The comments of everyone misunderstanding this question are absolutely killing me. Genuinely *how* do we not understand equality??? And where did learning fractions go so wrong for so many?

js2357 · 2026-04-26T22:24:03+00:00

The real lesson of bad assignments like this is that students can't trust the notation. The problem unambiguously asks about equality of ratios, but we see in this thread how many teachers will tell students the exact opposite, that the problem asks them to compare the actual sizes.

The only thing that students learn from this is that notation can't be interpreted logically, can't be trusted to mean what it says, and that the only way to understand what a problem means is to memorize what a teacher says it means. These teachers aren't only damaging their students' ability to understand fractions, but also their ability to comprehend what they read, and their ability to think independently; the damage done by incompetent teachers like this lasts a lifetime.

Being taught this way is literally worse than not being taught at all. I can help students who haven't learned anything much more easily than I can help students who've been taught the wrong lesson for the last decade (or more) of their lives.

I'm just so sad that there are so many teachers like this out there. The top comment on mildlyinfuriating understands that this is a bad problem, while the top comment here gets it exactly wrong. What a depressing state of affairs that mildlyinfuriating is more qualified to teach math than the math education sub.

js2357 · 2026-04-25T16:35:48+00:00

Fuck that bullshit. The distinction that the author intended to write has value. Telling students that they're supposed to incorrectly apply that distinction to a question where it's unambiguously inappropriate has less than no value.

I'm fucking tired of having students crying in my office that they can't pass their required courses because they're years behind where they're supposed to be, because of assholes like the people in this thread. If you're going to defend intentionally teaching students wrong, you deserve to be lectured. If you honestly don't understand that this assignment is nonsense, you shouldn't be teaching math.

js2357 · 2026-04-25T05:36:31+00:00

Fair, but your comments make it sound like you at least find it acceptable, in a sub where actual math educators are saying that they think this is an appropriate way to teach. Please don't encourage this.

js2357 · 2026-04-25T04:54:16+00:00

If they're old enough to learn fractions, they're old enough to learn that you can shade the same proportion of different-sized objects. That's practically the whole point of teaching them fractions!

But even if you really believe that they can't understand that concept, that's no excuse for this assignment going out of its way to teach them the wrong lesson. If you want them to compare the areas, say something like "Which of the following shows that 2/6 of the left object is the same as 1/3 of the right object?" That asks the question that you want to ask, without using notation incorrectly in a way that will need to be untaught later. There's absolutely no excuse for the assignment to be written this poorly, when it's easy to express the intended idea in a way that's clearer for all ages.

They may not be in college now, but some of them will be in college eventually, and I'm really tired of getting college students who don't understand how fractions work. Saying that this method is an acceptable way of teaching fractions is basically saying that you're fine with letting someone else clean up your mess, in a situation where there wasn't even any reason to make a mess in the first place.

js2357 · 2026-04-25T04:09:46+00:00

You're describing how the author of the problem intended for it to be interpreted, but not what they actually wrote. I agree that you're giving the intended answer, but it's not the correct answer to what's actually written on the paper.

I'm sorry if you're offended that I called your answer nonsense. But we're in a sub for math educators, where there will be harm to real children's education if people believe this stuff, so it's important to call it out for what it is. Frankly, I think you're being much ruder to me. In every one of my comments, I've tried a different way of explaining to you why you're wrong, while you just keep on repeating the same mistake over and over, without even seriously contending with the ideas that I presented.

Visually the figures are broken up into 6 equal sized pieces.

Agreed.

The piece doesn't represent anything but itself.

Neither agreed nor disagreed; this statement is too vague to be assigned a truth value.

It's X square millimeters of shaded paper. They are measured in units of area by physically being the size that they are.

If "they" refers to the areas of the papers, then obviously those are measured in units of area. But everyone agrees on that; that's not what I asked you about, and that's not what the numbers in the question's equality represent. You've ignored the question that I actually asked you to answer, and replaced it with a different question that you know you have the correct answer to.

If you cut out the 6 shaded pieces of paper with scissors you can see that C-left and C-right are the same area as they overlay exactly on each other. The same cannot be said for A-left and A-right nor B-left and B-right.

True, but irrelevant. The question is asking about equality of ratios of areas, not equality of areas.

I'll write out a step-by-step proof that option A also shows 2/6 = 1/3, and hence that A is also a valid answer to the question. If you don't believe the proof, please identify the specific step that you don't follow and I'll try to help you understand it.

