0Unit Lesson Plan‐ Suzanne Crabtree Introduction: Unit: Exponential Functions Subject: Precalculus Teaching Method: Inquiry based as well as traditional lecture style; scaffolding and Discovery as well Motivation: The main motivation for this whole unit is how many wonderful applications there are for exponential equations from population growth to radioactive decay, it is easy for the students to see just how useful these equations really are. DPI Standards: 2.01 Use functions (polynomial, power, rational, exponential, logarithmic, logistic, piecewisedefined, and greatest integer) to model and solve problems; justify results.
a. Solve using graphs and algebraic properties.
b. Interpret the constants, coefficients, and bases in the context of the problem.
2.03 For sets of data, create and use calculator-generated models of linear, polynomial,
exponential, trigonometric, power, logistic, and logarithmic functions.
a. Interpret the constants, coefficients, and bases in the context of the data.
b. Check models for goodness-of-fit; use the most appropriate model to draw conclusions or
make predictions.
Textbook: ‐Precalculus: Functions and Graphs – Swokowski‐ Cole ‐Technology: The main use of technology is through calculators however there are a couple of times where computer generated games are used in review sessions. ‐Discourse/Writing: Before each lesson I have given the students 5 minutes to ponder a mathematical topic in their journals. Some are shorter than others and require less thought however most encourage the students to think deeply about the topic on which they are about to discuss. ‐Problem Solving: This section is a little heavy on the problem solving however, there are also a lot of word problems which not only requires the students to translate from English to mathematical language, but also for them to have a firm grasp on the concepts at hand in order for them to apply them properly. ‐Assessment: There is a test at the end of this unit that will cover everything in the unit and each day, the students group work or warmup is collected. ‐Special Needs: If I had a class with special need students I would try to apply instructional practices that work for these students. Providing a higher level of support than necessary for most students is also important. Structure and guidance is essential to the learning of students with special needs. Lesson 1: Journal question: In words describe what an exponent is. (5 min.) Give them one side of the equation and see if they remember the rule for the other side of the equation: axay=ax+y =ax‐y (aP)x=aPx (ab)x=axbx = (25 min.) Then do the exponent review game: EXPONENT REVIEW
Set Up: Work in groups of 2-3
Items need: Die and pencil
Rules:
1. Roll the die to determine who plays first.
2. First player rolls the die and chooses any expression from the appropriate
column. (For example, if you roll a 2, use Column 2)
3. Simplify your expression and mark an X in the box. The exponent tells you how
many points you receive.
(For example, you would get 7 points for the
following: x7, 2c7, or
2
)
x7
4. Each expression can be used only once. If you roll a number in a column with no
available expressions, you lose your turn.
5. Watch your opponents: If you catch a mistake in their simplifying, they lose 10
points.
6. When time is called, the player with the highest score wins!!!
1
2
3
4
5
6
(n4)2
n4 • n-8
(x-2 • x4)3
p-1 • p-7
z-6 • z4
y8 • (y6)-2
(d-4)3
(a-2)5
(c-3)3
(f-2 • f3)5
n-4 • (n4)2
e-4 • e-3
j2 • j3
(h4 • h5)2
u2 • u-7
g-6 • (g-3)-2
(y-1 • y-2)3
(w-6 • w7)-2
k3 • k2
k7
⎛z ⎞
⎜⎜ 3 ⎟⎟
⎝z ⎠
s −3 s 7
s4
⎛m ⎞
⎜⎜ − 4 ⎟⎟
⎝m ⎠
2
4
⎛w
⎜⎜
⎝ w
−2
⎞
⎟⎟
⎠
2
−2
4
p
p4
Source: http://www.ilovemath.org/index.php?option=com_docman&task=cat_view&gid=52&Itemid=31 (5 min) Come back together as a class and discuss the following question: ‐What happens if we switch the constant and the variable of a quadratic equation, or any polynomial for that matter? We get an exponential equation! ‐Basic definition of exponential equation: f(x)=ax, a>0, a does not 0 ‐Explain the restrictions‐ if a<0, the function would oscillate and if a=0, f(x)=0. Split students up into assigned groups of 3 (randomly assigned):
(30 min.) Work on this explorations worksheet:
Graph y = x 2 and y = 2 x by hand. Put both functions in your graphing calculator and “fix the
window” in order to see a complete picture of the graph.
1. How are these graphs similar? In what ways are these graphs different? What are the
domains of the functions?
