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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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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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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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 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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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.
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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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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.
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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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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. $$P(C(20)) = \text{floor}(C(20)) + 1 = \text{floor}(20/10)+1 = \text{floor}(2) + 1 = 2 + 1 = 3\ \text{pieces} $$
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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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You can actually do it in ten minutes but your saw must look like this:
| |
| |
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| | <- cutting edges
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+--+--+
| <- handle
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:-)
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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).
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Answer = 20
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Answer = 15
That being said, it is a terribly poorly framed question.
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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.
On
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.
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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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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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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
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!
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!