Is 1/4 of the circle shaded? Most students get this wrong

Fractions are hard for many students. Special thanks this month to: Kyle, Lee Redden, Mike Robertson, Daniel Lewis. Thanks to all supporters on Patreon! / mindyourdecisions
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[@rich_in_paradise]

The problem isn't that the students don't understand fractions. The problem is that they didn't think about the question - they just saw 1 out of 4 slices is shaded, it must be 1/4 and didn't think about whether the 1 shaded slice was actually 1/4 of the total area.

[@JLvatron]

You are likely right. If it were presented as grab 1 piece of pizza, everyone would want the bigger piece.

[@femsplainer]

Well that ultimately comes down to a problem with the question. It never asks if the AREA of the shaded circle is 1/4, it just says 1/4th of the circle. There are 4 parts of the circle, if you take away any one part then it is no longer a circle, so each part is precisely 1 out of the 4 parts, thus 1/4 of the circle IS in fact shaded if you go by that perspective.

[@peterdegelaen]

The question is just badly formulated. It should ask if 1/4 of the AREA of the circle is shaded. Kids are not mathematicians that asume the things a math teacher does.

[@femsplainer]

@@peterdegelaen Exactly, but that's a good thing, because children are able to solve many puzzles that adults struggle with because we insert our own preconceived notions and presumptions into the mix. For example, there is a puzzle to fit a bunch of jigsaw pieces together into a square. I've seen adults spend hours trying to figure it out and failing to solve it. The kids usually solve it in under 10 minutes. The trick is that the pieces are not actually jigsaw pieces that are meant to fit together as we generally expect them to. You can have holes that require two arms to fill it and some holes are not meant to be filled at all, so a piece that normally looks like would go in a corner actually goes in the center.

[@glensmith491]

Reason many riddles work. They depend on people not really thinking about the question. As a software engineer, I also have found that not really thinking about the question is the source of many bugs.

[@jihrijihri710]

As a student, you can draw a reflection of the shaded circle and easily see that it does not cover half of the circle. Therefore, if two shaded pieces are less than half of the circle, then one piece must be less than a quarter of the circle.

[@MegaLokopo]

Yes, but the question is poorly worded, 1 out of 4 pieces is shaded, the question never specifies the unit or area.

[@AirLancer]

@@MegaLokopo Yeah it does, the unit is the circle. The shaded area does not equal 1/4 of the circle, therefore 1/4 isn't shaded.

[@MegaLokopo]

@@AirLancer That doesn't counter anything I said. The question never specified the unit in terms of what it means by 1/4th, and never mentions area. 1/4th could be interpreted to mean the 1/4th of the circle based on the diameter of the circle adjacent to the shaded area. The question also never specifies the width of each section, so it is impossible to know if 25% of the area of the circle is shaded or not. So considering that the width of each section is not specified, and we don't know the width or exact location of the separating lines, it is impossible to know what the area of any part of the circle is, so the only logical conclusion is the question is asking about pieces not area. In the video he assumed the line was infinitely small and exactly half way in between the center and the edge. The question never specifies that.

[@neutronenstern.]

​@@AirLancerthe unit could be the diameter of the circle tho instead of the area

[@willemakkermans4067]

​@@neutronenstern. then the question would state "is 1/4 of the diameter of this circle shaded", even though that is a nonsensical question. Really the question is clear and only makes sense in one way.

[@chrissscottt]

One major problem I've observed first hand in New Zealand primary schools is that many of the teachers themselves don't understand the math they're supposed to be teaching.

[@JohnSmith-ux3tt]

Math is hard. Let's count to 10 in Maori and call it job done.

[@Doktor47]

That and DMIC has absolutely ruined a generation of kids

[@realityobservationalist7290]

@@Doktor47 , people who unironically use the word 'equity', aka communists, have ruined far more than one generation.

[@Skank_and_Gutterboy]

Yeah, I've had math teachers that taught straight out of the book and wouldn't answer questions. In a lot of schools, teaching math is a crappy collateral duty that nobody wants. If they have an ounce of integrity, the classroom teachers will draw straws. In many schools, they'll just dump it on the PE teacher because he doesn't do much.

[@thomasfevre9515]

It's far from being specifically a new Zealand problem. Speaking from litterally the other side of the world.

[@smilerbob]

The students’ explanations for their answers could be divided into three groups: - those who did not mention the need for four equally sized parts (67% said this); - those who did recognise the need for equally sized parts but who stated that the parts were indeed equal (20% argued this); - and those who recognised that the parts were unequal and therefore the circle was not 1/4 shaded (13%)

[@daforkgaming3320]

I don’t understand the 20% tho. It seems pretty easy to visually tell that the center parts have a greater area than the edge parts

[@Abhigyan103]

If the question says that they are cut by 3 equally spaced lines, then it can be 25 % if the space between the lines is decreased (still equally spaced)

[@Jerome616]

I initially said no, then thought longer about it and realized that the question was only asking about 4ths, not equal 4ths. The pizza was indeed divided into 4 parts and 1 of the 4 parts were shaded. This is how you fail math class. Going with your second answer. 😅

[@TheTonyMcD]

@@daforkgaming3320 I think they may be falling to an optical illusion. I initially answered with no. But after looking closer, it seemed to me that the outer two sections were wider than the middle two. I still would have answered no, but I was starting to question it. I had to measure it to be sure. The whole "equally spaced across the diameter" bit would have been helpful to include on the actual question.

[@willemakkermans4067]

@@smilerbob my hope for humanity rests on the few people that were part of group 1 or 2 but now see the logic AND are willing and able to change their mind accordingly as to move into group 3. Teaching logic is hard enough, as we can see. But it's possibly harder to teach people to be open to logical thinking in the first place, as their cognitive bias (and with it a whole range of other logical fallacies) happily kicks in at every given opportunity. Heck, at this point I even enthusiastically applaud those willing to switch between group 1 and 2.

[@pi_xi]

McDonald's once introduced a "Third-of-a-Pound Burger", but it sold badly, because people thought that the "Quarter Pounder" contained more meat.

[@marioman118]

..Not a good look lol.

[@David280GG]

Americans failed 2nd grade math lol

[@mrsillytacos]

​@David280GG it's because of how the burger is semantically called. If you're trying to order something most people generally won't think of the semantics behind it and "Quarter Pounder" sounds more appealing and filling.

[@orionspur]

There's an optical illusion here that is surely confusing students. The outer (curved) strips look wider than the middle ones. It is quite easy to believe the lines cut the area into fourths.

[@KnugLidi]

If you shade the 3rd slice instead of the 4th, the percentage of correct responses goes up dramatically.

[@AxillaryPower2]

I clicked on this video for clarity on the spacing because I could not tell just by the picture if they were even. Were students allowed rulers? Baring that, or the question spelling out the spacing, I could see why some may get this question wrong.

[@6yjjk]

Absolutely. "Three equally spaced lines" does not imply that they are spaced at 1/4 of the diameter, or even that line 2 bisects the horizontal diameter, only that the gap between lines 1 and 2 is equal to that between 2 and 3.

[@beeble2003]

Regardless of whether they look wider, he calculates that the shaded area is 19.6% of the circle. You're not going to be able to eyeball the difference between 19.6% and 25%.

[@beeble2003]

@@AxillaryPower2 You have to assume that the lines are equally spaced -- without that assumption, the question is unanswerable. Since the question is aimed at 12-year-olds, I think they're expecting "They look equal in the diagram so I'll assume they are."

[@dr_volberg]

0:23 "equally spaced" - is this assumption well-founded? Nothing on the figure says anything about the distance between the dividing lines.

