Solving Exponential and Log Equations Flow Chart

I guess I've been kinda into flow charts this year; I created another one for solving exponential and log equations.  The idea is that kids start with the top box, if they can't complete that task, then they go on to the next box.  We put examples in each box.  I used this for both Algebra II and Pre-Calc this year.

It's not flawless, because mathematics requires more creativity than a flow chart can provide.  But it gets the basic ideas across.

Solving Exponential and Log Equations Flow Chart

Newton's Law of Cooling :: A Murder Mystery

You know those lessons/projects you give your students that you look back on and you're like, "Wow.  That wasn't half bad"?  I had one of those recently in Pre-Calculus.

We've all seen those exercises in textbooks where students are supposed to figure out the time of a person's death using Newton's Law of Cooling and given certain temperatures and times.  I always liked those problems, but never really knew what to do with them, more than just present them and say, "See!  Math IS applicable to real life."

Then a colleague of mine showed me a literacy activity adapted from Key Curriculum Press.  I found a version online that I used (but it was a direct link to the Word document, so I don't know to whom to give credit!).  The first page is what I found online (I added Newton's Law of Cooling to the bottom); the second page is the instructions for the kiddos, which includes the rubric:

Sherlock Holmes Mystery by Rebecka Kermanshahi Peterson

To start out, we first had to watch a trailer for BCC's Sherlock (LOVE):

I gave the students about a half a day to figure out the math and solve the murder.  The next day, I loaned out a laptop cart from the school and the kids finished the story, working in groups of 2-3.  I had the students submit their posts on a blog I created for our class via kidblog.org.  My principal told me about kidblog, a class-friendly version of Wordpress, and I absolutely love everything about it (except its name).  Students don't have to register or sign in with an email account:  you just set up usernames and passswords (which can be done in a jiffy) and then they can log in.

On the blog, I posted a sample writing that I found here.  (The math is a little off, so be sure to fix it if you use this link--the final t should be negative.)  This really eliminated the "I don't get what you want us to do!" comments because the students had an example with which to model their writing.  In fact, I didn't get a single such comment (kuddos, kids).  However, I also protected this sample with an extra password:  students could not get into the post until they had solved the crime, as the password to the post was the time of death (see, kidblog is awesome).

Once all the posts were in, I gave the students a couple days to go back in and comment on their favorite posts.  The posts with the most comments received some bonus points.

I was honestly blown away by my students' response to this assignment.  Their stories were original, entertaining, and included the required mathematics.

There's obviously room for growth here on my part, but for the first go-around, I was incredibly pleased with this activity.  Next time, I may make the crime a bit harder to solve, and I may give different versions.  We'll see how motivated I am.

Out of respect to my students, I don't want to post the password to the blog here.  But, if you'd like to check out their stories or the blog for instructional purposes, feel free to tweet me (@RebeckaMozdeh) or email me (rebecka dot peterson at gmail dot com).

Is mathematics invented or discovered?

We've been solving some quadratic equations in Algebra II currently, and I've had an ulterior motive this whole unit.

Quadratic equations seem to lend themselves particularly well to math history lessons (or "math commercials" as my principal calls them--love that).  For example, when we talked about the Square Root Principle, I gave a mini-lesson on Christoph Rudolf (names that rhyme are the best, aren't they?) and the introduction of the square root symbol and how it's supposed to resemble a lowercase r, etc.

I asked them innocently here, "So, do you think mathematics is invented or discovered?"

If they said, "invented," I said something like, "So, the square root of two didn't exist until Rudolf came up with a name for it?"

If they said, "discovered," I said something like, "So, you're just going to ignore the contributions people like Rudolf made to mathematics?"

I let them hash it out a little, playing devil's advocate all the way.  And then I ended with, "Well, interesting conversation, guys," and proceeded to my next slide.  Which, inevitably had the effect of "WAIT!  Aren't you going to tell us?"

"No."

Which then had the effect of, "I'm going to Google it!"