2. Which graph has no x-intercept? Are the y-intercepts the same positions?
3. Compare the range of y = x 2 with the range of y = 2 x .
1
2
Now graph y = 2 x and y = ( ) x , both by hand and in your graphing calculator.
4. Find the domain and range of these functions.
5. Which function models exponential growth and which one models exponential decay?
6. Would you prefer a monthly allowance of $1000 or to start with 2 cents and double it each
day, i.e. on day 1 you would receive $.04, day 2 - $.08, etc? Give arguments for your choice.
Here is the graph of the Allowance problem. Note: f ( x ) =.02( 2 x ) where a = the initial amount
of allowance and b = the doubling effect.
Source: http://transitions.kennesaw.edu/modules/expo_function/exp.htm ‐(15 min.) Come back together as a whole class. Discuss the answers to the worksheet and make sure there are no misconceptions. ‐If there is time left over, let them start their homework. Homework: pg. 299 1‐9 (odd) Misconceptions: Possible confusion between the two different types of equations, getting the characteristics confused. Motivation: The beginning game, competition, social interaction within groups. Assessment: group work Questioning: The questioning of this lesson occurs mainly when the students are asked to switch the variable and the constant of the polynomial equation, the discussion that follows is also a part of the questioning. Lesson 2‐ Logarithms: Journal Question: In what ways are exponents used in your daily life? (10 min) Intro to logs: ‐How do we solve this equation for x? 100 = 10x ‐Although we can easily see that the answer is 2, what about a more complicated situation such as… 32x‐1 = 27 ‐In these situations we can use logarithms to solve for x. Basic rule: ‐ y = logbx if and only if by = x, where x > 0, b > 0, and b 1. ‐Explain restrictions. ‐So going back to our first problem where b=10 100=10x turns into log10100=x. ‐Show them how to do this on their calculator. ‐On a calculator log10100=2, so they can see that this is for real. ‐So now let’s try our second more complex problem: 32x‐1 = 27 turns into log327=2x‐1 ‐Solving for x we get x=2. ‐This is why logs are useful so we can solve equations we couldn’t otherwise solve. ‐‐Note short hand notation for log10x is logx. ‐‐Also note that logbb2 = 2 (10 min)‐Some practice problems (complete in pairs‐ whoever is next to you) 1) log(x) = 3 2) log(10,000) = x 3) log(1) = x –explain why the answer is 0 4) log(50) = x ‐Can you solve this by hand? Have particular students present their answer on the board, and discuss the answers. Other uses of logs in life: ‐decibel scale ‐Richter scale ‐pH scale (20 min.) Explain logarithmic properties: 1.
2.
3.
4.
logb(xy) = logbx + logby.
logb(x/y) = logbx - logby.
logb(xn) = n logbx.
logbx = logax / logab.
‐Explain why these each work and that these formulas are simply shortcuts that make life much easier. (20 min.) More practice problems in groups of 2: 1) log69 + log64
2) log34 - log3324 3) log4167 4) 3 log210 + log23 - log215 - 2 log25 5) 2 log515 - 2 log53 Have the students put up their answers again and discuss the answers.
(30 min) Play a game: Box-wad-of paper
A review game that is similar to Jeopardy.
Site: http://www.ilovemath.org/index.php?option=com_docman&task=cat_view&gid=52&Itemid=31
Found under ‘Review Game- Exponents and logarithms’
Homework: pg 336 1-13 odd
Pg. 325 1,2,3 (a,c,e), 5,7,9
Motivation- Competition to win the game. Applications of logs in real life.
Misconceptions: confusion over what logs are since it is the first time they will have seen
them.
Assessment: Participation in group work and game.
Questioning: The questioning of this lesson is not as significant as others simply
because the students are learning brand new material that they would have no way of
knowing or figuring out themselves. However practice problem no. 3 and 4 of the first
set of problems are designed to get the students thinking on that higher level about how
logs are used.
Lesson 3- Exploring Exponential Equations:
Journal Question: What is a log in your own words?
(15 min.) Warmup/Review:
A few problems that they should know how to do…
Write each of the following in terms of simpler logarithms. 1
1) 2) 3) Students can come up and present their answers.
(45 min.) Activity: Analyzing the Relationship between a Logarithmic Function and Its
Inverse
*Completted in Random
mly Assigned Groups of 3 This activiity will be don
ne after studyying algebraicc functions, their inverses and basic pro
operties of exponential and logarithmic functio
ons. PART 1: IN
NVESTIGATIN
NG A LOGARITTHMIC FUNC
CTION Consider tthe followingg function: f(x) = lo
og(x) 1. M
Make a graph of f(x) using yyour graphingg calculator.