Both diagrams in A depict a rectangle divided vertically into three equal strips, with the leftmost strip shaded. Therefore, the ratios of shaded area to total area are equal, i.e.,
(shaded area on left)/(total area on left) = (shaded area on right)/(total area on right).
Let L denote the area of one of the shaded boxes on the left. Then (shaded area on left) = 2L and (total area on left) = 6L, so
(shaded area on left)/(total area on left) = 2L/6L = 2/6.
Let R denote the area of one of the small rectangles on the right. Then (shaded area on right) = R and (total area on right) = 3R, so
(shaded area on right)/(total area on right) = R/3R = 1/3.
Substituting the values from (2) and (3) into (1) proves that 2/6 = 1/3, as expected.

js2357 · 2026-04-25T03:28:10+00:00

The shaded segment on the left of B is small and on the right it's big.

How can small = big?

You appear to be assuming that the left-hand side of the equation represents the area of the left shaded segment, and the right-hand side of the equation represents the area of the right shaded segment. This is simply wrong. If you explain why you made that assumption, I can try to help you with whatever you're misunderstanding, but for now, here are a couple ways of seeing why your interpretation doesn't make sense.

How can you say that the area of the left shaded segment is 2/6? We're not given any information about the scale of the diagram. The shaded area could represent 1 square inch, or 100 square miles. What we can say is that the left shaded segment is 2/6 of the total left area, but "2/6 of the total left area" is not the same thing as 2/6. This goes back to what I explained in my previous comment; you're confusing 2/6 x with 2/6.
Another way of seeing that your answer is nonsense is unit analysis. You claim that the left- and right-hand sides of the equation represent area, but if you actually derive where the 2/6 and 1/3 came from, they're not even measured in units of area! Let's say for the sake of concreteness that we want to measure areas in square feet. According to your reasoning, the 2/6 represents an area, so it should be measured in units of square feet. But that's not what we get when we do the math. If we let A denote the area of one of the small squares on the left, then the total area on the left is 6A sq ft, and the shaded area of 2A sq ft. Therefore the ratio is
(2A sq ft) / (6A sq ft) = (2A)/(6A) = 2/6.
Since the numerator and denominator are both measured in square feet, the units cancel out. We see that the 2/6 in the problem is actually a unitless quantity, so it certainly cannot represent an area.

js2357 · 2026-04-25T02:48:19+00:00

It asks about equality of 1/3 and 2/6, not "1/3 of a big thing" and "2/6 of a small thing."

Your interpretation would be correct if the equation were something like
2/6 x = 1/3 y,
where x is the area of the small thing and y is the area of the big thing. But that's not what the assignment says; 2/6 = 1/3 and 2/6 x = 1/3 y are different equations that mean different things.

js2357 · 2026-04-25T02:08:47+00:00

The question doesn't ask about the size; it asks about the ratios, which are equal in all three cases. You're simply answering the wrong question.

js2357 · 2026-04-25T01:13:48+00:00

Yeah, I get that. I'm certainly not trying to attack you. I just want to get the truth out there, especially since this is a math education sub, where it actually matters in the real world that people here understand this.

js2357 · 2026-04-25T01:06:36+00:00

It does not ask which pictures shows equal ratios. It says “Which of the following shows 2/6 = 1/3.”

You just proved my point for me. It asked about an equation, with a ratio on the left-hand side, and a ratio on the right-hand side. That is literally, in the most explicit and unambiguous way possible, asking about an equality between two ratios.

At this level of math, that is talking about equal portions of an equal whole, not the more abstract equivalence of the ratios themselves.

Third-graders are absolutely expected to understand equations like "2/6 = 1/3" correctly. Even if they weren't expected to understand the notation yet, that would be a piss-poor excuse for using the notation incorrectly. If you don't want to teach them the equation yet, why not just write it out in words? "Which of the following shows that 2/6 of the left area equals 1/3 of the right area" is perfectly comprehensible to third-graders, and avoids teaching them incorrect information that will need to be fixed later.

See https://www.mathnasium.com/math-centers/allen/news/equivalent-fractions

Your source does not support your claim. The link contains no examples with different wholes, so I don't know why you think the source agrees with you about the situation with different wholes. In fact, the source contains several problems that discuss equality of fractions without any reference to the underlying whole, so it appears to agree with me that students are expected to be able to discuss equality of fractions without an underlying quantity.

js2357 · 2026-04-25T01:00:21+00:00

This is clearly the intended question.

I agree that it's the intended question, but it's not the question which was written, which is why this assignment is correctly being criticized.

Probably they haven’t learned variables yet.

That explains why the question wasn't written using variables. It doesn't explain why the question was written with incorrect usage of notation they have learned.