"Go for it."[1]

The next day, Day 2, we continued with the square root principle, but now we tried to solve equations like x^2=-1.  I let them try to convince each other that there is no real solution to this equation (though their multiplication skills are still lacking, so...sigh).  Here's where we talked about imaginary numbers and a mini-lesson on Euler ensued.  I told them Euler couldn't stand not having an answer to this problem, as it--along with other problems like it--had been appearing in mathematics for nearly two thousand years.  So, Euler made his own solution, and called one of the solutions i.

"Now do you think mathematics is invented or discovered?"  We took a poll[2]:

3rd Hour

5th Hour

At the end of class, I had them write a letter to me defending their answer.  They were instructed to choose only one (invented or discovered).  Here are two really great letters, one from each point of view:

Dear Mrs. Peterson,
Mathematics was discovered, because just because a human didn't know the answer to something doesn't mean it doesn't exist.  Before the Pythagorean Theorem was invented, a right triangle still had an area.  Humans simply put words/letters/numbers and theorems to help us find the answer, and explain math, but the problem they solve, and answers they find, were always there.  Some species of animals haven't been discovered yet, but when someone finds them, they didn't invent the animal, they discovered it.

Dear Mrs. Peterson,
I believe mathematics is invented.  I believe this because invented means to create or design something that has not existed before, or make up an idea, name, story, etc.  You have to create a name for mathematics to exist.  One apple is not one apple unless you give a name to the number or quantity of the apple.

The next day, something happened that I think will go down as one of my favorite teaching moments of all time.  A student, who has said from Day 1 that she's not good at math, came up to me before class and looked at me with her precious, sincere, huge brown eyes:

"Mrs. Peterson?  I really need to ask you something."

"Go for it.  What's up?"

"Can you PLEASE tell me--is mathematics invented or discovered?  I can't stop thinking about it."

Cue burst of emotion and huge cheesy grin on my face.  Why was this so wonderful?  Because she just experienced what makes mathematics so addictive:  the deep longing to solve or to prove, and the pleasure that follows the accomplishment.

Luckily (or maybe unluckily) for her, on this day, Day 3 of our discussion, I wrote a letter to my classes, defending my point of view.  Now, I didn't give myself the same restrictions I gave them, and I typed this up the night before (I know, bad Rebecka), so it's pretty rough around the edges (hey, that's the great thing about teaching--now I have a whole year to make it better).  But, here's what I wrote:
Discovered or Invented

What was so, so cool about this whole discussion is that it really appealed to most of my students.  They were hooked.  They kept asking about it.  They wouldn't let it go.

What more could I ask for?

So, what do you say:  Is mathematics invented or discovered?  I'd really like to know your input...so I can make my letter better for next year.


[1]  When I checked in on these students, they seemed more confused than when they started.  Let's hear it for UnGoogleable Problems!

[2] Don't let the total numbers fool you.  Only 1/2-2/3 of my classes participated (don't want anyone thinking I have a class of 17!).  Not sure if the rest didn't want to commit to a single answer, if they didn't have access to a phone, or if I just didn't quite hook 'em...

We're getting closer with recursive definitions

Last semester was the first time I used a College Algebra curriculum that taught sequences and series.  The first section of this chapter was a nice little intro to sequences and series, just to get students used to notation.  I thought I did such a good job explaining sequences that are defined recursively.  Until I saw the test.

Not.  So.  Hot.

Then, as I helped students in our Math Lab, I realized something--recursive definitions are not that obvious to students.

I think what happened here was a classic case of it's-so-obvious-to-the-teacher-she-automatically-thinks-it's-obvious-to-everyone-else.  We've all had teachers like this.  My absolute favorite prof from grad school loved the phrase, "Oh, this is kindergarten stuff!"  Which usually had one of two effects on me:  (1) Ahhhh!!  This is NOT kindergarten stuff!  I just spent the majority of my weekend trying to figure this out!  (2)  Where in the world did you go to kindergarten?  Remind me to send my kids there.

But I digress.

What hit me was that when I see something like:

an=an−1+an−2

I automatically think, "If I want to find a certain term, I need to sum up the two previous terms."  Furthermore, I know that

an+1=an+an−1

means the same thing as the previous equation.


On the other hand, when my students saw a recursive definition, I'm pretty sure they thought, "WTF.  Skip it."

