# A “simple” 3rd grade problem…or is it?

So this is supposed to be really simple, and it's taken from the following picture:

Text-only:

It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her to saw another board into $3$ pieces?

I don't understand what's wrong with this question. I think the student answered the question wrong, yet my friend insists the student got the question right.

I feel like I'm missing something critical here. What am I getting wrong here?

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This is simultaneously wonderful and sad. Wonderful for the student who was level-headed enough to answer this question correctly, and sad that this teacher's mistake could be representative of the quality of elementary school math education. – Jared May 3 '13 at 3:38
I took five minutes per cut. One cut yields two pieces. Two cuts yield three pieces. And so on..... – Michael Hardy May 3 '13 at 4:40
I think the issue is that the language and image are incongruent. The question should have been "10 minutes to saw 2 pieces from a board" (2 cuts), then the teachers answer would be correct. As it is stated, it implies sawing a board in half (1 cut). – Vijay May 3 '13 at 5:23
The language and the image are in perfect agreement- the image shows two pieces resulting from a single cut. – JayL May 3 '13 at 8:01
By the way, originally this was posted as "Teacher Math Fail" at zerooutoffice: zerooutoffive.blogspot.com/2010/10/teacher-math-fail.html – azimut May 6 '13 at 20:21

Haha! The student probably has a more reasonable interpretation of the question.

Of course, cutting one thing into two pieces requires only one cut! Cutting something into three pieces requires two cuts!

------------------------------- 0 cuts/1 piece/0 minutes

------------|------------------ 1 cut/2 pieces/10 minutes
------|------------|----------- 2 cuts/3 pieces/20 minutes

This is a variation of the "fence post" problem: how many posts do you need to build a 100 foot long fence with 10 foot sections between the posts?

Answer: 11 You have to draw the problem to get it...See below, and count the posts!

|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
0-----10----20----30----40----50----60----70----80----90---100

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Wow, I feel completely stupid. That was so obvious, yet I didn't figure it out. I guess it isn't too late to start at square one – Enigma May 3 '13 at 3:36
In all fairness, neither did whoever graded this problem. This question made me smile :) – Gyu Eun Lee May 3 '13 at 3:37
This is an example of a red herring in a word problem. The number of pieces (2) distracts from the actual variable - the number of cuts (1). I think this is a very common (and important) technique to teach children, because it happens so often in solving real world problems – xdumaine May 3 '13 at 17:20
Agreed and not only that but the question does say "a board" into "two pieces." A board = one board. Going from one board to two boards takes one cut. I can't see how this could be interpreted any other way. They don't talk about sawing equal length pieces or cutting two equal pieces from a very long board needing 3 cuts (you would have 3 pieces by then). anyway, my wife and I home school our kids because of stupid crap like this. – Eric M May 3 '13 at 20:02
Well, it depends on the topology of the board. The teacher is right if the board has the topology of, say, a ring or a torus. ;) – Heidar May 24 '13 at 23:11

Well, the information is incomplete, so they're both right and wrong. Since the question is for $3^{rd}$ graders the correct answer should be $20$ minutes ($2$ cuts $\times$ $10$ min), though the teacher is right if you do cut it like this (first red, then green):

The problem is that the question doesn't say anything about how you have to cut it, so the blue cut would have been good enough too. That cut should only have taken a few seconds.

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However if you look at the drawing of the piece of wood being cut, this approach would be extremely difficult. – Mark Adler May 3 '13 at 6:42
@Mark Yes, looking at the drawing of the piece of wood, I think the answer would rather be close to an hour... – Axel May 3 '13 at 7:48
The text is indeed incomplete, and this square is exactly what came to my mind. But then I noticed the picture next to the question. – ugoren May 3 '13 at 8:01
The question is hypothetical, and doesn't give the information about board size. So I think we might interpret 'as fast' to be that it means Marie can cut any board into 2 pieces in 10 minutes regardless of its size and length of the cut. – tia May 3 '13 at 8:32
+1 for the blue cut – kritzikratzi May 4 '13 at 14:38

The student was correct:

Sawing a board into two pieces requires exactly one cut to be made. Sawing the board into three pieces requires exactly two cuts...

