Depends on the size and shape of the pieces. Notching out a teeny bit at either end of the second board would result in three pieces – two very small, one large. I bet she could do that in, say, a couple of minutes.
Or – “if she works just as fast…” – you could take that to mean “just as little time”, meaning: first board, one cut – 10 minutes. Second board, two cuts – 10 minutes.
From my experience in grading and TAing, the question is actually loaded with ambiguity. First of all, it never specifies that the two boards are identical, and we can’t extrapolate well from her cutting speed on the first board to her speed on the second board. This is what I tell students is a necessary assumption to be able to solve a multiple choice problem, but it’s surprising how many don’t make these assumptions. So the problem really should have said “an identical board” rather than “another board.”
The second ambiguity is in the shape of the board, and how she’s cutting it into pieces. The picture implies a bit here, but we still don’t know, for instance, what angle she’s making the cuts at. Again, if this differs between the two boards, we can’t extrapolate. Less students are likely to think of this one, though.
But here’s a fun hypothetical: What if it’s a circular board that she’s cutting into equal wedges? In this case, the teacher’s answer is in fact right, as the number of pieces will be equal to the number of radial cuts. But the picture rules this one out, at least.
Funny, I immediately thought of the radial cuts option and figured that both teacher and student were, in fact, right 😀
I totally agree about the multiple ambiguities in the question.
Not necessarily. The problem specifies that Marie “works just as fast,” which could be interpreted as “ten minutes per cut,” regardless of the size or composition of any subsequent board.
I agree that in a properly phrased problem, such ambiguity would have been corrected.
I’m guessing that the person who graded this didn’t write the test and just took a quick glance at the problem, assumed it was about division, and graded thusly without a second thought. I’m sure they’re not that stupid.
I’m thinking that this is exactly the kind of stupidity we’re talking about here. Grading papers without understanding the question (or even trying to understand it)
I feel bad it took me a second to get that. The question isn’t ideally worded. Though that should trip up the student, not the teacher.
You might be surprised how many math teachers have just memorized the procedures and don’t actually have very good conceptual knowledge. Next time you talk to your local math teacher ask about dividing fractions or negative numbers. See if you get a satisfactory answer. I’m not saying you won’t, but you might not.
Hmm, I’m curious what you think baton means (and where you are from) since I would never have thought to describe the picture as a baton. Perhaps plank would be a more accurate term since I suppose a board ought to be thinner than it is wide and the object in the picture appears to have a roughly square cross section. However in my own american english I would probably call the item depicted a board.
If the first cut were down the length of the board–so one cut for 10 meters, then the teacher’s answer would be right if the next cut (creating the third piece) took one 10 meter board and cut 5 meters–say from an end to the edge just short of the middle (because the cut is not straight down the middle), then it could justify the teacher’s answer of 15. This would then allow for any answer greater than 10 to be correct (up to infinity). Imagine the infinitely thin saw cutting back-and-forth in a continuous manner so that the cut of the slices are a few molecules thick.
The mistake the marker made is in assuming that the amount of work is proportional to the number of pieces rather than the number of cuts. The question itself doesn’t mention the number of cuts and after seeing many similar problems and being in that grading trance where one thinks through the problems only superficially it’s not unlikely that a competent person would make that mistake.
The grader probably reasoned about time per piece rather than time per cut. So instead of thinking “ten minutes per cut,” the grader thought “five minutes per piece.” So this isn’t really a math fail, it’s an English fail. This grader thought that piece was an activity that took time, which it isn’t. It’s not even a verb.
Being fair, anyone could make that mistake _once_. The real test is whether the teacher immediately accepted that 20 minutes was a better answer when the mistake was pointed out.
One of my colleagues made mistakes on her answer key for a final exam and is now facing a student grievance because she won’t face up to it. I wonder if she graded this test.
Absurd? Yes, I quite agree. And I checked the solutions myself. When the student files a grievance the student is sure to prevail. (Even math teachers can be irrational.)
That is a pretty bad question; due to how it was worded I almost came to the same incorrect assumption regarding what the question asked — that is, I intially thought it read “to saw two boards”.
And then even after taking that into account, there’s the question of “just as fast”, as previously noted. I wouldn’t assume that it means that it took 10 minutes to saw the board, but if it was about interpreting word problems…then maybe I could see that being the case.
