My Exponential Function Unit for Algebra II
Basic Context: This unit is coming right on the heels of function transformations. Students are familiar with translating functions up, down, left, and right; reflecting functions over the x- and y-axes; and vertically and horizontally stretching and shrinking functions.
Structure: The work on exponential functions is broken into four parts.
Part 0: Preliminary Diversion into Inverse Functions
Part I: Graphing exponential functions
Part II: Solving basic exponential function equations
Part III: Applications of exponential functions (carbon dating and compound interest)
Time: This took a total of 13 days — including an introductory activity day, a review day, a day where we did an exponential decay simulation as an entre to carbon dating, and two assessment days.
Nature of Class: I teach 15 students in a non-accelerated Algebra II class. The ability level of the students range the gamut. Many have a hard time thinking abstractly. All have graphing calculators and know how to use them at the basic to intermediate level. We meet 4 days a week for 50 minutes each day.
Broad Goal: The goal for this unit was to really drive home the concept of exponential functions.
Major Failures: I see two major failures. One is not seriously talking about how fast exponential functions grow. This would have been a really fun day, working on a problem like: “Would you rather have (A) $1,000,000 a day for the month of May, or (B) $1 on the first day, $2 on the second day, $4 on the third day, $8 on the fourth day, etc.” The second is just not having a lot of fun with this. The exponential decay simulation we did could have been so much more powerful, and changed in so many fun and really great ways. We could also have done an activity for exponential growth, using real data — population growth, Moore’s Law, or something to do with the Supreme Court. It would have been nice to finish off with a nice 2 day research activity. If for nothing else, to let my students produce something they could be proud of.
Major Strengths: In terms of getting students to understand exponential functions conceptually, I think I’ve done a pretty good job. My students can relate tables, graphs, and equations. They understand why the functions look the way they look. By the time we finished the exponential application days, students were coming up with the formula for the depreciated value of an object without any help.
Materials [NOTE: If you are opening these docs on a Mac, “Select All” and change the font to “Gill Sans.”]
Part 0: Preliminary Excursion into Inverse Functions
PDFs of My Smartboards before class: 1, 2, 3.
Part I: Graphing Exponential Functions
1. Introductory exercise introducing students to exponential growth and decay (.doc)
2. Introduction to exponential functions, and graphing basic exponential functions (.doc); HW (.doc)
Part II: Solving Basic Exponential Equations
1. PDF of My Smartboard before class: 1
Review Sheet on Part 0, Part I, and Part II to prepare students for the assessment (.doc)
Part III: Applications of Exponential Functions
1. Coin Drop Simulation for Exponential Decay (.doc); HW (.doc)
2. Carbon Dating (.doc); HW (.doc)
3. Compound Interest (.doc)
What I Want You To Know: Looking at just the stark documents, this whole unit seems like it might be a bit formulaic. However, particular moments of the guided notes, or the SmartBoards, or during the activities, were actually designed to be places where we have classroom discussions. For example, when one of the worksheets reads:
we actually had a great 5-7 minutes talking about the answers! So I’m afraid these resources make it seem like we might not have really interrogated exponential functions. But we did.
You can really see what I mean because… during this unit, my friend came to observe my class. (It was an assignment for a class she was taking for her Masters.) It happened to be the class where we first talked about exponential decay. While I was teaching, she decided to make a (partial) transcript of the entire class. The transcript is very rough and partial, and you can’t really tell what’s going on exactly, but you can get a sense of what the class was like:
Transcription (with student names redacted) after the Jump
today is the day that it’s all supposed to come together
let’s see if the click happens
I am collecting the homework
I am not grading it on correctness, so don’t freak out
we’ll talk about it a little bit, and then have the rest of the lesson to try and have it make sense
(some are late, he looks at the watch)
based on class yesterday: what made sense? what didn’t make sense?
on the HW
document camera set up, whiteboard ready
question 2, 1b, 1c
i didn’t think about it this way
(is it wrong?)
no! i didn’t think about it this way! this is the best way! i am getting giddy!
(is there another way?)
yes, there is another way
use the formula, [a student] explains
someone else plots it on the calculator
so excited that
what would be a resonable other answer? 87? 72?
what would not make sense? 4?
could I have started out with 5 coins (i coulc have in theory, but the chance of that happening would be achhhhhh!)
let’s start on 1b before we start on today’s lesson
calling on [student],
I know you don’t have your paper, but we can still do this?
she is unsure, having a tough time
how would this equation change if we started out with 100 experiments vs, 537
come up with the equation: labels the parts of the equation
then go to part c
OK, if you have feedback on this I was definitely serious about giving it to me whether you liked it or not (an email, or a note on my desk)
but not cut out letters: that will scare me?
