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.
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.
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.
@@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.
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.
@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.
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.
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.
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.
@@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. 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.
Way more people would get this correct if the question asked “Is 1/4 of the circle’s AREA shaded?”. Because 1/4 of the circle’s width is shaded and 1 of 4 pieces of the circle are shaded making the question easily misinterpreted at best and ambiguous with multiple correct answers at worst.
@@NowWeJustWinItEven if you read again, You'll have to interpret because not enough details are given. Doesn't matter mathematics or other language. @draze2296 is correct on this. With well defined reasoning, It can have two answers. Just reasoning must match with answer.
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%)
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)
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. 😅
@@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.
@@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.
@@iseeyou1067Non euclidean geometry surprisingly breaks a lot of math. It's a pretty interesting field and a quick google search can show the funky world of non euclidean math
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.
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.
@@mattc3581 Why would he show the problem but leave out part of the information? Where would that information that got left out even be? After the answers? That would be a weird place to put it.
@@frank_calvert To be honest, I was doubting myself. I mean, I was expecting a trick question, so I was thinking "maybe the end sections are wider, and they really are each 1/4 of the circle", but I doubt my ability to eyeball such things and give me proper perspective. If I were taking the test, I probably would've been confirming the sections were at least close to equally as wide by measuring them against my pencil or something.
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.
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.
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
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 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.
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 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.
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.
By asking “Is 1/4 of the circle shaded?” rather than asking more generally “What fraction of the circle is shaded?” the question encourages students to work backward from the statement, analyzing the wording instead of independently evaluating the fraction. This makes the task more about validating a claim in the problem than about discovering the actual math themselves.
My issue is that it revolves around words not in the problem shown. Nothing shown states that the sections have equal width. To me, this is "not enough information," though "no" is the better option of the two presented.
upon seeing this, i immediately assumed that students just didn't think the problem was asking about the area of the circle, but rather its length. exactly 1/4 of the circle lengthwise is accounted for, and in the vague wording not ever mentioning area, that is a completely correct interpretation. an unlikely one, but more likely considering that tests are scary and have time limits, so students are incentivized to just answer with the first thing they think of, and we tend to consider length before we consider area. i've gotten suspicious of "omg all these kids don't understand math!" scaremongering ever since i took a quiz in a college-level chemistry class that asked "An athlete has 15% by mass body fat. What is the weight of fat in pounds of the athlete?" i answered that it was impossible to determine the mass of body fat without knowing the total mass. my instructor with a PhD marked this as wrong, and explained that she expected an answer to the effect of "0.033lb per 100g." humorously, she insisted that the question never asked how much fat was on the athlete's body (??), only to convert the proportion "from grams to pounds" (??????)
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.
Here is a shaded circle. Is 1/4 of the circle shaded? It's very clear that "shaded" is referencing the area of the circle, you don't use that phrasing if you're not talking about area. Is it the best way for children to understand? no. Is it misleading" No, it's perfectly reasonable with an ounce of context clues.
0:23 The problem as written says nothing about the distribution of the lines, making the answer indeterminate. Was your verbal statement that the lines are equally spaced included somewhere, unseen, in the actual question? If only as shown on screen, it’s a terribly formed question.
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 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".
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.
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.
@@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.
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).
@@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.
@@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.
@@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.
Tbh they didn’t specify or state the length between the lines in the question,so it could be a quarter. however it is quite obvious what this question’s intention is,it’s suppose to be a trick question
I doubt it's SUPPOSED to be a trick question. I think it's more likely that omitting to state that the lines are equally spaced was careless, not intentional.
@@gavindeane3670 I mean if u do enough tests and quizzes u would probably see the intention right off the bat,and true it might just want to test ur general thinking it’s not a trick question.however to the ones who failed,it just feels like one.
The question is very poorly worded, it never specifies units or area. 1 out of 4 pieces is shaded in. If you insist on doing area, we would have to know the width of each section, and we don't, so there is no reason to assume that the questions is wanting an answer to consider area.
The question NEVER asked about the "pieces." It asked only about the CIRCLE and if 25% of this CIRCLE is shaded. If the question asked: Are 1/4 of these pieces shaded? Then yes would be an acceptable answer.
