## Two identical (yet not identical) Pythagorean theorem word problems

Before my mentor became a mathematics teacher he taught welding.

They needed the Pythagorean theorem, and he gave a quiz which included these two questions:

1. Suppose a hiker starts at a camp and walks three miles north, then four miles east. If he takes the shortest route, how far does he have to walk to get back to camp?

2. You are welding together a corner which needs a brace. You want the sides to be 3 feet and 4 feet. How long should the brace be?

These were (including the pictures) on the same page right next to each other on the quiz.

Nearly all the students missed question #1 and got question #2 correct. When asked why that happened, one student explained the second question was welding, not math.

### 24 Responses

1. That is crazy and interesting.

In that situation, did they use paper or calculator, or well-honed estimation skill? Were they aware of the formula?

2. If the camper starts at the South Pole, ‘he’ could go straight south, so his distance would be 3 miles. (The diagram would be different.) That’s what this math nerd thought of upon reading the first problem. I’m guessing the students had other issues. What sorts of answers did they give?

This is fascinating. It reminds me of the studies of nurses. They weren’t so hot at the math in their classes, but on the floor, they never made a dosage mistake. (Try this link. If that doesn’t work, google “nurses proportional reasoning”.)

• I don’t remember (and I don’t know if he remembers) how they got the problem wrong, but I’ll ask.

The Brazilian street mathematics mentioned in this article touches on a similar issue to that of the nurses (they did certain problems correct 98% of the time in the real world, but when tested they did much worse).

The difference here is the issue popped up solely in the context of a quiz.

3. The second question looks much harder to me because you need to deal with the thickness of the materials. Do you need some amount of overlap for the brace?

On first reading, I assumed that the point was that the welding question didn’t have the obvious answer.

• The second question looks much harder to me because you need to deal with the thickness of the materials.

My drawing is probably a little off from the original. You don’t have to worry about the thickness, just use the Pythagorean theorem.

4. I am a retired electrical engineer and I worked for a company which most of its production required welding steel and iron pipes and profiles, a few of which just by curiosity I saw and watched at . My reaction was similar to Mark’s.

A possible explanation for the students performance is that the second question seems to be more concrete than the first one, because the context is related with what they were learning.

On the contrary for a sailor (or a sailor to be) the first one might be more natural.

5. To be fair, the questions aren’t asking the same thing. The first one is asking for the length of the SHORTEST path back to camp and the second is asking for the length of the hypotenuse. Most numerate people will know the hypotenuse as the LONGEST side of a triangle but they may not remember that its length is shorter than the other two sides combined. Welders get paid to produce so they don’t tend to sit around committing mathematical facts to memory.

6. One difference is that someone who actually hikes would not be likely to use the hypotenuse, because trails are laid out according to topography and don’t tend to closely approximate right triangles. Conversely, someone might very well weld a right triangle.

7. math scores are receding because reading skills are not applied. The problem of education begins in the home and then through the 6th grade. Children are not taught how to learn and need to be taught this elusive skill at the earliest of ages. It is most important that parents read to their children. Read Read Read.

8. Jason, could you give us some of the wrong answers, and can you tell what the students who answered wrong were thinking?

bookguy wrote: Children are not taught how to learn and need to be taught this elusive skill at the earliest of ages.

I think human beings are hard-wired with a passion for learning. And teaching can actually get in the way. So don’t try to teach kids ‘how to learn’, just nurture their passion.

• Sue –
Being a homeschooling father, I would agree about the passion for learning, sometimes though, one needs to direct the learning to “applicable skills” as opposed to memorizing lines from Lord of the Rings.

That’s my goal – do not squash passion!

9. Obviously, in the first case it isn’t five because the earth is curved, and in the second it’s clear that there’s a large overlap between the bars, making the effective size of the welded triangle smaller 😛

10. that’s easy- answer is 6

11. I can guess why students gave different answers to the same problem: lack of abstraction.
From Wikipedia: “Abstraction is the process or result of generalization by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behaviour, eliminating the characteristics of that particular ball.”

I think that teaching how to abstract from real life things is pretty difficult…

12. In the first problem, the path may not be flat (sorry, don’t have a better word). It could be mountainous, which means that even if you’re traveling at a straight path, the distance could be longer than 5 miles.

13. In my experience, students don’t actually think of subtleties like mountains, curvature of the earth, or thickness of the materials. In fact they usually oversimplify problems. In this case, I would be willing to bet that the most popular wrong answer to the first question was 7.

14. I guess the students are very stupid. These are simple problems. X^2+Y^2=Z^2 Works in every situation:)

15. Haha. I spent 5 min re-reading the problem and reading the comments to figure out why the answer wasn’t 5 to both.

• jricesterenator:
haha me too, seriously, isnt it five? im really beginning to doubt my maths skills when i found out noone got it.

Of course there might be mountains, but a map with contour lines would be nice, if you really want us to be specific. This is maths modelling, not geography.

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17. Both are 5 , questions are identical but when put in different ways one seemed easier to solve than other that’s it.

question 1 was more descriptive in students mind north, east and the numbers seemed like too much info. and complicated.

while question 2 was put in simpler way.

In my opinion that’s all.

18. In the real world, the problems are not the same. In an idealized mathematics world, they essentially are. So in the first case (real world) I’d say that neither has an answer of exactly 5. In the second (idealized world), both have an answer of 5. Thus, I’m not so sure that it’s reasonable to judge whether those students really got either problem right or wrong. But the tasks would be very useful in mathematics and mathematics education classes. The conversations they engender would likely be extremely useful if the teacher has developed a classroom culture that promotes meaningful, respectful discourse.