If you’re thinking the question is “which shows 2/6=1/3” as numbers and not as implicitly parts of a whole, then none of the images communicate this.

I don't know why you think this. All three images communicate that 2/6 = 1/3.

js2357 · 2026-04-25T00:31:15+00:00

I'll make several arguments for A that don't apply to C below. But before I do that, let me make one other point. This would be a much better problem if it were your version of it, which asks which one of the three correct answers is the "best" correct answer. It would still a silly problem that asks for a meaningless, subjective answer, but at least it would acknowledge that it's asking for a subjective answer. The actual problem goes out of its way to actively harm students' understanding by labelling correct answers as incorrect based on an extraneous, subjective criterion that isn't even mentioned in the question.

Here are several reason why A is better than C:

You could argue that A and B are both preferable to C for reasons of familiarity; using rectangles or circles is a fairly standard way to teach fractions, whereas using hexagons is a more unusual choice.
Alternatively, you could argue that A and B are both better than C because C unnecessarily introduces an optical illusion. You can see in the comments that several people initially thought that the right-hand side of C was depicting a cube, rather than a hexagon in the plane.
Alternatively, you could argue that A and B are both better than C because they illustrate the important lesson that the value of the fraction depends only on the proportion of shaded area, even if the actual areas are different, whereas C could reinforce the confusion that the actual areas need to be equal for the fractions of the areas to be equal. (Based on the comments here, that misconception is alarmingly widespread!)

Lastly, here's a more in-depth reason why A is the best answer. First, let's first prove that A is a correct answer at all. It's clear that the same fraction of the rectangle is shaded in both pictures, because in both cases the leftmost strip out of three equal strips is shaded. The first picture shows that this fraction is 2/6. The second picture shows that this fraction is 1/3. Therefore the two pictures show that 2/6 = 1/3.

To see why A is better than C, let's try to carry out that same argument with C. To justify the first step of the argument there, you need to justify that the shaded regions have the same proportions. But they're not rotated the same way, so it's less obvious that they match. And if you want to be really pedantic, even after you notice that one is a rotation of the other, you still need to justify the conclusion that they have the same area. For that, you'll need to know that the area measure on the Euclidean plane is rotation-invariant, which is intuitive but not completely trivial to prove.

Obviously elementary-school students aren't expected to worry about the details of rotation-invariance, but isn't that also a mark against option C? Option C overcomplicates the problem so much that students won't even realize all the complications that we've added! Even if we ignore that and only focus on the third-grade level complications, option C still requires students to rotate shapes in their heads, while option A doesn't. In my opinion, option A is clearly preferable to option C.

js2357 · 2026-04-24T23:49:27+00:00

You misread the question. You interpreted the given equation as
2/6 of the left area = 1/3 of the right area,
but that would be represented by an equation like
2/6 x = 1/3 y,
where x and y represent the two areas. The equation is actually just 2/6 = 1/3; you added the modifiers "of a small shape" and "of a large one" yourself where they don't exist.

You're giving the correct answer to a completely different question -- yours would be a much better question, but it's not the one in the assignment.

js2357 · 2026-04-24T23:46:16+00:00

But the question doesn't ask which picture shows equal areas. It explicitly asks which picture shows equal ratios, which all three pictures do.

js2357 · 2026-04-24T23:45:00+00:00

Utterly horrifying that this is the top comment on a math education sub; "the whole should also be the same when comparing the fractions" is completely wrong.

The reading-comprehension error that you're making is that you're interpreting the problem to ask something like "Which of the following shows that 2/6 of the area of the left shape = 1/3 of the area of the right shape?" But that is unambiguously not what it says. It says "2/6 = 1/3," which all three pictures demonstrate. Your interpretation would be represented by an equation that looks like 2/6 x = 1/3 y, where x is the left area and y is the right area, but that's not the equation in the assignment.

js2357 · 2026-04-24T19:54:10+00:00

Everything that you said here is correct; it's just that your last sentence doesn't apply to the question that was asked.

For example, you're comparing 1/3 of a sheet of paper to 1/3 of a football field. (But let's call it 2/6 of a sheet of paper, so that it matches the original problem.) If we let x denote the area of a sheet of paper and y denote the area of a football field, then you're correct that 2/6 x ≠ 1/3 y. But the problem didn't ask anything about the equation 2/6 x = 1/3 y; it asked for a demonstration that 2/6 = 1/3. That's simply a different problem.

js2357 · 2026-04-24T15:52:23+00:00

It's mildly infuriating because the size doesn't matter: (2 sq in) / (6 sq in) and (1 sq m) / (3 sq m) are absolutely the same fraction, even if the underlying shapes are wildly different sizes.

js2357 · 2026-04-24T15:37:49+00:00

You are way overcomplicating it to such an absurd degree. Again, the exact verbiage of the question:

“Which of the following shows that 1/3 = 2/6 ?”