Today we are going to learn about carbon dating?
Practical example of this type of experiment
What do we know about carbon dating?
(any history channel buffs?)
OK…there’s a thing called carbon 14, and there’s a half-life (whatever that is)
makes them feel comfortable if they don’t know about it
[student]: every organism goes through a life cycle
plants go through the chlorophyl thing…what is it called? photosynthesis?
there’s a krebs cycle, i don’t even know that that is…
the amount of C14 you have in your body is constant, as soon as you die, the C14 starts decaying.
i think collectively with all of our knowledge together: that’s essentially carbon 14
if you don’t get it, it’s OK; I looked it up on line, it’s very complicated
1) it’s radioactive: it has a half life
2) when you die there’s 100% of the carbon in your body
get in groups of 3, only restriction is not to be in the same group as yesterday
float around the room
checking the groups
what i want you to do is fill out this chart
let me explain to you what this chart is saying
what would be a good number to fill in in the section that says “years since organic matter was alive”
we were thinking about using the half life?
why would that be a good thing to use?
DO you know any other information for the intervals?
checking in with [student]; asking this question
could you fill in the rest of this table using this number? try it and see if it turns out to be good
would 1 year make sense?
could you fill in the other stuff?
no, what do you need to choose
some groups checking in with other groups!
what did you guys do…we are doubling the half life
looks good to me, but then my question is what would be useful here?
I am going to come back to you
they are working but look confused
this looks good! but you need to fill in the middle column
I am liking these numbers…can you fill in what these would be?
why don’t you erase the second number
(gives them the example, after 5750)
I can’t go any further
discussion of what they use carbon dating for
Oh, it takes this amount of years
this is making sense to me, now explain it to this one
wait…what are you doing?
i thought we were supposed to go to zero!
will it ever go completely to zero?!
no, what do we know about this?
but the practicality of
what about this middle column?
he’s distracted, apologizes and focuses
Oh really, it’s so fascinating!
out of context, i can see that it would be a strange comment
i just care that there are no numbers! no 10 tongues!
finish this on task and then do your graph
are you finished?!
Fill out the graph!
you want to finish and then you want to graph!
[Student] still needs the example
you may just want to make one graph between the two of you?
What’s that old?
anything with any organic material
Oh really, i am so interested in UNC, but I am more interested in this…
I came back
why would you double it instead of halve it?
[Student], you need to understand the concept
you need to wait 5750 years each time, using his hands to explain
Now the question is do you connect the dots?
in the other one we didn’t? why?
does it make sense to connect the dots?
does it make sense to have a 1.25 experiments?
you guys have been dawdling, you need to have this graph done
make sure you fill in the years
instead of making them nice numbers, what would a really nice and easy thing to do with the x axis
no it’s not, I am horrified by this? What are these numbers?
[Student] gets it! Explain to [Student] and [Student]
keep on adding that number
Oh really? there’s over 100% of carbon 14? how is that possible?
how’s your graph?
then if your graph is done see if you can’t answer just by looking at the graph: part A and B!
15 min left in class
just an estimation. you don’t have to be super accurate.
how did you get that? Oh, that makes sense. I totally buy that!
Once you finish A and B call me over!
[Student] clarifying that the number/percentage is what is gone or what remains
you are getting it!
How do you figure this out?
I’m not sure, this graph is a little off
You do need to label it!
why are we drawing a line? In the last one we didn’t!
OK? We’ll talk about it
there is a better reason!
I think you should plot the point 0,100
Since you started it, you might as well finish it
what’s going on over here?
I’m not sure I get that!
you can connect it with a line.
Why didn’t we connect the last one?
no! (cheerful, but dismissive)
hey hey language!
back to [student]’s group
OK, last time: why are you getting that?
Let me ask you: what does this column represent
[Student], you want to listen and focus, I am helping your group here
you want to think about part c a little more: I’m not sure I like your answer
what happens when t is zero…
how are you going to modify your equation so that you have something that makes sense for t to = zero
read question B very carefully to answer this
it’s not that 22.3% remains, it’s that 22.3% is lost
If you start with 100% how much is left after 22.3% is lost
sounds like they are hearing what is going on with the other groups, and
still working on their equation
what if t were 1?
you are very close! i like the division!
OK, I am going to bring us all together so we can go over parts A and B!
