[Pseudocontext Saturdays] Smoke Jumper

This Week’s Installment



What mathematical skill is the textbook trying to teach with this image?

[poll id=”7″]

(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

Current Scoreboard

Bad trend here. I do not like it.

Team Me: 4
Team Commenters: 1

Pseudocontext Submissions



Cathy Yenca


And no fewer than three people — Bodil Isaksen, Jocelyn Dagenais, and David Petro — sent me the following problem, created by a French teacher.


And I don’t know. The jist of the problem is that two soccer players are arguing about the perfection of one of their dabs. They consult a universal dabbing rulebook which says that in a perfect dab those triangles above must be right triangles. And it’s all pretty winking, right? It can’t be pseudocontext if it isn’t actually trying to be context in the first place, right? The judges give it a pass.


Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)


The commenters win a second straight week.


The judges rule that this problem satisfies the first criterion for pseudocontext:

Given a context, the assigned question isn’t a question most human beings would ask about it.

A question that might neutralize the pseudocontext is: “Can all of these smoke jumpers ride in the same plane together? How would you arrange them so the plane is properly balanced?”

Instead, the task here is to find mean, median, mode, standard deviation, first quartile, third quartile, the interquartile range, the maximum, the minimum, the variance, etc, etc.

Do you get my point? Yes, all of those operations could be performed on those numbers. We often assign all of the math that could be done in a context without asking ourselves, what math must be done in the context? What math does the context demand?”

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. The wildfire is growing in the general shape of a circle out from the point of a lightening strike by (insert reasonable fire growth rate) square miles per hour. The fire fighters are working to create a containment zone by clearing brush around the circumference of a 4-mile radius circle. They can clear a quarter mile per hour. If the fire is currently burning an area of 2 square miles, will the fire fighters be able to create the containment zone before the fire reaches them?

    Adjust my numbers for reasonable wildfire fighting statistics. :)

  2. I actually like the boiling point one! I’ll admit the picture seems to be illustrating a different set of X-y pairs, but I can totally imagine that a) some team of (pre-digital age) people might be exploring a new mountain, taller than they’ve traveled before, and need to know the boiling point based only on old data, and b) that they would use linear regression to do so.

    • I’m pretty sure people do actually do Maths based on such situations as the boiling point one. There are several web sites that talk about how to calibrate thermometers based on the boiling point of water at a given altitude.

      One even gave this piece of information “The boiling point lowers about 0.6ºC (1ºF) for each 168 meters (550 feet) above sea level” which I think is a nice context for linear graphs.

  3. I just want to know why the French use the adjective “rectangle” to refer to any polygon that has at least one right angle. :p

    From le-dictionnaire: Définition du mot :

    Adjectif singulier invariant en genre
    (géométrie) qui possède au moins un angle droit (triangle rectangle)

    Nom masculin singulier
    (géométrie) figure possédant quatre angles droits et quatre côtés égaux deux à deux

    • Nevermind, I answered my own question. Dang Romans. (I knew “rectangle” literally means “right angle”, but I didn’t know the below.)

      Carry on, sir.

      From etymonline:

      rectangle (n.) Look up rectangle at Dictionary.com
      1570s, from Middle French rectangle (16c.) and directly from Late Latin rectangulum, from rect-, comb. form of Latin rectus “right” (see right (adj.1)) + Old French angle (see angle (n.)). Medieval Latin rectangulum meant “a triangle having a right angle.”

  4. I’m willing to give the cube animals one a pass, almost.

    How stuff scales for biology is pretty damn interesting regardless of the shape of the animal, so you can do some simplifications in the modeling. Sure, it would have been nice if you left that part out, and had the students do it as part of their modeling, but its’ not the worst case of pseudocontext I’ve seen.

    The chapter on modeling with logarithmic functions from my Algebra II textbook, on the other hand, was complete ass. There wasn’t a *single* problem where the kids had to actually figure out a model – they just verified that the numbers in a given table matched a provided function. I skipped it, because ass. Will hunt down actual real world data sets and have them decide on the models instead.

    • We should just make the animals be gelatinous cubes — then the right kind of model is pretty obvious but at least it’s a more entertaining problem: You’re on a dungeon crawl and you encounter two gelatinous cubes, one 4m on a side and one 6m. You have {insert name of coldness spell here — Cone of Coldness I think?}, which will decrease the temperature of the air around one of the cubes to {some temperature} for {some number of seconds}. Which cube will get the most damage from this spell and why? How much damage?

  5. An interesting follow on to these might be to invert the question: What context would make that task necessary?

    Are there any circumstances when you care about the mean, median, mode, range, and standard deviation of a set of numbers? I can think of contexts where you care about a couple of those, but all of them? That’s hard.

    And if I can’t think of a context where the task is necessary, does that mean the task is bad in some way?

    • You could also invert the question the other way say “In xyz scenario, which one of these would be most helpful: mean, median, mode, range, or standard deviation?” If you’re lucky, you’ll start an argument during which the students have to explain why their choice is best.