[Pseudocontext Saturdays] Spaghetti Bridge

This Week’s Installment

Poll

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

Pseudocontext Saturday #8

  • Identifying supplementary angles (42%, 186 Votes)
  • Calculating angle measures in a regular polygon (36%, 158 Votes)
  • Proving triangles are congruent (22%, 99 Votes)

Total Voters: 443

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(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

I’m kicking the number of options back up to three. Two options simply doesn’t give y’all the challenge I know you need.

Team Me: 4
Team Commenters: 3

Pseudocontext Submissions

John Gibson

I don’t know if this is pseudocontext, but I for sure don’t know under what circumstances anyone would wonder about resultant momentum. In my head right now it’s like wondering about the middle names of the people who manufactured that car. It feels like trivia! I’m not saying it is trivia, but I am wondering if someone can put me in a position where knowing how to calculate resultant momentum would feel like power rather than punishment.

Rules

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.)

Answer

The commenters took this one right on the nose. The pseudocontext was in the last place they looked.

The judges rule that this violates the first rule of pseudocontext:

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

Moreover, I just don’t see any congruent triangles in the picture. None. I know I’ll see some if you widen the camera’s angle, but there aren’t any in the frame right now, which makes this a uniquely poor context.

The only way I can think to neutralize this pseudocontext:

Show students four spaghetti bridges. They have to decide which ones are fragile and which ones are strong. Understanding congruency somehow (waves hands) makes them more accurate in their decision-making.

Featured Comment

Dick Fuller:

I like physics. And math. One without the other is school.

About 

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

17 Comments

  1. The collision problem is a mess. The units for momentum are wrong. The momentum after the collision is clearly zero; the momentum of the cars is not conserved, the road exerts a force on them. As for the ¨power¨ felt in understanding dynamics, depends on what you feel when you can solve real problems. I like physics. And math. One without the other is school.

    • Clearly an inelastic collision. Hence momentum is not conserved and the question is pointless. And, Dan, being able to apply conservation of momentum is critical in being a crash investigator. By examining the aftermath and applying c.o.m. you can determine the velocity vectors of the vehicles prior to the collision.

    • And, Dan, being able to apply conservation of momentum is critical in being a crash investigator. By examining the aftermath and applying c.o.m. you can determine the velocity vectors of the vehicles prior to the collision.

      I believe it. But every time the crash investigator context appears, I’m left to wonder how students can interact with that context beyond a) recalling a formula, b) plugging in the right numbers, c) crunching the formula.

      As I asked farther down this thread:

      So is there a situation – even a simplified one – where one attorney might argue one thing and the attorney we work for as math consultants might use math to disprove her.

      Any thoughts?

  2. Dick Fuller, that’s right about the units! Good catch. But concerning the momentum after the collision, there will be a momentary resultant momentum vector that has to do with the sums of the two vectors; that is why the cars might bounce back, or in the case of the picture the truck bounced over, because the collision changed the vectors of both cars. But like you said the momentum will eventually dissipate from the force the ground exerts on the car.

    Dan, I could see this application being used in crash investigations because I do know they investigate velocity vectors of the cars, but that’s just a conjecture.

    • Dan, I could see this application being used in crash investigations because I do know they investigate velocity vectors of the cars, but that’s just a conjecture.

      Super. This is a start. Now I need to know how knowing resultant momentum will help me become a more accurate, faster, or more certain crash investigator.

  3. You’re definitely upping the game! All of these are reasonable questions to ask based on the picture, though none of them are interesting in the context of the picture.

  4. The car question follows a fascinating pattern that shows up in lots of physicsy work: it begs the question. Physicists like to measure things. Sometimes measuring something directly is tricky (or impossible), so we measure other things, and then calculate the thing we actually want.

    Questions like that have as their givens the thing we *can’t* measure and ask us to calculate the thing that we *can* measure. It’s absolutely backwards.

    And it’s really common. How on earth would anyone looking at that collision know the momenta of the vehicles prior to the collision? Sorcery? Being an automotive crash test lab running the collision under controlled circumstances?

    Even if you wanted the resultant momentum for some reason, the way you’d get at it is totally different.