[@SadSettings]

Measuring it.. 👀

[@greyrifterrellik5837]

​@@SadSettings Math test diagrams at this grade level are notoriously uncaring about precise scale; they focus more on clear, simplified representative imagery, to isolate and focus on the specific concepts being learned. A perfect example of this, though at a slightly higher grade level, is those triangles that provide only a few measurments, and ask you to use them to find a specific other measurement; the triangles shown are very rarely 100% accurate to the given values. Typically, the only information needed for the *expected* answer is provided by the question and diagram; if there is information not directly specified, it is not relevant.

[@peterbaruxis2511]

The distance doesn't matter, what matters is that the distances are equal.

[@beeble2003]

@@greyrifterrellik5837 The reason for not drawing triangles to scale is that, if they did, students could just measure things, instead of using the geometry or trigonometry that they're supposed to be tested on.

[@beeble2003]

@@peterbaruxis2511 But nothing in the question says that they're equal. We just assume that, because they look about equal and their being equal is the only reasonable assumption. If we don't assume they're equal, the question isn't really answerable.

[@BruceRicard]

Technically, the question never specifies that the 3 lines split the circle in 4 sections of equal width. You are assuming that it is the question, but I don't know that it is. One could argue that the lines were drawn specifically where they need to be in order for the shaded area to be 1/4th of the total circle. Which would make "Yes" the correct answer.

[@frank_calvert]

great substitute to it stating it: your eyes. they're the same distance.

[@lammahater9153]

​@@frank_calvertscales in diagram should not be used unless "diagram to scale" is stated.I've seen question where calculation show that a triangle is isosceles but the diagram shows an irregular triangle

[@mattias2576]

​@@lammahater9153same, ive had calculations where the orbit of a planet was shown as an ellipse, but we were told it was a circle

[@tjallingdalheuvel126]

That is how we was schooled. Pictures where om purpose not correct at times. Needed to be markings in it for the data to count. In this question, one could say it was asked wrong. Left room for interpretation. Equal in wich sense? Yes it looks like the diameter is cut in four equal lengths. But no marking that says it is. So it could be cut across the diameter to make equal in surface area.

[@beeble2003]

You have to assume that the sctions are of equal width as, without it, the question is just "I've shaded some random portion of a circle -- is it a quarter?" The only way to answer that would be by carefully measuring, which just isn't interesting.

[@jcortese3300]

At that age, I think they see one slice out of four slices total and not think about the area. The question should have specified of 1/4 of the total area was shaded. And it wasn't explicitly stated that the vertical lines were equally spaced.

[@calholli]

Exactly.. it's always a trick question that is forcing you to make assumptions.

[@jcortese3300]

@@calholli Yep. This is what happens when people who can't do math or have no training in it try to make up math questions. There is a rigor to math that the vast majority of K-12 teachers do not appreciate or understand.

[@zmgehlke]

Imagine trying this hard to explain why it's okay you're ignorant. This is why Asian education systems beat Western ones: their students wouldn't be this extra making excuses about how their failure is the test writer's fault, they'd just learn to do better. What "1/4" means is literally 25% of the circle, which is a 2D figure. You made the mistake of thinking that 1 in 4 segments was the same thing as 25% of the area. This question was exactly designed to catch that mistake and you failed. What you're calling a "trick" is literally the whole point: distinguishing two concepts you have confused. Stop blaming other people for failing to master a skill.

[@rogergeyer9851]

It depends on their education. Age 12 is the typical age of formal reasoning so they SHOULD think about area in a math question ABOUT area.

[@snared_]

@@rogergeyer9851 A prior question could be "which circle has each segment of equal area?" A is where the vertical lines are equally spaced, B is where the vertical lines are placed such that 25% of the circle is in each segment. That is the fairer way to ask this question, and most would get it right at that point I would assume.

[@hatimzeineddine8723]

I will say that there's nothing to indicate that the four sections have equal width. If the central sections are thinner than the outer ones, they could be of equal area.

[@MegaLokopo]

The only thing we can conclude here is the question is very poorly worded. It doesn't specify units and it doesn't specify area.

[@nwblader6231]

⁠@@MegaLokopoyou don’t need units or area because the question is relative, the shaded area would be the same fraction of the area regardless of the size. It is poorly worded as it doesn’t specify the relative distances between the lines

[@kristofer9776]

You would be correct, but he did mention at the beginning that the vertical lines are equally spaced across the diameter (0:22), so the slices do indeed have the same width.

[@mehill00]

@@kristofer9776Yes he mentioned the equal spacing along the diameter verbally, but the written problem didn’t mention it, at least as represented here, so presumably the NZ kids didn’t receive this clarification. On tests it is usually taught not to assume diagrams are proportionate and/or to scale.

[@steventagawa6959]

This would be an interesting problem. Bisect a circle. Then, within one of the resulting semicircles, construct a line parallel to the bisecting line such that the areas of the two resulting regions of semicircle are equal. On a radial line drawn perpendicular to the bisecting line, where does the constructed line intersect, and what is the ratio of the lengths of the two resulting segments of the radial line?

[@mareksroka5629]

Meanwhile me: "answer is NO anyway because a circle can't be shaded, only a disk can" (I had teachers that would automatically fail you if you dared speak about the area of a circle)

[@user-xj8wy4uu1q]

Huh

[@christophersayrs907]

LOL! Love this. 100% correct!

[@beeble2003]

@@user-xj8wy4uu1q Technically, a circle is just a curved line; the disc is the area bounded by that line.

[@kingjamesfmvp]

Exactly haha!

[@denelson83]

Draw a square that circumscribes this circle, with one side perpendicular to the line segments in the circle, then extend the line segments to the sides of the square, and you will have an easy visual explanation for why less than a quarter of the circle is shaded.

[@mayorb3366]

That was my first thought also, as to clearly visualize the difference in the sizes.

[@notafraidofchange]

Agh! Came here to say that. At least _somebody_ else said it too.

[@AA-100]

Thats only if you assume the diagram was drawn to scale, which wasn't stated in the question, so there actually isnt enough information

[@willemakkermans4067]

​@@AA-100 there is no 'diagram' in the question, representing something to a certain scale. The question is directly about the circle drawn.

[@Sauvenil]

@@willemakkermans4067 The circle is the diagram...

[@Vex-MTG]

My thought on the shaded circle question is that there's nothing in the text or image that tells students that the four slices are of equal width. Is it possible that many of those with the wrong answer were tricked by an optical illusion that the pieces on the ends were wider than those in the middle?

[@willemm9356]

Maybe, but a rough guesstimate is that for the shaded area to be 1/4 of the circle, the side bits would have to be almost twice as wide as the middle bits.

[@Kyrelel]

BOTH answers are correct, which is why the teacher added a follow up question regarding their thinking.

[@Ringcaat]

@@willemm9356 Not true. To me it looked like the ratio of widths was 6:7. I did the math and got that 23.6% of the circle is shaded, which seemed within the bounds of visual error to me. Until I measured the widths and found that (counterintuitively) the widths of the slices were equal, it seemed plausible to me that 25% was shaded.

[@mpetersen6]

No excuses. They are far enough in their education they should be able to reason that the shaded area is not 1/4th or 25% of the area of the circle.

[@robertveith6383]

*@ MindYourDecisions* -- The diagram lacks the necessity of explicitly indicating that the lines are meant to be evenly spaced. The question is wrong/does not make sense. A circle cannot be "shaded." The *"area of a circle"* or the *"area inside of a circle"* can be shaded. One of the acceptable ways to ask the question is, "Is 1/4 of the area of the circle shaded?" In conjunction with that, the diagram needs to indicate that the lines are evenly spaced. As it is, this question should not have been given, not counted, and/or thrown out. This problem is in a growing list of faulty problems you have been presenting to your viewers.

[@j100j]

​@@verkuilbAnd where was that told to the students?

