harrington uprlessonschedule

8/2/2019 Harrington UPRLessonSchedule

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Unit-Level Objectives

1. Students will conjecture about the relationship between the symbolic form of an exponential function

and its domain, range, asymptotes, and limit.

2. Students will use the structure of equations or expressions to identify useful ways to rewrite problems.

3. Prove that exponential functions grow by equal factors over equal intervals.

4. Construct linear and exponential functions, including arithmetic and geometric sequences, given agraph, a description of a relationship, two input-output pairs (include reading these from a table), or a

contextual problem.

Schedule of Lessons

* Note: Class periods are 95 minutes each so there may be multiple lessons linking to another topic per

day.

Lesson TopicLesson Learning

Objective(s)

Description of howlesson contributes tounit level objectives

Tasks/Activities for the Lesson

Exponential Growth1. Growth Factor2. Growth Rate

(1 Day)

Students will developan exponential functionfrom a series ofmeasurements.

Students will recognizethe difference betweengrowth factor andgrowth rate.

Observe using graphsand tables that aquantity increasingexponentially eventuallyexceeds a quantityincreasing linearly,quadratically, or (moregenerally) as apolynomial function.

Students organicallyconstruct exponentialform using the contextfocusing on initialamounts and the waythe disease isgrowing/mutating.

Students identify thefactor and the amount oftimes the factor appearsas the exponent on thebase. The base is nolonger the variable, butrather the exponent isthe number which varies

the number of timesthe base factor appears.

Looking at the growthpatterns, students willbegin to generalizeabout minima/maximaand their existence.

(1/3/4)

See Tasks/Activities sectionat end of document.

Exponential Decay1. Decay Factor2. Decay Rate

(1/2 Day)

Students will developan exponential functionfrom a series ofmeasurements.

Students will conjecture

Using the exponentialform generated from theprevious day, studentswill link the presence ofa fraction to decaylosing an amount each

See Tasks/Activities sectionat end of document.

Task finishes with discussionof situations identified bystudents to be exponential


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about the relationshipbetween the base valueand whether thefunction is exponentialgrowth or decay.

Students will relateexponential growth toreal-life phenomena oftheir own choosing.

day.

Since the radioactivematerial is breakingdown, by losing halfeach day, students canbegin to conceptualizewhy the decay will neverreach zero.

The decay causes thedifference betweensuccessive points todecrease or, in otherwords, the numbers aremultiplied by a constantfractional factor.

Students also start tobreak down thedifference between

percentincrease/decrease vs. aconstant integer factor.

Students are generatingthese conclusions froma multitude of contexts.

The introduction of thelimit will occur here asstudents begin toconceptualize how the

graph is approaching acertain value though notreaching it.

(1/3/4)

growth or decay.

Graphing ExponentialFunctions

Skills-basedsummative assessment

of last two class periods

(1/2 Day)Part of ExponentialDecay

Students will conjectureabout the relationshipbetween the base valueand whether thefunction is exponential

growth or decay.

Students will movebetweenrepresentations offunctions algebraic,contextual, graphical,and etc.

This is simply anassessment of theprevious activities.Students will be given agraph, a table, and a

function in which theyneed to discuss domain,range, asymptotes, andgrowth/decay.

(1/3/4)

Assessment:Moving betweenrepresentations focusing onfrom a ________ to a graph:

1. Function

2. Table3. Description/Context

Compound Interest Students will use the Students will develop See Tasks/Activities section


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* Actual lessonschedule flip-floppedcompound interest

and same-baseequations to coincide

with the work studentswere producing;however, this was theidea.

(1 Day)

properties of exponentsto transformexpressions forexponential functions.For example theexpression 1.15t can berewritten as(1.151/12)12t 1.01212t to reveal theapproximate equivalentmonthly interest rate ifthe annual rate is 15%.

the notion ofcompounded-nessthrough conversationand explorationsumming amountbefore applying interesteach period.

Using same factors,students can useexponents to begin toapproximate investmentopportunities.

Connection to same-base equations.

(2)

at end of document.

Same-Base Equations

The end serves as thelaunching point forlogarithmic functionsan example where thebases cannot easily beseen as equal.

(1 Day)

Students will use

properties of exponentsto find equality inproblems with differentbases bases wherethey are multiples of thesame prime number.

Students breakdown

different factors intotheir prime (or mostbasic) numbers andreason that same baseallow for equality ofexponents. Students willalso hypothesize wheresuch assumptions maybe false.

Showing equalitythrough exponents all

questions ease into thenotion of rewriting forsimplification. Studentscan conceptualize howvisually differentmutations can actuallyresult in the samenumber of victims.

Throwing in unequalbases can allowstudents to begin to

approximate exponentvalues and lead into theintroduction oflogarithms and thechange of baseproperty.

(2)

See Tasks/Activities section

at end of document.


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Task/Activities

EXPONENTIAL GROWTH

Name: __________________________________ Class: __________________ Date: ______________

Day 1 Research: The Disease Wont Stop Growing!

Student X has just passed the virus to one other classmate. Current research has placed the number at 4 for the

number of humans each infected human can infect. Should we begin panicking or is the spread slow enough to

avoid a catastrophe?

1. Think about the situation and create a visual aid to show the spread across a number days (for example:

chart, graph, picture, diagram).

2. Will our whole school be suffering (or deceased) from Coughing Veins in 12 days? What should be

our panic level? (Assume our school has 500 persons.)

3. Come up with a description that will help us figure out how fast this disease can spread for a given

number days, such as n days.

