Where inquiry and methods intersect

Had a nice, quick twitter conversation with Anna (@borschtwithanna) yesterday morning. Anna reached out with a question about providing methods in an inquiry-based classroom.

Anna was conflicted due to her students’ unwillingness to deviate from their inefficient problem-solving strategy. Rather than setting up an equation…

Setting aside for the moment that this is actually a pretty good problem to have (students willing to draw diagrams to solve a problem, even at the cost of “efficiency”), it does circle back to the age-old question when it comes to a classroom steeped in problem-solving: “Yeah, but when do I actually teach?”

The answer to that particular question is “um, kinda whenever you feel like you need to or want to?” The answer to Anna’s question is pretty interesting though, and I’d be curious what you think about it. Personally, I never had students that were so tied to drawing diagrams to solve a problem, that they weren’t willing to utilize my admittedly more prescriptive method. I do have a potential ideas though.

Consider Systems of Equations. This is a topic that is particularly subject to the “efficient” method vs. “leave me alone I know how to solve it” method spectrum. Substitution, elimination, and graphing were all methods that students “had” to know (I’ll let you use matrices if you’d like, I’m good with just these three for now).

Anyway, so I’m supposed to teach these three different methods for solving the same genus of problems. I want kids to know all three methods (generally), but also want to give them the agency to solve a problem according to their preferred method. Here are a few possibilities to tackle this after all three methods are demonstrated:

1) Matching: Which method is most efficient?

OK so matching is kind of my go-to for any and all things scaffolding. It’s my default mode of building conceptual understanding and sneaking in old material (and sometimes new material!).

In this activity students cut out and post which method they think would be the most “efficient.”

Students could probably define “efficient” in several ways, which is ok in my book. Also, it’ll necessitate they know the ins and outs of all three methods.

2) Error finding and samples of work

This is another go-to of mine. Either find or fabricate a sample of work and simply have students interpret. If you’re looking to pump up particular methods, consider a gallery walk of sorts featuring multiple different methods to solve a particular problem. The good folks at MARS utilize this in several of their formative assessment lessons. These are from their lesson on systems.

Screen Shot 2014-10-24 at 7.47.37 PM

Screen Shot 2014-10-24 at 7.47.43 PM


Students are asked to discuss samples of student work and synthesize the thinking demonstrated, potentially even to the point of criticism.

That’s a couple different ways to address methodology and processes that may turn out to be more efficient, while still allowing for some agency and inquiry on the part of the student.

What do you have?

Larry Ellison, billionaire CEO, makes unsound business decisions with regards to his basketball playing on his yacht.

Larry Ellison, co-founder and CEO of Oracle, has gobs and gobs of money. How much money? Well enough that he can do this.


Boy that seems wasteful, doesn’t it. I mean, when I’m playing basketball on my yacht and I lose a ball into the ocean I just purchase an extra basketball. Wouldn’t it make more sense for Ellison to just buy a bunch of basketballs and grab a new one every time he loses one overboard? So my question is this: How many basketballs would Ellison have to lose in order to make the expense of basketball retrieval worthwhile?

Here is some of the board work we generated during the initial Problem Defining and Know/Need-to-Know process.


It’s critical that we understand our ultimate goal here: we want practice developing a mathematical model based on a given scenario. A model should, among other things, simplify a complex situation. We wound up focusing on only two variables: the cost of the annual salary and the cost of a basketball. A couple variables that folks tossed out ended up not being explored mathematically. As you may have experienced, when given a modeling scenario, students might throw out potential variables to tack on in perpetuity. There comes a tipping point where the mathematical model ceases to simplify a complex scenario and only confuses it further. I find this pretty typical of “make a budget” tasks or other accounting-type tasks (“what about sunscreen costs? what about health insurance? what about the yacht food? etc etc etc.”). When you’re facilitating the brainstorming process, I’d suggest you restrict the number of variables you’re including to two or three. This way, the entire class is focusing on the same few variables, keeping the focus on the model development, not the number of ingredients you can toss into the stew.