The question isn't specific about what it's asking. "Is 1/4 of the circle shaded" Yes, for it has been divided into 4 pieces where in which one of those four are shaded. Regardless of if the parts are equal or reflective if the total area, 1/4th of it is shaded" is a perfectly valid answer based on the question as written. If a bunch of people are failing your question, it's worth revisiting how it's being asked. Most of the "this question stumped people" are just poorly worded from the intent
The question is specifically asking about "the circle." If this question asked: 'Are 1/4 of these segments shaded?' Then the answer yes would be acceptable, as 1 of the 4 segments is, in fact, shaded. However, it asks about the circle and if 1/4 = .25 = 25% of that circle is shaded. 25% of this circle is not shaded. Therfore the answer is no. It's really that basic and that simple. But people just keep overthinking it.
@Dhoulmegus those segments are of the circle. You have divided the circle into 4ths even if they aren't exact or even. So unless you further specify it is not unreasonable to infer that the question is about those 4ths and not the total area of the circle. You have to be specific and clear about what is being asked, people can't read minds. If I asked "what color is a polar bear" and say "white" is wrong because actually the hair is transparent and the skin is more of a Grey color. That's me wording the question poorly at best and intentionally setting people up to fail at worst. They aren't wrong for not knowing what I meant.
@@C4MG1RLNo, it is not divided into 4ths. It is divided into 4 parts. Those are objectively different things and the wording of the question would have to be different if it was asking about the latter.
@@gavindeane3670 The question never specifies what it's asking other than "is 1/4 of the circle shaded". It doesn't specify if it's asking about the total area or the sections it is divided into. With that prompt, the students aren't wrong for not giving the intended answer. The question is worded poorly. If they asked "Is 1/4 of the area shaded" then sure, but that's not what they asked. The circle is divided into 4 sections. One of those four is shaded. 1/4, so 1/4 of the circle is shaded. So the answer is Yes. The circle is divided into 4 sections. One of those four is shaded. But the shaded area does not cover 1/4 of the total area of the circle. So the answer is No. Those are both right with how it's presented.
@@C4MG1RL They are not both correct at all. One quarter of the number of pieces into which a thing has been divided is not the same as one quarter of the thing.
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
"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.
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.
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.
The only part of the question that isnt clear is whether the distance between the lines (and the distance between the lines and the edge) is the same. Everything else is perfectly clear. The problem isnt the test makers. Its the test takers
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)
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.
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.
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.
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.
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.
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%.
@@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."
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
@@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.
@@JustMeAP1Just keep in mind the unalive rate in asian countries for students is way higher that in western countries. Their learning methods are still flawed, also how fast you can solve a problem doesn't mean you're good at solving problems, as you could simply have memorized the solution.
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.
@@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
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.
@@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.
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?
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?
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.
@@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.
@@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?"
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.
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.
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.
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.
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%
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.
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).
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_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
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.
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.
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 💯👍
This isn't really a test, it's more of an experiment. It's kinda deceitful, you're not really expecting, basically, a trick question, you'd expect it to be "is this a quarter of a circle" for what this question actually is
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%.
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.
We're missing some context: often when learning about fractions students will be given shapes with equal sub-areas and they just need to count how many are coloured. This question looks incredibly similar to those diagrams, so it triggers the same sort of response. When asking about areas, usually area is mentioned and measurements are given. On top of that, the unshaded area wouldn't usually be divided into parts, and usually the question would be to determine the area, not to compare it to 1/4 without determining what it was. So all the context clues point towards the wrong approach.
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"
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.
"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!
@@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
@@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.
@@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.
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
0:18 There's no statement in the question that all divided by the same width. If we assume that the shaded part width is 30% of diameter. then the answer is yes (rounded).
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.
There's an optical illusion. The outer segments appear wider. If you have a ruler you'll get the right answer, but if you eyeball the width you won't. We don't know whether rulers were permitted in the exam or whether most student would have been advised to bring one.
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.
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
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.
@@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
@@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.
@@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.
@@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.
*@ 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.
@@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.
@@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.
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.
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.