[@Acetyl53]

It shouldn't need to be explicitly stated. Part of not being an NPC golem is being able tow ork out the intention yourself.

[@j100j]

@@Acetyl53 When schools teach that in maths tests you can't assume anything that has not been stated. There shouldn't even be an expectation that the center one is in the middle even though it looks like it is. I've actually had questions where stuff had looked like they are arranged perfectly neatly but when you do the math, you realize that they are barely off.

[@Acetyl53]

@@j100j You essentially just repeated what I said in different words, you've described a golem that can't think for itself. I am aware of that same tired line, I was told the same. I didn't find it compelling then, and I don't find it compelling now.

[@Acetyl53]

@@j100j As far as I'm concerned the test is psyche profiling in disguise.

[@j100j]

You gave the viewer additional information notpresent in the sheet. You can't just go assuming the exact positions of lines in quesrions like these.

[@leonfa259]

Yes & visually 30% is not that far off from 25% if you only have a small picture.

[@mattc3581]

This wasn't the original paper, he redrew the question and gave that information verbally. Given he provided that info we can probably assume that is was stated on the original question paper.

[@jimi02468]

Do you really think the lines divide the diameter of the circle something like 25.0001% and 24.999%, etc, but not exactly into 25% sections? No you don't. So you can always assume this stuff just visually.

[@mattc3581]

@@jimi02468 Tbh, it doesn't really matter if they aren't supposed to be the same width. Maths problems are sometimes ambiguously drawn to force the solver to work out the correct shape/properties from other information provided. In this case however there is nothing that allows us to work out the relative widths, so the diagram itself is the only truth, and the diagram is visually perfectly clear that the outer sections are smaller in area then the inner sections, the exact width doesn't really matter for the question the kids had, they can assume they are equal width or not.

[@rogergeyer9851]

Exactly. I think we should have seen the problem with the SAME information as the original, and without that information, the students could NOT have answered it unless told the drawing was to precise scale AND given a good measuring device. In my day such questions would be stated precisely AND we'd be told NOT to assume any such drawings were to scale (they wanted only MATH to be used to solve the problems).

[@tokencivilian8507]

20 plus years ago, a friend had a small lot concrete truck out to his place for a pour he needed - the guy you hire when you need more than it makes sense to get bags from big box home center, yet less than a full typical concrete truck's worth, so between about 1 and 7 yards. I was helping frame up the forms, move dirt, etc. Anyways, in talking with the truck operator, it turned out he was the company owner. He noted that as part of his interview of any potential employee, he put a cardboard box on the desk, handed the candidate a ruler, pad and pencil and had them calculate the volume of the box. Concrete is sold by the cubic yard. I relate this as we all were sitting there in math class asking "where will we ever use this in real life?". Selling concrete or sticking the tank of the fuel truck, in this case, are two practical examples.

[@Yonkage-ik5qb]

Doing it with a box is serious easy-mode. Try, the customer is installing a chain-link fence and is going to have four post-holes that are 24" deep and 6" across, and the pipes themselves are 2" across, how much concrete to fill them? I had to do that one IRL.

[@rogergeyer9851]

Yeah, it's hard to see how someone who can't calculate the volume of a box (with all perpendicular sides and a ruler) can get through life, much less do a job involving mixing the right amount of concrete

[@Solrex_the_Sun_King]

They are asking if 25% of the circle has been shaded. The answer to that is no. However, if you take a more loose and less objective approach to this question, the circle is divided into 4 parts. One of those 4 parts is shaded. Thus, 1/4th of the circle being shaded is both correct and incorrect at the same time, depending on the lens used to view it with. This is rare in math.

[@brianvernaglia9449]

4 people are in a room and one leaves. 1/4 of the people left. 4 people are in a room and the skinniest is asked to leave. Less than 1/4 of the mass of people has left.

[@smilerbob]

But what if _some_ were pregnant? 🤔 (All in jest)

[@brianvernaglia9449]

​@@smilerbobdang. My example foiled because people are more complicated than circles. ;)

[@smilerbob]

@@brianvernaglia9449 Apparently some circles are more complicated than people 😁

[@JLvatron]

But the person left doing the Moonwalk, so it looked like he was entering the room. So while 1/4 of the people left, it is not visually to scale!

[@williamniver6063]

@@smilerbob Pregnant with how many? Twins, trips, more? And how far along on the pregnancy? That issue goes to "personhood" question AND the added mass of these additional "persons".

[@RobertBlair]

I feel test questions like this say more about the quality of test makers than test takers. I can imagine a host of ways to write the question that help folks understand better. e.g. Ask if 4 people split a cake this way, do they each have an equal amount of cake? do this test 20 times with different phrasing, see which form of the question has the most accurate response rate.

[@gamisa]

Teacher: is 1/4 circle shaded? Student: yes Teacher: No! Youre terribly wrong Student: Jokes on you, I painted it on a non euclidean surface

[@Songfugel]

Man, I love how you changed this ridiculously depressing stat into a great educational video with a cool real life practical example! I will have to keep this example in mind the next time I am teaching this 💯👍

[@Kyrelel]

"ridiculously depressing stat" !? There is no right/wrong answer to this question, so the "stat" is meaningless.

[@bobrienzi6305]

11:00 Hey that's me! Thanks for the shout out. Great content as always.

[@kenmore01]

I certainly wouldn't expect 4th grade students to know that math, but I was able to read that question two ways. Area or linear. The question doesn't mention area, that's implied. One could make the point though, that 1/4 of the horizontal distance is shaded.

[@femsplainer]

You could also read it as component parts, like blocks. If you take away any of the 4 blocks then the shape ceases to be a circle, so in that regard 1 out of 4 necessary components was shaded. In this way you can ignore overall size and shape and still conclude it to be a correct statement.

[@kenmore01]

@@femsplainer Agreed!

[@willemakkermans4067]

The question clearly takes the circle ("this circle") as a unit. It doesn't ask about 1/4 of the horizontal distance, or 1/4 of the parts of the circle. It specifically asks about 1/4 OF THIS CIRCLE. But I'd love to take you out for a pizza. I'll eat 1/3 of the pizza, and pay for 1/3 of the price, it's only fair. I realise you might live far away, but that's OK too. We have a pizzeria here in town, which is some distance from me and also some distance from you, so we can meet 1/2 of the way.

[@mattc3581]

The original question is fine for a 4th grade student, you don't need the math extension. Just observe that the circle is cut into four parts and two of the parts are smaller than the other two, therefore those parts aren't a full quarter of the total.

[@kenmore01]

@@mattc3581 Yes, I agree with that. Not the part about calculating the actual area of the shaded part though.

[@agnichatian]

It's 100% shaded - mostly black, less than 1/4 red, and a bit of white.

[@lukasjetu9776]

true, don't know why they didn't make it transparent, it would make more sense, instead of some black white red mix

[@briant7265]

@@agnichatian The glass is completely full. Half water and half air.

[@christopherwellman2364]

Red? Is there red I'm not seeing? I'm a little colorblind.

[@agnichatian]

@@christopherwellman2364 in the thumbnail/title picture it is red

[@lukasjetu9776]

@@christopherwellman2364 you don't see the red?

[@christopherwellman2364]

Given no information about length, width, etc., makes the answer "not enough information". 🤦‍♂️ The question doesn't specify that the vertical lines are evenly spaced. If they are evenly spaced, of course the answer is "no". However, given the right spacing, the shaded area most certainly could be 1/4 of the circle.

[@prbmax]

We had a tank on its side with two truncated hemispherical end caps filled with anhydrous ammonia. This same issue was bought up of what percentage full the tank was. A little different when you have to factor in the spherical end caps.

[@Nate-r3f]

So a capsule then.

