Open-Ended Math Questions?

[ReFLeX]

I'm just starting a Bachelor of Education and my teacher for Gr. 1-8 Mathematics told us in the first class that she had a math problem with no answer next week. She shared the anecdote that a student from last year had written her afterwards asking what the "real" answer was, which was laughable because there wasn't one. Now I'm sure there are good questions like that, but I was shocked when this turned out to be the question:

A woman bought a horse for $50 and sold it for $60. She then bought the horse back for $70 and sold it again for $80.
What do you think was the financial outcome of these transactions? The woman . . .
- Lost $10
- Earned $10
- Lost $20
- Earned $20
- Came out even
- Other (Describe)
Explain your reasoning.

I saw it and immediately saw the answer was +20. Because of what she'd said in advance, I tried numerous strategies and always got the same answer. So I was confused. Most people in the class ended up getting +20 as well, which puzzled the professor until one girl admitted to getting +10. Apparently all her other sections got several different answers to this question (scary!!). She tried to explain how she'd done it, and she'd used a confusing verbal approach, which to me isn't a valid strategy when you talk yourself into a wrong answer. So I asked what exactly what was open-ended about this question, because she'd been presenting it as though a variety of answers was a good thing. Apparently she'd done it herself individually and with a team of other adults and gotten different answers. But I argued that since we were dealing with concrete numbers, this was not a question with different possible answers. I recognize the value of questions with multiple solution strategies, but I told her I was sure I could find some better "open-ended" problems. I mentioned the Missing Dollar and Monty Hall and forwarded those to her, but I was wondering if anyone else knew some good ones that aren't so pathetically easy. Or even better, that have multiple valid answers. Is that even possible in say, high school level math?

 

[Grundar]

Interesting question. I'm studying to be a math teacher myself. At the moment i can't figure out any really open ended questions i math. There is questions that makes the one solving them use some approximation and reasoning to get the answer. Like Showing a picture of a climbing wall with two people standing at the bottom, one about to start climb and the other holding the safety rope. The question then is how long is the rope?

Then there is the problems of the type: I have two kids the product of my kids ages are 36 how old are the kids? Thats kind of open ended since you don't have enough information to give a right answer only a list of answers. There is also the trick questions like how much dirt is there left in the hole?

I hope someone can think of better examples because i want to see them too.

/Hans

 

[drkitten]

Apparently all her other sections got several different answers to this question (scary!!). She tried to explain how she'd done it, and she'd used a confusing verbal approach, which to me isn't a valid strategy when you talk yourself into a wrong answer. So I asked what exactly what was open-ended about this question, because she'd been presenting it as though a variety of answers was a good thing. Apparently she'd done it herself individually and with a team of other adults and gotten different answers. But I argued that since we were dealing with concrete numbers, this was not a question with different possible answers.

What makes it an "open-ended" question is the fact that most people don't really understand how "profit" and "loss" work. It's possible, for example, for me to think that I got a "loss" in the middle of the transaction (when I sell an item and then buy it back for more money), and you really need an accountant's mind and the ability to keep your metaphorical eye on the metaphorical ball to trace the money through.

That's more or less true of most "open-ended" mathematical problems. If you define the terms carefully enough, the open-endedness vanishes. But most people don't recognize the need for definitions.

I recognize the value of questions with multiple solution strategies, but I told her I was sure I could find some better "open-ended" problems. I mentioned the Missing Dollar and Monty Hall and forwarded those to her, but I was wondering if anyone else knew some good ones that aren't so pathetically easy. Or even better, that have multiple valid answers. Is that even possible in say, high school level math?

Of course, but they all involve fallacies of equivocation. The easiest one is probably "average." A millionaire is taking his staff on a learjet. Beyond the millionare himself, with an income of $100 million a year, he's taking his pilot ($50,000/yr), his secretary ($30,000/yr) and his two bodyguards ($20,000/yr). What's the average income of the people in the plane?

Depending upon whether you take the mean, the median, or the mode, you will get three different answers. Depending upon whether you take the arithmetic mean or the geometric mean, you will get two different answers. Which meaning of "average" you use should depend on the context.

Benjamin Hoff's excellent book How to Lie with Statistics has a number of other examples.

 

[sphenisc]

What's the first uninteresting number?

 

[Horatius]

Part of the problem is she gets an extra $10 in the middle step with no indication of where it came from. This complicates things. Essentially we don't really know how much she started with. Let's call it X, which has to be greater than the $50 she initialy spent.

Before the first purchase, she has $X, and 0 horses.

Step 1) she has $X-50, and 1 horse.

Step 2) she has $(X-50)+60==$X+10, and 0 horses

Step 3) she has $(X+10)-70==$X-60 and 1 horse

Step 4) she has $(X-60)+80==$X+20 and 0 horses

So she ends up with $20 more than what she started with.