Hence, if it took $\bf 10$ minutes to make one cut, then cutting a board twice, at the same pace, would take $\;2 \times 10 = \bf 20$ minutes.

The instructor should receive tutoring from the student, I'm afraid!

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I'm curious: where did you find this image? – amWhy May 3 '13 at 3:47
@amWhy Diagonally across the image is a watermark that says zerooutoffive.com, where the image was at some point submitted. – Karl Kronenfeld May 3 '13 at 9:43
Retrosaur: Oh! I was just wondering if this came from actual work submitted by a student... Sometimes we see samples of work or questions/solutions where the educator went awry...that's all. ;-) $\;\;$ I didn't mean to question its authenticity. Just wondering if this actually happened to a student. – amWhy May 3 '13 at 13:11
@amWhy: This reminds me of the Monty Hall Problem and the firestorm that it raised! :-) +1 – Amzoti May 4 '13 at 0:22
100th upvote! Haha, enjoy the gold! – ABC May 6 '13 at 4:42

You can actually do it in ten minutes but your saw must look like this:

|     |
|     |
|     |
|     |      <- cutting edges
|     |
|     |
+--+--+
|         <- handle
|


:-)

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Well of course... but why 2 blades, why not 10 or 100. Obvously it could not be done in 10 minutes. Adding a second blade doubles the friction, so to complete the cut with 2 blades in the same time she'd have to increase the force applied, and/or the rate she moved the blade. But, if she could increase the force and rate, she could make a single cut with 1 blade in perhaps 5 minutes, or even 1 minute, invalidating the relationship of time, and any two cuts. So, it's clear she must work with the same force and rate. Therefore, with 2 blades it would take 20 minutes. Still 10 minutes per cut. – Kevin Fegan May 4 '13 at 20:37
This is definitely a mathematician's answer! – bob.sacamento May 5 '13 at 17:38
@PyRulez: Nope. What's involved here is surface energy. To create new surface area, you need to supply this energy to the material by doing work on it. As the blade thins out, the closer the work you expend becomes to the minimum surface energy. That's asymptotic behavior with asymptote most definitely not at zero. – Kuba Ober May 6 '13 at 22:53
Use a circular saw: 1 minute. – naught101 May 7 '13 at 3:03
And, bods, for the love of $DEITY, please don't vote this up so far that it exceeds the correct answers. It was just a bit of humor (hence the community wiki) and I never expected it to be quite this popular - if it starts to look like it's threatening the correct answers, I'll have to delete it, and the world will be a drabber place :-) – paxdiablo May 7 '13 at 5:51 Another correct answer would be 10 minutes. One could infer, "If she works just as fast," that "work" is the complete amount of time to do the job. - That's how I understood it at first. – Johannes May 3 '13 at 16:48 I once took a test that had the question "What is the last thing you should do before handing in a test?" and I answered (incorrectly, sadly), choice (c) which was "Staple a$20 bill to the test" which is the last thing I would ever consider doing. I tried to explain my reasoning to the teacher but they just laughed and said I was an existential philosopher... – Michael May 3 '13 at 20:59
@Michael That is why mathematics puzzles as english words seldom often have ambiguous answers. The english language is rich with ways to say one thing and mean another. – Jonathan May 3 '13 at 21:42
@Jonathan or, apparently, to say two opposite things at the same time…! (“seldom often”?) – Aant May 4 '13 at 17:33
@AAnt Whoops! I was going to say "words seldom have only one meaning", but when I changed the comment to refer to the puzzles, I left in an unintended "seldom". Well spotted. :) – Jonathan May 7 '13 at 7:15

The topologists among us may perhaps enjoy the following defense of the teacher's answer: if the board is in the shape of a ring, it will take two cuts to get two pieces, and three cuts to get 3 pieces.

