EAST ALLEN COUNTY SCHOOLS

Bundle 6

Algebra II

an

Big Idea: Inverses Exponentials and

Logarithms

Enduring Understandings Essential Questions Exponentials are graphed by picking points and translating.

Logarithmic functions can be grouped by picking points and translating.

Use the properties of logarithms to expand, simplify, or solve an expression or equation.

Use logarithms to solve exponential growth and decay problems.

Exponentials and logarithms are inverses of each other.

What are the relationships between an exponential and logarithmic equation?

How are functions and their inverses related?

What are the applications of an inverse function?

How are exponential and logarithmic equations used in real world applications?

From where do we get the irrational number e?

CC/Learning Targets Core Vocabulary Links to Technology A.SSE.1b A.SSE.2 A.APR.6 A.APR.7 A.REI.2 A.REI.11 F.IF.4

F.IF.7b F.BF.1 F.BF.4 A2.6.1 A2.6.2 A2.6.3 A2.10.1

exponential/logarithm equation continuous compound interest natural logarithm asymptote

Coffee Problem - iFactor (app) - MathRefFree (app) - AlgebraGenie (app) - KhanAcademy (app) - FreeGraphingCalculator (app) - LogarithmSolved (app)

Bundle Performance Task(s) You work for MacBurger Queen fast food restaurant that has just been sued because a customer burned themselves on your coffee. The ideal temperature for

brewing their coffee is 170 F. But that’s a bit too hot to drink. At that temperature, a customer who spills it on him or herself may get burned. They need a special

container that will heat the water to 170, brew the coffee at that temperature and then cool it rapidly to a more drinkable temperature at 140 and then keep it above

120 for a reasonable time without further cooking. There are three companies that are vying for the chance to sell you their new and improved containers. Your task is to recommend to MacBurger Queen which container to purchase. Each group will make a video to showcase their mathematical discoveries. All students will watch these and determine who had the most convincing presentation in the video. The teacher will determine the correctness of the mathematics. (See the attached PDF document to hand out to students.)

Algebra 2 Math Bundle 6

Quarter 3 Feb - Mar

Trigonometry

Algebra II Algebra II – Bundle 6

CC/Learning Targets Resource of Ideas Evidence of Learning A.SSE.1

(A2.2.4a,b) Interpret expressions that represent a

quantity in terms of its context.

-Prentice Hall Algebra II Lessons 8.1, 8.2, 8.3 - Pearson Algebra II iBook Lesson 7.2, 7.3, 8.4

-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments

A.SSE.2 Use the structure of an expression to

identify ways to rewrite it. For example,

see x4 – y

4 as (x

2)2 – (y

2)2, thus

recognizing it as a difference of

squares that can be factored as (x2 –

y2)(x

2 + y

2).

-Prentice Hall Algebra II Lessons 9.3, 9.4, 9.5 - Pearson Algebra II iBook Lesson 8.4

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A.APR.6

(A2.5.2a) Rewrite simple rational expressions in

different forms; write a(x)/b(x) in the

form q(x) + r(x)/b(x), where a(x), b(x),

q(x), and r(x) are polynomials with the

degree of r(x) less than the degree of

b(x), using inspection, long division, or,

for the more complicated examples, a

computer algebra system.

-Prentice Hall Algebra II Lesson 6.3 - Pearson Algebra II iBook Lesson 8.6

-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments

A.REI.11 (A2.2.4c,d) (A2.3.7a,b) (A2.5.4a) (A2.5.5b) (A2.6.4a)

(A2.7.4a,b)

Explain why the x-coordinates of the

points where the graphs of the

equations y = f(x) and y = g(x) intersect

are the solutions of the equation f(x) =

g(x); find the solutions approximately,

e.g., using technology to graph the

functions, make tables of values, or

-Prentice Hall Algebra II Lessons 8.1, 8.2, 8.3, 8.5, 8.6 - Pearson Algebra II iBook Lesson 8.6 First Concept Byte after Lesson 8.6 Second Concept Byte after Lesson 8.6 Concept Byte (after lesson 7.6)

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Trigonometry

Algebra II Algebra II – Bundle 6

find successive approximations.

Include cases where f(x) and/or g(x)

are linear, polynomial, rational,

absolute value, exponential, and

logarithmic functions.

