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Consider the following problems:

A) You have 20 problems for your math homework this week, and you want to do 1/5 of them today. How many problems do you need to do today?

B) You need to read a 20 page book this week, and you want to read the first 1/5 of the book today. How many pages do you need to read today?

C) You need to run a total of 20 kilometers this week, and you want to run 1/5 of that today. How many kilometers do you need to run today?

Question:

Are these problems conceptually different enough that an educator needs to carefully deal with each of these scenarios, or will a child that can (actually) solve one of these problems also be able to solve the others?

Edit:

To explain why I consider these problems different:

A) You have 20 problems. You can create 5 groups with 4 problems in each.

B) You have 20 pages. You can create 5 groups with 4 pages in each. However, in this case there is an implied order of the pages. This feels more like a scaling of a discrete set, than placing objects into groups of equal size.

C) The kilometers are not really objects and so it doesn't really make sense to group them. One could of course imagine them as "objects", but arguably we are really performing some kind of scaling operation.

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    $\begingroup$ Can you add a bit, explaining how you feel these questions are different? (And I don't believe the secondary tag should apply, this should be learned long before high school) $\endgroup$ Nov 2, 2021 at 10:25
  • $\begingroup$ I was going by the tag description which says "approximately ages 10-18", and I was considering the lower end of that range. $\endgroup$
    – Improve
    Nov 2, 2021 at 11:02
  • $\begingroup$ My apologies. I did not realize that tag actually includes high school. I mistakenly thought primary ended at 8th grade (age 13-14) and secondary was the 4 years of High School (14-18). Now I know, thanks. $\endgroup$ Nov 2, 2021 at 12:10
  • $\begingroup$ This sort of thing is taught around age 7-8 in the UK, in a way that assumes integer answers, or easy fractional answers like 3½ (@JTP), and that would treat all these questions the same. By early secondary school I'd expect it to be extended both to arbitrary non-integer answers (fractions or decimals) and to touch on concepts around precision in the continuous (km) case. But secondary is the gap in my recent knowledge of maths teaching. BTW terminology and grouping of school years varies hugely, so it's as well to check. $\endgroup$
    – Chris H
    Nov 3, 2021 at 14:37
  • $\begingroup$ Just a minor nit-pick that won't matter for a personal classroom test, but something that shines out like a searchlight beacon to me as a result of many years of editing items like this -- there's a huge difference between wanting to do something and needing to do something . . . $\endgroup$ Nov 3, 2021 at 19:36

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i don’t think they are. In fractions there are (at least) 3 analogies: set (discrete objects), area (or volume), and length.

Your 1st is set, 2nd maybe area (more like length) and 3rd is length.

You could re-write #2 so that your filling a glass or jug or jar with something (or using 1/5 of an amount)

But, i think they are all conceptually the same. a fraction of a whole.

Why not ask about missing parts or what is the whole, Then these would be conceptually different thus modifying / increasing the rigor and assess a students understanding of fractions (here they may just be applying an algorithm / process):

why not ask that after the 1st day you left 4/5 of the problems left. If there are 16 problems left, how many problems in total are there (or you completed 4 or 1/5 of the problems, how many problems are there in total). You could also ask if you completed 4 problems out of 20, what is the fraction of the problems you completed or have not completed.

You could even ask an open question. You’ve completed 1/5 of the problems. How many problems are there in total? How many problems did you complete? (would be challenging because student would wonder: where are the numbers to work with , and you’ll have to tell them. make them up!)

Each one of these suggestions asks a different aspect of the problem (generate a fraction of a part , a missing part, or the whole).

These are just some possibilities (and analogies) that can really modify what you are asking for in these problems.

You could even work with fractions greater than 1 also ;)

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  • $\begingroup$ In how far are area and length different analogies? $\endgroup$
    – Jasper
    Nov 4, 2021 at 8:03
  • $\begingroup$ “and you’ll have to tell them. make them up!” that's a really bad idea. Let‘s consider I have 0 exercise to do… $\endgroup$
    – Laravel
    Nov 4, 2021 at 15:06
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This depends on when in the "fractions curriculum" this happens.

If the children know that "a fraction of" really means to multiply by this fraction, then all problems are equal enough that one single student will be able to solve either all or none of them.

If the problems are used to teach "fraction of is multiplication by", it will be useful to consider them separately.

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One point where C is different from A and B is that a distance can scaled (or split) by any fraction. So 20km can be split into 6 equal parts, even if the result cannot (yet) be represented by the numbers the students know.

But 20 problems or 20 pages cannot be split into 6 equal parts. At least nothing simple comes to mind.

