# How long will it take Marie to saw another board into 3 pieces?

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?

• 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. May 3, 2013 at 3:38
• I took five minutes per cut. One cut yields two pieces. Two cuts yield three pieces. And so on..... May 3, 2013 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). May 3, 2013 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, 2013 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 May 6, 2013 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

• 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 May 3, 2013 at 3:36
• In all fairness, neither did whoever graded this problem. This question made me smile :) May 3, 2013 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 May 3, 2013 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. May 3, 2013 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. ;) May 24, 2013 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.

• However if you look at the drawing of the piece of wood being cut, this approach would be extremely difficult. May 3, 2013 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, 2013 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. May 3, 2013 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, 2013 at 8:32
• This answer, my friends, demonstrates what we call "being a smartass". Apr 9, 2014 at 12:05

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.

• @amWhy Diagonally across the image is a watermark that says zerooutoffive.com, where the image was at some point submitted. May 3, 2013 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. May 3, 2013 at 13:11

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

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


:-)

• 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. May 4, 2013 at 20:37
• @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. May 6, 2013 at 22:53
• Use a circular saw: 1 minute. May 7, 2013 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 :-) – user1324 May 7, 2013 at 5:51 • Come on, guys, even in Physics class you can assume 0 friction, 0 air resistance, infinitesimal widths etc. etc. Sep 25, 2013 at 4:33 Another correct answer would be 10 minutes. One could infer from: If she works just as fast that "work" is the complete amount of time to do the job. • 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... May 3, 2013 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. May 3, 2013 at 21:42
• @Jonathan or, apparently, to say two opposite things at the same time…! (“seldom often”?)
– Aant
May 4, 2013 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. :) May 7, 2013 at 7:15
• @Michael: (c) was not one of the answers, it was a directive. Apr 22, 2014 at 16:17

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.

• Of course, if you look at the board itself, it clearly isn't in the shape of a ring... Nov 28, 2013 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. Nov 28, 2013 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.) Jan 17, 2014 at 8:20
• But if your saw was long enough, you could just make one cut to cut it in half Oct 12, 2017 at 0:51
• I'm late to the party, but you can saw the ring to make 2 pieces with 1 cut if you saw through the plane part. Jun 18, 2019 at 10:59

Let P : pieces

Let m : minutes

Let C : cuts

Let t : time per slice = 10

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

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

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

• Notation overflow! May 5, 2013 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? Jan 10, 2014 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. Jan 11, 2014 at 3:23
• Couldn't you find a way to throw some tensor calculus in there? Jan 29, 2017 at 3:41

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.

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.

• How much time does it take to cut the board into 0 pieces? Jan 29, 2017 at 3:43
• @richard1941 Are you allowing negative numbers? Apr 22, 2017 at 2:30
• Everybody has the wrong idea. It took ten minutes to make two pieces. That is five minutes per piece. So three pieces take fifteen minutes. That is how the teacher seems to be thinking (but not me). Aug 7, 2019 at 6:47
• @richard1941 I'll bet pretty much everybody understands that interpretation. The issue is that it is physically absurd. Feb 13, 2022 at 18:35

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.

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:<br

Time taken to cut it into one piece = $$0$$ minutes
So Time taken to cut it into $$3$$ pieces = $$0 \times 3$$ 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 what 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.

• 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. Sep 15, 2013 at 22:35

Teaching children, you have to be fair to them:

• Think as they do.

• Third graders are budding topologists, just very far from graduation.

• 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

• While this is a nice insight on teaching practices, this doesn't actually answer the question :C Sep 11, 2013 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? Sep 11, 2013 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 ) Aug 7, 2014 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 Aug 7, 2014 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.

• I'd argue it would take at least another four hours. May 8, 2013 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. May 18, 2013 at 13:07
• @daviewales Plus the 28 seconds to chime. 4 hours 16 seconds. May 19, 2013 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 Aug 28, 2013 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. Jan 17, 2014 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

• The student is actually right. The teacher is wrong... May 3, 2013 at 12:22
• why this answer has down vote. it he correct also like student. May 3, 2013 at 14:20
• I'm assuming it was down voted because it said the student was wrong then they weren't. May 3, 2013 at 18:14
• Since all of their logic is correct, I assume it was just a mistake in typing 'student' instead of 'teacher'. May 4, 2013 at 1:40
• Absolutely, it was a typo in the haste of things. The logic was simple enough Dec 21, 2014 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.