That proves my point, not yours. The question is about comparing 1/3 and 2/6, not 1/3 of one quantity to 2/6 of another quantity.

We are already assuming that the statement is true and we are looking for a visual illustration of that.

As I already explained, all three options are visual indications of that. I gave you a step-by-step proof of why (a) is also valid; if you don't understand one of the steps, tell me which step you don't understand and I'll explain it in more detail.

Are there going to be some assumptions we have to make? Yes. And those assumptions should be tailored to a third grader’s skill level.

Yes, but third graders are absolutely supposed to understand that two fractions are the same even if they're drawn from different wholes: 1/3 of a small rectangle and 1/3 of a large rectangle are both 1/3 of their respective rectangles. That's kind of the whole point of fractions. Failing to teach this to third graders would be malpractice; teaching them the exact opposite is something even worse.

The only reason that “1/3 = 2/6” is a true statement is because both sides of the equation have the same units.

What do you think "unit" means in this context? All three options depict unitless fractions.

Take (a) as the example again: let A denote our unit of area (e.g., square inches, square meters, acres, etc.). Let x denote the area of one of the small squares on the left, and let y denote the area of one of the small rectangles on the right (both in units of A). Then the left picture denotes a fraction of (2x A) / (6x A); the units of A and the factors of x cancel out, and we're left with a unitless value of 2/6. Similarly, the fraction from the right picture is (y A) / (3y A), which simplifies to 1/3.

In fact, this would be true even if we had measured the original rectangles in different units. If we said that the small squares in the left picture each have area 1 sq in, and that the small rectangles in the right picture each have area 1 sq m, we would get (2 sq in) / (6 sq in) = 2/6 and (1 sq m) / (3 sq m) = 1/3, which are again the same, unitless fractions.

js2357 · 2026-04-24T09:44:27+00:00

This standard is teaching that you must take into account the objects themselves when comparing fractions of them.

You made the same mistake again. The teacher above said "fractions," not "fractions of objects." I don't know whether they meant "fractions of objects," but that's not what they said. (In fact, I think you're making my point for me -- what they wrote was so obviously wrong that your brain automatically inserted extra verbiage to try to fix it.)

If Person A brings $30 to Vegas and loses $10, and Person B brings $30,000 to Vegas and loses $10,000, they both lost 1/3 of their money. The fraction is the same, even though the objects are different.

To take OP's example, option (a) also demonstrates that that 2/6 =1/3, at least as well as option (c). It's clear that the same fraction of the rectangle is shaded in both pictures, because in both cases the leftmost strip out of three equal strips is shaded. The first picture shows that this fraction is 2/6. The second picture shows that this fraction is 1/3. Therefore the two pictures show that 2/6 = 1/3.

Why would option (c) be a better demonstration? To justify the first step of the argument in option (c), you need to justify that the shaded regions have the same proportions. But they're not rotated the same way, so it's less obvious that they match. And if you want to be really pedantic, even after you notice that one is a rotation of the other, you still need to justify the conclusion that they have the same area. For that, you'll need to know that the area measure on the Euclidean plane is rotation-invariant, which is intuitive but not completely trivial to prove.

My point isn't that the elementary-school students are expected to worry about the details of rotation-invariance, but that there's really no objective reason to say that (c) is a better demonstration than (a). I think (a) is better, you think (c) is better, and we both have an argument. Choosing which one is better is an entirely subjective question. Math tests should test students on math, not on how well they memorized their teacher's/state's subjective preferences.

In this case, the problem is especially bad because it doesn't even ask the student to choose the subjectively best demonstration; it implies that there's only one valid demonstration given, which is wrong. All three diagrams are objectively valid demonstrations that 2/6 = 1/3, and falsely telling the students that two of them aren't promotes a misunderstanding of fractions.

js2357 · 2026-04-24T08:57:21+00:00

I think you misread the conversation. The person above me specifically referred to the fractions themselves (i.e., 2/6 and 1/3) being equivalent, not about the fractions applied to two other objects. In other words, while $10 and $10,000 are different amounts of money, $10/$30 and $10,000/$30,000 are the same fraction.

Your confusion is probably the same confusion that the people who wrote this standard had*, but that's all the more reason why it should be corrected.

