# Body Graphing

Here’s another wonderful activity I learned at the Dana Hall Math Conference.  I’ve written about more activities I learned there HERE and HERE.

Set up a Cartesian plane with x and y-axes wherever you have outdoor space at your school.  You can use chalk if you have a blacktop, or string, or cash register receipt rolls.  Label the axes, and label the scale on each one.  Ask students to choose a spot on the x-axis, (and make sure that some choose negative numbers).  Take a portable whiteboard with you (or you could prepare some paper ahead of time), and write a function of x (e.g. y=2x+3).  Ask students to “act out” their x-coordinate on the plane.

My students got the hang of this quickly, but they also appreciated the challenge.  You can make this a mental math practice by giving targeted, or more complicated equations.  Or you can give them paper or whiteboards on which to calculate.  It worked well to illustrate functions – whenever one student was not “in line” with the others, it was obvious that something was wrong, and there was rich discussion about how to fix the problems.  It also showed in a memorable way where linear functions got their name (That was my exit ticket question for the day).

I had really good success with this, and the kids very much appreciate getting to go outside.  I recently did some professional development with an expert on brain development, who suggested that kinesthetic activities like this really cement concepts in kids’ memory as well.  An outside location was important because we could effectively illustrate how quickly some functions grow (compared to others) by allowing real distance for exponential functions.

I came up with an extension on the fly that seemed like a good idea, but didn’t work out so well.  I’ll share it in case someone has a thought about how to make it work better.  I asked one kid to come up with a function, and bring the other kids to x,y coordinates that fit their function.  Then I asked other students to try to identify the function.  It seemed like it should have been rich, but it fell sort of flat.  Please share any idea that might improve this.

There are definitely other variations on this lesson floating around, including this example, specifically geared toward slope-intercept form.

In a related post, Michael Pershan writes a nice post about functions HERE.  I especially love the yarn.  There is something wonderful about the aesthetic sensibility; about how that slightly sad piece of yarn is elevated by giving it “function” status.  While we’re on yarn, check out the yarn work from Minneapolis artist, Hot Tea.  I’m not sure how yet, but I’m confident that these artworks will have a classroom use at some point.

# Balloons +Tangerines

I totally stole adapted Sam Shah’s beautiful and inspired Related Rates and Balloons Lesson for my 8th grade math class.  I remade the graphic organizer to be a little friendlier looking for my middle school students, and to give them a chance to practice drawing with a compass and ruler.  They particularly liked the language of the degenerate circle.  From hence forth, that’s how we’ll be referring to all points.

After wrangling about the balloons not really being round, and about the fact that circles only exist in your mind anyway (you really need to enjoy these moments if you’re going to be spending time with 13-year olds), we measured the circumference, surface area, volume, and weight of balloons inflated with one, two, three, four and five breaths, and then calculated radius and diameter.  To connect with our study of functions and graphs from last month, we plotted and graphed our results and compared the linear and not-so-linear functions.  This was a great hands-on way of comparing rates.  I saw light bulbs go on for several kids when they realized that one function was getting bigger faster than another.

Two things I didn’t foresee:

• Some of my kids are actually small enough that they couldn’t blow up the balloons.  Not a big deal, but interesting to remember.
• Many kids wanted to inflate the balloons with more than 5 breaths.  I encouraged this, but somewhere around 8-10 breaths, my dollar-store balloons began to give out, and we were all a little on edge from popping balloons by the end of class.  Next time, limit to 5 breaths, or buy better ballons!

I like using the highly technical tangerine peel method to help explain the surface area of a sphere, (Thanks to Miss Quinn for reminding me of this!) though mapmakers might have something to say about trying to flatten the sphere in this way…

# Target Number Game

There are lots of iterations of numbers games like this, but this one is especially nice because it’s open-ended and repeatable, and can be extended to your grade level.  Marilyn Burns has published several variations of this game like this one, or many more from her book, About Teaching Mathematics among others.  I’m sure that I’m not alone when I say that Marilyn is my hero.  These games are full of thinking and practical number sense, and require almost no prep.  This one is a perfect back-pocket activity if you have a few idle minutes (though I can’t actually remember that ever happening – we always seem to have too much to do!), or a nice way to regularly open or close class.