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Of course, if you look at the board itself, it clearly isn't in the shape of a ring... – Joe Z. Nov 28 '13 at 9:46
Actually an examination of the figure suggests that the question may originally have been phrased in terms of "cutting off" such and such number of pieces (with the other end being off the board). This formulation would make the teacher's answer correct. One can speculate that the phrasing was carelessly changed by someone somewhere along the way. – user72694 Nov 28 '13 at 19:16
(Also, if the board is in the shape of a ring, you do indeed have to "saw it into one piece" first, as some people were jokingly pointing out.) – Joe Z. Jan 17 '14 at 8:20

Let P : pieces

Let m : minutes

Let C : cuts

Let t : time per slice = 10

$$C(m) = m/t , \{m| m < Life(Marie)\}, \{C < length(board)\}$$

$$P(C(m)) = floor(C(m)) +1 , \{m| m < Life(Marie)\}$$

You're right that clearly isn't a simple grade 3 problem, but the answer is still 20. $$P(C(20)) = floor(C(20)) + 1 = floor(20/10)+1 =floor(2) + 1 = 2 + 1 = 3\ pieces$$

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+1 That's perfect! – Kevin Fegan May 4 '13 at 1:34
Notation overflow! – user13107 May 5 '13 at 16:39
I don't understand... How can you have an inequality C < length(board) where the LHS is an amount of cuts and the RHS is the length of the board? They're completely different units. Isn't this a case of comparing apples and pears? – Alderath Jan 10 '14 at 15:40
A ==|==|== B The length of the board is the distance from A to B, the length of C is a real number, the set of C has an integral length. The granularity of the length of the board is infinite, but the number of times Marie is capable of subdividing it is finite. The length of a cut occupies some arbitrary amount of space which is of the same unit in which you are measuring the board. – awiebe Jan 11 '14 at 3:23
lmao­­­­­­­­­­­ – Derek 朕會功夫 Aug 2 '14 at 22:21

The student answered the question the most correct way possible. First it is stated that Marie spends 10 minutes on sawing a board into two pieces. And then the student must answer how long it will take to saw another board into three pieces.

So we are not talking about chopping off pieces from an undefined source. We are talking about splitting a board.

However, it's poorly phrased because it's not explained how the board must be cut. It can be cut in infinite ways. Also, we don't know if the two boards are identical, so we must rely on assumptions here.

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The teacher would be correct if the question was "... to cut two pieces from the end of a board ...", implying more strongly that the pieces were being cut so as to leave another remaining piece.

I don't think that a reasonable person would interpret the question in that way, though.

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The student is absolutely correct (as Twiceler has correctly shown).

The time taken to cut a board into $2$ pieces (that is $1$ cut) : $10$ minutes
Therefore, The time taken to cut a board into $3$ pieces (that is $2$ cuts) : $20$ minutes

The question may have different weird interpretations as I am happy commented:

Time taken to cut it into one piece = $0$ minutes
So Time taken to cut it into $3$ piece s= $0 * 3$ minutes = $0$ minutes.

So $0$ can be an answer. but it is illogical just like the teacher's answer
and as Keltari said

Another correct answer would be 10 minutes. One could infer, "If she works just as fast," that "work" is the complete amount of time to do the job. -Keltari

This is logical but you can be sure that this is not waht the question meant.

but the student has chosen the most relevant one. The teacher's interpretation is mathematically incorrect.

The teacher may have put the question for the students to have an idea of Arithmetic Progression and may have thought that the students will just answer the question without thinking hard. In many a schools, at low grades children are thought that real numbers consists of all the numbers. Only later in higher grades do they learn that complex numbers also exist. (I learned just like that.) So the question was put as a question on A.P. thinking that the students may not be capable of solving the answer the correct way.

Or as Jared rightly commented:

This is simultaneously wonderful and sad. Wonderful for the student who was level-headed enough to answer this question correctly, and sad that this teacher's mistake could be representative of the quality of elementary school math education. – Jared

Whatever may be the reason, there is no doubt that the student has been accurate in answering the question properly and that the teacher's answer is illogical.