What i did is take only the right half of the circle, and it's a yes, if at least half of that half is shaded, or in other words if the right slice is bigger than the left slice. Then just overlay them.
The students aren't given the information that the three lines are equally spaced, the middle line is the diameter and that the shape is to scale. That means they have to judge the areas by eye. And colouring the piece in question grey makes it look bigger then it is - just generally but also because the eye bleeds the grey into the edges. So then they are left with judging whether something that is 20% looks like something that could be 25%. A lot of the kids gave the wrong answer with the right reasoning i.e. they said the shaded shape was 1/4 because the four components had equal areas and the shaded area was one of them. There was another question that was similar but with pieces more obviously being different areas and the kids overwhelmingly said that the shaded piece wasn't 1/4. So, this is more that the question was incorrectly specified to get a correct response than the kids didn't know what they were doing.
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
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.
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
The fact that you calculated how much it actually was and at what point would it have to be to actually be 25%. Two random questions I had earned you a subscriber lol
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?
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.
Lots of easy, intuitive, elegant answers to show this is easily less than 1/4 of the circle. But to overkill it, you can make it a circle of radius 2 centered at the origin, then compare 2*integral of sqrt(4-x^2)dx over the interval 1 to 2 (doubling to take care of the bottom half of the circle). And you can clearly see this value is not 1/4 of 2*integral of sqrt(4-x^2)dx over the interval -2 to 2. Again, way more advanced method than is needed, but still fun to provide yet another solution.
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.
Better question would be at what angle through the equally spaced diameter points do I have to draw parallel lines, so that each part is 1/4th of the total circle area. Yes, such an angle exists. You can prove it as well. If the angle is theta where theta tends to zero. At this point the 2 outer parts are half of the circle each and the 2 inner parts are zero each. As I increase the angle from zero degrees, the area of the outer sections starts reducing from A/2 each in a smooth fashion. Now look at the case shown in this problem (90 degree angle). The outer parts are less than 1/4th the area each and the inner parts are more than 1/4th each. Hence at some point between those two angles, the outer parts would have to be 1/4th of the area each. Then remains 1/2 of the area for both the inner parts. By symmetry then, since both inner parts always have the same area, each must have an area of 1/4 of total area. So at some angle between 0 degrees and 90 degrees, each of the sections will be exactly 1/4th of the total area of the circle. You can also compute that angle by determining the chord length such that Area of outer section = 1/4 th of total area = Area of sector - Area of triangle. Then compute what angle it will subtend.
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.
0:27 if this is from New Zealand the question is actually “is one quarter of this circle shaded”. I’ve only ever heard Americans say “fourths”. (Which raises a question, why don’t Americans say “seconds”?)
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!!!
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
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.
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.
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
Watching these videos, you are inclined to read every question as a trick question. To give the kids the benefit of the doubt, if this was a trick question, it could go as follows: The question is not if 1/4 of the circle's area is shaded. Just if 1/4 of the circle is shaded. There are four clearly marked sections, and one is shaded. The fact that the sections have different areas is not necessarily relevant to the question - it's like asking if one out of four cars is blue, you don't measure which car is bigger or heavier, you just count if one out of four is blue. That could apply here - the areas may be different, but one out of four parts of the circle is shaded.
the question is worded badly. obviously what is ment here is asking ifa fourth of the volume is shaded which it isn't but even if it were, we don't have any measurements so we could just argue that the width of the shaded area is larger and therefore the total shaded area is 1/4.
New Zealand Year 9 student here (would be year 8, skipped a year) This is very common because kids my age and especially a year younger are barely literate in mathematics and English. I go to a private school, with one of the best teachers for my year level in the country. We are extremely lucky to be able to learn stuff that plenty of public schools fail to teach. Also, we're usually left out of those studies.
Draw horizontal lines where the outer vertical intersects the circle. Draw vertical lines tangent to the circle. This creates rectangles that are strictly smaller than the middle segments but strictly larger than the outer segments. Therefore the middle segments are larger. The vertical diameter does cut the circle in half, but that half isn't divided equally, so it's not divided into ¼s.
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.
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 "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.
@@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.
@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.
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?
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.
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.
You are likely right.
If it were presented as grab 1 piece of pizza, everyone would want the bigger piece.
