0.0368�0.0368𝑡

with the base

replaced with . 𝑝(𝑡) = 16.3𝑒0.0368𝑡

This is the formula for any growth rate compounded continuously. It is also used in the financial business to calculate continuous compounding of money. PROBLEM STATEMENT: If Niger’s population is currently 16.3 million, what will its population be one year from now? One year from now, Niger’s population will be 16.911 million people.



7) Use your model to try to solve for when Niger’s population will reach 25 million. SPOILER ALERT: You need to know that loge(x) = ln(x) and is called the natural log. Find the calculator button and then you can finish solving the problem. Niger’s population will reach 25 million people in about 11.6 years.



BUILDING FLUENCY

Using the definition of ln and technology, solve the following equations for t. Discuss your methods and compare your answers with your partner as you progress through each section. Section 1: a) et = 2 b) et = 20 c) et = 55 d) et = d With your group compare the above problems with the base 10 problems you did after the pH problems. Section 2: With your group members, compare the algebraic processes of section 1 and 2 noting similarities and differences in the structure of the equations.

a) Is te⋅3 the same as te)3( ?

b) What would be the first step in solving the following exponential equations?

3·et = 900 5·et = 400 21 ·et = 44 a·et = d

Answer Check: Section 3:



a) With your group members, review your work in section 1. Solve the exponential equations and record any similarities and differences in the structure of the equations.

Equation WORK Similarities and differences between equations, the relationships they represent, and their solutions

11·e2t = 2560

74

·e2.56t = 641

1507 32

=⋅t

e

a·ect = d

EXIT TICKET

Using a graphing calculator, graph the two functions xey = and xy ln= .

Make a sketch for each and identify the important characteristics of both graphs. What do you notice about the two graphs?



Assessment: Using the Natural Log to Solve Exponential Equations

1) Bacteria populations grow continuously. One bacteria cell will split and create new cells continuously at a rate of 4% per minute. If we start with 726,431 cells, the equation for the bacteria populations as a function of t hours is P(t) = 726431•e0.04t Find how long it would take for the population to reach 35,238,456. Show your work and check your answer.



Curriculum Embedded Performance Assessment (CEPA) Superintendent Dilemma: There’s a flu epidemic! Will we have to close the schools?

The CDC has put out an advisory that this season’s flu virus may lead to an epidemic. You work for a risk analysis firm and a school superintendent has hired you to predict the outcome of a flu epidemic and recommend a prevention plan with the goal of keeping the district schools open. The town school committee and board of health have given you the following information:

• The district has 3200 students. • The district will close schools when 24% of the students are out sick. • Students recover from the flu 7 days from when it was contracted and return to school. • Mathematically, this means the number of students out sick must stay below 24% of the

total population for at least 7 days. • The plan will be implemented once 10% (320) students are ill. Therefore, the starting

population of sick students is 320. • According to the research we know that:

o With no prevention, the sick student population growth rate is 100%. o If every student wears a facemask, the sick student population growth rate is 50%. o If every student uses hand gel, the growth rate is 24%. o If students use both a facemask and hand gel, then the growth rate is 50% of 24%.

• A successful plan is the most inexpensive. • The cost to provide one new face mask to each student for 7 days is $2016. • The cost to allow each student to apply hand gel twice a day for 7 days is $1008.

Your task is to generate the model equations for each scenario, (1) no prevention, (2)facemask only, (3) gel only and (4) facemask and gel combined and present a report to the superintendent and board of health detailing when each model predicts the district would need to close.

You will create a presentation (e.g.,, report, podcast, script, PowerPoint/Keynote, etc.) using precise mathematical language for the superintendent and board of health that includes for each scenario a model equation, tables, graphs and algebra. The presentation must clearly identify the time when (or if) the superintendent would have to close the district schools. The report should have a professional appearance and include an executive summary with your recommendation for the best preventative measures.



Solutions

t = # of days, P = total number of sick students

Plan Model School closes in

No prevention P = 320et .8813 days

Facemask P= 320e0.5t 1.7563 days

Gel P=320e0.24t 3.6438 days

Facemask + Gel: P = 320e0.12t 7.2938 days



Curriculum Embedded Performance Assessment (CEPA) Superintendent Dilemma: Flu epidemic, will we have to close the schools?

Exceeds Standards

Meets Standards Partially Meets Standards

Does not Meet Standards

STANDARDS

F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior


Uses mathematical properties and relationships features of logarithmic functions, using them to accurately graph and identify clearly the correct solution to when (if) the school district should close.

Uses mathematical properties and relationships of logarithmic function, using them to graph the function and identify the correct solution to when (if) the school district should close.

Uses key features of graphs and tables, and uses mathematical properties and relationships logarithmic function, using them to graph the function and identify the correct solution to when (if) the school district should close.

Uses provided mathematical properties and relationships to sketch the graph.

STANDARDS


SMP. 4 Model with Mathematics


Uses mathematical properties and structure of exponential expressions to rewrite the as the natural log to determine when (if) the number of ill students reaches 750. Work is detailed and accurate, in all representations: graphically, algebraically and numerically in tables.

Uses mathematical properties and structure of exponential expressions to rewrite the as the natural log to determine when (if) the number of ill students reaches 750. Work is accurate, in at least two representations graphically, algebraically and numerically in tables.

Uses mathematical properties and structure of exponential expressions to rewrite the as the natural log to determine when (if) the number of ill students reaches 750. Work is accurate, in at least one representation graphically, algebraically and numerically in tables.

Uses mathematical properties and structure of exponential expressions to determine when (if) the number of ill students reaches 750. Work is not accurate, in any representations graphically, algebraically and numerically in tables.



CEPA Report

Mathematical Concepts


Work shows evidence of in-depth understanding of using exponential and logarithmic functions to address when (if) the school district should close

Work shows evidence of full understanding of using exponential and logarithmic functions to address when (if) the school district should close

Work shows evidence of some understanding of using exponential and logarithmic functions to address when (if) the school district should close

Work shows evidence of limited understanding of using exponential and logarithmic functions to address when (if) the school district should close

CEPA Report

Precision of Language


Complex mathematical language is accurately used (exponential, logarithm, natural log, end behavior etc.) throughout the report to communicate mathematical reasoning and recommendation for when (if) the school district should close.

Appropriate mathematical language is accurately used (exponential, logarithm, natural log, end behavior etc.) in much of the report to communicate mathematical reasoning and recommendation for when (if) the school district should close.

Some mathematical language is accurately used (exponential, logarithm, natural log, end behavior etc.) in much of the report to communicate mathematical reasoning and recommendation for when (if) the school district should close.

Limited mathematical language is accurately used (exponential, logarithm, natural log, end behavior etc.) in much of the report to communicate mathematical reasoning and recommendation for when (if) the school district should close.


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Modeling Exponential and Logarithmic Functions

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