Dancing with the Functions

I'm honored and humbled to have been a (small) part of Christopher Danielson's online course The Mathematics in School Curriculum: Functions. There were some great tasks, discussions, and contributors. I now have a better misunderstanding of functions. However you interpret that last sentence, let me assure you that this two week course broke me down in order to give me a better perspective and idea of functions. Professor Triangleman moderated the course well, provided challenging tasks and opportunities that took me out of my comfort zone, encouraged us to think differently, and didn't hesitate to whip us into shape as you can see here:

He's referring to beating me down while informing the teacher's pet (Fawn) of his tactic.

Our choices for our final project were:

• a blog post,
• a lesson plan,
• an interpretive dance,
• a work of visual art,
• etc.

I thought writing a blog post was "too easy" in the sense that I could blog about anything ordinary at anytime. This class wasn't ordinary though, and I felt I'd rather try and give something back to the class, professor, and community in exchange for what I have received. No Fawn, not because I'm "too lazy." Therefore, I'm going to give you a lesson idea I have, based on an interpretive dance, which might be a work of visual art, all wrapped up in a blog post.

Interpretive dance really got me thinking. I thought back to the handful of dance lessons my wife (fiance at the time) and I took to practice for the First Dance at our wedding. My wife was a natural. As for me, well let's just say all the dance lessons in a lifetime wouldn't have helped. Here's a dance photo I have of us where it actually looks like I'm doing something worthy. Don't be fooled.

Don't worry, I won't torture you with video. Anyway, our dance instructor taught us many helpful tips and gave us a glimpse of dances like swing, salsa, the waltz, and the two-step box. We only did a few moves in our wedding dance, but it mainly revolved around the two-step box. We had a short song, thank goodness. I'm sure our guests would have taken their gifts back had they seen me dance any longer.

Here comes my lesson idea. I'd like to see the relationship between the number of steps taken in a dance over time. So let's make it a graphing story. Here's the first 30 seconds of a dance. Write a story for it. Even better, can you write the functions (along with any intervals, domains, ranges, etc)? Go here to Desmos to check your answers. I give you my interpretive dance.

Thanks to Sadie, Timon, and Michael Pershan for inviting me to their hangouts. I was honored to collaborate with you guys during one of the hangouts and bounce ideas off of each other. Thanks to Fawn for getting me in trouble, ratting me out to the teacher, and reminding me to submit my final project. Where would I be without you? Probably in class and not in the principal's office.

I would love some feedback on this lesson idea. Would you have your students dance? If so, what dance(s)? Would you have students come up with a function for each type of dance? What kind of relationships would you have your students look for? Would you consider "dancing" an applicable use of functions? I leave you with this clip. You must give these guys (Sean and John Scott) some crazy respect. They're insanely fantastic at tap-dancing. Just watch the first minute. Then make a graphing story.

Dance,
933

Buses

I just returned from taking my son (turns 3 in a month) to preschool. One of the many perks to Spring Break so far! On the way to preschool, my son spotted a city bus.

Son: Ohhhh! A bus!

Me: Right. What type of bus? 

Son: A city bus. 

A little background knowledge here: He loves trucks! Let me rephrase that. He loves anything larger than a car, has an engine, is big, makes a lot of noise, is big, does construction, is big, intakes diesel, etc. I've seen many kids share these interests, especially when the garbage man does his rounds in our neighborhood. You'd think the garbage man was passing out ice cream or something (and he doesn't need that silly Ice Cream truck music either). We have multiple truck books that get frequent use before nap and bedtime. We have a surplus of toy trucks and Legos, recently added to by the most generous, wonderful, and great Fawn. You're the best!

Back to our drive. We had just stopped at a red light and on the other side of the street he spots a school bus.

Son: Ohhh! A school bus!

Me: You're right!

Son: We've seen two buses!

Me: I know. Look what's coming up behind the school bus.

Son: Another bus.

Me: And what type of bus is that?

Son: A city bus.

Me: So how many buses have we seen?

Son: Three!

Me: That's right. We've seen 2 city buses and 1 school bus, so we've seen a total of...

I pause for him to fill in the blank.

Son: Three!