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.
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.
@@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.
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.
A&W once introduced a "Third-of-a-Pound Burger", but it sold badly, because people thought that the "Quarter Pounder" contained more meat.
..Not a good look lol.
Americans failed 2nd grade math lol
@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.
actually it was A&W who did this, not mcdonald's
Guarntee the third pounde did not actually have 1/3 more mean and it wast just a quarter pounder anyway.
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.
Math is hard. Let's count to 10 in Maori and call it job done.
That and DMIC has absolutely ruined a generation of kids
@@Doktor47 , people who unironically use the word 'equity', aka communists, have ruined far more than one generation.
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.
It's far from being specifically a new Zealand problem. Speaking from litterally the other side of the world.
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.
Yes, but the question is poorly worded, 1 out of 4 pieces is shaded, the question never specifies the unit or area.
@@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.
@@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.
@@AirLancerthe unit could be the diameter of the circle tho instead of the area
@@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.
Way more people would get this correct if the question asked “Is 1/4 of the circle’s AREA shaded?”. Because 1/4 of the circle’s width is shaded and 1 of 4 pieces of the circle are shaded making the question easily misinterpreted at best and ambiguous with multiple correct answers at worst.
Or you could just, yknow read.
There is nothing to interpret in mathematics if you can actually read
@@NowWeJustWinItEven if you read again, You'll have to interpret because not enough details are given. Doesn't matter mathematics or other language.
@draze2296 is correct on this. With well defined reasoning, It can have two answers. Just reasoning must match with answer.
There is only two spaces between the lines, those spaces look more narrow than the wide of the left and right shapes.
The assumption is that the question is asking about the circle as a whole not the circle’s width
Technically correct.
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%)
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
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)
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. 😅
@@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.
@@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.
11:00 Hey that's me! Thanks for the shout out. Great content as always.
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
Can you tell this joke to someone with no context
@@iseeyou1067Non euclidean geometry surprisingly breaks a lot of math. It's a pretty interesting field and a quick google search can show the funky world of non euclidean math
@@Brixster thank you Patrick!!! I'll search it up
@@iseeyou1067Non Euclid 3d game
Jokes on you, I shaded it with a white colored pencil!
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.
He verbally stated that they were equidistant. Presumably it was written on the original paper, but he isn't showing that.
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.
you could use your eyes???
@@mattc3581 Why would he show the problem but leave out part of the information? Where would that information that got left out even be? After the answers? That would be a weird place to put it.
@@frank_calvert To be honest, I was doubting myself. I mean, I was expecting a trick question, so I was thinking "maybe the end sections are wider, and they really are each 1/4 of the circle", but I doubt my ability to eyeball such things and give me proper perspective. If I were taking the test, I probably would've been confirming the sections were at least close to equally as wide by measuring them against my pencil or something.
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.
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.
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
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.
Exactly.. it's always a trick question that is forcing you to make assumptions.
@@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.
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.
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.
@@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.
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.
That was my first thought also, as to clearly visualize the difference in the sizes.
Agh! Came here to say that. At least _somebody_ else said it too.
Thats only if you assume the diagram was drawn to scale, which wasn't stated in the question, so there actually isnt enough information
@@AA-100 there is no 'diagram' in the question, representing something to a certain scale. The question is directly about the circle drawn.
@@willemakkermans4067 The circle is the diagram...
I love how the problem revolves around the words in the sentence and not the actual math.
That is the nature of mathematics word problems.
By asking “Is 1/4 of the circle shaded?” rather than asking more generally “What fraction of the circle is shaded?” the question encourages students to work backward from the statement, analyzing the wording instead of independently evaluating the fraction. This makes the task more about validating a claim in the problem than about discovering the actual math themselves.
@benjaminmorris4962it is mainly a word problem as it would be easy without the words being confusing
My issue is that it revolves around words not in the problem shown.
Nothing shown states that the sections have equal width.