[@wombat4191]

That gets quite complicated, I'd be happy to just approximate the ends as flat if a practically good enough answer will suffice, instead of a completely accurate one.

[@briant7265]

@prbmax That does make it more fun. From a practical point though, if you're looking for the closest 10%, the caps don't really make a difference.

[@paulgreen9059]

Could you fill a tank a quarter full and measure that and just keep the stick handy?

[@Skank_and_Gutterboy]

I saw this when monitoring tank levels in the propulsion plant of a ship. We had some cylindrical tanks with spherical ends and a sight glass. The sight glass didn't quite reach the top and bottom. If level was just to the top of the sight glass, it was about 95% full. If the sight glass was at the halfway point, the tank was clearly half full. If the sight glass was at the 3/4 mark, it was 3/4 full, right? No. Because the cross section varies with depth, a graph of level vs. tank-quantity is not linear (for flat ends or cylindrical end caps). If you need tank quantity vs. level to be a linear relationship that you can log accurately, better get yourself a rectangular tank.

[@adrianscarlett]

The question did not explicitly state that the lines were equidistant, and when faced with a drawing with no dimensions, I was taught that you can not assume the drawing is to scale. Otherwise, draw 2 vertical lines at a tangent to the circle and join them with horizontal lines intersecting the endpoints of the two shorter internal lines.

[@mattc3581]

He verbally stated that they were equidistant. Presumably it was written on the original paper, but he isn't showing that.

[@mattc3581]

Though also if there are no measurements given then it is impossible for it not to be drawn to scale. The actual diagram provided is the only version of it there can be and the outer sections are clearly smaller in area than the inner ones, so no need to overthink it.

[@frank_calvert]

you could use your eyes???

[@smylesg]

Does the question state the lines are evenly spaced or the drawing is to scale? If not, there is not enough information given.

[@verkuilb]

Yes, it does state they are evenly spaced.

[@robertveith6383]

The diagram he is submitting does not indicate explicitly the lines are evenly spaced.

[@smylesg]

​@@verkuilbJust because Presh said those words doesn't mean the problem states it. I don't see it in the problem description. It only reads, "here is a shaded circle. Is 1/4 of the circle shaded?"

[@JLvatron]

The question doesn’t say it. But at that grade, the diagram is probably indeed to scale. So evenly spaced.

[@secr3t7]

I mean, it also doesn't state that the circle is in euclidean space, but at this age/grade, it would be assumed.

[@prasoon7916]

Is the distance of the chord given from the circumference? Because it may alter the answer, but if the 2 chords are equidistant from the diameter and the circumference (which i assume is the case here, because it would be too difficult for children to solve), hence the answer is no.

[@Patrik6920]

ur correct, i doubt very much a 8 year old would be able solve it matematically ...if as u say thay are indeed at a equal distance, say r=2 and the lines are at +-1 from center the shaded regoion is 19.55 .. % given by ( πr^2 * (1/3) - ( (r/2)*(r√3/2) ) ) / ( π*r^2 ) r can ofc be any arbitrary number, so if r=1 we get ( (π/3) - ( (√3) / 4 ) ) / ( π ) ≈ 19.55%

[@AA-100]

@@prasoon7916 It isn't given. Nor is it explicitly stated the diagram was drawn to scale, so technically speaking, there isnt enough information

[@sebbbi2]

This image has a visual illusion. Smaller segment seems wider. Student might believe that it’s wide enough to compensate for the vertical area loss. I don’t like unclear questions lile this. It should be somehow stated that all segments are equal width. Did the students have access to a ruler?

[@verkuilb]

PRESH states @0:22 that, “They are equally spaced across the diameter of the circle.” But the PROBLEM as presented in the visual problem DOES NOT. So you can’t ASSUME that the vertical lines are evenly spaced across the diameter!!!

[@Paul_Bedford]

For 8 year olds, you can safely assume that they are evenly spaced. Because they have not gone over the higher order math required to understand that nuance

[@JLvatron]

At that grade, the diagram is probably indeed to scale.

[@ebanavorio]

Come on

[@DragonBoiYeah]

in the visual the lines are equally spaced

[@gafjr]

Trouble with coming to a Presh video 3 hours late is that all of my comments have already been made.

[@spockis51]

"Circles" are a set of points. You cannot shade a set of points, points have no dimension. The word "area" is never used. "Here is a shaded circle." is an non-sensical statement, mathematically. So the correct answer is no but not for the reasons assumed.

[@adipy8912]

I like when you extend the content not by just asking what fraction is shaded, but also asking how to get 1/4 of the area.

[@quantumbuddha777]

I can see some ambiguity in this question. The way it's worded, it never actually mentions area. "Is 1/4 of the circle's area shaded?" would be clearer. Are we talking 1/4 of the area or 1/4 of the pieces? I'm a math guy, so my brain immediately goes to 'area'. I can see where others might go to 'pieces'. Look at the way we divide the year into quarters. Q1 has 90 days, Q2 has 91, Q3 and Q4 each have 92. Even though the length of time for each quarter is different, we still call them quarters because they are each 1/4 of the 12 months in a year. Different size pieces, still 1/4.

[@rogergeyer9851]

To me, the implication is CLEARLY that this is about area, and I'm a math and science person. When they're talking about diameters and a quarter of the circle, I think counting pieces would be a ridiculous context. OTOH, area would be more precise AND as poor as education standards have become, at least they should make math problems clear and unambiguous. (It seems to me that they took care to do that in the standardized tests like the SAT back in my day 5ish decades ago).

[@JoeBorrello]

My house is supplied from a horizontal cylindrical tank of propane. There’s a gauge at the top, presumably attached to a float. The numbered part of the scale only goes between 20% and 80%. I now realize that this is probably because over this range the relationship between height and % full doesn’t deviate from linear as much. This also explains why they tell me to have it filled when it gets to 20% and they only fill it to 80%.

[@TheEulerID]

I am astonished anybody would think one quarter of the circle is shaded if the "slices" are the same width. You can clearly see that if you slide the righmost position one "slice" to the left that it all fits in that space of that 3rd from right space leaving a large amount to spare. Consequently the right most slice must be less than one quarter. A better question would be what proportion of the circle is shade, although that might be a bit much for a 12 year old.

[@AA-100]

The true correct answer is there's not enough information. Nowhere did the problem state the lines were drawn to scale, and that the distance between the lines were equal, meaning its possible the shaded area is actually 1/4 of the total area, if the line was moved further left

[@Twigpi]

Where do you think the numbers come from? Measure it. But the point of the question was not to calculate the areas, but to demonstrate basic logical reasoning that the shaded part can completely fit inside the piece next to it, no contest - it's not even close - and so cannot be 1 / 4.

[@AA-100]

@@Twigpi Measuring doesn't work for this problem because the problem didn't say the diagram was drawn to scale, nor did they specify the lines were the same distance apart from each other

[@Twigpi]

@@AA-100 I think it's pretty obvious that the line does not come close to the 1/ 3 mark. There exists a threshold of reasonableness. Very rarely in real life is anything going to have an exact measurement, so as a 12 year old we're going to have to approximate the pizza to determine that it wasn't sliced fairly, and back out up with sound logic. That was the point of the exercise. And the point of the video was not to come down on the people who answered cautiously and cited mathematical rigor, but rather on the people who answered that, yes, it was 1 / 4, based on a flawed understanding of geometric areas.

[@jimijenkins2548]

@@Twigpi And yet, this question was placed on a test, despite being horrendously ill-defined. Another commenter pointed out that the question states "Here is a shaded circle, is 1/4 of this circle shaded?" The information you are given is that the circle is shaded, and does not state any relation to the diagram. It is a perfectly reasonable answer to say no, on the grounds that the entire circle is shaded.