 

[RenaissanceBiker]

Then there is the problems of the type: I have two kids the product of my kids ages are 36 how old are the kids? Thats kind of open ended since you don't have enough information to give a right answer only a list of answers.

The list:
9, 4
6, 6
18, 2
12, 3
1, 36

In case anyone else was interested.

 

[sir drinks-a-lot]

What's the first uninteresting number?

It depends on if you are limiting your answer to integers, or if you are looking for real numbers as well.

The first (lowest) uninteresting integer is 41, and the first uninteresting real number is 17.41. However, this is in dispute. To some, the fact that the string "41" is included in the numeral "17.41" is in and of itself interesting. To these people, the first uninteresting real number would be 17.92.

 

[AmateurScientist]

I'm just starting a Bachelor of Education and my teacher for Gr. 1-8 Mathematics told us in the first class that she had a math problem with no answer next week. She shared the anecdote that a student from last year had written her afterwards asking what the "real" answer was, which was laughable because there wasn't one. Now I'm sure there are good questions like that, but I was shocked when this turned out to be the question:

I saw it and immediately saw the answer was +20. Because of what she'd said in advance, I tried numerous strategies and always got the same answer. So I was confused. Most people in the class ended up getting +20 as well, which puzzled the professor until one girl admitted to getting +10. Apparently all her other sections got several different answers to this question (scary!!). She tried to explain how she'd done it, and she'd used a confusing verbal approach, which to me isn't a valid strategy when you talk yourself into a wrong answer. So I asked what exactly what was open-ended about this question, because she'd been presenting it as though a variety of answers was a good thing. Apparently she'd done it herself individually and with a team of other adults and gotten different answers. But I argued that since we were dealing with concrete numbers, this was not a question with different possible answers. I recognize the value of questions with multiple solution strategies, but I told her I was sure I could find some better "open-ended" problems. I mentioned the Missing Dollar and Monty Hall and forwarded those to her, but I was wondering if anyone else knew some good ones that aren't so pathetically easy. Or even better, that have multiple valid answers. Is that even possible in say, high school level math?

At the risk of being accused of employing the No True Scotsman fallacy, no mathematician would find that question ambiguous or open-ended.

I can understand some educators talking themselves into a circle over it, but not a mathematician. Sorry.

AS

 

[drkitten]

At the risk of being accused of employing the No True Scotsman fallacy, no mathematician would find that question ambiguous or open-ended.

I don't care about your Scotsmen -- but you overestimate the ability of your average mathematician to read a balance sheet.

 

[roger]

I got -10.

here's why. Post number #5 shows the reasoning that leads to an end profit of $20. But she could have made $30, since obviously the market price of the horse was $80.

Yes, this all hinges on what you mean by "lost" - no need to post a 'correction' of my reasoning. She was clearly $20 ahead at the end, but she could have been $30 ahead. Profit, or loss? Undefined by the problem statement.

 

[Rasmus]

I've never been very good at math, but how can there be a question that the woman ends up with 20$ more than what she started out with?

And you got that from a teacher for math teachers no less?

You should start to propose little bets to her throughout your course ...

 

[AmateurScientist]

I don't care about your Scotsmen -- but you overestimate the ability of your average mathematician to read a balance sheet.

Bah. A mathematician has no use for a balance sheet. That's for accountant weenies. The answer is always "0." How exciting.

AS

 

[AmateurScientist]

I got -10.

here's why. Post number #5 shows the reasoning that leads to an end profit of $20. But she could have made $30, since obviously the market price of the horse was $80.

Yes, this all hinges on what you mean by "lost" - no need to post a 'correction' of my reasoning. She was clearly $20 ahead at the end, but she could have been $30 ahead. Profit, or loss? Undefined by the problem statement.

This is the kind of reasoning that causes you to miss the question on the SAT.

It doesn't hinge on anything except simple arithmetic skills, reading comprehension, and the ability to grasp the simple concept that there are two separate purchases and sales here.

This is a question for Grades 1-8. Fair market value is not a concept for 8th graders, nor is it required to solve the problem. Bringing in outside concepts that aren't defined in the problem always results in the wrong kind of reasoning on multiple choice questions like this (unless you are examining a test that is designed specifically not to have defined "correct" answers, like the Microsoft interview test questions, for example).

AS

 

[fribble]

To be honest, you won't find many "open-ended" maths questions, because if a question has more than one way of being interpreted then it's usually considered a bad question, for being imprecise.

If you're looking for fallacies like the Monty Hall Problem, however - my favourite such fallacy is the "All Triangles are Isoceles" proof, which you can find halfway down the page at http: //euler.slu.edu/~clair/puzzlers .html (I can't post links, so you'll have to delete the spaces). You may also be interested in proofs that 1=2; google "1=2 proofs" and you'll get a whole stackfull of 'em.