But yes, I’ve read in the past about the sad state of conceptual mathematics knowledge among teachers in the US. It seems to me as though some elementary school teachers choose that field of study even partially *because* they really don’t understand math.
Still, there are lots of schoolteachers who don’t understand multiplication and division of fractions themselves. Considering that coming to understand what a certain operation means (rather than focusing on applying operations to model situations and do problem solving) is the brunt of grade school mathematics, it’s particularly frustrating.
This is another example why the system of filling in blanks is a bad one. If the pupil had been asked for a detailed explanation of how he arrived at the result, the teacher might have learned something.
This question is not designed to test mathematics skills. It is designed to catch people out, and in this it has succeeded.
The mistake is a very easy one to make, particularly if you are in the middle of a mathematics exam and not thinking about how to saw wood. To see how easily this mistake can be made, let’s change one word.
It took Mary 10 minutes to work a board into 2 pieces. If she works just as fast, how long will it take her to work another board into 3 pieces.
Depending on the definition of work(i.e. including finishing), the answer could be either 15 minutes or 20 minutes.
There’s a more serious problem. While this question is designed to catch people out, it only accepts a final answer. The student has put down 20 here, but we have no real indication of whether the student actually understood the question or not. They could have simply multiplied the first two numbers they saw, 10 and 2. What is this students reasoning? Are we sure they understood the question? At least we know what the examiner’s reasoning was, even if it was flawed.
If this question was a single word problem as part of a larger exam, then it is fine. But if this is the kind of question that mathematics students are expected to answer, then there is little wonder why they would regard mathematics as an inscrutable academic con game designed to impede their education. You can’t throw about too many questions like this in an exam.
Mathematics is not about navigating a series of artificial intellectual pitfalls. It is about understanding the world through reason and systematic methods. Perhaps this questions was designed to test those skills, but I have my doubts.
This is a completely understandable off-by-one error. I have a hard time faulting the teacher for this when I make them in the regular course of programming.
Now, if the student brought it up and the teacher refused to budge, that would be “argh!” worthy.
Now, if the student brought it up and the teacher refused to budge, that would be “argh!” worthy.
Reminds me of many moons ago as a high school sophomore (age 16) when I found an error in the Plane Geometry textbook I was being taught from. I pointed it out to my teacher, and she saw the error and agreed with me. She took it to the department head who dismissed my observation out-of-hand (i.e., with NO explanation) and insisted the textbook was right.
Of course, I did learn something from the experience …
Almost as annoying are the equations on the bottom. 10 = 2? I know what the grader means, but students get into the habit of thinking of ‘=’ as synonymous with ‘the answer is’ and this causes conceptual trouble in algebra.
This question is a less tricky variant on “If it takes 6 seconds for a grandfather clock to strike 3 bells, how long will it take to strike 12 bells?” Although it is easy to give the wrong answer if one is thoughtless, the correct answer is clear and not all that hard if one thinks carefully. As a teacher myself I find it inexcusable for a teacher not to carefully think through and check all solutions on a test key.
I don’t see how “just as fast” is ambiguous. If I run at a rate of 6 miles an hour and my friend also runs at a rate of 6 miles an hour, then my friend runs just as fast as I do. If I run at a rate of 6 miles per hour and my friend runs at a rate of 8 miles per hour, then my friend runs just as fast as I do, but I do not run just as fast as my friend. So, if a is “just as fast” as b, it means that b is at least equal to and possibly more than a. The problem here, I don’t think, to be ambiguity. The problem here comes as that “just as fast” does *not* mean exactly equal.
and that is basicaly the definition of ambiguity. if there is more than one acceptable interpetration, then it’s ambiguous. if ‘just as fast’ does not mean exactly equal, but equal or greater, then there is more than one answer and therefore it’s ambiguous
I agree with commenters who say it’s partly an English fail. It definitely took me a second go-round to see the correct answer, because I immediately gravitated to the numbers within the paragraph and saw the same thing the teacher saw: 10 is to 2 as x is to 3.
It would have been clear if it had said, “It took Marie 10 minutes to make 1 cut in a board. How long would it take her to make 2 cuts?”
But perhaps the point of the problem is to have the test taker work through that extra interpretive step? Still, it comes off to me like a trick question. Most people who answered as the teacher did would see their flaw immediately, but it doesn’t prove they don’t know how to do math.