[@Twigpi]

@@jimijenkins2548 And yet, it was good question. You will find it true to life that you will have to make decisions at certain points without enough information to be 100% certain. But, come on. Given the choices of yes, no, or not answering, a reasonably educated, intelligent individual could infer what the question is actually asking, deduce that it is not a mathematical rigorous question, and determine with 98% certainty that this ISN'T a drawing so poorly done that it could possible that the shaded part is the same size as the piece next to it when it's obvious that it can fit inside that piece.

[@Tiqerboy]

How many of us watched this end to end without hitting the pause button? Presh gets so passionate about the videos he makes, you can tell.

[@patelk464]

From the context it is difficult to know if the vagueness of the question is deliberate or an oversight. The question does state that the diameter is equally split so whether or not the correct answer is no depends on if the student can deduce that the shaded cord is less than a quater of the area of the circle. If they had access to something that measure the gaps then they could deduce that the answer is no. If they didn’t they could not but could argue that the circle does appear to be equally split so that the answer is no. From the answer as to when the shaded area does represent ¼ area, this is approximately when the segmentation is ~⅖r from the center as opposed to ½r. So the difference is 0.1r. As such the answer may not be as visually obvious as suggested.

[@rogergeyer9851]

Since the question DOES state that the diameter is equally split, it is not vague at all. And if a student understands the properties of a circle, looks at the diagram, and uses a little logic, no calculation is needed. Now, when they learn what, re math, might vary wildly by country, school district, etc.

[@therealsuper5828]

This question is poorly worded because it doesnt mention the relative width of each piece if the outer 2 pieces are longer in width, they could very well be 25% each.

[@markholm7050]

As usual with word problems, this is not a geometric or mathematical problem. It’s a language problem. Geometry and mathematics require a level of precision that natural human languages achieve only with some effort. The “correct” answer here is not unambiguously defined by the simple natural language question. At least one extra assumption has to be added to complete problem in the “correct” fashion. That assumption is that one is to measure the circle by its area. While that seems obvious to those of us who have been tortured by several years of math instruction, it is not at all obvious to children who have only been drilled in linear thinking, with number lines, line segments and such. A question that is stated and evaluated without stating its assumptions is an invalid question.

[@Nezuji]

Absolutely. To play devil's advocate, Presh clarified by using the word "area" all through his discussion, although this word doesn't appear in the actual question. Most of us would assume that the question meant "1/4 of the circle's area", but it isn't specified. And I doubt that any students used this justification, but 1/4 of the circle's horizontal diameter is indeed shaded. It's interesting, given how Presh has occasionally pointed out other poorly-worded questions in the past, although this one is a bit more subtle than most.

[@lazyvector]

Modern people when asked what's 2 plus 2 (it's a question that may have different answer due to language problem)

[@Sauvenil]

@@lazyvector "2+2 = 4" doesn't exist in binary. It's 10+10 = 100 instead. Joking aside, "2" and "4" are defined so there's no ambiguity between languages.

[@moth5799]

@@lazyvector That's not the point they're making. 2 plus 2 is rather unambiguous because you're not phrasing it as a sentence, it's still a mathematical expression, you've just written out + as a word. And obviously we can derive 2+2 = 4 from the ZFC axioms.

[@juliegreen7604]

@Sauvenil @moth5799 @lazyvector Actually 2 + 2 is not necessarily well defined, if we assume that it's in decimal (or hexadecimal, octal.... etc.) then it's correct, but you have to assume that. It could be in base 3 or base 4 ( answers 11, or 10 respectively) Nevertheless, the overall point of this thread is correct, it requires suitable assumptions, not necessarily available to children, yes is a valid answer if you don't automatically assume the quarter represents area, it could represent diameter - in many ways more valid as we would generally describe a circle by diameter and not area - children would automatically relate a quarter to diameter ahead of area as they may not have got to the concept of pi x r^2 yet. Only (some) adults would understand this.

[@Zmit]

If the depicted problem is the complete description, I will argue that both answer options are valid based on the rationale given for the choice of "Yes" or "No", There is no mentioning of area in the problem description, and the premise that the division is equally spaced is an assumption made by the problem solver. In linguistic logic the answer is "Yes": one of the the four parts of the circle is shaded.

[@cougar1861]

The question "When will we ever need this" points to a problem EVEN worse that the (horrifically dreadful) statistics of students' answers to the 1/4 pizza issue: that is, some societal forces/influences that seem to severely repress both curiosity and imagination?!?

[@jimijenkins2548]

I would dismiss you as a conspiracy theorist if there wasn't so much cultural sketchiness going on with religion, media, and tradition. Really feels like there's a Mao style cultural revolution underway.

[@cougar1861]

I doubt one could justifiably refer to 4+ decades of optimally avaricious neo-liberalism a conspiracy, by any stretch of the imagination ...more like a "Mao style revolution," in-reverse, that will have scuttled the US, at current rates of descent, by its 50th anniversary.

[@username17234]

The question doesn't ask if 1/4 of THE AREA OF the circle is shaded. It's the most natural interpretation, sure, but it shouldn't be implicit in a math question.

[@Crowsinger]

An easy way I solved it was by constructing a square around the circle and seeing that the rightmost 1/4 of the square had much less area occupied by the circle that the 1/4 next to it.

[@rogergeyer9851]

Yes. I just drew a mental horizontal line for the top segments to make sure I was visualizing it right, but that's an easy way to get the answer once you KNOW the segments are of equal widths.

[@aididdat1749]

You started answering my questions two seconds before I even realized I wanted to ask them.

[@raulvidal2343]

This is how my mom cuts 1/4 of a cake.

[@marcelosalgado9729]

The problem is trivial. One just needs to focus on 1/4 of the circle (1st quadrant). Both "stripes" have the same base, but the interior stripe is longer than the exterior one (the one with much curved perimeter). So, the shaded area is less than 1/4 of the total area. The question does not demand precision. For that you can be as sophisticated as you want, and generalize the question, but it gets boring.

[@ronen44444447]

edit: i'm guessing that you just didn't include that info, I'd personally prefer to have seen it myself, otherwise I can't really know for sure. 05:52 *"So we have a circle that's devided up by equally spaced vertical segments"* UH WOAH THERE?? Where did you get that information?? who said they are equally spaced? who said that they are parallel or vertical? since when are we relying on the diagram like that?? as far as I know these lines are arbitrarily placed along the circle, with differing lengths and orientations, unless explicitly stated otherwise. Besides, the question says: "here is a shaded circle". Id say that my best bet is that the entire circle is shaded, as the question stated. Visually? sure. it's less than 1/4. But the question is inherently terribly worded, and should be discarded. what is this?? the correct answer is "Not enough information"

[@Twigpi]

Interesting opinion. It wouldn't be terribly difficult to hold one's finger up to the parts and see that the shaded part can fit inside the piece next to it.

[@mattc3581]

"Visually? sure. it's less than 1/4" I think that's the crux right there. Doesn't matter if they are supposed to be equal width or not, the sections are clearly not the same area therefore the question is answerable. The question doesn't ask about the properties of a circle which it describes how to construct, it asks about the circle and shaded section that is drawn on the page in front of them, visually is fine!

[@ronen44444447]

@@mattc3581 since when are you supposed to rely on the likely faulty printing of a diagram in standardized exam? That's why numbers exist, to eliminate the ambiguity. I don't recall ever encountering the need to visually approximate the answer on a geometry question, let alone on a graded exam. Maybe as an example in a textbook or on the board itself, but never on an exam paper

[@mattc3581]

@@ronen44444447 In basic tests designed for 8-12 year olds like this I would expect there to be many situations where you look at the picture and assume it is sufficiently correctly printed to rely on. 'What shape is printed above, how many sides does it have etc.' In this case you are hardly approximating a numerical answer, it only requires you to confirm that it is obviously smaller. There would be a region of uncertainty. Eg I can't tell visually whether the sections are supposed to be the same width, they look very similar but may not be. The areas of the sections though are fundamentally very different on this diagram, they aren't close enough to be uncertain, the shaded one is significantly less area than the section next to it. I think it was clearly intended by the setter that the widths looked the same but the areas weren't to see which children understood the difference.

