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!

Systems of Linear Equations Activity: 3 Cases

So, this isn't anything super fancy, but it worked quite well with my Algebra II kiddos (without much prep on my part, which doesn't happen often), so I wanted to archive the idea and hopefully get some feedback/ways to improve it.

I gave four lines in slope-intercept form and had my students get out a clean sheet of paper, fold it twice to create four quadrants, and write one of the lines at the top of each quadrant.

Then they were to write four categories (in each and every quadrant--oh my!):

• Given line (Y1)
• No solution line (Y2)
• Infinitely many solutions line (Y3)
• One solution line (Y4)

The given line (Y1) is the line I gave them.

For Y2:  we talked about what would need to be true about the second line in order for it to never touch the given line.  The kids were pretty quick to tell me that the lines would have to be parallel, and for that to be the case, the lines would need to have the same slope (and different y-intercepts, btw, cherubs).  A-ha!  We do remember some things from Algebra I!  So, we decided on a line that was parallel to Y1 and wrote it in the category of "No solution line."

For Y3:  we discussed what would need to be true about a line in order for it touch the given line at each and every point on that line.  Well...it's gotta be the same line!  Write that in the category of "Infinitely many solutions line."

For Y4:  the typical case, but I love that this activity made them think a little deeper about this case.  "So...what has to be true for a line to touch the given line once and ONLY once?"  Pause.  Pause.  Pause.  

Still, small voice:  "Different slopes?"

Oooo...

"So, a line with ANY slope other than that of the given line will intersect with the given line somewhere?"

Pause.  Pause.  Pause.

Unanimously:  "YEAH!"

Wohoo!  So, we made up a line with different slope and wrote it in the category of "One solution line."[1]

Graphing calculator time...

For the given line of y=2x+1, our y= screen may have looked something like this:

We changed the features for Y1 and Y3 so we could distinguish between the lines and actually see the calculator graph the given line again for the special case of infinitely many solutions.  I had them sketch these lines at the bottom in addition to stating the point of intersection for Y1 and Y4 (they could use their calculator).

I did one of these exercises with them and then had them do the same thing for the remaining three given lines on their own/with their partner.

About half-way through the period, I had them turn their papers over.  Using the same quadrants and the same lines they created, we solved each of the three cases algebraically.  That's twelve systems they solved in half a lesson.  The goal was to get them to see that algebraically a false statement is related graphically to two lines that never intersect (no solution); that a true statement is related to two lines that always intersect (infinitely many solutions); and that a conditional statement is related to two lines that intersect once (one solution).

Again, it didn't take lots of prep and I think it really brought together the geometry with the algebra.  I hope you're proud, Descartes.

[1]  Lots of students would just change the slope of the given line but keep the y-intercept.  Then, when they solved the system, they noticed that x always turned out to be zero.  "Mrs. Peterson!  I keep getting x=0!  What's going on?"  "What did you keep the same?"  "The y-inter...oooo..."  Light bulb.  One kid was so excited about this I truly thought he was going to pee his pants.  It's the little things in life.

Math Teachers at Play #54

Welcome to the fifty-fourth edition of Math Teachers at Play!  We have a great roundup of articles this month...

Literacy            

• Yan Kow Cheong at Singapore Math gives some ideas for journal writing in math class.
• Terrance Banks at So I Teach Math and Coach? gives a great literacy strategy that encourages students to be the teacher.
• Bowman Dickson at Bowman in Arabia shares how to start class with whiteboarding (reading a graph is a literacy component, if you ask me!).
• For more graph reading, I posted two AP Calculus problems I think are great.

54:  The number of countries in Africa

Instruction

• Guillermo Bautista at Mathematics and Multimedia explains common errors in MDAS operations.
• Dan MacKinnon at mathrecreation shows beautiful patterns that arise in the multiplication tables that we were all told to memorize in grade school.
• Glenn gives a pretty snazzy visual representation of quadratic equations that have complex solutions.
• Jeff Brenneman  at Trust Me-I'm a Math Teacher shares a project idea in which students identify the key issues to winning the 2012 Presidential election.