I got -10.

here's why. Post number #5 shows the reasoning that leads to an end profit of $20. But she could have made $30, since obviously the market price of the horse was $80.

Perhaps the price of the horse went up due to inflation? Hmm?

 

[AmateurScientist]

Perhaps the price of the horse went up due to inflation? Hmm?

Right, and for Ed's sake, what happens when we bring capital gains tax issues into the picture? We don't know in what tax year the woman first bought the horse, how long she held onto the horse before first selling it, in what tax year she later re-purchased the horse, or how long she held it that time. We don't even know the woman's marginal income tax rate, so we cannot compute the amount of net gain she has each time because we don't know at what rate she pays ordinary income tax.

What if she races horses for profit and can expense the entire cost of the horse in the year of the purchase?

Wow, these simple 8th grade problems sure can get complicated. I'm glad I'm not in the 8th grade.

AS

 

[ReFLeX]

If you're looking for fallacies like the Monty Hall Problem, however - my favourite such fallacy is the "All Triangles are Isoceles" proof, which you can find halfway down the page at http: //euler.slu.edu/~clair/puzzlers .html (I can't post links, so you'll have to delete the spaces). You may also be interested in proofs that 1=2; google "1=2 proofs" and you'll get a whole stackfull of 'em.

Thanks, fribble, although I remember the 1=2 proofs and they aren't really questions. I might try to bring them up in class for fun though.

The isosceles triangle puzzle didn't exactly fit into the author's supposed requirements:

# Are short and easy to tell people
# Drive you nuts thinking about them
# Have answers that make you kick yourself

http://euler.slu.edu/~clair/puzzlers.htm

 

[ReFLeX]

It depends on if you are limiting your answer to integers, or if you are looking for real numbers as well.

The first (lowest) uninteresting integer is 41, and the first uninteresting real number is 17.41. However, this is in dispute. To some, the fact that the string "41" is included in the numeral "17.41" is in and of itself interesting. To these people, the first uninteresting real number would be 17.92.

What definition of interesting is this based on?

At the risk of being accused of employing the No True Scotsman fallacy, no mathematician would find that question ambiguous or open-ended.

I can understand some educators talking themselves into a circle over it, but not a mathematician. Sorry.

Hell, that's what I said! "10 math professors would get the same answer," I told her. And she responded, "but 10 ten-year olds wouldn't." And I kept getting cut off for more of other people's questions.

I got -10.

here's why. Post number #5 shows the reasoning that leads to an end profit of $20. But she could have made $30, since obviously the market price of the horse was $80.

Yes, this all hinges on what you mean by "lost" - no need to post a 'correction' of my reasoning. She was clearly $20 ahead at the end, but she could have been $30 ahead. Profit, or loss? Undefined by the problem statement.

I would still disagree, because the question simply asks, "What is the financial outcome of these transactions?" You are adding information to the question. That said, open-endedness shouldn't necessarily equal ambiguity or need for interpretation.

 

[roger]

I would still disagree, because the question simply asks, "What is the financial outcome of these transactions?" You are adding information to the question. That said, open-endedness shouldn't necessarily equal ambiguity or need for interpretation.

Yes, but everyone is assuming that financial outcome = profit. What the heck does financial outcome mean? I don't think it's defined in any economics test. Was she up 20 bucks? yup, trivally. Was she down 10 compared to where she could have been at the end? yup. The problem is not properly specified.

I suspect the problem the teacher had was the previous times she had presented it she had used even more ambiguous language than 'financial outcome'. Or maybe the teacher was just wrong/dumb. Certainly there is only one right answer if you are asking how much more money she has at the end compared to the start.

AS, the question was not for 8th graders, it was posed to college students and adults.

 

[Rasmus]

What definition of interesting is this based on?
Hell, that's what I said! "10 math professors would get the same answer," I told her. And she responded, "but 10 ten-year olds wouldn't." And I kept getting cut off for more of other people's questions.

Just out of curiosty again, in that school system that you're being prepared for - who's doing the teaching? The teachers or the 10 year olds?

 

[Rasmus]

Yes, but everyone is assuming that financial outcome = profit. What the heck does financial outcome mean? I don't think it's defined in any economics test. Was she up 20 bucks? yup, trivally. Was she down 10 compared to where she could have been at the end? yup. The problem is not properly specified.

If you go down that road, the question becomes pointless, though. She could have bought a goose for the money the first time around, and guess how much money she could have made then.

What tells you that the horse was worth the same amount of money throughout, or that the woman would have been able to keep the horse for the time that she had sold it away?

If you work with what you got, she made 20 bucks.