[@hat_sauce3846]

@@ronen44444447 I was told that diagrams I used in secondary and third level were explicitly not to scale to avoid people using Protractors/ relying visually. For me personally, it appears the outer segments are wider than the inner (due to some sort of optical illusion) so without access to straight surface I wouldn't be able to tell if the segments were equal with.

[@willjames7841]

Posit radius = 10. Therefore each vertical cord is separated by 5. The shaded area equals the total arc angle area of the section drawn from the center minus a triangle consisting of two radii (10) and the length of the smaller vertical cord. Half small vertical cord length (l) squared equals 15 times 5 so l = 5 radical 3. Arcsin of 5 radical 3 divided by 10 is 60 degrees meaning for an arc of 120 degrees the arc area from the center is 1 / 3 the total circle area. Total area of the central triangle is 2 times 1 / 2 times 5 times 5 radical 3. So in this case the quantity (1 / 3) times 100 pi minus 25 radical 3 then divided by 100 pi would be just under one fifth, not one quarter the total circle area.

[@Ken-er9cq]

A university I was at, first year physics students were given a question on the lens equation, which is if given values of o and i, determine f. Most students correctly gave the equation 1/f=1/i+1/o A third decided that is easy just invert and calculate f=i+o

[@AA-100]

f = 1/(1/i + 1/o) is not the same as f = i+o though

[@Ken-er9cq]

@@AA-100 Yes it is not. I thought that would be obvious.

[@mattc3581]

@@Ken-er9cq Well not obvious to at least 1/3 of degree level physics students, so I guess not obvious enough.

[@Ken-er9cq]

@@mattc3581 The silliest thing is that it is easy to do. Given i=20, o=10, just enter in calculator as 20 1/x+10 1/x = gives 1/f then just push 1/x

[@GrnXnham]

One thing my geometry teacher drilled into my head back in high school was that unless some numbers are given showing lengths of lines within the circle, you can't just assume that any of those lines are what they appear to be. There is no proof, for example, that the middle line is exactly in the middle of the circle, dividing it into two perfect half circles. It APPEARS to do so, but we have no proof of this. There is also zero evidence that the three lines within the circle perfectly divide the circle into 4 equal parts. It might--it might not. Answer is no.

[@gblargg]

I don't blame students. In school things can be very warped, and questions within a context of answering what the teachers had in mind rather than approaching the problem generally. Also having to quickly answer lots of questions and not give much attention unless it seems like a trick question.

[@Hal-rn2qm]

How is this level of inteligence even human?

[@illinois_b]

For engineers, 19.6% ~ 1/4. Thus, these students are destined to become engineers.

[@sparkleyard]

☝

[@AlphaGaming-iw2xg]

how 19.6 is approximately 1/4 its approximately 1/5 which engineering have this type of approximation

[@sparshsharma5270]

​@@AlphaGaming-iw2xg Since there was no option as 1/5 in question, so the value of shaded region is more closer to 1/4.

[@programaths]

@@AlphaGaming-iw2xg When I was doing electro mechanics, we did things like PI=3 and 11=10. Depends where those numbers pops in you calculation, sometimes, even 10=1 introduce an error at the sixth decimal, so we didn't even bother doing a sum. As an example, if you've sqrt(x+10000)/100 and you evaluate for x=7. You 7 is close to 0 in that context, so you evaluate with x=0. and the whole thing is equal to 1. It's really equal to 1.0003, but if there is no scaling after that, who cares ? You can land on the moon with that precision. And a resistor has a chunky 10% tolerance. And basically, x x=0. If you are unable to do this, then it quickly becomes a nightmare, you don't always have a computer next to you. If numbers are not nice, make them nice.

[@Sauvenil]

​@@AlphaGaming-iw2xgAccording to measuring with eyeballing it, it's around 30%. Depends on how accurate you measure ;)

[@ConReese]

Ive always loved fractional visual questions like this because theres a few ways to solve for them without actually using any math at all. This one obviously being quite easy but i always found the 'trick' was to cut out the shapes of whatever fraction you want physically. Weigh the totality of the shape and then calculate the mass relationship between the X area over the total area based on weight

[@WideCuriosity]

Your pupils probably interpreted the question differently. Ignoring the fact that the equally spaced information is not in the shown question; many, but especially the young, are merely going to see a figure divided into 4, one of which is shaded, and think, 1 out of 4 is a quarter of the bits. Excuse me as I opt out at 1:05

[@JeffGoris]

As the question is presented on the paper, strictly speaking, there is insufficient information to answer. I notice that in many school maths questions they often are expected to assume things that are not necessarily true. Eg, it appears to be expecting you to assume that the lines are parallel and equally spaced. What is more concerning is the bigger mystery of how 1/2 + 1/4 was correctly answered by just 45% of year 8 students in 1997, but most recently only 32% answer correctly. 45% seems very low, but 32% is woeful. Are other countries reporting the same downturn?

[@chandrasivaram]

I think Finding areas by Integration is easier than Trigonometry.

[@carultch]

Technically, you do use trigonometry, because this ends up being an integral you solve with trig substitution. Integral sqrt(1 - x^2) dx, from 1/2 to 1, which calculates half the area in question, for the unit circle. Double this value and divide by the area of the unit circle (i.e. pi) to find the fraction. Let x = sin(u). Thus, dx = cos(u) du Using sin(u)^2 + cos(u)^2 = 1, we get: 1 - x^2 = 1 - sin(u)^2 = cos(u)^2 Thus: sqrt(1 - x^2) = cos(u) sqrt(1 - x^2) dx = cos(u)^2 du Rewrite cos(u)^2 in reduced trig form: cos(u)^2 = 1/2 + 1/2*cos(2*u) Which allows us to integrate directly as: 1/2*u + 1/4*sin(2*u) + C And translating back to the x-world: 1/2*arcsin(x) + 1/4*sin(2*arcsin(x)) + C Plug in limits: 1/2*arcsin(1) + 1/4*sin(2*arcsin(1)) - [1/2*arcsin(1/2) + 1/4*sin(2*arcsin(1/2))] pi/4 +1/2*sin(pi) - [pi/12 + 1/4*sin(pi/3)] pi/4 - [pi/12 + sqrt(3)/8] Double and divide by pi: 1/2 - 1/6 - sqrt(3)/(4*pi), which is approx 0.1955

[@raypalmer7733]

A brilliant way to show an analogy of maths to a practical question many people can relate to.

[@matthewedwards9423]

Nice. I'm well happy with my handwavey "around but not exactly 20%" answer.

[@JonTheDisciple]

Same! I looked at it and thought it was probably a fifth of the circle

[@Finity2010-ud2rl]

It wasn't exactly 20%. It was about 19.6. . .%

[@JonTheDisciple]

@@Finity2010-ud2rl correct. But with a visual look, I thought it about one fifth. I didn't do the math to check. This one is above my pay grade

[@anthonymaalouly1202]

For the quarter circle problem, a proof by contradiction: Assume hypothesis is true, that is the shaded area is indeed 1/4 of the total area. Consider the circle in the beginning but with the other supposedly quarter area shaded as well. Consider also another circle but with the lines drawn horizontally. Now add the two circles together. If the hypothesis is true, then we should be able to totally shade the combined circle. When combining the circles together, we get a 3x3 grid with the top left, top right, bottom left and bottom right counted twice, and 4 squares in the middle not shaded. If we bring the top left, top right, bottom left and bottom right in the middle of the grid, it will form another circle that will be bounded by the 4 squares in the middle, which makes it impossible to fully shade the circle -> contradiction.