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You are probably right that the question is not looking for the answer I gave. However, you most certainly cannot be sure. You are making a decision based on assumptions. The age of the test takers and the wording of the question are points to consider. However, there is nothing to prove or disprove that my answer is, or is not, what they are looking for. – Keltari Sep 15 '13 at 22:35

One part of what the teacher suggests is possible. Four pieces can be obtained in twenty minutes, because this takes only two cuts: cut it in two, then parallel the pieces and cut again, such that the saw goes through both at the same time. (The assumption is that the extra energy doesn't take more time, just more effort per stroke: not realistic, but let's go with it).

The mistake is interpolating between the two possibilities. If two pieces takes ten minutes, and four can be had in twenty, it does not follow that three pieces can be had in fifteen. However, six pieces can be had in thirty minutes which averages out to three in fifteen.

Suppose two workers are put on the job, and suppose it is somehow possible for them to divide a cut between themselves by attacking it from opposite sides without hindering each other, so they can meet in the middle in five minutes and complete the cut. They can execute this at the beginning to make one board into two. Then they double up the board, and each makes a ten minute double cut through both boards: six pieces in fifteen minutes, so basically three pieces per worker per fifteen minutes.

So if we think about just a one-off job carried out by a single person with a saw, then the student is right. However, if we were talking about productivity over multiple pieces, and possibly with multiple workers, then the teacher would also be right; the problem is, nothing of the sort is suggested in the way the question is posed.

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+1 for pointing out the incorrect interpolation. – l0b0 May 5 '13 at 19:01

Teaching children, you have to be fair to them:

• Think as they do.

• Children are honest and direct in their assessments - it would not likely occur to them to "cut into 3 pieces of equal length" because that was not in the question. Nether would they likely think of any of the alternative cuts offered here in the various answers --- precisely because:

• People (especially children) tend to be very visual. DUH there is a picture of the saw cutting the board. The board IS a board, not a paper cut out, not a piece of rectangular plywood; and the way the saw is positioned very strongly implies the next cut would be made in a similar fashion. Honestly, how many of you looked at that picture and almost unconsciously imagined moving the saw to the right (or maybe to the left) of the current cut ? I did - and I bet the children would too ... because:

• Children are hands on.

When I read the problem, I thought the test grader just muffed it misreading the scoring sheet. Wow - I guess I'm childish :-P

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While this is a nice insight on teaching practices, this doesn't actually answer the question :C – Enigma Sep 11 '13 at 2:48
@Retrosaur thx. Oh sorry, I tried to address the underlying cause of the problem. You say you "don't understand what's wrong with this question. " Take it at face value, as a child would. I bet every kid would get the answer correct! You ask, "What am I getting wrong here?" My answer is: The grader or teacher is being too much an adult, thinking too much into it. :)) And when I read "I feel like I'm missing something critical here," I was prompted to give the answer above. The missing link is knowing when to throttle back the intellect, before it overpowers playfulness. Is the answer clearer? – Howard Pautz Sep 11 '13 at 18:33
And a follow up @renegadeballoon aka Retrosaur - by the looks of it months later, very few other people 'actually answered the question' too :C (Just proves math people are as incorrigible as engineers ouch :-P ) – Howard Pautz Aug 7 '14 at 2:16
Didn't intend to come of that way, just that it didn't directly answer the question. But still very informative and thorough, which is why I gave it a +1; it'd do you some good to add just a small bit to the bottom explaining the actual answer – Enigma Aug 7 '14 at 2:19

There is a similar problem that needs an argument quite analogous to what the student seems to have used:

A clock takes 12 seconds to strike 4 o'clock, how long will it take to strike 8 o'clock?

The interpretation is that the time is spent between the strikes, so the answer is 28 seconds instead of 24.

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I'd argue it would take at least another four hours. – Marcks Thomas May 8 '13 at 13:47
@Marcks That would be my argument too. Except, by the time 4 o'clock has been struck, we are now already 12 seconds into the first hour. So it would be 3 hours 59 minutes and 48 seconds. – daviewales May 18 '13 at 13:07
@daviewales Plus the 28 seconds to chime. 4 hours 16 seconds. – Paul Fleming May 19 '13 at 13:17
Are we assuming the clock being measured is accurate? Does the time we use to measure the clock fall on a Daylight Savings day? When you say "how long" are you referring to time or distance the hands traveled? :D – Keltari Aug 28 '13 at 3:23
@Keltari Since Daylight Savings switches at 2:00, which is not between 4 o'clock and 8 o'clock, it's irrelevant to the question. – Joe Z. Jan 17 '14 at 8:18

The answer will have to be 20. If it takes 'Marie' 10 minutes to cut the board into two pieces then that means it has taken her 10 minutes to make that chop.