To me, this is "not enough information," though "no" is the better option of the two presented.
upon seeing this, i immediately assumed that students just didn't think the problem was asking about the area of the circle, but rather its length. exactly 1/4 of the circle lengthwise is accounted for, and in the vague wording not ever mentioning area, that is a completely correct interpretation. an unlikely one, but more likely considering that tests are scary and have time limits, so students are incentivized to just answer with the first thing they think of, and we tend to consider length before we consider area.
i've gotten suspicious of "omg all these kids don't understand math!" scaremongering ever since i took a quiz in a college-level chemistry class that asked "An athlete has 15% by mass body fat. What is the weight of fat in pounds of the athlete?" i answered that it was impossible to determine the mass of body fat without knowing the total mass. my instructor with a PhD marked this as wrong, and explained that she expected an answer to the effect of "0.033lb per 100g." humorously, she insisted that the question never asked how much fat was on the athlete's body (??), only to convert the proportion "from grams to pounds" (??????)
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.
If the question was 1 out of 4 then sure but it's not. It's 1/4, 25% and it's obviously false.
@@2EntireLegs linguistically, 1/4 can be interpreted as 1 of 4
@@2EntireLegs "1 out of 4" literally means the same as "1/4".
Here is a shaded circle. Is 1/4 of the circle shaded?
It's very clear that "shaded" is referencing the area of the circle, you don't use that phrasing if you're not talking about area. Is it the best way for children to understand? no. Is it misleading" No, it's perfectly reasonable with an ounce of context clues.
@CodeAlpaca No, it just references a part of the circle, not that the area is shaded
0:23 The problem as written says nothing about the distribution of the lines, making the answer indeterminate. Was your verbal statement that the lines are equally spaced included somewhere, unseen, in the actual question? If only as shown on screen, it’s a terribly formed question.
Agreed, answer could be anything
You're over thinking it, that's why you got it wrong.
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.
But what if _some_ were pregnant? 🤔
(All in jest)
@@smilerbobdang. My example foiled because people are more complicated than circles. ;)
@@brianvernaglia9449 Apparently some circles are more complicated than people 😁
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!
@@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".
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.
Yes & visually 30% is not that far off from 25% if you only have a small picture.
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.
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.
@@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.
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).
0:23 "equally spaced" - is this assumption well-founded? Nothing on the figure says anything about the distance between the dividing lines.
Measuring it.. 👀
@@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.
The distance doesn't matter, what matters is that the distances are equal.
@@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.
@@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.
Tbh they didn’t specify or state the length between the lines in the question,so it could be a quarter.
however it is quite obvious what this question’s intention is,it’s suppose to be a trick question
I doubt it's SUPPOSED to be a trick question. I think it's more likely that omitting to state that the lines are equally spaced was careless, not intentional.
@@gavindeane3670 I mean if u do enough tests and quizzes u would probably see the intention right off the bat,and true it might just want to test ur general thinking it’s not a trick question.however to the ones who failed,it just feels like one.
The question is very poorly worded, it never specifies units or area. 1 out of 4 pieces is shaded in. If you insist on doing area, we would have to know the width of each section, and we don't, so there is no reason to assume that the questions is wanting an answer to consider area.
The question NEVER asked about the "pieces." It asked only about the CIRCLE and if 25% of this CIRCLE is shaded.
If the question asked: Are 1/4 of these pieces shaded?
Then yes would be an acceptable answer.
The question isn't specific about what it's asking. "Is 1/4 of the circle shaded"
Yes, for it has been divided into 4 pieces where in which one of those four are shaded. Regardless of if the parts are equal or reflective if the total area, 1/4th of it is shaded" is a perfectly valid answer based on the question as written.
If a bunch of people are failing your question, it's worth revisiting how it's being asked. Most of the "this question stumped people" are just poorly worded from the intent
The question is specifically asking about "the circle." If this question asked: 'Are 1/4 of these segments shaded?'
Then the answer yes would be acceptable, as 1 of the 4 segments is, in fact, shaded.
However, it asks about the circle and if 1/4 = .25 = 25% of that circle is shaded. 25% of this circle is not shaded. Therfore the answer is no. It's really that basic and that simple. But people just keep overthinking it.
@Dhoulmegus those segments are of the circle. You have divided the circle into 4ths even if they aren't exact or even. So unless you further specify it is not unreasonable to infer that the question is about those 4ths and not the total area of the circle. You have to be specific and clear about what is being asked, people can't read minds.
