If the sun is an 8 foot diameter balloon, what is Pluto?


The following clips are cribbed from Nova: The Pluto Files in which Neil deGrasse Tyson sets up a model of the heavenly bodies of our solar system, comparing their sizes relative to the sun and each other. Not all of the clips were able to be chopped to give the appropriate bleep sound. So we just have a few. More on that in a moment.

Side note: for the record, if you want to see a kid from the age of 4 to 14 get animated about something, tell them that Pluto isn’t a planet and/or let them watch this episode of Nova. They’ll go berserk. Neil’s right: people are crazy when it comes to Pluto’s planetary status.

Anyway, on to the entry events.

Intro & Mercury



Bonus!: Diameters of Uranus vs. Pluto

I’m not sure you need all three or four of these for kiddos to get the point. And I’d get some predictions on the board before having students explore this on their own or make and calculations.

The Process

  1. Show the Intro & Mercury clip
  2. Get some predictions.
  3. Reveal just the Mercury solution. Show some of the calculations involved. You can find all the heavenly body sizes from our solar system here or here.
  4. Show the Saturn clip.
  5. Let students make some predictions and do some research on the actual sizes of the heavenly bodies. More predictions on the board.
  6. The big reveal. For the solutions, you can just watch the clip straight from the home site linked above.

Here’s a potential accompanying worksheet.

Possible Extension

Neil says that we can’t properly represent the distance of the planets from the sun on this scale of a field. So my question is, how could we represent a scale model of the planetary orbits and distances from the sun?

Here we have Mercury five yards away from the sun. If Mercury is 5 yards away, how far away would Pluto be then?

field ss

Let them get it wrong: Caloric Quandary

Artifact & Facilitation

I must have cylinders on the brain. Maybe because they’re actually one of the few traditional geometric shapes that we actually interact with on a regular basis? Maybe it’s because they’re readily measured?

Anyway, here have a Coke can and one of those mini-Coke cans. Though it’s dependent on you exactly what information you’d like to black out.

You could black out one of the calorie counts and compare it to the fluid ounces.


You could black out one of the fluid ounces counts and compare it to the calorie counts.


You could eliminate the fluid ounces and one of the calorie counts to get at a really nice volume comparison (though, you’ll need additional dimensions – that’s good! Ask the kiddos what other dimensions you’ll need to procure?). 


While you’ll need other dimensions, I would actually withhold the dimensions of the base at the beginning. Why? Because students of all ages have a real tough time with scale factor and volume. Like, REAL tough. As in, I tell them straight up “when you increase the dimensions by a factor, the volume increases by that factor cubed” and then they totally forget that by the time I’m done saying it out loud.

So let students solve it using a simple proportion.


4/5=90/x –> x=112.5 calories

Then when you reveal the actual calorie count, we’re all like “wha?”




“WHAAAAAA??!?!??!! Math is wrong! You lied to us!” Or maybe they’ll claim corporate conspiracies to get us all fat. Either way: win-win.

This is the part when you swoop in with some additional dimensions to save the day. Find the volume relations of the two cylinders, the calorie counts, and you’re home free.



I also feel like there’s some way we can leverage this into some additional follow-ups/extensions: 

Or this.


I like having calorie counts as the final measuring stick for this task instead of volume.

Like I said, scale factor and volume (and area) were something my students would consistently get wrong. I think it’s indicative of the problem with front-loading instruction. Students don’t need to think deeply about the content because I’ve showed them how to do it in the “Scale Factor Unit” when it’s applicable, of course. Then, three months later, when we’re not in that unit any more, it’s out the window.

I’d suggest you read Frank’s post and watch the embedded Veritasium (@veritasium) video for more on allowing students to swim in their misconceptions a bit to enhance learning in the end.

What? How do YOU spend your two-hour school delays?, Water Content in a Snow Cylinder

As anyone in town for NCTM in Denver know, it’s been a bit snowy here this week. In fact, Fort Collins just had its biggest snowfall of the year. But how big?

We had a two hour school delay this morning as my daughter and I were greeted by this on our back doorstep.


“Wow that’s a lot of snow!” she says. But how much snow is it?  Go go gadget EmergentMath!


I got this ridiculously large [cola] mug at a white elephant gift exchange last Christmas. And now I have a chance to use it!


I asked her to make a prediction on how full the mug would be after it melted. We each made a prediction using her hair ties (hers on top, mine on bottom).


We took a couple measurements just for posterity’s sake.



I dunno, we might want them later. For now though, we just stuck with the predictions.

We then watched it melt. Slowly.

Sure enough, we were both way off:

013 015

Wow. All that snow and only that much actual moisture. I have some questions:

  • Is this typical? What if we redid this in the afternoon after the snow had packed a little more? 
  • What if we used different shapes? Could this be a sort of alternative to the how-full-is-the-weirdly-shaped-glass problem?
  • Going back to the original photo, how much water was on that table?

I also have a couple comments:

  • Want an easy way to build buy in? Have kids make predictions on something and make sure it *takes a long time* for them to see if they’re right. Like I said, our delay was a couple hours and this pretty much took up the entire time. This was sort of analogous to Dan Meyer’s now-famous water tank filling task.
  • This seems ripe for Estimations 180.
  • I’m not sure what you could do if you live in a non-snow state. What would Texas use? Sand? Cicadas?

My daughter and I could have gone into the volume of the near-cylinder, which dimensions were useful and that sort of thing. But our two hours were up. It was time to go to school.