I do not buy that the teacher should get a pass. The teacher fails on two counts. The first is having a trick (or tricky) question on a test that appears to be a short answer answer test with a lot of questions, especially if the student isn’t going to have much time to interpret what is being asked. The second is that the teacher is not under the time or stress pressure of the student and should have been able to correctly answer the question.
Of course there is some ambiguity in the exercise, just as there will always be some ambiguity in all ‘realistic’ word problems. Part of that ambiguity is resolved by the picture of the board & saw to the right of the problem. In most cases its absolutely clear, even with some mild ambiguity, what the mathematical problem is that is supposed to be solved.
In this case, however, all the mild ambiguities about cutting times, shape of the board, whatever… fade to nothing compared to the fact that the teacher clearly did not see the reasoning of the student (the correct reasoning in my opinion.) Part of being a good teacher is the ability to recognize how students think, and even if they get an incorrect answer, be able to realize by what crooked reasoning that answer was obtained. Such an analysis enables teachers to identify trouble spots in their own teaching. This teacher clearly did not attempt that, or if he/she did, failed miserable.
The fact that there are 3 stars behind the question should also be an indication, to both student and teacher, that this requires a bit more than just standard equal-ratio reasoning.
Thank you for reminding me why I am not being too harsh in supporting the death penalty for innumeracy 🙂
The “bad phrasing” is on purpose. The pupil is supposed to have to work out that cutting a board into 2 pieces requires 1 cut, therefore cutting a board into 3 pieces requires 2 cuts.
When I was taking my O-level maths, calculators were not allowed and half the marks would have been awarded just for showing working — i.e., we would have been expected to write something to the effect of “2 pieces = 1 cut = 10 mins, 3 pieces = 2 cuts = 20 mins”. Even “2 pieces = 1 cut = 10 mins, 3 pieces = 2 cuts = 30 mins” would have earned something, even though 2 * 10 != 30, for correctly surmising how to answer the question.
The practice of showing all intermediate steps was drilled into us. Even when not required for the test, it is a useful habit to get into; because if {when} you do make an elementary mistake with the figures, you can go back and see where you went wrong.
That’s still not a board, though. It’s not even a plank. It looks like a length of 50×50 PAR.
Response to comments on the new real numbers:
The field axioms are listed in Royden’s Real Analysis, pp. 31 – 32. One of them is the field axioms that says: given two real numbers x, y one and only one of the following holds: x y. L. E. J. Brouwer and myself constructed two different counterexamples to it in Benacerraf, P. and Putnam, H., (1985), Philosophy of Mathematics, Cambridge University, Press, Cambridge, 52 – 61 and Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84, respectively. The completeness axiom, a variant of the axiom of choice, also leads to a contradiction in R^3 known as the Banach-Tarski paradox. Its contradiction comes from itself but its use of the universal or existential quaantifier on infinite set. Nonterminating decimals are ambiguous, ill-defined, because not all its digits are known or computable. Thus, division of an integer by a prime other than 2 or 5 is ambiguous, ill-defined because the quotient is nonterminating, e.g., 2/7. That is why it is impossible to add or multiply two nonterminating decimals; we can only approximate their quotient by a terminating decimal. Moreover, it is improper, i.e., noesense, to apply any operation of the real number system to the dark number d* because d* is not a real number; therefore, the result will be ill-defined. BTW, foreign writers of English sometimes write better English because they learn the language the right way.
line 10: “Its contradiction comes from itself but its use of the universal or existential quaantifier on infinite set.” should read “Its contradiction comes not from itself but from the use of the universal or existential quantifier when applied to infinite set.”
I just noticed that posts sometimes do not come out correctly. This error happened twice in the statement of the trichotomy axiom. It should read as follows: given real numbers x, y one and only one of the following holds: x is less than y, x equals y, x is greater than y. Please check this out, Administrator.
Now, to the more substantive issue: I consider this blog one of the better ones along with Free Library and Larry Freeman’s False Proofs. Wikipedia is the worst. Not only does it delete posts and block this blogger to keep unanimity, its links also post outright lies. For example, WikiPilipinas posted lies about Filipino inventors, who are long gone, and this scientist. Moreover, it printed a fake apology purportedly coming from and blocked this blogger. Only public pressure forced deletion of both posts.
Next in the worst category are Halo Scan and Don’t Let Me Stop You. They delete posts they disagree with or don’t like.