Here’s the initial PDF file if that embedded version looks goofy.

Basketball Overboard – Problem Solving Framework (pdf)

This High School football coach plays “Would You Rather” Math, and so should you

Add “Would You Rather?” to your bookmarks. Phrasing math problems in terms of “Would You Rather” is simple and brilliant. I love this framework for three reasons:

1) It’s relatable. We’ve all wondered whether it’s more efficient to mow the lawn in concentric rectangles or in stripes. We’ve all run. We’ve all argued with other people.

2) It allows for immediate estimation. Students will immediately have a conjecture. As I’ve mentioned that Dan has mentioned before, that’s one of the best pound-for-pound ways of getting kids to learn math in new ways.

3) It hammers – I mean just crushes – CCSS Standard of Practice 3: Construct viable arguments and critique the arguments of others, which Steve Leinwald has called “the most important nine words in the common core”. *

So yeah, bookmark it, give it to your kids tomorrow. I’ll be sprinkling them throughout my curriculum maps very soon.


Speaking of a “Would You Rather” approach to math, I was watching a video on Grantland about a High School football coach in Little Rock, Arkansas who never punts and always onside kicks. While statheads have been clamoring for less punting for years, he (and the video produces) articulates the math quite clearly and attainable (a mathematical skill).

Would you rather go for it on 4th and 7 from your own five yard line (about a 50% success rate)….



punt and give the opponent the ball on the 45 yard line?


Here’s where it gets interesting though. By going for it on 4th down that close to his own goal line, the opponent would score 92% of the time. By punting to the 45 yard line, the opponent would score 77% of the time.


There you have it. Coach Kevin Kelly played “Would You Rather” Math and used some compound probabilities to determine that they’ll never punt. While there’s more to it (i.e. it’s not a singular event: once you get the first down, you have to get the next first down and so on), just imagine if you could get kids to think mathematically this way under the Friday Night Lights. Coach Kelly uses similar logic regarding always onside kicking.


Go check out the full video and full article if you like. Shoot, try to get your class to convince your schools’ football coach they should never punt. It probably won’t work but it’s worth a shot. Wouldn’t you rather give it a shot?

*Thanks to Chris Robinson for helping me track down the author of that quote.

Update 11/20/13: Really wonderful interview with Kelly on this weeks’ Slate: Hang Up and Listen podcast. You can also find lots of bunny-trails in their links section to further elucidate the topic of never punting in football.

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

Hot Rod Quadratics: Let’s jump this jump!


Hot Rod is one of those movies that’s incredibly dumb the first time watching it. The second time watching it, it’s still incredibly dumb, but it gets funnier with each passing viewing. It’s basically just an excuse for Andy Samberg to to Andy Samberg things for 90 minutes. I’m ok with that.

Anyway, this isn’t Rotten Tomatoes. Let’s get to our Entry Event:

The final stunt of the movie, cut ever-so-slightly short.

Suggested questions

  • The most obvious one: will Hot Rod land the jump? Or maybe better if you know how it ends: will he clear the jump?
  • You could get all physical on this question. Considering how high he is and how fast he’s going at liftoff might be some other options.


In a nod to Dan’s “Will it hit the hoop” task, I’d go with Geogebra here as well. In fact, let’s just use pretty much the same idea (side note: if anyone can rip a better quality video, I’d be interested).


In fact, I’m going to go ahead and toss it in after the Basketball task in my Algebra 1 curriculum map. I’d consider using the Hot Rod task as a way of solidifying conceptual understanding by removing some of the sliders on the Geogebra task, or removing it from Geogebra entirely and having groups do some hand-to-hand combat with it on paper. It’s times like this that a problem taxonomy could help: do you want to assess student learning or enhance prior learning?

The goods

The video (above)

A couple screen shots

Fig 1:

jump1 jump2

Figs 1 & 2 mashed together


The Geogebratube student worksheet

Aaaand the “reveal”:

On final monkey wrench:


Conservation of momentum?

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.