So, this semester I paid much more attention to these types of sequences.  The very first thing I did regarding recursive definitions was show a slide with this at the top:

I asked students to fill in the blanks and then asked them three questions:

1. What do the dot, dot, dots mean?
2. What do we call the term before a_n?
3. What about the term after a_n?

Maybe this is an obvious starting point, but it was such a revelation to me.

We then did some examples with

an=an−1+an−2

I had them tell me what they thought it meant (with a lot of guidance from questions like, "a_(n-1) is related to a_n how?").  Then we wrote a few equations in symbols and in words.  For example for,

a4=a3+a2

an=an−1+an−2

I made them write "The fourth term is equal to the third term plus the second term."  And so on.

Then came

an+1=an+an−1

which everyone was convinced was a totally new problem (darn you, indices!).  But, once we did the same examples (finding a_4, etc.), I think/hope all minds were changed.


After working some specific examples, where initial values were given, I gave them an exit ticket of something like:

List the first five terms of the sequence defined by:
a_1 is the number of boys in the room; a_2 is the number of girls;

an=an−1+2an−2

Finj 

I think about 80% of students got it with zero help from me.  Not perfect, but I'll take it this time around!

Someday I'll figure out how to teach circles

Oh, circles.  How can I do you justice?

Circles present a dilemma for me.  I think it's because the way I typically introduce the standard equation of a circle is so abstract for something that is rather straight-forward.

The way I see most textbooks present circles, and the way I've presented circles in the past is very similar to this:

 

Aye aye aye.

Every semester I think, "Maybe I can make this derivation make sense to them this time around!"

And every semester I fail.  Miserably.  I mean, you can practically taste the glazed-over looks.  It doesn't matter how many times I say, "It's just the Pythagorean Theorem, guys!"  I've lost them.

So, I tried something different this time around.  It's still far from perfect, but the equation of a circle came much faster and from many more mouths than usual.  I did what most of us probably do when trying to verify a general concept:  apply it to a specific example.

I showed them this picture and asked them to relate the lengths of a, b, and r using the Pythagorean Theorem, and then find the exact lengths of a and b.  And, most importantly, how did they arrive at the lengths for a and b (without just counting units).

Then I showed them the same picture as before and asked for the same information, but this time--no grid, no numbers.  I told them to call the center (h,k) and the other point, which represents any point on the circle, (x,y).

They came to the result pretty quickly.

Some concerns I have with this strategy:  Do they understand the significance of x and y?  Did they just regurgitate the example from before without thinking about the fact that they're deriving a general equation? I desperately want to teach my students how to think abstractly and generally because, well, that's what pure mathematics is about.  But is it more important for some ideas than others?

Like I said, I'm happier with how things went this time around.  But not yet satisfied (hopefully I never fully will be).  Any suggestions are gladly welcomed!

Extraneous Solutions of Log Equations--A Graphical Representation

Last semester when I taught log equations, a curious student asked, "Why is it that we sometimes get extraneous solutions?"  A fantastic question that, at the time, I answered horribly.  I gave some mumbo-jumbo explanation about how extraneous solutions can appear when we combine logs. For example, for log(ab) to be defined, we need only ab>0, but that can be the case when a and b are either both positive or both negative.  If they're both negative, then they wouldn't satisfy the original expansion log(a) + log (b).

Not my finest moment as a math teacher.

This semester, I was determined to come up with a better explanation.  I gave my classes the following true/false question.  (I'm mean and didn't give them the option of "sometimes true.")

True or False:

logx+logx=logx22

The students said it's true (with no hesitation) and justified it with the Product Rule (Power Rule works, too).

Then I showed them this:

f(x)=logx+logx

f(x)=logx2

What the #$%@...?

"So, what's going on here?"  I asked innocently.

We had a good discussion about how the left graph is the same as the right if you restrict them to x>0 and how the right side would be defined for both positive and negative values for x since any real number squared is non-negative.  This (I hope) led to the powerful conclusion that the properties for logs have to be used with caution since logs are only defined for positive arguments.  This was stated when we first introduced the rules, but it's easy to forget.  Furthermore, when solving log equations, we don't know if the arguments are positive or not...so we have to use the properties and then come back and correct ourselves if need be.