Re: A math question ...
Re: A math question ...
My third grader got a multiple choice question in relation to a set of figures in a Math benchmark test. Because I think that the question was flawed, I am creating this poll to see how many agree with me. To prevent me from biasing or prejudicing the poll, I will refrain from posting my point of view until a couple of days (at least)!
You are provided two two-dimensional figures that are mirror images which cannot be rotated in a manner that they match - for instance, the number 7 and its mirror image. With that information, please answer the poll ...
You are provided two two-dimensional figures that are mirror images which cannot be rotated in a manner that they match - for instance, the number 7 and its mirror image. With that information, please answer the poll ...
Last edited by an_asker on Mon Nov 18, 2013 8:39 am, edited 1 time in total.
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Re: A math question ...
Wow... blast to the geometry past.
Congruency refers to an object that can be transformed into the other by methods such as rotating, flipping, etc
Similarity refers to an object that can be scaled into the other. Only changing of size.
Congruency refers to an object that can be transformed into the other by methods such as rotating, flipping, etc
Similarity refers to an object that can be scaled into the other. Only changing of size.
Re: A math question ...
Well, looks like I was wrong. I didn't think mirror images were congruent. I've always been good at math, but this is a vocabulary question. I never remembered associative, commutative, etc. either. Just arbitrary definitions.
Re: A math question ...
They can be flipped (reflected), so they are congruent. Since they are mirror images (same size) they are not similar.
Re: A math question ...
7 and it's same-sized reflection are both similar and congruent according to wikipedia:
http://en.wikipedia.org/wiki/Congruence_(geometry)This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.
http://en.wikipedia.org/wiki/Similarity_(geometry)More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other.
Last edited by tadamsmar on Mon Nov 18, 2013 8:36 am, edited 2 times in total.
Re: A math question ...
I would have said they are similar but not congruent because of how I read your prompt, but looking at other web sites the answer is congruent but not similar (similar = resizing).
Assuming your child was taught these terms per this link (http://www.mathsisfun.com/geometry/congruent.html) or the like, then the problem is well defined.
Assuming your child was taught these terms per this link (http://www.mathsisfun.com/geometry/congruent.html) or the like, then the problem is well defined.
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Re: A math question ...
This is more an English Language test than a math test, imo...
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Re: A math question ...
Could you provide a citation for similar not including reflection?JamesSFO wrote:I would have said they are similar but not congruent because of how I read your prompt, but looking at other web sites the answer is congruent but not similar (similar = resizing).
Assuming your child was taught these terms per this link (http://www.mathsisfun.com/geometry/congruent.html) or the like, then the problem is well defined.
Could all the rest of those posting include references for their claims.
I cannot find citations that contradict the wikipedia, I'd like to see where you are getting your info.
Re: A math question ...
How should I rephrase it to better help you?!The Wizard wrote:This is more an English Language test than a math test, imo...
Re: A math question ...
Textbooks do not agree on whether 7 and its same-size reflection are are similar:
If non-unity scaling is required for similarity, then 7 is not similar to 7
http://en.wikipedia.org/wiki/Similarity_(geometry)This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar.
If non-unity scaling is required for similarity, then 7 is not similar to 7
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Re: A math question ...
You phrased it fine.an_asker wrote:How should I rephrase it to better help you?!The Wizard wrote:This is more an English Language test than a math test, imo...
But speaking as an engineer who's encountered a lot of math, this aspect of geometry is primarily definitional...
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Re: A math question ...
Sounds hard for 3rd grade.
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Re: A math question ...
One could argue that all math is completely definitional.The Wizard wrote:You phrased it fine.an_asker wrote:How should I rephrase it to better help you?!The Wizard wrote:This is more an English Language test than a math test, imo...
But speaking as an engineer who's encountered a lot of math, this aspect of geometry is primarily definitional...
Re: A math question ...
The correct answer depends on the definition the child was taught.
Aside: there was a story in the Washington Post about the common core in which the writer asked a PhD friend about a question given to a 4th grader. The PhD couldn't answer because the PhD didn't know the vocabulary. This was supposed to illustrate that the test question was not appropriate for a 4th grader. But a 4th grader can learn vocabulary. Heck, even two year olds can learn vocabulary, otherwise they couldn't talk! The 4th grader could answer the question the PhD couldn't because the 4th grader knew the vocabulary.
Aside: there was a story in the Washington Post about the common core in which the writer asked a PhD friend about a question given to a 4th grader. The PhD couldn't answer because the PhD didn't know the vocabulary. This was supposed to illustrate that the test question was not appropriate for a 4th grader. But a 4th grader can learn vocabulary. Heck, even two year olds can learn vocabulary, otherwise they couldn't talk! The 4th grader could answer the question the PhD couldn't because the 4th grader knew the vocabulary.
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Re: A math question ...
Even a PhD can learn vocabulary.