My kids love this game, and ask for it if we haven’t played it for a while.  I heard about it first from Tom at the Dana Hall Math Workshop.  I mentioned how much I loved this conference in an earlier post.

Here’s how the game works

• Choose 4 numbers between 1 and 10
• Choose 1 multiple of 10 between 20 and 100
• And choose one 3-digit number between 200 and 1000 (Your Target Number)
• Use the first five numbers and any operations to get as close as you can to the 3-digit number.

Example: 4,8,3,2,70      Target 497

First Try: 70 x 8 = 560-4x3x2 = 560-24 = 536

A Better Attempt: (4+3)x70 = 490+8-2 = 496

I’ve opened this up to include exponents or roots, with pretty good success.  Do let me know if you use this, or have any variations up your sleeve.

# Concepts Of Circles And Volume Of Spheres Through The ORBEEZ Lens

It’s nice when there is the coincidence of a Common Core Standard, and the perfect experiential activities for your grade level.  To explore concepts of circles and volume of spheres, (CCSS 8.G) I adapted Dan Meyer’s Orbeez lesson.

The Orbeez marketing materials are so wonderfully sickening.  I made my Orbeez Orbinizer to match the color scheme.  Many of my students do better with a set of boxes like these to give them some boundaries, and I create an extra version with lines inside the boxes for a few.  Some do all of their work on graph paper, and staple it to the organizer, and one or two turn in much of their work electronically.  Good thing I’m easy.

My kids love being skeptical almost more than they love being sarcastic, and when I suggested that the marketers of Orbeez might be having us on when they claim that “Orbeez grow to 100 times their original volume…” they got all riled up, totally outraged, and were off and running.

What I Did

• I asked them to work in pairs, played the Orbeez advertisement video, and put up the above image of the Orbeez materials (I blurred out the word “volume” and changed it to “size” so that kids would have to decide what the makers meant by “size.”  See below for the doctored image).  We brainstormed questions.  I fanned the flames of outrage, and then passed out the Orbinizer.  I ask kids to fill these out individually.  Even if they’re working in pairs, they each have to have individual writing as an artifact of what they’ve done.

• Once we decided on the things we would measure, kids needed to strategize about how to go about this.  They thought of some ingenious methods of measuring the tiny beads; my favorite was lining up 10 dark colored Orbeez on white tape, measuring with a ruler, and dividing by 10 to get an “average” Orbee (Can that really be the singular for Orbeez?) size.  At a certain point, they realized that they needed to be able to calculate volume of a sphere to find the truth.  “Hey Nat, how do you find the volume of a sphere?”  Gotcha! Join me at the whiteboard…
• Some direct instruction here.  There is interesting math in deriving the volume of a sphere, and if kids know the Pythagorean Theorem, they can wrap their heads around this – even if it did cause a little smoke to come out of their ears by the end.  I used THIS Archimedean logic as the basis for my discussion.  Even though my kids aren’t ready for calculus, they could understand conceptually what was going on here, and I think that it will help them to remember the ideas if not the formula.
• I gave them the sphere volume and surface area formulas, and let them go for it.  They measured diameter, converted their measurements, and came to individual volume averages.  We averaged the whole class’ data as well, and got the Orbeez soaking for the next day.

Next Day

• Kids got to class bursting to check on their Orbeez expansions.  Ensue fierce calculating.  Actually ensue “What was the formula again?” and, “Eww these things are GROSS.”  Then fierce calculating. Exclamations as they mostly confirmed the claims.  Some unexpected outrage that a few Orbeez actually grew BIGGER than they were supposed to.
• Brought the class back together, and showed them how the makers had come to their claim.  The discussion on Dan’s blog made for very authentic experience for them.  We averaged the class’ findings, and sure enough our results came to almost exactly 100 X the original volume.  Usually the world is a little grungier, but it worked for us this time.  I’ll take it.
• We settled down for some traditional work on a problem set about sphere volumes.  I suppose you have to at least expose the kids to what a textbook looks like.  :)

THE MATERIALS:

Orbeez Orbinizer

Orbeez Ad: Manipulated