Finally, I commend L. P. Cruz for not hiding behind username which means he is confident his post makes sense and ready to stand behind it.
The teacher is actually right. For one cut it took 10 minutes, which resulted in two pieces. Say the board was a square, when one cut has happened, you’re left with two half boards. To cut the other half board it will take half the time it took to cut the first one. Thus it only takes 15 minutes to cut it into three pieces = two cuts = 10 minutes plus half of it.
Mike, you’re assuming that the board is square, that it is being cut into two perfect halves, and that one half is being cut into two squares and not lengthways. Each assumption you have made is questionable.
I’ll have to admit it took me a minute to think that one through. It seems logical to think that if you saw one board into two and it takes 10 minutes, it would take 15 minutes to saw a board into 3 pieces.
The flaw is that it is not the number of pieces that matters it is the number of times you have to saw the board to get those pieces. 2 pieces = 1 saw = 10 minutes. 3 pieces = 2 saws = 20 minutes. That is a little bit tricky at first, but easily understood if you think it through.
Ugh!!!!!!! That person teaches!
Yes, Bora, that’s exactly the cause of this title. It’s enough to inspire dry heaves!
Does it take 5 minutes to leave the board alone?
That’s a math WIN! (and a construction FAIL)
Well she practiced when she was doing the first cut, and so was able to do the second two faster.
Or the second board was only 75% as wide as the first board!
Second two? If it takes Marie two cuts to saw a board in three pieces, and she has already completed one cut, how many more cuts must she make? 🙂
Ah, never mind, she starts with a new board. I’m the bonehead here, not Sean 🙂
The extra BONUS fail is to follow the link to the FailBlog site and see a whole bunch of commenters saying, “Where’s the fail?”
(But expecting internet commenters to be intelligent is really asking too much — present company excluded, of course!)
Depends on the size and shape of the pieces. Notching out a teeny bit at either end of the second board would result in three pieces – two very small, one large. I bet she could do that in, say, a couple of minutes.
Or – “if she works just as fast…” – you could take that to mean “just as little time”, meaning: first board, one cut – 10 minutes. Second board, two cuts – 10 minutes.
How long is a piece of string, anyway?
I can saw a board into one piece in no time. I just did it to a billion of them while you were reading this.
From my experience in grading and TAing, the question is actually loaded with ambiguity. First of all, it never specifies that the two boards are identical, and we can’t extrapolate well from her cutting speed on the first board to her speed on the second board. This is what I tell students is a necessary assumption to be able to solve a multiple choice problem, but it’s surprising how many don’t make these assumptions. So the problem really should have said “an identical board” rather than “another board.”
The second ambiguity is in the shape of the board, and how she’s cutting it into pieces. The picture implies a bit here, but we still don’t know, for instance, what angle she’s making the cuts at. Again, if this differs between the two boards, we can’t extrapolate. Less students are likely to think of this one, though.
But here’s a fun hypothetical: What if it’s a circular board that she’s cutting into equal wedges? In this case, the teacher’s answer is in fact right, as the number of pieces will be equal to the number of radial cuts. But the picture rules this one out, at least.
Funny, I immediately thought of the radial cuts option and figured that both teacher and student were, in fact, right 😀
I totally agree about the multiple ambiguities in the question.
@infophile
Not necessarily. The problem specifies that Marie “works just as fast,” which could be interpreted as “ten minutes per cut,” regardless of the size or composition of any subsequent board.
I agree that in a properly phrased problem, such ambiguity would have been corrected.
I’m guessing that the person who graded this didn’t write the test and just took a quick glance at the problem, assumed it was about division, and graded thusly without a second thought. I’m sure they’re not that stupid.
I’m thinking that this is exactly the kind of stupidity we’re talking about here. Grading papers without understanding the question (or even trying to understand it)
I feel bad it took me a second to get that. The question isn’t ideally worded. Though that should trip up the student, not the teacher.
You might be surprised how many math teachers have just memorized the procedures and don’t actually have very good conceptual knowledge. Next time you talk to your local math teacher ask about dividing fractions or negative numbers. See if you get a satisfactory answer. I’m not saying you won’t, but you might not.
Are there are some language differences in different countries? The question talks about a board, but the picture shows a baton/plank.
Hmm, I’m curious what you think baton means (and where you are from) since I would never have thought to describe the picture as a baton. Perhaps plank would be a more accurate term since I suppose a board ought to be thinner than it is wide and the object in the picture appears to have a roughly square cross section. However in my own american english I would probably call the item depicted a board.
