Function Transformations/Domain and Range: Day 2 (and a bit more)

Day 2:  Domain and Range of Parent Functions

I started the year with domain and range of a finite set of points, because, it's an easy concept and I wanted my students to know--Algebra II is totally conquerable.

Fast forward a month, and I think they might just ready to handle a continuous case.  We broke it up sloooooowly and built towards finding the domain and range of the four functions they found the day before.

Domain and Range

Obstacle #1:  Closed circle v. Open Circle
I really enjoyed this part of the lesson because the students were totally in to the closed circle v. open circle, which is awesome...but not before we jumped over a few hurdles.  When I asked them what an open circle denotes in mathematics, a few people proudly reported, "Parenthesis!"

No, sweethearts, an open circle does not mean parenthesis, last time I checked the dictionary.

So, we got to talk about my favorite branch of all--analysis.[1]  How could we write all the numbers between -5 and 4, but not including -5 (see fourth slide above)?  Of course, [-4.9, 4] was suggested.  But then poor -4.99 (just to mention one) gets left out!  Any number they suggested that was close to -5, but bigger, I could always find a number that was even closer (thanks, density!).

Hmmm...

And then a kid suggested something that was absolutely brilliant.

Incorrect, but brilliant nonetheless.

"Could we use -4.999...?"

Cue look of How far do I dare take this subject with these kids?

I wasn't quite sure.  And still am not so sure.  I decided to write this on the board:

-4.999... = -5

"This is what we know, and can prove, mathematically:  that -4.9 repeating is equal to -5.  So, sadly, writing [-4.999..., 4] does not help our quest because that's the same as [-5, 4], which is what we were trying to avoid in the first place."  BRILLIANT thought though, loquacious kid in the front row.

So, new notation is all that we could come up with in order to fix this dilemma.  Parenthesis.  As was wildly suggested before.

Now if only they could remember a few of the deeper ideas as opposed to just "open circle=parenthesis."

Obstacle #2:  Domain of Deceiving Functions
Here's what I mean by a deceiving function.  I took y=x^3 and showed a graph like this:

Most students were convinced that the graph would never pass x=-3 on the left, and x=3 on the right; hence the domain must be something like [-3,3].  And who can blame them?  They haven't developed a good sense for what the graph of a function is yet.

But that's ok.  Because we have Desmos.

So, to abettercalculator.com we went to graph the function, along with the line x=-3:

The kids' case was looking good.  Until we started scrolling down:

And down...

Aw, isn't that a lovely linearization...

And changing the y-axis view even more...

Gah!

And this convinced many (though not all)...the domain is indeed all real numbers.

Days 3-5 were spent focusing on vertical and horizontal translations.  We tried vertical stretching/shrinking, too, but I started to lose several of them, so I decided parent functions, domain/range, and translations were plenty for now.  We can come back for the rest later.  That's the beauty of 180 days, as opposed to the 48 that I'm used to.

Before the unit test, we played Kate Nowak's Speed Dating Game, but I adjusted it for these topics.  You can find the game cards I created here.

[1]  I claim analysis/advanced calculus as my emphasis in grad school...mostly because that's what I took the most classes in and that's what I took my written comprehensive exams in.  In any case, while I can't say I've read that many texts on analysis, I can say, that Understanding Analysis by Stephen Abbott is, by far, the best text I've ever seen on introductory analysis.  All calculus teachers should be required to read it.  Truly!  It's the best.  Get the whole thing for free here.  Go.  Read.  ENJOY.  He's a master teacher.

Function Transformations/Domain and Range: Day 1

As I've said before, I'm all for the motto of "The person doing the work is the person doing the learning," and I fully believe in making the students do the work in class.  However, I do think something gets overlooked a lot with this motto:  if I want my kids to be doing the work in class, I usually have to prepare a crapload outside class.  I'm willing to do this (most days).  But, I don't think that gets stated enough.

Being new to this age group (and never having attended a public high school myself), the trick for me is to anticipate the students' every move:  to come up with activities and lessons that are challenging enough to keep 36 students at 36 different levels engaged for an hour, but that are not too difficult so students just give up (or call your name so many times that by the end you're dizzier than a Turkish whirling dervish).

Some days these lessons flow out a lot faster than other days.

This was not a lesson that came quickly.  My insomnia from grad school is back and running, so this is a lesson that got started around 5 in the morning on a Saturday (I repeat, 5 in the morning, on a Saturday), and got finished sometime in the late morning.  For all that work, it has a lot of flaws.  But it has some good aspects, too.

So, here's Day 1 of Parent Functions, Domain/Range, and Transformations

Day 1:  Introduce Parent Functions

I really believe one of the most important skills I can teach my students is to read, comprehend, and subsequently follow directions.  There are so many cool things I've learned in my life, and so many more cool things I hope to learn.  But I couldn't have learned most of those cool things if I hadn't taken the time to read, comprehend, and apply my knowledge.  With that in mind, we had a big-time literacy day in Algebra II.