Three pieces would require two chops therefore the teacher is wrong:

2 * 10 = 20

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The student is actually right. The teacher is wrong... – Michal B. May 3 '13 at 12:22
why this answer has down vote. it he correct also like student. – Mowgli May 3 '13 at 14:20
I'm assuming it was down voted because it said the student was wrong then they weren't. – CramerTV May 3 '13 at 18:14
Since all of their logic is correct, I assume it was just a mistake in typing 'student' instead of 'teacher'. – Kevin Fegan May 4 '13 at 1:40
Absolutely, it was a typo in the haste of things. The logic was simple enough – kaiten65 Dec 21 '14 at 22:31

Considering they show a drawing of the piece of wood on the problem itself, the assumption is the cut would be made in the same way, hence the student was right to start with.

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Well, i have something else in my mind.I know it is not that practical still i want to share my views about the question.

Assuming the board is of 1 meter long and we have to cut it into 3 pieces of equal length i.e. finally each of the pieces should be length of 1/3 meter long (see the picture given in the question). So we have to make two cuts at length 1/3 and at length 2/3. Now note that after cutting it first time(which will take 10 minutes), we will have two piece. One is of 1/3 meter long and the other portion is of 2/3 meter long.Now we have to make a cut to the last portion(which is of 2/3 meter long) and make it in two pieces.

Now the interesting part comes. If we assume the board has a uniform resistance against the saw, then after losing its one third end, board will lose its resistance uniformly (assuming resistance depends on length proportionally). In that case, it will take another $10*(2/3)=6.67$ minutes to get another two pieces.

hence we need total $16.67$ minutes.

I know it is not practical, still...

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Since you are cutting across the width of the board, the resistance to the saw is a function of the width of the board, not the length of the board. – Kevin Fegan Sep 4 '13 at 0:47

I would think of it this way: how long would it take to cut it in to 1 piece... 0 minutes because it is already in one piece. The model is: time = cuts x 10. As 1 cut = 10 minutes.

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Quite a random question. The answer would depend on where and how the cuts are made.

e.g. Let's say the $2$ blocks are identical (assumption on my part) each of $10$cm $\times$ $5$cm $\times$ $1$cm Let's say first block is cut lengthwise, i.e. into 2 blocks of 10cmx5cmx0.5cm. This means it took 10 mts to cut through and area of $10\times5 = 50cm^2$

Now let's look at the second block. Cut along width to create 2 blocks each of 5cmx5cmx1cm. Then you take one of these and cut off along the thickness to create two pieces 5cmx2.5cmx1cm. In this case you have just gone through an area of 5x1+5x1 = $10cm^2$ so it should only take 2mts.

Of course, if this was a question in a $3^{rd}$ grade exam, none of the above is relevant. The way the question is written, the answer could be 20 (if you are cutting pieces off a long piece of wood as the diagram indicates) or 15 (if you cut a block into half and then use the second cut to cut one of the halves into half).

+++++++++++ | +++++++++++ | +++++++++++
+++++++++++ | +++++++++++ | +++++++++++
+++++++++++ | +++++++++++ | +++++++++++
+++++++++++ | +++++++++++ | +++++++++++
+++++++++++ | +++++++++++ | +++++++++++


++++++++++ | +++++++++++++++++
++++++++++ | +++++++++++++++++
-----------| +++++++++++++++++
++++++++++ | +++++++++++++++++
++++++++++ | +++++++++++++++++


That being said, it is a terribly poorly framed question.

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Given the length of your second cut there, I'd say it would take 25 minutes, not 15. – Joe Z. May 22 '13 at 18:28

Sawing once takes 10 minutes and obtains 2 pieces. So, since we obtain 3 pieces when we saw twice, it takes $2 \cdot 10 = 20$ minutes.