If the first cut were down the length of the board–so one cut for 10 meters, then the teacher’s answer would be right if the next cut (creating the third piece) took one 10 meter board and cut 5 meters–say from an end to the edge just short of the middle (because the cut is not straight down the middle), then it could justify the teacher’s answer of 15. This would then allow for any answer greater than 10 to be correct (up to infinity). Imagine the infinitely thin saw cutting back-and-forth in a continuous manner so that the cut of the slices are a few molecules thick.
“just as fast” is ambiguous.
The mistake the marker made is in assuming that the amount of work is proportional to the number of pieces rather than the number of cuts. The question itself doesn’t mention the number of cuts and after seeing many similar problems and being in that grading trance where one thinks through the problems only superficially it’s not unlikely that a competent person would make that mistake.
The grader probably reasoned about time per piece rather than time per cut. So instead of thinking “ten minutes per cut,” the grader thought “five minutes per piece.” So this isn’t really a math fail, it’s an English fail. This grader thought that piece was an activity that took time, which it isn’t. It’s not even a verb.
Being fair, anyone could make that mistake _once_. The real test is whether the teacher immediately accepted that 20 minutes was a better answer when the mistake was pointed out.
I think my daughter had this teacher for algebra.
Ow!
One of my colleagues made mistakes on her answer key for a final exam and is now facing a student grievance because she won’t face up to it. I wonder if she graded this test.
P.S.: Hey! There are two Zenos here!
How is that possible? It the mistake demonstrable? If so, how could she possibly expect to have her decision survive a grievance? That’s absurd!
Absurd? Yes, I quite agree. And I checked the solutions myself. When the student files a grievance the student is sure to prevail. (Even math teachers can be irrational.)
That is a pretty bad question; due to how it was worded I almost came to the same incorrect assumption regarding what the question asked — that is, I intially thought it read “to saw two boards”.
And then even after taking that into account, there’s the question of “just as fast”, as previously noted. I wouldn’t assume that it means that it took 10 minutes to saw the board, but if it was about interpreting word problems…then maybe I could see that being the case.
But yes, I’ve read in the past about the sad state of conceptual mathematics knowledge among teachers in the US. It seems to me as though some elementary school teachers choose that field of study even partially *because* they really don’t understand math.
Still, there are lots of schoolteachers who don’t understand multiplication and division of fractions themselves. Considering that coming to understand what a certain operation means (rather than focusing on applying operations to model situations and do problem solving) is the brunt of grade school mathematics, it’s particularly frustrating.
This is another example why the system of filling in blanks is a bad one. If the pupil had been asked for a detailed explanation of how he arrived at the result, the teacher might have learned something.
This question is not designed to test mathematics skills. It is designed to catch people out, and in this it has succeeded.
The mistake is a very easy one to make, particularly if you are in the middle of a mathematics exam and not thinking about how to saw wood. To see how easily this mistake can be made, let’s change one word.
Depending on the definition of work(i.e. including finishing), the answer could be either 15 minutes or 20 minutes.
There’s a more serious problem. While this question is designed to catch people out, it only accepts a final answer. The student has put down 20 here, but we have no real indication of whether the student actually understood the question or not. They could have simply multiplied the first two numbers they saw, 10 and 2. What is this students reasoning? Are we sure they understood the question? At least we know what the examiner’s reasoning was, even if it was flawed.
If this question was a single word problem as part of a larger exam, then it is fine. But if this is the kind of question that mathematics students are expected to answer, then there is little wonder why they would regard mathematics as an inscrutable academic con game designed to impede their education. You can’t throw about too many questions like this in an exam.
Mathematics is not about navigating a series of artificial intellectual pitfalls. It is about understanding the world through reason and systematic methods. Perhaps this questions was designed to test those skills, but I have my doubts.
This is a completely understandable off-by-one error. I have a hard time faulting the teacher for this when I make them in the regular course of programming.
Now, if the student brought it up and the teacher refused to budge, that would be “argh!” worthy.
Reminds me of many moons ago as a high school sophomore (age 16) when I found an error in the Plane Geometry textbook I was being taught from. I pointed it out to my teacher, and she saw the error and agreed with me. She took it to the department head who dismissed my observation out-of-hand (i.e., with NO explanation) and insisted the textbook was right.