-Bundle Task Assessments

F.IF.4

For a function that models a

relationship between two quantities,

interpret key features of graphs and

tables in terms of the quantities, and

sketch graphs showing key features

given a verbal description of the

relationship. Key features include:

intercepts; intervals where the function

is increasing, decreasing, positive, or

negative; relative maximums and

minimums; symmetries; end behavior;

and periodicity.

-Prentice Hall Algebra II Lessons 8.1, 8.2, 8.3, 8.5, 8.6 - Pearson Algebra II iBook Concept Byte (after lesson 7.3)

-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments

F.IF.7 (A2.3.4a-f) (A2.7.1a-c)

Graph functions expressed

symbolically and show key features of

the graph, by hand in simple cases

and using technology for more

complicated cases.

b. Graph square root, cube root, and

piecewise-defined functions,

including step functions and

absolute value functions.

e. Graph exponential and logarithmic

functions, showing intercepts and

end behavior, and trigonometric

functions, showing period, midline,

and amplitude.

-Prentice Hall Algebra II Lessons 8.1, 8.2, 8.3, 8.5, 8.6 - Pearson Algebra II iBook Concept Byte after Lesson 8-1 Concept Byte after Lesson 8-3 Lesson 6.8, 7.1, 7.2, 7.3 Concept Byte (after lesson 7.5) - MathGraph (app)

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Trigonometry

Algebra II Algebra II – Bundle 6

F.LE.4 (A2.7.2a-c) (A2.7.3a,b) (A2.7.4a-d) (A2.7.5a) (A2.7.6a,b) (A2.7.7a) (A2.7.8a,b)

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.

-Prentice Hall Algebra II Lesson 8.1, 8.2, 8.3 - Pearson Algebra II iBook Lesson 7.4, 7.5, 7.6

-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments

A2.6.1 a. Simplify an expression with negative

exponents.

b. Simplify and expression with

fractional exponents.

-Prentice Hall Algebra II Lesson 8.2

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A2.10.1 a. Use a variety of problem-solving

strategies, such as drawing a

diagram and writing an equation.

b. Use a variety of problem-solving

strategies, such as drawing a

diagram, guess-and-check, solving

a simpler problem, writing an

equation, and working backwards.

-Prentice Hall Algebra II Lessons 8.1, 8.2, 8.3, 8.4, 8.5, 8.6

-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments

Correlating CC/Learning Targets Teacher Notes

A.SSE.1a A.APR.1 A.CED.1 A.CED.2

A.CED.3 A.CED.4 F.IF.5 F.IF.8 F.BF.3

Don’t Burn Yourself

Your task force has been called in to solve a problem encountered by a local restaurant,

MacBurger Queen. The owners of the restaurant believe that their coffee should be

brewed at 170°Fahrenheit. However, at that temperature it is too hot to drink, and a

customer who accidentally spills the coffee on him or herself might receive third-degree

burns which could result in a lawsuit against the company.

What they need is a special container that will heat the water to 170°, brew the coffee at

that temperature, then cool it quickly to a drinkable temperature, like 140°F, and hold it

there, or at least keep it at or above 120°F for a reasonable period of time without further

cooking as to not burn the coffee. To cool down the coffee, three companies have

submitted proposals with these specifications:

(a) The TherMosDef Company has a container that will reduce the temperature of a

liquid from 200°F to 100°F in 90 minutes by maintaining a constant temperature of

70°F.

(b) Therman Munster Incorporated has a container that will reduce the temperature of a

liquid from 200°F to 110°F in 60 minutes by maintaining a constant temperature of

60°F.

(c) The HottIce Corporation, has a container that will reduce the temperature of a liquid

from 210°F to 90°F in 30 minutes by maintaining a constant temperature of 50°F.

Your job is to make a recommendation as to which container to purchase. For this you

will need Newton's law of cooling which follows:

In this formula, T represents the temperature of the surrounding medium, u0 is the initial

temperature of the heated object, t is the length of time in minutes, k is a negative

constant, and u represents the temperature at time t.

Question 1: Use Newton's Law of Cooling to find the constant k of the formula for each

container.

Question 2: Use a graphing utility to graph each relation.

Question 3: How long does it take each container to lower the coffee temperature from

170°F to 140°F?

Question 4: How long will the coffee temperature remain between 120°F and 140°F?

Question 5: On the basis of this information, which company should get the contract

with MacBurger Queen?

Question 6: Define "capital cost" and "operating cost." How might they affect your

choice?