[@Shizuna560]

Don't blame the players, blame the game Don't blame the students, blame the poorly designed question

[@beeble2003]

I don't think it's very poorly designed. Most twelve-year-olds are going to look at the lines and just assume they're equally spaced because they look like they are. More mathematically sophisticated people will realise that, without the assumption of equally-spaced lines at 1/4, 1/2, 3/4 of the diameter, the only way to answer the question would be by careful measurement, and that's a silly question.

[@willemm9356]

@@beeble2003 You can do a quick back-of-the-envelope estimation to see how much wider the side bits would have to be than the middle bits, for them to have equal area. I haven't done it carefully but my rough estimate is that the lines would need to be at around 1/3, 1/2 and 2/3 of the diameter. Which is way off the given drawing.

[@paulhammond6978]

this feels like something I heard before - if you move the maths to a more concrete situation, you are more likely to get the students realising that they can do it. I heard they did something like this with street kids somewhere who weren't used to being in school - if you gave them abstract maths problems they couldn't do it, but if you made it a money problem, they could work it out in the context of buying things and making change.

[@SASA_maxillo]

Long story short, the shaded area is not the same area as the other (advanced math explanation in the video)

[@WashiAmano]

a) Yes, 1/4 of the circle is shaded. b) Explanation: The circle is visually divided into 4 sections by lines, and one of these sections is shaded. If the interpretation focuses solely on the number of parts (not their areas), then one out of the four parts is shaded, making the answer 1/4

[@spockis51]

"Circles" are a set of points. Points are dimensionless. You cannot shade points. Area is never mentioned. So the answer is definitively, no. But not for the reason Presh stated. The "area" bound by the circle is divided into 4 "areas" bound by "line segments" as shown, where one area is shaded. Math is pedantic.

[@stanieldev]

Doesn't specify the gaps between the lines of sections, therefore it could be argued it's 25%. I.e. Bad Question, nothing on the students.

[@BKNeifert]

That's a tough one. A circle is a fundamentally different shape than a square. The thing I thought about first was Intersecting Chords Theorem, the reason why it works. Then I just visualized the slices converged into the center and saw they were less, so they couldn't possibly represent a fourth, as if they did, they should perfectly cover it.

[@wfukfm]

Just at first glance it kinda looked like a fifth of the circle shaded .

[@briant7265]

It would be interesting to try some seemingly trivial variations of the problem to see if the results change. 1. Rotate 90 she's to the right, so the bottom section is shaded. 2. Use a diamond (square one) instead. 3. Give a set of different figures and ask which have 1/4 shaded. Would the results change?

[@psychette8846]

Or you could just talk to an old surveyor and they will bring out the planimeter.

[@desertdarlene]

Well, finally got one right and totally understood this.

[@verkuilb]

New Zealand is home of “Lord of the Rings”…unless one of those rings is divided into vertical slices…

[@MichaelPiz]

Lord of the ¼ Rings.

[@Adammmmn]

I love how the problem revolves around the words in the sentence and not the actual math.

[@gavindeane3670]

That is the nature of mathematics word problems.

[@karyne826]

Not 1 quarter of the area.

[@CharlesPage-lb6dm]

I just watched a video you put out 7 years ago and commented on it. I hope you read the response. Because your answer was wrong and not 1 person in comments found your error. Times below are how long each would take to do job. Alice .5 hours Bob 1.5 hours Charlie 2.5 hours total: 4.5 hours (between the 3 of them) divided by 3 (workers) equals 1.5 hours (1 hour 30 minutes)

[@krabkrabkrab]

I was a pretty good student, but I would have to say I'm not sure I would have answered it correctly when I was 12. The reason is that it is poorly worded. A circle is a 1-dimensional object, not 2D. The question should ask "Is the shaded area 1/4 of the area enclosed by the circle?".

[@Steve_Stowers]

I admire your pedantry.

[@Hacker1o1]

a circle is 2d brother are you okay. 1d is a line.

[@skatex261]

how's a circle not 2d lol

[@Steve_Stowers]

@@skatex261 Mathematically, a circle is a one-dimensional space, in the sense that it only takes one coordinate specify a position on the circle. If you live on a circle, you can only move forwards or backwards. (Because, to a mathematician, a circle is only the points whose distance is exactly R from its center, not the interior.)

[@mattc3581]

@@Steve_Stowers Well true, but since the question involves shading areas of the circle we are clearly talking about the circle and it's interior here. You can try saying 'No, because you can't have an area of a line' and pedant yourself all the way to zero marks if you really want to, not sure that's a battle worth fighting.

[@johnsimpson2096]

I grew up in NZ & looking back, the maths in high school at least left a lot to be desired. There was no real geometry. Just a bit of basic trig & triangle stuff. Also next to no practical application of maths. The question is not precise enough because it does not specify a fraction of the area. It is a 1/4 of the width. So most of the students were right.

[@larscandland4072]

Our generations cooked

[@vk3dgn]

They should ask the teachers the same question; most primary teachers in Australia/NZ aren't up to teaching how to think about such questions.

[@JustMeAP1]

I've only met one K-6 teacher who could solve 1/2 - 1/3. Also, only one who could solve 24 x 0.49 (it took her over 5 minutes). Asian countries ask their 5th graders "how long will it take to fill a hole with two hoses where one can do it in 30 mins and the other in 45 min" and stop any kid who cannot answer in one minute from continuing their education. But a 1990's study found 90% of Japanese Blue-Collar workers knew the answer. Many claim that college is a waste of time. In truth, 7th to 12th is a waste for most children AND K-12 teachers. American teachers are the worst at math and science. ZERO American middle school science teachers know why the Greenhouse Effect does NOT explain why a Greenhouse gets hot. Few American Kindergarten teachers simply understand negative numbers. My own Kindergarten teacher asked what was 4 minus 5 where she thought the right answer was zero because numbers just can't go negative. After two years of me and my younger brother strongly disagreeing, she ended up institutionalized. The FACT that teachers lack ANY knowledge of teaching is an even greater problem. What's the equation for degrees of freedom and what's its importance? I knew by ten. How about telling us the benefits of "systems thinking" as taught to MIDDLE SCHOOLERS and provide us an example? Finally, is it a good idea to state your conclusion before taking data? I'm asking if you have the thinking tools of a child. As Marva Collins once asked, "How can teachers teach what they do not know?" English: What are the three classifications of the "ch" sounds? (Marva's question) Art: How do modern art texts ignorantly claiming that a montage and collage are synonyms threaten the future of the 20th century collage art revolution? Compare montage vs collage vs assemblage vs decoupage vs pointillism vs photo-mosaic. Cooking: Why is it called shortening? Hint: it's about defeating the matrix. Teaching: What did Harvard's Rosenthal prove about children in 1962? And finally, how is organizational learning so much better than traditional "eyelash" learning? Most of my Boomer friends developed their signature in Second Grade, so they still sign like a struggling Second Grader. I have been told by vendors Millennials often sign by drawing stick figures. Many don't know how to produce a signature because they were never taught script handwriting. UK citizenship tests routinely ask, "Where is Cockney spoken?" Might you know that it is usually places within earshot of the Bow Bells of St. Mary-le-Bow Church in Cheapside, London? LOL

[@vk3dgn]

@@JustMeAP1 Yes, it's amazing how education has declined. My, now deceased Grandmother could read and write beautifully even though she came from a dirt poor rural family. Her handwriting put mine to shame; arithmetic was no problem. Somebody with too much influence is ruining education - maybe it's a foreign scheme; maybe our governments like having ignorant populations who are incapable of critical thinking.