54:  The number of playing cards
(with 2 jokers)

Gamification

• Bon Crowder at Math is not a Four Letter Word shows how to get children thinking about graphing using a Lite-Brite.
• Lilac at Learners in Bloom shares some great ideas for introducing inequalities and pattern-recognition to your preschooler (whoa)!
• Mrs. M at The Daily Cupcake shares eleven awesome activities for kindergarten students learning numbers 0-5.
• @reminoodle at The MathSmith describes a group project she gives every trimester for students to create their own review games. Be sure to check out the impressive pictures!
• Christy at Just Another Step to Take gives five ways to create homemade board games for practicing math facts.
• Jennifer Smith-Sloane at Live. Teach. Create. shares an awesome foldable for converting fractions, decimals, and percents.

54:  The number of colored squares
on a Rubik's Cube.

Great Advice and Insight

• Denise at Let's Play Math! gives tips on how to teach math to your children, even if you experience math anxiety.
• Peter Glynn at Nature Math has started a discussion on why some kids hate math.
• Colleen Young at Mathematics, Learning, and Web 2.0 gives some great back-to-school advice.
• Jimmy Pai at The Pai Intersect discusses both his interests in and his concerns about flipping a classroom.
• Cindy Johnson at The Conic Card Lady's New Adventure shares fantastic advice to new high school teachers.

+54:  The international phone code for Argentina

The next installment of MTaP will be held at Mathematical Palette!  Submit your responses here.

Also, be sure to check out the latest Carnival of Mathematics and Mathematics and Multimedia Carnival.

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:

Submit your posts for Math Teachers at Play #54!

Bloggers--

The September 2012 issue of Math Teachers at Play is just around the corner! Please submit your posts (old  or new) through the MTaP Submission Form or send them directly to me via my Contact Page.  If you send them to me, include the following:

• Submission URL
• Submission title
• Email address

In addition, feel free to include your name and any comments you may have about the post.  You're encouraged to submit either your own posts or others' posts you've come across recently that deserve more attention!

Tune back here next week for the 54th Carnival!

The MTaP is a monthly blogging round-up shared at a different blog each month. We welcome posts from parents, teachers, homeschoolers, and students -- anyone who is interested in playing around with school-level (preschool to pre-college) or recreational math. Each of us can help others learn, so in a sense we are all teachers. --Let's Play Math

Week 3 :: A math quote I love

A prompt for this week's New Blogger Initiation was to pick one of our favorite math quotes and write about it.

I just got done with Open House at our school, and I shared a slide with the following to my parents:

In mathematics, the art of proposing a question must be held of higher value than solving it.

Georg Cantor, 1867

I asked them if they were ok with a brief math history lesson.  They said yes as long as there wouldn't be a test afterwards (I like this group).  I gave them a few tidbits that I find really fascinating about Mr. Cantor:  he was one of the forerunners to formalize the rigorous definitions we use in calculus; he classified the levels of infinity (WHOA); people thought he was so crazy, he got thrown into an insane asylum.

So a genius beyond genius, if you ask me.

But one of the things I love most about Cantor is the quote from above.  Because it summarizes my thoughts not just on mathematics but on teaching mathematics so beautifully.

I told the parents, as a teacher, I'm much more interested in developing students who can ask deep questions, than students who know how to take a multiple-choice test.  I'm interested in helping students learn when to ask, "Mrs. Peterson...why do we do it this way?" or "What if we considered this case instead?"  I want to help develop curious students; and central to curiosity is the desire to ask good questions.

One challenge that I face, however, is that my students are used to their curiosity being satiated so quickly and easily.  If they want to know the answer to something, they can just Google it.  On their phone.  Right there.

Don't get me wrong, I love that we have access to so much knowledge.  The downside is...many of us are not used to really, truly working hard to find a satisfying answer.

And the downside to that?  We never experience the joy of a hard-fought discovery.

And that's one of the coolest things in life.

Hence, my love of this quote.