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Teachers method: 10 minutes per 2 pieces, hence 5 minutes per 1 piece 3 pieces implies 5(3) = 15.

Common sense: 10 minutes for 1 cut of a board (which makes 2 pieces) Therefore 3 pieces requires 2 cuts hence 2(10)=20

Too much missing info,I just asked my boss this question and he said that the answer would make sense if you were slicing a square board in half and that took 10 minutes, then slicing one half in half would make it 3 pieces in half the time as the original cut since it is half the size... 10 + 10/2 = 15

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Both the teacher and the student are right. It depends on how you look at the problem. We have the rate: 10 minutes to saw 2 pieces.

First way to understand the problem (from the student's perspective): We have a board. We make one cut and get 2 pieces (the sawed off part and the remaining part). From this point of view, every cut yields 2 pieces. So, first cut (takes 10 minutes) and we get 2 pieces. The second cut (takes another 10 minutes - same rate) and we get 3 pieces Total time = 10 + 10 = 20 minutes

Second way to understand the problem (from the teacher's perspective): We have a board. We make one cut and get 1 piece (the sawed off part. The remaining piece of the board is not counted) From this point of view, every cut yields only 1 piece. So, the rate is 10/2 = 5 minutes/1 piece So, first cut (5 minutes), we get 1st piece Second cut (5 minutes), we get 2nd piece Third cut (5 minutes), we get 3rd piece Total time = 5 + 5 + 5 = 15 minutes

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Perhaps if it was "can marie saw the board into 3 pieces in 10 minutes?" then it would be correct. Maybe it was a misprint.

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The problem is If she works just as fast, that means that if she cuts with the same speed...

$$\vec{v} = \frac{\triangle \vec{x}}{\triangle \vec{t}}$$

If Marie saw a board into 2 pieces in 10 minutes, that means 1 cut in 10 minutes ($\vec{v} = \frac{1}{10}$ cuts per minute ).

So to perform 2 cuts to obtain 3 pieces, we have:

$$\triangle \vec{t} = \frac{\triangle \vec{x}}{\vec{v}} = \frac{2}{\left (\frac{1}{10} \right )} = 2\times 10 = 20$$

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Well, it took her ten minutes to cut the board into two pieces. To cut something into two pieces, you have to slice it only one time. Also, the number of pieces needs to be one more than the number of cuts if it's cut evenly enough. This means to get three pieces, you need two cuts, so $10\times2=20$. Remember that she works at the same speed.

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Think about it, how many cuts do you need to make 2 pieces?

======|======

One cut.

How many cuts do you need to make 3 pieces?

====|====|====

Two cuts.

So 10 minutes for the single cut means 20 minutes for the double cut.

I'm surprised at your teacher. I think you should seriously hunt them down and duke it out verbally.

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All the answers there seems to point already on this? – nicael Nov 8 '15 at 19:54
I needed to vent. – Daniel Nov 9 '15 at 1:05

The student gave the correct answer since it takes 2 cuts to make 3 pieces-each cut using up 10 minutes!

That's hilarious,the reason the student got the answer "right" and the teacher got it "wrong" is because the student actually thought through the question from first principles-realizing 2 cuts,each eating up 10 minutes,would be needed-and the teacher,used to thinking in "tricks", assumed an equal amount of time would be needed for each piece rather then each cut!

I'm rather ashamed to say I thought that was the answer,too-until I tossed out all my preconceptions and looked at it from scratch using only what I was given-as the student clearly and correctly did!And this is really the essence of correct mathematical thinking,which the student has and the teacher has obviously lost laboring with inferior educational methodology: The correct approach to any mathematical problem is to begin from first principles knowing only what is given. The power of mathematics is to produce a method of solution to a problem where none existed before.Sadly,most of our school mathematics-especially in America-is designed to produce purely practical thinking with spoon fed algorithms-and original thinking is not only discouraged, but indirectly punished.

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## protected by robjohn♦May 3 '13 at 19:04

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