If I asked "what color is a polar bear" and say "white" is wrong because actually the hair is transparent and the skin is more of a Grey color. That's me wording the question poorly at best and intentionally setting people up to fail at worst. They aren't wrong for not knowing what I meant.
@@C4MG1RLNo, it is not divided into 4ths. It is divided into 4 parts.
Those are objectively different things and the wording of the question would have to be different if it was asking about the latter.
@@gavindeane3670 The question never specifies what it's asking other than "is 1/4 of the circle shaded". It doesn't specify if it's asking about the total area or the sections it is divided into. With that prompt, the students aren't wrong for not giving the intended answer. The question is worded poorly. If they asked "Is 1/4 of the area shaded" then sure, but that's not what they asked.
The circle is divided into 4 sections. One of those four is shaded. 1/4, so 1/4 of the circle is shaded. So the answer is Yes.
The circle is divided into 4 sections. One of those four is shaded. But the shaded area does not cover 1/4 of the total area of the circle. So the answer is No.
Those are both right with how it's presented.
@@C4MG1RL They are not both correct at all. One quarter of the number of pieces into which a thing has been divided is not the same as one quarter of the thing.
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
"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.
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.
Yes but the picture shown is clearly spacing the lines properly and 1/4 is the most probable answer.
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.
The only part of the question that isnt clear is whether the distance between the lines (and the distance between the lines and the edge) is the same. Everything else is perfectly clear. The problem isnt the test makers. Its the test takers
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)
Huh
@@YunxiaoChu Technically, a circle is just a curved line; the disc is the area bounded by that line.
Exactly haha!
@@YunxiaoChu A circle is an infinitely thin line. Try shading an infinitely thin line, I dare you.
@@mareksroka5629 hmm, you CAN speak of the area - it’s just zero.
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.
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.
@@rogergeyer9851 Does it? Not in the visual presented at 0:29
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.
If you shade the 3rd slice instead of the 4th, the percentage of correct responses goes up dramatically.
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.
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.
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%.
@@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."
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.
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
@@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.
@@JustMeAP1Just keep in mind the unalive rate in asian countries for students is way higher that in western countries. Their learning methods are still flawed, also how fast you can solve a problem doesn't mean you're good at solving problems, as you could simply have memorized the solution.
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.
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.
@@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
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.
@@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.
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?
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.
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?
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.
BOTH answers are correct, which is why the teacher added a follow up question regarding their thinking.
@@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.
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.
I think they are allowed to use a ruler to check the sizes of the areas and see they are of equal horizontal sizes but not vertical sizes, no excuses
Does the question state the lines are evenly spaced or the drawing is to scale? If not, there is not enough information given.
Yes, it does state they are evenly spaced.
The diagram he is submitting does not indicate explicitly the lines are evenly spaced.
@@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?"
The question doesn’t say it. But at that grade, the diagram is probably indeed to scale.
So evenly spaced.
I mean, it also doesn't state that the circle is in euclidean space, but at this age/grade, it would be assumed.
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.
So a capsule then.
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.
@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.
Could you fill a tank a quarter full and measure that and just keep the stick handy?
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.
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.
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%
@@prasoon7916 It isn't given. Nor is it explicitly stated the diagram was drawn to scale, so technically speaking, there isnt enough information
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.
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).
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.
great substitute to it stating it: your eyes. they're the same distance.
@@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
@@lammahater9153same, ive had calculations where the orbit of a planet was shown as an ellipse, but we were told it was a circle
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.
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.
Well, finally got one right and totally understood this.
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 💯👍
"ridiculously depressing stat" !?
There is no right/wrong answer to this question, so the "stat" is meaningless.
This isn't really a test, it's more of an experiment. It's kinda deceitful, you're not really expecting, basically, a trick question, you'd expect it to be "is this a quarter of a circle" for what this question actually is
Sorry but, are you against or do you agree with the vídeo?
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%.
You started answering my questions two seconds before I even realized I wanted to ask them.
Year 8 in Australia is approximately 14.
A brilliant way to show an analogy of maths to a practical question many people can relate to.
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.