Update 4/16: I’ve got my Facebook friends eating out of the palm of my hand. *maniacal laugh*



Sort of related: a couple atmospheric scientist friends of mine started a Facebook page crowdsourcing, archiving, displaying, and discussing clouds: Community Cloud Atlas 

You should join their Facebook page and tell them to get a twitter account.

Red Solo Cup / I Lift You Up / Let’s Find the Volume! / Let’s Find the Volume!

Hat tip to @mathhombre.


The red Solo cup, apparently.

Guiding Questions

  • What are those little markings between 12 oz and “Top Me Off, Bro”?

  • Can you use proportions to find it? Similar polygons? Volume? Help me out here. Can I have a ruler? Can I use Geogebra to diagram this? Wait, can’t I just physically get my hands on one?

Actually, let’s do this in a manner that A) won’t get us fired, and B) doesn’t have that obnoxious “BroBible” stuff down on the bottom right.

Or even better than that, have an actual one on hand with some rulers hanging about, just for good measure.

The good news is, this problem has a real nice, testable solution: measure out the ounces for the attempted solutions and fill her up. Were your students right? If not, were we too low? Too high? Where did you go wrong?

Just make sure that when your students are working your next school fundraiser, they don’t overfill the red Solo cups with too much orange drink. That’s where the money is made and you want to be precise, after all.

Can we make an even “edgier” brownie pan? What about the “perfect” brownie pan?


This, my friends, is part math, part food, part art, all deliciousness:

It’s the all edge brownie pan, which I found from my new Favorite Website of All Time, Reasons to Go Broke. Here’s the description from the Amazon page (perfect 5-star rating):

“For corner brownie fans and chewy edge lovers, it’s a dream come true — a gourmet brownie pan that adds two chewy edges to every serving!”

2012 just became the best year ever.

Guiding Questions

  • How can we measure the “edginess” of this brownie pan?
  • What would happen if you added a couple more horizontal partitions?
  • What if you liked the center brownies? Could we make a pan to cater to these monsters?
  • Similarly, what if you like brownies with three or four edges?
  • Can we make an even “edgier” brownie pan by adjusting the partitions?
  • Does the edginess change if we increase or decrease the dimensions of the pan?

Suggested activities

  • Develop a metric for the “edginess” of a brownie pan. I’m thinking surface area-to-volume ratio should do the trick.
  • Plot the number of partitions against the “edginess”.
  • Use Google Sketch Up to make a model of this brilliance.
  • (Just go with me on this one) Take a poll. Figure out how many people like 1-, 2-, 3-, 4-, or zero-edged brownies, then challenge the class to make the “ideal” brownie pan.
  • Make awesome brownies.

I’d also be willing to bet that someone more skilled than I at Geogebra could make a construction of this, complete with a diagram and a plot of partitions vs. edginess.

The more I think about it, the more I like that “ideal” brownie pan idea. But here’s my question: are there people out there than think two is not the ideal number of brownie edges? My fear is that the “ideal” brownie pan has already been made. And it’s available for $34.95 at Amazon.

‘Readymade’ Suitcase Project (Part 2)

If you didn’t see the intro to this project, check out Part 1. Briefly, I want to make this:

Now that I’ve categorized all the items for my travel needs, I need to measure the dimensions to determine the smallest travel case I can purchase.

Here’s that list of items again.

4 pairs of socks
4 T-shirts
4 pairs of boxer shorts
1 dress shirt
1 Toiletry Bag
2 pairs of pants
1 pair of workout/sleeping pants/shorts

1 Phone charger
1 set of keys
1 book, which I probably won’t read
1 winter hat & 1 pair of gloves for colder seasons/locales
1 swimsuit for warmer seasons/locales

Out comes the measuring tape. Who’s ready to see picture of my socks??

(note: my apologies for the unreadable numbers on the tape measure. The close up resolution on my camera is probably worse than the one on your phone. So I put the dimensions on there (using Inkscape) for clarity’s sake. Click on the picture if you want to enlarge. For some strange reason.)

Here are the dimensions in table form.

Length (inches) Width (inches) Depth (inches) # Volume (in3)
4 boxer shorts 6 3 2 4 144
4 pairs of socks 8 2 3 4 192
4 T-shirts 11 8 1.5 4 528
1 dress shirt 11 9 4 1 396
1 Toiletry Bag 9 5 4 1 180
2 pairs of pants 11 11 2 2 484
1 pair of workout/sleeping pants/shorts 11 5 4 1 220
1 Phone charger 3 2 2 1 12
1 set of keys 2 1 1 1 2
1 book, which I probably won’t read 6 9 2 1 108
1 winter hat & 1 pair of gloves for colder seasons/localesor, 1 swimsuit for warmer seasons/locales 8 4 4 1 128
Total volume 2396

So if I had the ability to mash all my objects up into a perfect rectangular prism, I would require 2396 cubic inches. This begs some additional questions:

I’m in love with a suitcase; ‘Readymade’ Suitcase Project (Part 1)

My job affords me the opportunity to travel all over the country and observe teachers putting their best feet forward. Problem is, I’ve never been much of a traveller. I’m terrible at packing. Therefore, I could really use something like this:

This is a project suggested by Readymade magazine. I don’t want to gush too much, but I absolutely love this suitcase. Seriously, I would marry this suitcase like that girl did the Eiffel Tower if it were legal. I want one. Unfortunately, the whole point of Readymade is to get you to make your own stuff. So it looks like I’ll be tackling this project.

I don’t have a cool, classic hard cased suitcase like this, so I’m going to have to buy one at a thrift store or something. How big of a suitcase will I need?