The concepts are more important than the vocabulary, after all you can do math just as well in French or German as in English.
Unfortunately too many educators think that knowing a particular vocabulary is more important than knowing the concepts, and don't get me started about the ones who think they were appointed to set the one and only true definition. Sometimes the most important thing to bear in mind is that a word can mean different things to different people.
The concepts are more important than the vocabulary, after all you can do math just as well in French or German as in English.
Unfortunately too many educators think that knowing a particular vocabulary is more important than knowing the concepts, and don't get me started about the ones who think they were appointed to set the one and only true definition. Sometimes the most important thing to bear in mind is that a word can mean different things to different people.
Re: A math question ...
Even the French and the Germans use words. The words they use have definitions. Heck, even the words you use have definitions. Words without definitions wouldn't be of much use in communication.Epsilon Delta wrote: The concepts are more important than the vocabulary, after all you can do math just as well in French or German as in English.
If you tell me what you mean by similar, then I will understand you when you use the word. As you say, there doesn't have to be one true definition, but for us to communicate, we have to agree on one for the time being.
Re: A math question ...
http://upload.wikimedia.org/wikipedia/c ... es.svg.png
reversed 7's are congruent. Their appearance is not similar.
reversed 7's are congruent. Their appearance is not similar.
Re: A math question ...
Now I get The Wizard's point ...tadamsmar wrote:One could argue that all math is completely definitional.The Wizard wrote:You phrased it fine.an_asker wrote:How should I rephrase it to better help you?!The Wizard wrote:This is more an English Language test than a math test, imo...
But speaking as an engineer who's encountered a lot of math, this aspect of geometry is primarily definitional...
However, that itself was my premise in asking the question! If every country (or classroom or generation) has its own definition of basic mathematics terms such as congruent figures, similar figures, etc. then (not just) my poor kids are in a world of hurt when time comes for standardized testing (SAT, ACT, GRE, GMAT ... throw your pet exam acronym)
Re: A math question ...
Not if they learn the vocabulary used by the SAT, ACT, GRE, GMAT. That's what study guides are for, so they can learn the vocabulary.an_asker wrote: However, that itself was my premise in asking the question! If every country (or classroom or generation) has its own definition of basic mathematics terms such as congruent figures, similar figures, etc. then (not just) my poor kids are in a world of hurt when time comes for standardized testing (SAT, ACT, GRE, GMAT ... throw your pet exam acronym)
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Re: A math question ...
Yes, you can define anything to be anything, so of course definitions matter, but using traditional Ecludian definitions,
these would be both congruent and similar.
The traditional proof that the sides of an isosceles triangle (opposite the equal angles) are the same length (congruent)
involves flipping the triangle over (reflection) and showing that the triangles are congruent due to the
Angle-Side-Angle test for congruence of triangles. (Pf: Consider two isosceles triangles, one and its reflection. The corresponding
base angles are equal by definition, the base is congruent to itself, therefore by ASA, the triangles are congruent.)
Congruent implies similar in any definitions I have seen. Two triangles are similar if all three angles
(only two really needed in Euclidiean Geom. since the other is 180 deg - the sum of the other two).
If one side is the same, then you have ASA which is a conguence criterion.
I would agree that this seems ridiculous for a 3rd grader. Typically teachers at that level should be concentrating
on getting the reading/language skills improved, and math skills to be conentrated on would be arithmetic. If you want
to go more advanced, show them things that will make arithmetic easier or more accurate. Commutative, associative, and
distributive laws. ( e.g. 7*8 is the same as 8*7, you can multiply by 9 simply by multiplying by 10 and subtracting the
original number)
these would be both congruent and similar.
The traditional proof that the sides of an isosceles triangle (opposite the equal angles) are the same length (congruent)
involves flipping the triangle over (reflection) and showing that the triangles are congruent due to the
Angle-Side-Angle test for congruence of triangles. (Pf: Consider two isosceles triangles, one and its reflection. The corresponding
base angles are equal by definition, the base is congruent to itself, therefore by ASA, the triangles are congruent.)
Congruent implies similar in any definitions I have seen. Two triangles are similar if all three angles
(only two really needed in Euclidiean Geom. since the other is 180 deg - the sum of the other two).
If one side is the same, then you have ASA which is a conguence criterion.
I would agree that this seems ridiculous for a 3rd grader. Typically teachers at that level should be concentrating
on getting the reading/language skills improved, and math skills to be conentrated on would be arithmetic. If you want
to go more advanced, show them things that will make arithmetic easier or more accurate. Commutative, associative, and
distributive laws. ( e.g. 7*8 is the same as 8*7, you can multiply by 9 simply by multiplying by 10 and subtracting the
original number)
Re: A math question ...
My kids had math teachers that marked them off if they didn't put the answers in a column on the right hand side. I taught, and I didn't give full credit for a correct answer without the work to show where the answers were coming from. You have to play by the rules of the person who is testing or grading you. I don't see how this should be surprising.