[@nikitakucherov5028]

Given the circle question didn’t include any numbers I think the answer is inconclusive

[@wiseoldfool]

I totally agree, and that would have been MY answer!

[@mattc3581]

You honestly think that the shaded area might be a quarter of the total area? Did you actually look at the diagram?

[@christopherwellman2364]

​@@mattc3581If you've ever studied geometry in class, you should know that diagrams are usually not to scale. Given no measurements of width, length, diameter, or circumference in the diagram, the question is unsolvable.

[@mattc3581]

@@christopherwellman2364 The point is that in those cases you are given information about how the diagram should actually be constructed and hence you are expected to rely on that information being accurate and not necessarily the diagram they have drawn (for illustration purposes) which may not be to scale. In this case however the diagram is the actual thing we are looking at. "Here is a shaded circle, is 1/4 of this circle shaded?", not 1/4 of the circle that this diagram may or may not be a fair representation of. This actual literal circle before you.

[@byronwatkins2565]

The word "area" is not in the question. I predict that most of the students who said yes simply misunderstood the vague question. One fourth of the total slice count IS shaded.

[@stare1716]

Exactly, the question should've been "Is the shaded region equal to 1/4th the area of the circle"

[@PugganBacklund]

As the question doesn't mention area, so saying that 1/4 of the horizontal diameter of the circle is shaded is true, so 1/4 of the circle is shaded.

[@prasoon7916]

1/4 th of the circle means 1/4 of its area or circumference, depending on the context, which is area here.

[@magmabot1016]

agreed

[@tsparks4133]

​@@prasoon7916 Nope.

[@atulmalhotra2303]

No it's not Diameter divides the Circle into 2 equal parts. Each half has two sections. There is no visual way to know what is the size of each of these sections.

[@Androecian]

Did you finish the video?

[@chinareds54]

I think the pizza discussion was completely irrelevant. If they substituted a pizza for the circle almost everyone would have answered the same as they did. Students know that a pizza is a circle. The confusion here is that the object in question has already been pre-divided into four discrete parts. Therefore in some contexts taking 1 of the 4 discrete parts (whether equal or not) could be correctly considered 1/4. For example, let's say you have a family with two parents and two toddlers. If the father gets into the car, would you say 1/4 of the family is in the car? I think most would. But of course, 1/4 of the total mass of the family is not inside the car. I think if on the other hand the problem was changed to add the word "area", there would be a dramatic increase in correct answers.

[@jasonpatterson8091]

The first question is just a bad question. If presented as shown, it doesn't say that the lines are evenly spaced and it doesn't ask if 1/4 of the area is shaded. It DOES ask if 1/4 of the circle is shaded, and the circle has 4 parts, with one shaded.

[@zmgehlke]

That's the whole point of the question: to see if students understand the difference between "1 in 4 segments of a circle shaded" and "1/4 of a circle shaded". A lot of people don't understand the distinction, so find this question confusing. But it's not a trick: it's accurately assessing if students understand the concepts. Those who don't understand the distinction (such as yourself) find the question "bad" or a "trick" -- when it's accurately assessed your math ability.

[@Qermaq]

If you want to find two points that form a chord with 1/4 of the circle's area, call the angle at the center of the circle that subtends these 2theta, and it's (r^2)(theta - sin(theta)cos(theta)). Theta rounds to 69 degrees. Nice.

[@mainakkamakar599]

Me dividing the pizza among my siblings be like:

[@0dd701]

the question didn't mention that the 3 vertical line are equidistant so I think we can't simply say the answer is yes or no

[@HYEOL]

Not enough information

[@Sauvenil]

I was taught that pictures can be inaccurate and the lines might be badly drawn and maybe they were going for actual quarters but there is no proof of any numbers or areas in this picture so we can't assume that's what they were going for.

[@HYEOL]

@@Sauvenil exactly. Thank you for writing out what I was thinking

[@tiger5869]

The lines could be evenly spaced from each other, but not from one end of the circle to the other

[@TJ-hg6op]

Based on the picture, it is very safe to assume that the shaded portion is less than a fourth. You don’t need hundreds of numbers or an exact explanation to figure this out.

[@HYEOL]

@@TJ-hg6op very save is not enough. In math it's normally a clear yes or no.

[@kwelchans]

You could also compute the area by taking 1/3 of the area of the circle ((pi x r^2)/3) and then subtracting the area of the equilateral triangle with the side length r.

[@akuunreach]

The 1/4 pounder beat out the 1/3 pound burger in the US, because the average American thinks 1/4 is greater than 1/3 because the number is bigger (their reasoning)

[@Steve_Stowers]

Solution: the 3/9 pound burger. kzbin.info/www/bejne/e36xon2Hlstjbqc

[@HarbisonelII]

9:43 I can confirm that this is far from the most practical solution. Weight can't be measured easily, but 99% of tanker trucks have capacity gauges somewhere, with the fluid leve displayed there. Okay, but let's say that's not the case. It broke or something. Humans have 3d vision, so generally we can get a decently accurate measurement just looking through the hatch. Only in the rare case the measurement tools broke and you need a very percise measurement, this method has any use. But even then, that measurement will be wrong, because the front and the back are not perfectly flat, so it defeats the point. Might as well just eyeball it.

[@kurtclark8560]

not so sure, as a former. manager of a gas station, We had to ‘stick the tanks’ daily, and after every fuel delivery. We had a chart for each tank to convert depth to gallons, but it got lost and we had to create a new one using a simple computer program similar based on an equation very similar to this.

[@HarbisonelII]

@@kurtclark8560 Hmmm... You do have a point.

[@ThatOpalGuy]

this is the result of worrying about cat litter boxes and ten commandments in classrooms

[@Pocketfarmer1]

The circle isn’t shaded at all . The circle is the set of points that are equidistant from a point not in the set or the circumference. The disk that the circle is around has some shading. It is arguable that circles do not have area. They only describe area. It is only as a product of casual speech the we say the area of a circle. You wouldn’t say the area of a fence, but more likely a fenced in area.

[@spockis51]

You are very correct. The question is much too vague to answer.

[@sirspaghetti8318]

Is 1/4 of the circle shaded? YES Is 1/4 of the circle AREA shaded? NO

[@arb1ter543]

"Area" would be redundant here

[@brianvernaglia9449]

Yes. The circle has four sections. 1 of them is shaded. 1/4 is shaded. Of course not 1/4 of the area.

[@arb1ter543]

@@brianvernaglia9449 So if a circle is made up of two sections, one clearly bigger than the other, and the smaller section was shaded, you would say half the circle is shaded? Great.

[@melody._.3251]

2:23 Casual sibling rivalry: dividing the bottle into to equal glasses MUST BE EXACT

[@mathematician369]

Indian students be like: I have learnt it at the age of 6

[@kaustubhgupta168]

haha all countries have different styles lets respect all

[@Ggdivhjkjl]

Year 8 in Australia is approximately 14.

[@drlongwind]

As always, math is rendered more difficult by asking something obvious in an ambiguous way. If you ask the question, "Is one of these four slices shaded?" the answer is yes. One of four is shaded. A language-oriented child is likely to read such a question in that way. If you ask the same child, "Is slice A bigger than slice B?" they're not struggling. It's obvious to most that they're not the same area; it's just not obvious that that's what's being asked in that question.

[@francis6888]

Presh, both answers are correct. It's the presentation of the problem that needs correcting and the intent made clear. To prove that "yes" is a correct answer, the argument that out of 4 sections/pieces/parts only one is shaded is all you need. In this example not every piece has to be equal measure.