Students were given these xy tables, graph paper, and the directions below.  Not a whole lot else.  At first, they were livid.

"I don't get it!"

"Read the directions."

"What do you want us to do?!"

"Read the directions."

"You haven't taught us this!"

"Read the directions."

"UGH!"

"Read the directions."

It was an exhausting day, I'm not going to lie.  But they eventually caught on, and I learned that some of them are great at reading, comprehending, and applying, and some are not.  Here are the directions.  Many thanks to @Fouss for the subtitle. ;)

Parent Function Directions

Not everyone got to Part II, which was actually nice as it allowed for differentiated instruction.  I posted the best graphs from Part II in the front of the classroom so that students have these four parent graphs in front of them at all times for now.

What I liked

• Students READ.
• Students did the work.
• Students focused on a small amount of information:  four rather important graphs.

What I didn't like/Questions I still have

• Do they really understand that the graph of an equation is the representation of every single solution of that equation?  I feel like I say that a lot, but that doesn't mean anyone actually understands what I'm saying.
• It's not super exciting.  I know these graphs have a lot more interesting aspects to them then just "Draw an xy-chart and plot the points," and I feel like maybe I stripped them of a lot of their intrigue.
• I let the students pick their own groups.  I still don't know if that was good or not.  The complainers tend to be friends with each other.
• I only printed one set of directions for each group because I'm more than a little frugal with my copies.  Also, I wanted them to work together.  However, I think the students would have benefited from everyone having his/her own set of directions.

I will post more on the unit soon!  Hopefully!

Week 4 :: Writing Piece-wise Functions

A prompt for the final week of the New Blogger Initiation was to write about another new blogger's post.

Maggie (@pitoinfinity8) posted an awesome activity for piece-wise functions in which students literally cut up the different pieces of a given function and then puzzle them together.  Brilliant.  In response, Bowman Dickson mentioned that it might be useful to go the other way, too; in other words, give the students the graph and have them write the equation.

I love both these ideas:  the first gets the students to read; the latter gets them to write.  I didn't read Maggie's post until after I introduced piece-wise functions this year in Pre-Calc, but I did read it in time for our first test review.

So, here's what we did...

I gave them a few minutes to answer these questions and then we used their answers along with the restrictions to write the function.

Onto another one:

They were rockin and rollin, so I asked them...

This time, I gave them the problem in the traditional manner:

Success!  Finally!  Many thanks to Maggie and Bowman.  What a great way to review both piece-wise functions and function transformations.

Speaking of function transformations, a twitter conversation in which I laughed out loud:

Week 2 :: A Warm Up I'm proud of

One of Week 2's prompts for the New Blogger Initiation is to pick something--anything--that we've created that we're proud of.  If I read it correctly, it can even be just a problem.

So, here goes: a Warm Up/Bell Ringer/Do Now that I'm quite proud of.  It's the pre-cursor to a lesson on transformations for Pre-Calculus:

WARM UP (Do Now)

1. Write f(1)=2 as an ordered pair.
2. If you know the point (1,2) is on the graph of y=f(x), what point do you know has to be on the graph of y=f(x)+3?  Why?  (Hint: What is f(1) equal to?)
3. What about on the graph of y=f(x+3)?

Plotting these transformations gets us thinking about translations and (I think) reveals the "opposite" behavior of the "inside" transformations.  (Though it's really not opposite at all, is it?!  Yay for math!)

That's all I have for this week!  Thanks to the wonderful people at the blogging initiative for encouraging continued writing.

Transformation Scavenger Hunt

Last week was the wonderful function transformation lesson.  I say wonderful with a bit of sarcasm as this section tends to overwhelm me a little every semester.

Let me tell you about symmetries.  Let me tell you about translations.  Let me tell you about reflections.  Let me tell you about stretching and shrinking.

All in under an hour.

This semester, the lesson did go a bit better, I have to say.  For one, the students seemed totally into transforming graphs of functions to make them look completely different than what they started out as.  Kudos to them.  For two, I adapted Kate Nowak's buried treasure idea to fit into our lesson.

I took the twenty-five desks in my classroom and handed out a worksheet that had this on the top:

So each desk corresponded to a point on the Cartesian plane.  I gave my students a set of points as well as a function transformation.  They were to locate the points and transform them correctly.  I told them the treasure was at the point furthest to the left, for example.  Underneath the desk, I had taped an index card that said something like, "Congratulations!  Hint 1 out of 3 correctly found."  Once they found all three cards I rewarded them by playing a favorite YouTube video of mine (which, I admit, was not math-related).

All in all, the scavenger hunt was a success!  This semester, I had the class split into groups.  Once a group felt like they had the correct point, I had them tell me their answer.  If they were wrong, I told them to try again; if they were right, I had them wait for the rest of the class to get to the right answer.  This of course forced the faster students to wait a little, but that was my only complaint about the activity.

I plan to use this again next semester when we come across function transformations again, as well as incorporate Desmo's graphing calculator, which has (wait for it) SLIDERS.