We're missing some context: often when learning about fractions students will be given shapes with equal sub-areas and they just need to count how many are coloured. This question looks incredibly similar to those diagrams, so it triggers the same sort of response. When asking about areas, usually area is mentioned and measurements are given. On top of that, the unshaded area wouldn't usually be divided into parts, and usually the question would be to determine the area, not to compare it to 1/4 without determining what it was. So all the context clues point towards the wrong approach.
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"
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.
"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!
@@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
@@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.
@@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.
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
0:18 There's no statement in the question that all divided by the same width. If we assume that the shaded part width is 30% of diameter. then the answer is yes (rounded).
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.
There's an optical illusion. The outer segments appear wider. If you have a ruler you'll get the right answer, but if you eyeball the width you won't. We don't know whether rulers were permitted in the exam or whether most student would have been advised to bring one.
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.
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
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.
@@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
@@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.
@@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.
@@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.
Great video, loved thw derivation of the segment formula and the practical application with fuel tankers.
It just amazes me how many people here are making excuses for getting the question wrong.
Assuming that the circle is divided and shaded as it’s exactly shown in the diagram, it’s a no both in circumference and area.
*@ 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.
@@verkuilbAnd where was that told to the students?
It shouldn't need to be explicitly stated. Part of not being an NPC golem is being able tow ork out the intention yourself.
@@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.
@@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.
@@j100j As far as I'm concerned the test is psyche profiling in disguise.
I really enjoyed watching this. I think there's a lot to be learned from grass-roots geometry questions like this.
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.
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.
What i did is take only the right half of the circle, and it's a yes, if at least half of that half is shaded, or in other words if the right slice is bigger than the left slice. Then just overlay them.
This is one of your best videos yet.
Just at first glance it kinda looked like a fifth of the circle shaded .
The students aren't given the information that the three lines are equally spaced, the middle line is the diameter and that the shape is to scale. That means they have to judge the areas by eye. And colouring the piece in question grey makes it look bigger then it is - just generally but also because the eye bleeds the grey into the edges. So then they are left with judging whether something that is 20% looks like something that could be 25%. A lot of the kids gave the wrong answer with the right reasoning i.e. they said the shaded shape was 1/4 because the four components had equal areas and the shaded area was one of them. There was another question that was similar but with pieces more obviously being different areas and the kids overwhelmingly said that the shaded piece wasn't 1/4. So, this is more that the question was incorrectly specified to get a correct response than the kids didn't know what they were doing.
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
f = 1/(1/i + 1/o) is not the same as f = i+o though
@@AA-100 Yes it is not. I thought that would be obvious.
@@Ken-er9cq Well not obvious to at least 1/3 of degree level physics students, so I guess not obvious enough.
@@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
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.
It doesent say 1/4th of the area so it is still valid to say it 1/4 of an equal width cut
It says 1/4th of the circle. 1/4th of the circle means 1/4th of the area
Nice job Presh, loved it. Thank you
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
The fact that you calculated how much it actually was and at what point would it have to be to actually be 25%. Two random questions I had earned you a subscriber lol
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?
You went further into the problem and should us real world use for those extra calculations
That was awesome!!!
This is how my mom cuts 1/4 of a cake.
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.
Long story short, the shaded area is not the same area as the other (advanced math explanation in the video)
Lots of easy, intuitive, elegant answers to show this is easily less than 1/4 of the circle.
But to overkill it, you can make it a circle of radius 2 centered at the origin, then compare 2*integral of sqrt(4-x^2)dx over the interval 1 to 2 (doubling to take care of the bottom half of the circle).
And you can clearly see this value is not 1/4 of 2*integral of sqrt(4-x^2)dx over the interval -2 to 2.
Again, way more advanced method than is needed, but still fun to provide yet another solution.
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.
Better question would be at what angle through the equally spaced diameter points do I have to draw parallel lines, so that each part is 1/4th of the total circle area. Yes, such an angle exists. You can prove it as well.
