1 comment on “Just how high was the Big Thompson Flood? And how often will a flood like that occur?”

Just how high was the Big Thompson Flood? And how often will a flood like that occur?

Recently, the family and I were taking in an afternoon in Boulder, CO. After taking in a lunch at the lovely Dushanbe Tea Room we took a stroll along Boulder Creek. Right by a retaining wall stands this object.


This monument demarcates how high the waters rise for a flood of various magnitude. Zooming in a bit to the demarcations we see the following (from bottom to top):

Version 2

Version 2Version 3

The thing that makes flood levels so interesting mathematically is that in addition to height, they’re measured in probabilistic time. That is, every 100 years we can expect one flood to reach as high as the demarcation of the “100 Year Level”. Every 500 years we can expect one flood to reach as high as the “500 Year Level” and so on.

So… what does that suggest for the marker near the tippy top of this monument, marking the height of the Big Thompson flood of 1976?

Version 4


Suggested Facilitation

Provide the following (enhanced) picture.


Follow up with your favorite problem kicking off protocol. I’d suggest either a Notice/Wonder or a Know/Need-to-Know.

Potential Questions

  • How high did the water level get during the Big Thompson flood?
  • How often does an event like that happen?
  • How high are these markers off the ground?

For this last one, you’ll probably need some sort of base level unit to measure the heights, for perspective’s sake. Allow me to provide one additional picture.


(Hey, if it’s good enough for Stadel, it’s good enough for me.)

4 comments on “If the sun is an 8 foot diameter balloon, what is Pluto?”

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

2 comments on “Let them get it wrong: Caloric Quandary”

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.

2 comments on “CNET has some TV viewing size/distance recommendations.”

CNET has some TV viewing size/distance recommendations.

Feels like there’s a similarity (and a lot of other stuff) type problem in here.


From CNET:

 In a perfect videophile world, you’d want to sit no closer than 1.5 times the screen’s diagonal measurement, and no farther than twice that measurement to the TV. For example, for a 50-inch TV, you’d sit between 75 and 100 inches (6.25 and 8.3 feet) from the screen. Many people are more comfortable sitting farther back than that, but of course the farther away you sit from a TV, the less immersive feeling it provides.

I’m wondering if you could pair this with Tim’s TV 3 Act problem. Perhaps even Brian’s Holiday Shopping problem. There’s honestly a lot of stuff going on here from CNET: proportion, distance, maybe even a system of equations or linear programming problem (what with the upper and lower bounds suggested above, then toss in cost constraints).

Update (6/12/17): CNET has apparently redirected the original article to a generic TV buying guide, so the above text is no longer viable. However, here’s something from The Home Cinema Guide.

A good rule of thumb is that the ideal viewing distance for a flat screen HDTV is between 1.5 and 3 times the diagonal size of the screen – and we can use this to calculate both approaches.

Still, the work for the rest of this article reflects CNET’s original viewing recommendations.

Guiding Questions

  • How big a TV should I buy based on the above guidelines and my particular living room?
  • Could we develop a mathematical model to illustrate these guidelines? With, like, variables and stuff?
  • Alternatively, how could I set up my living room in order to fit the kind of TV I purchased?

Suggested Activities

  • Have students develop a model (or “rules” to follow) to express the above recommendation mathematically. (This one’s partially answered below)
  • Students could optimize viewing experience given a floorplan and a TV.
  • A Consumer Reports-ish type TV buying guide? We’re veering here…

Attempted Solution

So the initial model for the constraints listed by CNET aren’t terribly complex.

Constraint 1) “you’d want to sit no closer than 1.5 times the screen’s diagonal measurement”

Constraint 2) “no farther than twice that measurement to the TV”

So lower bound: d>1.5x ; upper bound: d<2x ; and there you have it.

Surely we could ramp up the complexity of the problem with some of the above floorplanning activities and additional cost constraints. How would you modify this situation to serve our mathematical purpose here?

0 comments on “Red Solo Cup / I Lift You Up / Let’s Find the Volume! / Let’s Find the Volume!”

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.

3 comments on “The Pizza Casbah 30-inch pizza challenge”

The Pizza Casbah 30-inch pizza challenge

This is a picture of a single slice of pizza from my favorite pizza place in Fort Collins, Pizza Casbah.

My god that looks amazing. I’m getting hungry just looking at it.

This is a picture of an entire Pizza Casbah, 18-inch pizza (presumably, that means the diameter of the pizza is 18 inches):

It seriously is amazing pizza. And I can eat a lot of it.

A couple weeks ago, a friend and I were ordering a pizza from the online Pizza Casbah menu and we saw this (click to enlarge):

Now, I don’t mean to brag too much, but I really think I can eat anyone under the table when it comes to Pizza Casbah. I’ve never really gone head-to-head, or truly pushed my limits, but I’m fairly confident I can eat an entire 18-inch, 5-topping pizza by myself.

I really want my picture up on their wall-of-fame. I also really want a gift card for more Pizza Casbah.

5 comments on “Pythagoras and Pele; Gooooooooooooaaaaaaaa… (to be continued)”

Pythagoras and Pele; Gooooooooooooaaaaaaaa… (to be continued)


There’s about a hundred different ways you could use the following artifacts to construct a lesson around Pythagorean’s Theorem. So I’ll just toss out all the artifacts and let you, esteemed teacher, take it from there. I’d love to get feedback and suggestions on how to implement these materials in the comments below.


Use any combination of the following.

This video from This Old House in which two small girls assist with the construction of a pint-sized soccer net: How to Build a Soccer Goal | Video | Family Projects | This Old House.

The screen shot of the girl holding up one of the 5 most beautiful right triangles I have ever seen.  (note: before math geeks go berserk, I know it’s technically not a right triangle with the extra bit off to the side, but still.)