I need someone to volunteer their graph!
Why did we connect them, when in the last one we didn’t ([Student])
so everyone gets the answers that many answered in their groups
guesstimating: close to 5750: 4500, 4000?
part b: [student] help me out! If you start out with a dollar and you lose 22.3% of it, how much is left?
it’s about halfway in…what akeim did was take about half of 5750
I think it’s like the other one we did
Gives an equation: why the 5750 here? Oh, it doesn’t make sense
we know that we’re wrong, but I like where we are going, this is a great start!
so watch this! fills in the division in the exponent!
if you didn’t see it this time, that’s OK. I didn’t mean for you all to see it the first time, but hopefully you will be able to see it and understand it for next time
what I need you to do for HW is part D and E
and then I am changing the HW to only parts 1 and 2, not 3 and 4, I will post it on the conference
(I am grading it for a serious attempt)
little “labs” you could do for half-life data:
1) start with 20 pennies in a cup, shake and dump out on table. (n = 20)
2) remove any pennies that are tails. count # of heads pennies. (n = 10 or so)
3) put the n “heads pennies back in cup, shake and dump out on table.
4) remove any pennies that are tails. count # of heads pennies. (n = 5 or so)
5) repeat steps 3-4 until n = 0
6) graph data
(also try using with die, where you remove only when face value is 5 or 6… this gives a longer half-life)
supposedly, the foam head on beer “decays” more or less exponentially. i haven’t tried this yet, but a friend told me that you can have kids take data on the foam of non-alcoholic(!) beer poured into a graduated cylinder. the mL markings offer a metric for amt of foam. use a stopwatch for time. measure every x seconds. graph… calculate half-life of beer foam head.
@mjs: In fact, I did that penny drop exercise (https://samjshah.files.wordpress.com/2009/05/2009-04-27-exponential-functions-applications-c14.doc) — but instead of getting the pennies, I had students run a calculator simulation. I thought we could get the same benefit, but without all the time spent counting. Plus, I wanted students to drop 100 pennies. As for the die, I actually had that as a homework problem this year!
The beer thing is new to me… looking online, there’s an IgNobel that went to it, but for some reason I can’t access the paper…
I know why everything on your blog went italics. You put an em tag (for the italics) in this post right before the fold, but when you page normally loads, it doesn’t load the below the fold part where the tag is closed. That’s my read. Basically, close the em tag before you add the option to expand the post.
@Nick: You fixed it! Wow, good troubleshooting. Thanks a zillion. It was annoying me to no end!
Enjoyed reading this – it gave me some good ideas for next year’s exponential functions unit in Algebra 2.
One question, though – why do you just meet 4 days/week? Just curious. :)
@Kristen: we have a rotating schedule at my school. And the way it rotates, students meet for classes 4 days a week, instead of 5. And the classes meet at different times of the day, each day.
Note: The exponential equation y = a x, where a > 0 and a is a one-to-one function and has an inverse which is defined implicitly by the equation x = a y.
1. The log of a number is the exponent when written in exponential form.
2. If the base of a logarithmic function is the irrational number e, then we have the natural
logarithm function. This function is given a special symbol. That is .
3. are inverse functions.
4. Logarithms to the base 10 are called common logarithms. This logarithm is abbreviated as
y = log x.
5. Change of base formula:
. Equations that contain logarithms are called logarithmic equations. In solving this type of equation be sure to check each apparent solution in the original equation and discard any that are extraneous. One of the most powerful notions in mathematics is the idea of approximating a function with other functions. Students’ first exposure to this concept typically is Taylor approximations at the end of second semester calculus where a function f(x) is approximated by a polynomial, which can be thought of as a linear combination of power functions with non-negative integer exponents. Thus, these power functions can be thought of as a basis for the vector space of Taylor polynomial approximations.
The next exposure to this concept for those students majoring in mathematics and some related fields is the notion of Fourier series in differential equations or a more advanced course. Here, a function f(x), usually a periodic function, is approximated by a linear combination of sinusoidal functions of the form sin (nx) and cos (nx). In this case, the sinusoidal functions can be thought of as a basis for a vector space. However, by the time students Finally, when we speak of the agreement between a function f and an approximation En of order n, we will use the interpretation that f and En agree in value at the indicated point and that all derivatives up to order n also agree at that point. Thus, at x = 0, say, we require that
f ’(0) = En’(0), f “(0) = En”(0), …, f (n)(0) = En(n) (0).
and prof dr mircea orasanu