In other words, C is a problem from a somewhat different family than A and B, in that they generalise in different ways (in the sense of varying the problem "parameters"/constants/numbers).

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    $\begingroup$ I think this is a key difference between #1-2 and #3 which has not been brought up: discrete versus continuous visuals of fractions. If a student's only visual of a fraction is "divide objects into equal groups," then the notion of "divide a segment of the number line into equal lengths" may not be super far off but it still might make them uncomfortable. They need exposure to both. $\endgroup$
    – Opal E
    Nov 3, 2021 at 18:37
  • $\begingroup$ @OpalE And conversely, if familiar with "divide a segment of the number line into equal lengths", or pizza, or dough, water, spaghetti, then when "divide [discrete] objects into equal groups" comes around, it would pose some difficulties. :-) $\endgroup$
    – Pablo H
    Nov 4, 2021 at 13:06
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Yes, the second and third have an implied order to them that the first doesn't, but it doesn't impact the basic operation (look at the words and translate them to 20 times 1/5.) I think myself (and most students) would sense something a little different, but still perform the operation. After all you're not asking a question that requires some different use of this additional information. Maybe, to be fair to you, we could say the latter problems are slightly harder since there's a slight distraction in the order.

My advice is to avoid worrying about what confusions might occur and deal with them more as they occur. Lots of theoretical people want to anticipate every eventuality and then end up teaching with masses of caveats, explanations, etc. But math is hard enough. It's easier to get a simple explanation than an elaborate one. (Kids are not computers, they have limited processing capacity.) I actually don't think the order thing will flummox the kids but even if it does, I would deal with it afterwards rather than ahead of time.

Also, can't you just observe and report your experience? Why the need/desire to consider the problem sans kids, rather than looking at what actually happens? Surely the actual results in teaching setting would inform our answers more than just looking at the problems without recourse to that information.

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I'll point out that the OP's question really contains two independent questions:

Are these problems conceptually different enough that an educator needs to carefully deal with each of these scenarios...

It would be a bad idea for an educator to spend time presenting each of the given questions independently and analyzing them. First, we don't have that much time available. Second, trying to identify and clarify some subtle distinction is more likely to further confuse students on those nuances than anything else. Third, it seems to be just the tip of the iceberg to how many iterations could be made to this and any other problem, and I don't see why the discussion would end with just these 3 variations for this, or any other problem.

... or will a child that can (actually) solve one of these problems also be able to solve the others?

Well, that's a separate issue, and I wouldn't guarantee that all students actually will be able to perfectly transfer their abstract knowledge from case to another. In fact, it's pretty well established with numerous examples that that's not, completely generally, the case. Maybe a student just gets distracted by a word or phrasing they don't know in one instance.

My understanding is that it's been pretty well established that the key to student reading proficiency is a legacy of reading wide and deep subject matter.* If students have read on the subject in question, they'll be more comfortable and proficient dealing with the word problem. If not, they'll likely to be distracted and unable to parse the sentence. We can't know in general which camp they'll fall in when they get to these problems, so: no guarantees.

But nonetheless, it's not a good use of time trying to analyze and distinguish the nuances of these problems; that's not really where any difficulty will reside, and would increase confusion rather than decrease it. Time in the math class is best spent on the abstracted math itself, not other issues; if subject-matter difficulties arise, then you might have to assist with that on a case-by-case basis.

(*) Can't find a good link at this writing; if anyone can share one in comments I'd edit it in.

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It would depend on the student. If this is the first time they are seeing fractions of a set, then I don't think one problem would be enough for them to get it unless they are very quick. If it is the first time they are seeing fractions of a set, it would be worthwhile for them to see more problems. Of course it might be better to change the numbers in the problems in that case. On the other hand some children benefit from seeing the application of the same numbers in different contexts.

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My state has a standardized test for math given in the second year of high school. It has a low bar for passing, about 30%, but it's a requirement for graduation.

I teach a class to help the students who are struggling to pass. In this class, I use a similar strategy as I use for the SAT or ACT (college entrance exams). That is, to read a question and be able to articulate the math problem being asked. I've encountered students who can calculate the area of a circle, but have trouble when the problem is posed as, say, a radio tower with a certain transmission distance, asking how many square miles of coverage the radio station has.

Often, I'll project a problem onto the board and cross out the extra details that don't really matter. In many cases, the students are English language learners, and they may be fine math students, but have difficulty with long word problems where the vocabulary really has nothing to do with the math. In this case, the minor differences in your 3 problems get crossed out and "1/5 of 20" is all that remains.

In my opinion, there are slight changes you can make ala jmg's suggestion and others, which would change it up enough to not feel redundant to the typical student. The difference among the 3 feels too slight to matter.

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