We're still stopped at the red light and have a little time before the green light. I turn around and illustrate this again with my fingers, because lately he's been doing really well identifying the numbers one through five on a single hand. Since we saw two different types of buses, I use two hands. I hold up on hand with two fingers up and say, "We've seen two city buses" and on the other hand I hold up one finger saying, "and we've seen one school bus. So we've seen a total of how many buses?"
I expect him to say "three" because it's fresh in his mind, but he pleasantly surprises me and I can see his eyes moving across my fingers and mentally counting the fingers to verify the word 'three' matches up with Dad's fingers. "Three." he says.

The light turns green and we're on our way. We have less than five minutes until we get to preschool. Of course, I'm keeping my eyes peeled for more buses. No buses. Shucks. However, we pull into the parking lot of the preschool and park. Before I exit the car to get him out, I turn around and want to try something. Simply for fun.

Me: So we saw two city buses [I'm holding up two fingers on one hand]. What if we saw two school buses [I hold up two fingers on my second hand]. How many buses would we have seen?

Son: Three.

Me: Are you sure? Count my fingers.

Son: One... two... three... four, five, six, seven, eight, nine, te...

Me: Okay, silly. [holding up two fingers on each hand again] If we saw two city buses and two school buses, how many total buses would we see?

Son: Three.

I left it at that. He's content with the concrete. It's not about future counting for him. It's about what he just experienced, what's relevant, what's applicable and what's associated with his interests. I made a small attempt at the abstract, just for fun. There's no need to push this any further. He's convinced we saw three buses and he's right. He doesn't care about a fourth bus that we didn't see. Plus, it's time for preschool. Man, I love vacation. I get to have conversations like this with my son. It doesn't get any better than that.

Buses,
1047

[UPDATE]
*Read what happened two days later in Buses [Day 2].

Not Drawn to Scale

I hope you'll allow me to vent for a bit. I have been encouraging my students to be in tune with the 8 Mathematical Practices by Standard of the CCSS for some time now. It's pretty safe to say that my students know I really favor Mathematical Practice Standard 6, Attend to Precision. However, some of the resources I occasionally use in class are beginning to play tricks with everyone's minds, including mine. Here's a resource I have, Cooperative Learning and Geometry by Becky Bride.

Don't get me wrong, I like this book. It has some great explorative exercises that have appropriately challenged my students. For example, look at this exercise to help students derive the 30-60-90 triangle relationships. Take an equilateral triangle, its altitude, and the Pythagorean Theorem to find out the special relationships between the shorter leg, longer leg, and hypotenuse. Great.

Here's where I start to beat my head against the wall. The book uses diagrams that simply shouldn't be used, especially in the context of 30-60-90 triangles. Look closely...

That's right, the 30 degree angle is opposite the longer (drawn) leg for questions 1, 3, and 4. My students get bothered by this contradiction. I do too. I have no problem admitting this to them. I'm honest with them saying, "I know guys. It goes against everything we strive to do in here. I encourage you guys to attend to precision and check for reasonableness. Yet, I give you this. I'm sorry. It says at the top 'not drawn to scale', but they should be drawn to scale. Right guys?!"

I think this about sums it up. Students will come up and ask about the dimensions they've solved for and whether or not they're reasonable. I'm proud of my students for making sense of their answers and checking for reasonableness.  I know something is a skew when my response to those students is,

"I never assume those things are drawn to scale." 

I feel rotten saying this to students. I feel like I've just provided them with a worthless and menial task. I've let them down. I feel dirty. Mr. Stadel's quality control group hasn't done their job. What message are we sending students? Do they think we're out to trick them? Do the directions read, "Find the mistakes?" They should. It's times like these that force me to (gladly) keep a closer eye on the content I provide my students with. Don't just throw some triangles at them with random angles and units. Make sure they're reasonable.

Have you ever felt this way? Have you ever been caught in this situation? What did you do? How do we avoid these situations again? How do we demand better quality content from publishers? How do we make sure we provide our students with content that matches the CCSS and Mathematical Practices? Maybe you're okay with these types of diagrams, so please explain why. I want to hear from you all on this.

nOt tO sCAlE,
939