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 $\cdot$ $10$. As $1$ cut $= 10$ minutes.

• How log to cut it into zero pieces? negative ten minutes? Jan 29, 2017 at 3:47
• @richard1941 how long ago was the board cut from the tree? Apr 5, 2020 at 16:21

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.

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.

• Given the length of your second cut there, I'd say it would take 25 minutes, not 15. May 22, 2013 at 18:28

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...

• 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. Sep 4, 2013 at 0:47

Ten minutes. $\ \ \$ $\ \ \$ $\ \ \$ $\ \ \$

• Cryptic. $\_\_\_$ Feb 28, 2017 at 4:28
• @pjs36. The dotted line is meant to be one sawcut = 10 mins. Feb 28, 2017 at 4:42
• @Leucippus. Why does it not answer the question? There's nothing in the question posed that says the cut must be a straight line (or are we meant to teach children to think in straight lines?). Feb 28, 2017 at 5:14
• +1 for creativity. But we must be sure the path length of your cut is the same as that of the (presumably linear) reference cut taking 10 minutes. Apr 22, 2017 at 2:40
• True, but if the time is taken to be proportional to the length of the cut there is no definite answer, as noted by AndSoYouCode above. In the cut shown both the start and end points and can be as close as you please to the bottom left hand corner and the loop as close as you please to zero (saw permitting), so the length of the cut can be any positive number. Apr 24, 2017 at 10:20

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

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:

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

• In the absence of any indication of how she made the cut, I think you have to assume that just as fast is to be measured in cuts per unit time rather than speed, or even velocity. (By the way, I've heard of the arrow of time, but shouldn't $t$ be a scalar?) Feb 28, 2017 at 4:59
• The cutting speed is not a distance speed (perhaps proportional to one). Speed is not a vector anyway, it is a scalar. And you have a time vector? In the denominator? It's rare to see someone know how to use a language like Latex so well with so little respect for the conventions of the symbols. You can define vector division and such any way you like and I will happily accept it, but to offer it as if it should be understood is an abuse of notation. en.m.wikipedia.org/wiki/Abuse_of_notation?wprov=sfla1 Apr 22, 2017 at 2:36

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

• This shows how weak an appeal to "common sense" is. No way should the listener have to know what the speaker has assumed to be true or assumed to be false in arguing that something is "common sense". Mar 25, 2018 at 8:45

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.

Simple question, simple answer:

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 $$2\times10=20$$ minutes for the double cut.

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

• All the answers there seems to point already on this? Nov 8, 2015 at 19:54
• I needed to vent. Nov 9, 2015 at 1:05

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.

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.

Well this was a incredibly difficult question considering the teacher got it wrong!

Indeed it seems like this question is slightly ambiguous, however the student's answer is definitely correct!

$1$ cut from the saw take 10 minutes, as $1$ cut from a saw should cut a board into $2$ pieces.

In order to cut another board into $3$ pieces however, $2$ cuts are needed.

As $1$ cut takes $10$ minutes, $2$ cuts will take $10 \times 2 =20$ minutes!

$\mathcal Therefore,$ the student is correct with an answer of 20 minutes!

Assume the board is flexible. Fold the board in half and cut. It takes 10 minutes.

Something is wrong, certainly. Several answers have debated the question of whether it was the student or the teacher who was wrong. I see another possibility: the question is wrong.

The student's been taught, in a maths lesson, about proportions or "rule of three" or something similar. So it's natural for the student to assume that the question's purpose is to test the student's ability to use that technique. It also makes the student assume that the question is sound, and that the student is not responsible for looking for errors in it and dealing with them (if there are any). So even if the student spots the out-by-one error, they might assume that that is merely an error in the way the question was presented, and then go on to answer the proportions question that they think was intended.

"$$15$$ minutes" and "$$20$$ minutes" are both reasonable responses to a question which was unsound in what the circumstances seem to be. The correct thing would've been to present the underlying proportions question in a different and clearer way.

• And what way do you believe would have been best appropriate in presenting the question, considering how this question was for third graders? Mar 25, 2018 at 9:06
• Some way that doesn't involve an out-by-one error. So, not in terms of cutting a long thing into pieces, or of making a fence. Mar 25, 2018 at 10:01
• Downvoter, please would you care to say why you downvoted? Note that I wrote "I see another possibility" and did not mean to imply that this must've been the truth. Jan 8, 2020 at 12:07