If the angle is theta where theta tends to zero. At this point the 2 outer parts are half of the circle each and the 2 inner parts are zero each. As I increase the angle from zero degrees, the area of the outer sections starts reducing from A/2 each in a smooth fashion. Now look at the case shown in this problem (90 degree angle). The outer parts are less than 1/4th the area each and the inner parts are more than 1/4th each. Hence at some point between those two angles, the outer parts would have to be 1/4th of the area each. Then remains 1/2 of the area for both the inner parts. By symmetry then, since both inner parts always have the same area, each must have an area of 1/4 of total area. So at some angle between 0 degrees and 90 degrees, each of the sections will be exactly 1/4th of the total area of the circle.
You can also compute that angle by determining the chord length such that Area of outer section = 1/4 th of total area = Area of sector - Area of triangle. Then compute what angle it will subtend.
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.
Nice. I'm well happy with my handwavey "around but not exactly 20%" answer.
Same! I looked at it and thought it was probably a fifth of the circle
It wasn't exactly 20%. It was about 19.6. . .%
@@Finity_twenty_ten 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
I used quarter circle integral to calculate the area. Your answer is much more intuitive and elegant per usual haha
86% got this wrong, oh damn. 8 year old, that explains alot....
9 or 10 year old no idea where this guy got the school ages from
0:27 if this is from New Zealand the question is actually “is one quarter of this circle shaded”. I’ve only ever heard Americans say “fourths”.
(Which raises a question, why don’t Americans say “seconds”?)
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!!!
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
At that grade, the diagram is probably indeed to scale.
Come on
in the visual the lines are equally spaced
Trouble with coming to a Presh video 3 hours late is that all of my comments have already been made.
Excellent journey, with a lot to see. Thanks!
1:48 Why tf is the pizza filed with tomato slices???
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.
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.
What other kind of "one-fourth" would it be?
@@robertloveless4938 The horizontal length, as if it was a progress bar
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.
I think Finding areas by Integration is easier than Trigonometry.
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
Watching these videos, you are inclined to read every question as a trick question. To give the kids the benefit of the doubt, if this was a trick question, it could go as follows:
The question is not if 1/4 of the circle's area is shaded. Just if 1/4 of the circle is shaded. There are four clearly marked sections, and one is shaded. The fact that the sections have different areas is not necessarily relevant to the question - it's like asking if one out of four cars is blue, you don't measure which car is bigger or heavier, you just count if one out of four is blue. That could apply here - the areas may be different, but one out of four parts of the circle is shaded.
New Zealand is home of “Lord of the Rings”…unless one of those rings is divided into vertical slices…
Lord of the ¼ Rings.
Very nice explanation of the math involved.
the question is worded badly. obviously what is ment here is asking ifa fourth of the volume is shaded which it isn't but even if it were, we don't have any measurements so we could just argue that the width of the shaded area is larger and therefore the total shaded area is 1/4.
@@marc_so Area, not volume. Area is two-dimensional, volume is three.
@@verkuilb Screw it, cylinder time.
New Zealand Year 9 student here (would be year 8, skipped a year)
This is very common because kids my age and especially a year younger are barely literate in mathematics and English.
I go to a private school, with one of the best teachers for my year level in the country.
We are extremely lucky to be able to learn stuff that plenty of public schools fail to teach.
Also, we're usually left out of those studies.
Or you could just talk to an old surveyor and they will bring out the planimeter.
Draw horizontal lines where the outer vertical intersects the circle. Draw vertical lines tangent to the circle. This creates rectangles that are strictly smaller than the middle segments but strictly larger than the outer segments. Therefore the middle segments are larger. The vertical diameter does cut the circle in half, but that half isn't divided equally, so it's not divided into ¼s.
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.
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.
Modern people when asked what's 2 plus 2 (it's a question that may have different answer due to language problem)
@@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.
@@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.
@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.
Brilliant episode!
None of this suprises me; teaching is so often incompetent these days.
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?
It's 100% shaded - mostly black, less than 1/4 red, and a bit of white.
true, don't know why they didn't make it transparent, it would make more sense, instead of some black white red mix
@@agnichatian The glass is completely full. Half water and half air.
Red? Is there red I'm not seeing? I'm a little colorblind.
@@christopherwellman2364 in the thumbnail/title picture it is red
@@christopherwellman2364 you don't see the red?
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.
Not 1 quarter of the area.