Of course, I did learn something from the experience …
Almost as annoying are the equations on the bottom. 10 = 2? I know what the grader means, but students get into the habit of thinking of ‘=’ as synonymous with ‘the answer is’ and this causes conceptual trouble in algebra.
This question is a less tricky variant on “If it takes 6 seconds for a grandfather clock to strike 3 bells, how long will it take to strike 12 bells?” Although it is easy to give the wrong answer if one is thoughtless, the correct answer is clear and not all that hard if one thinks carefully. As a teacher myself I find it inexcusable for a teacher not to carefully think through and check all solutions on a test key.
I don’t see how “just as fast” is ambiguous. If I run at a rate of 6 miles an hour and my friend also runs at a rate of 6 miles an hour, then my friend runs just as fast as I do. If I run at a rate of 6 miles per hour and my friend runs at a rate of 8 miles per hour, then my friend runs just as fast as I do, but I do not run just as fast as my friend. So, if a is “just as fast” as b, it means that b is at least equal to and possibly more than a. The problem here, I don’t think, to be ambiguity. The problem here comes as that “just as fast” does *not* mean exactly equal.
and that is basicaly the definition of ambiguity. if there is more than one acceptable interpetration, then it’s ambiguous. if ‘just as fast’ does not mean exactly equal, but equal or greater, then there is more than one answer and therefore it’s ambiguous
interpretation*
“Just as fast” could mean:
Same number of boards cut per time.
Same number of linear board-inches cut per time (same as above?).
Same number of cubic board-inches cut per time.
Same number of saw strokes per time.
I agree with commenters who say it’s partly an English fail. It definitely took me a second go-round to see the correct answer, because I immediately gravitated to the numbers within the paragraph and saw the same thing the teacher saw: 10 is to 2 as x is to 3.
It would have been clear if it had said, “It took Marie 10 minutes to make 1 cut in a board. How long would it take her to make 2 cuts?”
But perhaps the point of the problem is to have the test taker work through that extra interpretive step? Still, it comes off to me like a trick question. Most people who answered as the teacher did would see their flaw immediately, but it doesn’t prove they don’t know how to do math.
I do not buy that the teacher should get a pass. The teacher fails on two counts. The first is having a trick (or tricky) question on a test that appears to be a short answer answer test with a lot of questions, especially if the student isn’t going to have much time to interpret what is being asked. The second is that the teacher is not under the time or stress pressure of the student and should have been able to correctly answer the question.
I guess I fail at proof reading.
This. I honestly wasn’t able to find the mistake.
Of course there is some ambiguity in the exercise, just as there will always be some ambiguity in all ‘realistic’ word problems. Part of that ambiguity is resolved by the picture of the board & saw to the right of the problem. In most cases its absolutely clear, even with some mild ambiguity, what the mathematical problem is that is supposed to be solved.
In this case, however, all the mild ambiguities about cutting times, shape of the board, whatever… fade to nothing compared to the fact that the teacher clearly did not see the reasoning of the student (the correct reasoning in my opinion.) Part of being a good teacher is the ability to recognize how students think, and even if they get an incorrect answer, be able to realize by what crooked reasoning that answer was obtained. Such an analysis enables teachers to identify trouble spots in their own teaching. This teacher clearly did not attempt that, or if he/she did, failed miserable.
The fact that there are 3 stars behind the question should also be an indication, to both student and teacher, that this requires a bit more than just standard equal-ratio reasoning.
Thank you for reminding me why I am not being too harsh in supporting the death penalty for innumeracy 🙂
The “bad phrasing” is on purpose. The pupil is supposed to have to work out that cutting a board into 2 pieces requires 1 cut, therefore cutting a board into 3 pieces requires 2 cuts.
When I was taking my O-level maths, calculators were not allowed and half the marks would have been awarded just for showing working — i.e., we would have been expected to write something to the effect of “2 pieces = 1 cut = 10 mins, 3 pieces = 2 cuts = 20 mins”. Even “2 pieces = 1 cut = 10 mins, 3 pieces = 2 cuts = 30 mins” would have earned something, even though 2 * 10 != 30, for correctly surmising how to answer the question.
The practice of showing all intermediate steps was drilled into us. Even when not required for the test, it is a useful habit to get into; because if {when} you do make an elementary mistake with the figures, you can go back and see where you went wrong.