2. Th
he table below lists a seriees of nine outtput values from f(x). Fill in the table byy using the grraph an
nd the TABLEE to estimate tthe missing xx values. TABLE 1
X F(x) .05 .33 .89 1.2 1.5 1.75 2.1 2.3 2.5 Using a taable of y‐value
es to find thee corresponding x‐value: Put f (x ) = log(x ) into Y1
In TBLSE
ET, let TblStartt = 0 and ∆Tbl = 10. As you
can see, 1.75
1
is the y-vaalue somewherre between x =
50 annd x = 60.
Now, let TblStart
T
= 55 an
nd ∆Tbl = 1. You
Y can see thaat
1.75 is
i a y-value bettween x = 56 and
a x = 57.
Let TblSttart = 56 and ∆Tbl
∆
= 0.1. Oncce you can be
accurate to the tenths place, choose the x-value.
3. Create two lists L1 and L2 in the calculator to store the x and f(x) values. Put the x values in L1 and the f(x) values in L2. 4. Generate a scatter plot of (x, f(x)) using the STAT PLOT feature and the two lists, L1 and L2. Do your x values appear to accurately represent f(x)? If not, modify the values so that they better represent the curve. 5. Generate a new STAT PLOT using L2 and L1 instead of L1 and L2. Hypothesize at least two different families of functions to which this new curve might belong: family 1: __________________________ family 2: __________________________ 6. Using L2 and L1, test your hypothesis with the different regression analysis options and record your results, including the regression equations and R2 values, below. 7. Which of the above selections do you think represents the best fit? Why? 8. If you selected Exponential, you were CORRECT! Congratulate yourself. 9. Look at the exponential equation that you generated. Using your mathematical knowledge and some rounding, estimate g(x), the actual equation of the inverse of f(x): Source: www.math.uakron.edu/.../Inverses/Logarithmic-ExponentialInverseRelat_84.doc
*-Come back as a class and discuss the results of the activity and the students thought on it.
(5 min) Lead the following activity in class as a whole, explaining the constants along the way
and asking the students what they think about the questions. In order to make the transition as
smooth as possible, there can be a brief break where homework and upcoming tests and such
are explained.n order to make the transition as smooth as possible, there can be a brief break
where homework and upcoming tests and such are explained.
(25 min.)-Below is an interesting example of exponential decay. It shows the voltage from a
capacitor over time after removing the power source from the circuit.
The function is
per second
, where a is the charge at t=0 and b is % charge remaining
1. What effect does the constant b have on the graph of the function with the rule f ( x) = b x ?
2. What effect does the constant a have on the graph of the function with the rule
f ( x) = a (2 x ) ?
3. What effect does the constant c have on the graph of the function with the rule f ( x ) = 2 x − c ?
4. What effect does the constant d have on the graph of the function with the rule
f ( x) = 2 x + d ?
5. Are there any restrictions on a, b, c, or d?
Source: http://transitions.kennesaw.edu/modules/expo_function/exp.htm
Homework: pg. 300 #11 and 12 (a-j every other letter), 13-23 odd
Misconceptions: Some of the students might become confused with the exponential decay graph
because they have, up to this point only seen exponential growth models.
Motivation: Social pressure in groups, students don’t want to look unintelligent in front of their friends.
Assessment: Collection of group work
Questioning: Both of the activities presented in this lesson are inquiry based, asking questions of the
students, trying to prompt higher level thinking. Lesson 4‐ Analyzing the graphs of exponential equations: (35 min)Present the following word problem to the class: Š There are 4 bacteria cells in your petri dish in biology class. Each cell divides into to two once a minute. Analyze the growth of the population of this colony of bacteria. ‐Discuss with the class what they think it means to ‘analyze the growth’ and what they think the first step should be. ‐Lead them through the process of making a table and graphing the ‘data’. ‐What kind of function does this look like? Exponential! ‐And the general form of an exponential equation is … y=A*B^(cx) ‐Next lead them through how to find the constants. They should already know the domains for the constants and have a clue how to find them. ‐After finding the constant A by using the starting value (A=4) find the relationship between B and c and explain what they can do with this relationship. (45 min.) A fun activity to reinforce these ideas… Split them up into groups of 3 randomly assigned. Teacher’s Initials ________
Student Activity #1 m&m
M & M Activity Supplies Needed: Regular M&Ms, paper towel, and a paper cup
Procedures:
a. Count and record the number of M&Ms you received. This is trial number 0.
b. Put the M&Ms back in the cup.
c. Pour the M&Ms in a single layer.
d. Remove any M&Ms that do not land “m” up.
e. Record the trial number and number of M&Ms remaining.
f.