4. Describe how you came up with your solution.

Name: __________________________________ Class: __________________ Date: ______________

Day 1 Research: The Fall of the Human Race?

So far, we have looked at the spread of the disgusting Coughing Veins disease and found that we are in a

world of trouble if we cant find a way to stop it. But what is happening to our race? Are we on our way out of

existence? Lets assume the outbreak began in Michigan which has a population of 9,969,727 persons.

Days Number of Humans Left


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1. How many Michiganders are still alive after 5 days? What about 20 Days?

2. In what ways are the two situations mathematically related?

3. Write a generalization (in other words a function) for the decline of the humans.

Name: __________________________________ Class: __________________ Date: ______________

Day 1 Research: The Disease is Mutating!

Our research indicates that variations of Coughing Veins are beginning to appear throughout the state of

Michigan. The scientists provided us with the equations, but we need to figure out what the equations say about

that particular strain of Coughing Veins.

Describe each function in terms of the spread of Coughing Veins.

1.

2.

3.

EXPONENTIAL DECAY

Name: __________________________________ Class: __________________ Date: ______________

Day 2 Research: The Stone is Decaying

Scientists have stated that the meteorite is breaking down at a rate half its size per day. As of today, the weight

is 25 kilograms. As the radioactive material decays, the powerful effects begin to dwindle. Use your knowledge

of exponential growth to determine how long it will take for the Solanum to reach a stable statewhere it does

not affect humans any longer.

1. After the fifth day, how much radioactive Solanum is left in the meteorite? (Creating a chart may help,

there are also scissors and paper to help visualize exactly what is happening.)

2. Create a graph describing the situation. What does the y-intercept of the graph tell us?


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3. If the Solanum no longer affects humans at a weight of about 0.01 kilograms, how many days will it

take for stability to happen? [HI NT: If we quarantined the Solanum for 5 days, is that long enough?

What about 20 days?]

4. Apparently scientists also miscalculated the decay of the radioactive Solanum. Instead of halving each

day, the stone decays by 5% of its starting size each day. Write a function matching this description.

SAME-BASE EQUATIONS

Name: __________________________________ Class: __________________ Date: ______________

Day 3 Exploration: Meanest Diseases

A couple classes ago, we took some time to explain translations in exponential growth using mutations. Since

then, the disease has mutated what seems like an endless number of timeseach more sinister than the last.

Today, our job is to find on what day different mutations result in the same amount of infections. Each pair of

mutations is represented differently, so we will need to be creative with how we generate the equations.

1. Using the chart, write the function that describes the spread of new infections for this mutation:

Day Number of New Infections (Mutation A)

0 1

1 2

2 4

3 8

On what day will Mutation A infect new victims?

2. On what day will Mutations B and C infect the same number of new victims?

Mutation B:

Mutation C:

3. On what day will Mutations D and E infect the same number of new victims?

Mutation D:

Mutation E:

4. On what day will Mutations F and G infect the same number of new victims?

Mutation F:

Mutation G:


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5. Which mutation should the human race worry about the most? ______________________

How do you know?

6. On what day will Mutation H infect 724 people?

Mutation H:

How is this problem different from the previous 5 questions?

COMPOUND INTEREST

Name: __________________________________ Class: __________________ Date: ______________

Day 4 Exploration: Prepping for Doomsday

Many people prepare for hard circumstances by saving money; however, at the end of the world, money doesnt

really mean much does it? So, instead of saving for the future, as the head of your family, you started saving forthe date, 01 January 2012. Taking out all of your monetary gains from banks in Michigan on this date would

give your family enough time to spend the money in preparation before the 22 December 2012 end of the world

Lets say your doomsday fund began in 1997.

DIRECTIONS: Using the information sheet provided, answer all of the following questions.

Preliminary Question: What does compounded actually mean as in, The money was compounded

quarterly?

1. Consider the list of items provided. In order to create your fund, you sold all non-essential belongings.

Take some time and determine the non-essentials.

2. Add up the monetary worth of all your non-essential belongings. You have sold all of them at a silent

auction for the given valuesrecord the worth of your freed assets.

________________________

3. At first glance, which investment plan seems as if it will provide your family with the most monetary

gain? Why?

4. Using each plan, find the total worth of your investment from the time you opened the account.

5. Which investment opportunity provided you with the best financial position? Explain why you think that

investment gave the most?


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6. Why is it beneficial to be compounding as much as possible? Use growth rate in your explanation.

Doomsday Reference Sheet

List of Family Possessions and Values:

Possession Value ($) Possession Value ($)

Victorian-era Home 81,000 SUV 11 23,000

Mid-size Sedan 02 7,500 Antique Coin Collection 5,000

Refrigerator 500 Plasma TV 500

Furniture 2,000 Anti-Burglary System 200

Family Heirloom 300 Sports Memoralbilia 4,000

Canned Food 100 Dog 30

Cat (x5) 10 Social Security Number 5000

Baby Crib 100 Kidney (Black Market) 93,000

Cell Phone (x3) 300 Computer/Laptop (x2) 900

List of Investment Opportunities:

Institution Interest Rate (%) Compounded Min/Max Investment ($)

Prudential 11.7 Monthly 25,000/None

Vanguard 11.8 Bi-monthly 20,000/None

Edward Jones 11.8 Quarterly 35,000/None

Merill-Lynch 12 Annual 32,500/NoneYour Rich Uncle N/ADoubles Investment When You Need It None/7,000

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