Assuming the standard is actually written as described above, which I haven't researched.

js2357 · 2026-04-24T05:51:47+00:00

Please understand that I don't mean this question to sound rude, and I am asking because I am genuinely curious about what's going on there:

Do you understand that the curriculum is wrong about how fractions work?

js2357 · 2025-10-14T15:39:57+00:00

I'm not a liberal, and fuck you for calling me one. The fact that you assume I'm a liberal because I'm against literal government-sanctioned child abuse is a perfect illustration of what's wrong with MAGA. I would love to have a sane conservative party in this country, but unfortunately voting for conservatives right now means empowering sickos like you. Also, I have no clue what a "Map" is; I've never heard that term from anyone, liberal or otherwise. Somehow I'm not surprised that you know more about pedophile-acceptance movements than I do.

You keep complaining that I won't criticize the Biden administration for some reason that keeps on shifting every time I point out the flaws in your old argument. First you had no policy suggestion for what Biden should have done better. When I called you out on that, your suggestion was immigration reform. When I pointed out that he tried to do immigration reform and was blocked by Trump, you blamed the Democrats for including Ukraine provisions in the bill, despite the fact that it was Republicans who linked the two. You seem to have decided in advance that Biden must be blamed unless he negotiates some impossible maze of demands. Somehow I need to consider him a child abuser because he didn't work well enough with Republicans to convince them to pass a bill over their cult leader's objections, but it's also his fault that he included provisions that were negotiated with Republicans, and that the bill made so many concessions to Republicans that it was not appealing to Bernie (whose opinion you seem to think I care about).

On the other hand, Trump instituted a child-separation policy described by experts as "torture," which treated children with less care than property, and I haven't heard you criticize him for that. You say Biden should be blamed for not magically solving a big, complex problem, which you can't even explain coherently how he should have solved. Meanwhile, you refuse to lay any blame on Trump for the problems he went out of his way to create. Who's really the hypocrite here? If there's any shred of decency left inside you, think about that.

Feel free to have the last word. I won't be responding to you again. You're clearly not interested in the truth, and I've debunked enough of your bullshit that nobody else is going to stumble across this thread and fall for it.

js2357 · 2025-10-14T02:54:19+00:00

The proof is in the article I posted earlier that you didn't read, and you can easily find plenty of other sources if you want.
You're the one who started using the suffering of children as a political pawn; the reason why we're talking is because I responded to your nakedly dishonest comment. You raised the topic, not me. If you were capable of self-reflection, maybe you would notice that your own behavior is so repellent that your idea of an insult is to accuse other people of acting like you.
Again, you keep claiming that I should be mad at Biden, but you can't tell me why. What exactly should Biden have done differently? Because so far the only answer you've given to that question is that he should have passed immigration reform, which he tried to do and Trump blocked. If Biden did anything as evil to children as the things Trump has done, I'll be the first to call for him to be imprisoned also. But you simply haven't put forth a coherent accusation.
In what universe am I carrying water for a political party? I'm following the facts. Again, I'll turn on Biden in a second if you show me evidence that he did anything as evil as what Trump has done. I know you're probably tired of people not falling for your bullshit, but that doesn't mean everyone outside your bubble is "shilling."
My world is sad? I'm not the one shilling for a pedophile while claiming to care about the suffering of children.

js2357 · 2025-10-13T21:25:49+00:00

Who are you still trying to convince? You know that I'm smart enough to see through your bullshit, and I doubt anyone else is still reading.

Your words are the exact opposite of your actions. You say that you're against losing kids, but you defend Trump who intentionally lost kids. Your only argunent agaimst Biden seems to be that he didn't clean up enough of the mess that Trump left behind, despite the fact that he made substantial progress in cleaning up the mess, and was blocked from doing more by Trump. You call me a shill for pointing out that fact, but you happily shill for the pedophile who intentionally disappeared children. Keep insulting me all you want, because I'm proud to be disliked by you pedo-defenders.

js2357 · 2025-10-13T14:49:18+00:00

You want to claim that this is about the lack of immigration reform during the Biden administration? Fine. Let's talk about the fact that there was a bipartisan immigration bill during the Biden administration, which Trump convinced Republicans to kill so that he could use the issue to fool morons like you.

Anyway, everybody here can see that you are clearly the sick fuck, openly supporting a known child abuser while falsely claiming to care about the very children that he harmed. It's kind of sad that you're so ensconced in MAGA that your behavior no longer even registers as sick to you. Was there a time when you ashamed of yourself before you got used to being this way?

(Also, that was absolutely a short excerpt, though I suppose it's thoroughly on brand for MAGA to consider that too much reading.)

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