Put the remaining M&Ms back in the cup.
g. Repeat steps c through f until no M&Ms remain.
h. Plot the points in your table on your graphing calculator using a line graph plot. Do not
plot any points where the number of M&Ms left is zero.
i.
Examine the table for patterns.
j.
Compare your plot to a line. Is there a constant difference? ___________
k. Compare your plot to an inverse variation graph. Is the product of the Trial # and the #
of M&Ms left a constant? Explain why you should check
this.___________________________________________________________________
______________________________________________________________________
_____________________________________
l.
Approximately what fractional part of your M&Ms left from Trial #2 remain in Trial #3?
_______________ About what fraction of your M&Ms from Trial #3 remain in Trial #4?
_____________________ Continue this process and determine if a pattern exists.
Explain what you see.
______________________________________________________________________
________________________________________________
1
2
m. Finally, compare your graph to an exponential graph of the form M ( t ) = a( ) t where a
is a constant. Experiment to determine a constant, a, so the function appears to pass
through most of the points?
______________________________________________________________________
________________________________________________
n. Does this simulate anything in the real world? ______________________
o. Turn in both sheets and your completed table.
# of M&M’s left
Trial # 0
1
2
3
4
5
6
7
8
9
(10 min.) Talk about what they learned during this activity and let them eat the m &m’s. HW: pg 300 #25, 27, 29 Misconceptions: Some students might be confused with the exponential decay idea since they have mainly been looking at exponential growth problems. Motivation: M&M’s! Assessment: group work Questioning: Again both of these activities are inquiry based. In the first one, I am asking the questions of the students trying to guide them to where I want them to go and in the second activity they are doing this on their own with the guidance of a worksheet and group members. Lesson 5‐ The number e: Warmup Question: What is the fifth letter of the alphabet? (5 min.) Introduction to e: The number e is a mathematical constant named after Euler (a Swiss mathematician). One reason that this number is important is because of the properties of the function f(x)=ex. This function’s tangential slope is always the same as the value of x at that point. (explain with a graph) Because of this property one can imagine that exponential functions with e in them are very important. So let us begin working with them. New Notation: Logex=lnx This new notation (ln) is called a natural logarithm. It behaves the same way as any other logarithm does and the same properties apply. So… Ln1=0 Ln4=1.386 leads to e1.386=4… So now that we know all about the number e, what can we do with it? Well we can’t use e quite yet, but we will soon. You must walk before you can run!
(10 min.) So that being said, here is a simple interest problem to guide them through…
Suppose a bank is offering its customers 3% interest on savings accounts. If a customer
deposits $1500 in the account, how much interest does the customer earn in 5 years?
In this problem, we are given the interest rate (r), the amount put into the account (P), and
the amount of time (t). However, before we can put these values into our formula, we must
change the 3% to a decimal and make it 0.03. Now we are ready to go to the formula.
So after 5 years, the account has earned $225 in interest.
If we want to find out the total amount in the account, we would need to add the interest to
the original amount. In this case, there would be $1725 in the account. Keep in mind that
our formula is only for the amount of interest. The formula can also be solved for other
variables as in the examples below.
Source: http://www.algebralab.org/Word/Word.aspx?file=Algebra_InterestI.xml
-Hopefully this should be something they’ve seen before so one example should jog their
memory.
(20 min.) Next comes Compound Interest, a very important part of our society that requires
basic math knowledge and is a special (and very useful) case of exponential functions:
For working with compound interest problems, we will be using a formula that involves five
variables in an exponential equation. Four of the variables will always be given to you in the
problem. Your job will be to find the fifth variable. The level of difficulty in solving for that
variable will depend on whether it is located in the exponent or not. We’ll look at several
different types of problems that all use the same formula.
The formula for interest that is compounded is
ƒ
ƒ
ƒ
ƒ
ƒ
A represents the amount of money after a certain amount of time
P represents the principle or the amount of money you start with
r represents the interest rate and is always represented as a decimal
t represents the amount of time in years
n is the number of times interest is compounded in one year, for example:
if interest is compounded annually then n = 1
if interest is compounded quarterly then n = 4
if interest is compounded monthly then n = 12
Suppose Karen has $1000 that she invests in an account that pays 3.5% interest
compounded quarterly. How much money does Karen have at the end of 5 years?