  5. How on earth would anyone looking at that collision know the momenta of the vehicles prior to the collision? Sorcery?

    If you know the weight of your car and multiply it by the speed you have its momentum. Both are quite easily measured, so momentum is a piece of cake (especially if you don’t worry about converting to standard units, which a question like this doesn’t require) and I could tell you the momentum of my car by simply looking at my speedometer and knowing my car’s weight.

    My objection to the question is that it is a physics question and so has no place in a Maths book. I should not be explaining conservation of momentum to my students.

    I will sometimes show a problem to my students when explaining how we might use vector addition, but I won’t actually do it.

    • > If you know the weight of your car and multiply it by the speed you have its momentum.

      That gets you at best half way there. How are you going to know that for the other guy’s car? And realistically, finding out the weight of your car is non-trivial. How much fuel do you have in it? How much junk do you have in it? Unless you’ve just driven off a good scale, you’re going to be off by double, possibly triple digit kilograms.

      That doesn’t even touch on how you know that the angular difference between their velocity vectors is 15 degrees. Protractor? Mk. 1 Eyeball?

      And if you’re in your car such that you can be looking at the speedometer, and suffer a collision like that, vector mathematics will be the last thing on your mind.

      And if you’re not in the car, you don’t have access to the speedometer.

      There’s just no circumstance in which you could know all of those givens outside of a very well controlled auto collision testing facility.

    • Interesting comment on its place in a maths book. In the UK, we teach applications of mathematics as an integral part of the subject, especially in the 11th and 12th grades, when statistics, mechanics and discrete mathematics (algorithms etc) may all be part of the syllabus (the exact syllabus depending on the examination board and the choice of units). After all, much of mathematics has its origins in applications (especially in physics), so it seems a little bit strange to divorce our teaching of the subject from its wider meaning and application.

  6. Just to be nerdy about the “momentum” question:
    (1) units for momentum are mass * velocity, so kg * m/s would work.
    (2) One reference I found gave the unloaded mass of a typical passenger car for a variety of models. Eyeballing the range, I get roughly 1100 kg (2400 lb Hyundai Accent) to 2700kg (6000 lb Cadillac Escalade). See Car Weights.
    (3) loading the cars with people, or as my family likes to do, tossed aside food wrappers, let’s say we have an extra 300 lbs in the car.
    (4) My typical range of road speeds is roughly 30 kph to 80 kph, or 8 to 22 m/s.
    (5) That gives me a momentum range from about 10 000 kg m/s to 63 000 kg m/s.

    Surprisingly, the textbook at least gives reasonable values for the measures, at least at the bottom of our range from the plausibility investigation.

    Now, why would you ever want to do this calculation? I think the answer lies in the result of the google search where I tried to find out how much people typically decelerate in a crash (expecting to bring my range for “velocity at the moment of the crash” down). Most of the hits were accident lawyers, fishing for clients.

    Putting the pieces together, I realized that this skill will be crucial when attacking the presentations of the opposing attorneys in injury lawsuits our students experience.

    • Putting the pieces together, I realized that this skill will be crucial when attacking the presentations of the opposing attorneys in injury lawsuits our students experience.

      Cool, cool. Any interest in going farther down this road with me? It’s inconsistent, in my view, to wave my hands at a career that uses some math / physics, without also giving students a chance to feel that same power.

      So is there a situation – even a simplified one – where one attorney might argue one thing and the attorney we work for as math consultants might use math to disprove her.

  7. Can’t see the text book problem so writing based on the comments above. Collisions or impact problems are typically solved using impact-momentum relationship. The force of the impact is greatly determined by the time of deceleration. Could present some interesting problems to students. *Calculate the F of impact for two old steel cars vs two modern cars (lighter plus body designed to collapse to lengthen time of impact). *Calculate the force of impact of an F1 race car crashing into wall. How does the driver walk away?? *Have students be crash investigators and have them figure out the speed and direction of two crashed vehicles at moment of impact based on evidence from crash and mass of the vehicles. (Amount of damage to the two cars should be able to estimate impulse, *Students predict results of impact between vehicles of different masses ie pick up truck vs economy car. *Discuss and calculate effect of no braking prior to collision vs minimal braking etc *What will happen in collision between two rail cars travelling in the same direction *Etc.
    Students can look up related facts on their devices.
    Demonstrate plastic and elastic collisions in class & have students predict results then discuss.
    Could be a fun class while covering a few math topics.