That’s still not a board, though. It’s not even a plank. It looks like a length of 50×50 PAR.
But how long does it take to saw a board into one piece?
If it takes 1 programmer 12 months to develop an application, then surely it would take 12 programmers 1 month to develop the same application, right?
Response to comments on the new real numbers:
The field axioms are listed in Royden’s Real Analysis, pp. 31 – 32. One of them is the field axioms that says: given two real numbers x, y one and only one of the following holds: x y. L. E. J. Brouwer and myself constructed two different counterexamples to it in Benacerraf, P. and Putnam, H., (1985), Philosophy of Mathematics, Cambridge University, Press, Cambridge, 52 – 61 and Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84, respectively. The completeness axiom, a variant of the axiom of choice, also leads to a contradiction in R^3 known as the Banach-Tarski paradox. Its contradiction comes from itself but its use of the universal or existential quaantifier on infinite set. Nonterminating decimals are ambiguous, ill-defined, because not all its digits are known or computable. Thus, division of an integer by a prime other than 2 or 5 is ambiguous, ill-defined because the quotient is nonterminating, e.g., 2/7. That is why it is impossible to add or multiply two nonterminating decimals; we can only approximate their quotient by a terminating decimal. Moreover, it is improper, i.e., noesense, to apply any operation of the real number system to the dark number d* because d* is not a real number; therefore, the result will be ill-defined. BTW, foreign writers of English sometimes write better English because they learn the language the right way.
Cheers,
E. E. Escultura
Correction: “field axioms” on line 3 of my post should read: “trichotomy axiom”
Cheers,
E. E. Escultura
Sorry, more corrections to my post:
line 4: “x y.” should read “x > y.”
line 10: “Its contradiction comes from itself but its use of the universal or existential quaantifier on infinite set.” should read “Its contradiction comes not from itself but from the use of the universal or existential quantifier when applied to infinite set.”
Cheers,
E. E. Escultura
I just noticed that posts sometimes do not come out correctly. This error happened twice in the statement of the trichotomy axiom. It should read as follows: given real numbers x, y one and only one of the following holds: x is less than y, x equals y, x is greater than y. Please check this out, Administrator.
Now, to the more substantive issue: I consider this blog one of the better ones along with Free Library and Larry Freeman’s False Proofs. Wikipedia is the worst. Not only does it delete posts and block this blogger to keep unanimity, its links also post outright lies. For example, WikiPilipinas posted lies about Filipino inventors, who are long gone, and this scientist. Moreover, it printed a fake apology purportedly coming from and blocked this blogger. Only public pressure forced deletion of both posts.
Next in the worst category are Halo Scan and Don’t Let Me Stop You. They delete posts they disagree with or don’t like.
Finally, I commend L. P. Cruz for not hiding behind username which means he is confident his post makes sense and ready to stand behind it.
Cheers,
E. E. Escultura
Any guess on the grade level for which this is intended? 1st? 2nd? 3rd?
The teacher is actually right. For one cut it took 10 minutes, which resulted in two pieces. Say the board was a square, when one cut has happened, you’re left with two half boards. To cut the other half board it will take half the time it took to cut the first one. Thus it only takes 15 minutes to cut it into three pieces = two cuts = 10 minutes plus half of it.
Mike, you’re assuming that the board is square, that it is being cut into two perfect halves, and that one half is being cut into two squares and not lengthways. Each assumption you have made is questionable.
illustration?!
I’ll have to admit it took me a minute to think that one through. It seems logical to think that if you saw one board into two and it takes 10 minutes, it would take 15 minutes to saw a board into 3 pieces.
The flaw is that it is not the number of pieces that matters it is the number of times you have to saw the board to get those pieces. 2 pieces = 1 saw = 10 minutes. 3 pieces = 2 saws = 20 minutes. That is a little bit tricky at first, but easily understood if you think it through.
As usual I’m late, late for this blog, and late for this reply. I just hope this is not looked on as bad manners here.
I think eveyone here just failed badly.
There seems to be consensus that the teacher doing the scoring failed badly, mostly because of the ridiculous table he used as reason for the scoring.
The student also failed because he made assumptions that are just not supported by the wording of the problem.
But the one who failed the most badest was the person who worded the problem.
When I try to give an answer to the problem as worded, I can only come up with bounds. The lower bound would be 0 and the upper bound infinity.