Let’s look at our formula and see how many values for the variables we are given in the
problem.
The $1000 is the amount being invested or P. The interest rate is 3.5% which must be
changed into a decimal and becomes r = 0.035. The interest is compounded quarterly, or
four times per years, which tells us that n = 4. The money will stay in the account for 5
years so t = 5. We have values for four of the variables. We can use this information to
solve for A.
So after 5 years, the account is worth $1190.34. Because we are dealing with money in
these problems, it makes sense to round to two decimal places. Notice that the formula
gives us the total value of the account at the end of the five years. This is not just the
interest amount, it is the total amount. Since there are many variables in the equations,
there are several ways that problems can be presented. Let’s look at some other examples.
Source: http://www.algebralab.org/Word/Word.aspx?file=Algebra_InterestII.xml
(15 min.) Practice problem in pairs (whoever is sitting next to you):
-William wants to have a total of $4000 in two years so that he can put a hot tub on his
deck. He finds an account that pays 5% interest compounded monthly. How much should
William put into this account so that he’ll have $4000 at the end of two years?
(5 min.) Come back together as a class and discuss the answers that everyone got.
And now with the next problem we can finally use our new found knowledge of the number
e!
-Ask them ‘what if we wanted to continuously compound the interest instead of just monthly
or yearly?’ Hint: use e.
-After they speculate a bit lead them through this problem…
(20 min.) Interest that is compounded continuously seldom occurs at banks that you might
deal with on a regular basis. However it is very useful for finding the maximum amount of
money that can be earned at a particular interest rate. It is a very effective way to
demonstrate how powerful compounding interest can be.
In working with interest that is compounded continuously, the same formula is always used.
You should be careful to note that for interest compounded for any amount of time other
than continuously, there is a different formula. The following applies only to interest
compounded continuously.
The formula for interest that is compounded is
ƒ
ƒ
ƒ
ƒ
A represents the amount of money after a certain amount of time
P represents the principle or the amount of money you start with
r represents the interest rate and is always represented as a decimal
t represents the amount of time in years
e is not a variable. It has a numeric value (approximately 2.718) although we do
not usually use the value. We simply solve the problem using the “e” button on the
The letter
calculator. So there are four variables in the equation and the problem will give us values
for three of those variables and we will need to solve for the fourth.
Suppose $5000 is put into an account that pays 4% compounded continuously. How much
will be in the account after 3 years?
We know the original amount (P) to be $5000. We know the interest rate (r) is 4% which is
0.04 in decimal form. The amount of time (t) is 3. Since this is interest compounded
continuously, we will use the formula . Substituting the values we know into the equation
and then solving gives
So after 3 years, the account is worth $6107.01. Because we are dealing with money in
these problems, it makes sense to round to two decimal places. Notice that the formula
gives us the total value of the account at the end of the three years. This is not just the
interest amount, it is the total amount. The calculations should be done on the calculator by
using the “e” button and not the decimal approximation for e. One way to think of this
answer is that the most money that will be in an account after three years at 4% interest
will be $6107.01
Source: http://www.algebralab.org/Word/Word.aspx?file=Algebra_InterestIII.xml
(10 min.) And again practice makes perfect so in groups of 2 work out this practice
problem:
-If interest is compounded continuously at 4.5% for 7 years, how much will a $2000
investment be worth at the end of 7 years?
(5 min.) Come back together as a class and discuss the answers.
Homework: Pg. 311 #5, 7, 9, 35, 37
Motivation: The motivation for this lesson is difficult because this is mainly a
lecture based lesson due to the material. However hopefully the students will be
interested in this new number e and why it is so important in mathematics.
Misconceptions: Some students might find it difficult to understand the
differences between the different types of interest, which is crucial to their
understanding of this lesson.
Assessment: Collection of group work.
Questioning: The questioning of this lesson is also difficult because it is a lecture
based lesson however at the beginning of each new section, questions are asked
and at the end the students are asked to complete problems on their own.
Day 6- Review for Test.
First answer any questions they have right off the bat. Then begin review activity of
Jeopardy with all topics being covered on the next day’s test.
Leave some time at the end of class for any additional questions.
Day 7 –Test day