Engineers: when you were in school, didn't you have rules to follow on homework and tests? I remember having to fold my homework in half lengthwise. Not folded, not graded. It didn't take me long to learn. I ain't that stupid that I would keep turning in an unfolded homework paper.
Engineers: when you were in school, didn't you have rules to follow on homework and tests? I remember having to fold my homework in half lengthwise. Not folded, not graded. It didn't take me long to learn. I ain't that stupid that I would keep turning in an unfolded homework paper.
Re: A math question ...
That is not what is taught these days. Algebra and logic and geometry are all included in 3rd grade. The thing that bugs me is that the current practice is to give the students a homework problem on a concept that hasn't been taught. After the students struggle with the idea, a solution method is presented. That's the reverse of the way most of us were taught: concept, then practice. Now it is practice, then concept.MathWizard wrote: I would agree that this seems ridiculous for a 3rd grader. Typically teachers at that level should be concentrating
on getting the reading/language skills improved, and math skills to be conentrated on would be arithmetic.
Edit added: I have a granddaughter in 3rd grade this year; another was in 3rd grade two years ago. My daughter and granddaughters like to show me ill-defined questions. Of course, my concerns and experiences are local. Since we don't have a national curriculum with national materials, what you see depends on your school district.
Last edited by sscritic on Mon Nov 18, 2013 10:16 am, edited 1 time in total.
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Re: A math question ...
http://www.mathopenref.com/congruent.htmltadamsmar wrote:Could you provide a citation for similar not including reflection?JamesSFO wrote:I would have said they are similar but not congruent because of how I read your prompt, but looking at other web sites the answer is congruent but not similar (similar = resizing).
Assuming your child was taught these terms per this link (http://www.mathsisfun.com/geometry/congruent.html) or the like, then the problem is well defined.
Could all the rest of those posting include references for their claims.
I cannot find citations that contradict the wikipedia, I'd like to see where you are getting your info.
So I guess in this senario the answer would 100% be congruent. There is fuzziness on the requirement of the polygons to be different sizes.Definition: Equal in size and shape
Wikiaddresses this issue:
some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar.
Re: A math question ...
In my experience, many teachers/instructors make up their own definitions.
I once got into a heated debate with a college instructor who insisted that a set was NOT a subset of itself. By every definition I'd ever read, a set IS a subset of itself, but not a proper subset of itself. Differences like this lead to confusion and frustration for students.
So while people may debate the need for federal standards, perhaps issues like this provide justification.
I once got into a heated debate with a college instructor who insisted that a set was NOT a subset of itself. By every definition I'd ever read, a set IS a subset of itself, but not a proper subset of itself. Differences like this lead to confusion and frustration for students.
So while people may debate the need for federal standards, perhaps issues like this provide justification.
Re: A math question ...
I was home schooled by my mom who only has a high school education and did very poor in math, so If I couldn't get my answer to look exactly like what was in the book, i got it wrong. While it wasn't as bad as 4/10 != 0.4 it was pretty close, but typically with things like sin(), e, or pi. It got really bad with calculus, but at least I got plenty of math practice insscritic wrote:My kids had math teachers that marked them off if they didn't put the answers in a column on the right hand side. I taught, and I didn't give full credit for a correct answer without the work to show where the answers were coming from. You have to play by the rules of the person who is testing or grading you. I don't see how this should be surprising.
Engineers: when you were in school, didn't you have rules to follow on homework and tests? I remember having to fold my homework in half lengthwise. Not folded, not graded. It didn't take me long to learn. I ain't that stupid that I would keep turning in an unfolded homework paper.
Once in college, there really weren't any rules other than submit the paper, show your work, and make it legible.
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Re: A math question ...
I guess you are right on that. My youngest son practically begged me to get him out of the "Math in Context"sscritic wrote:That is not what is taught these days. Algebra and logic and geometry are all included in 3rd grade. The thing that bugs me is that the current practice is to give the students a homework problem on a concept that hasn't been taught. After the students struggle with the idea, a solution method is presented. That's the reverse of the way most of us were taught: concept, then practice. Now it is practice, then concept.MathWizard wrote: I would agree that this seems ridiculous for a 3rd grader. Typically teachers at that level should be concentrating
on getting the reading/language skills improved, and math skills to be conentrated on would be arithmetic.
as soon as I could, which in our district was 7th grade, at which time with a high enough score on a qualifying
test and teacher approval you could to start HS Algebra in the 7th grade. The "squishy" logic frustrated him. He was asked to come up with a "story" on how he got the answer, and fro some reason "following the rules in the
textbook" was never a good enough answer.
Even with homework every night, he was much more happy in Algebra. He actually is very good in logic, but
that was probably taught at home more than in school.
He is now a freshman in college, studying mechanical engineering.
I am concerned about the Math and Science (and corresponding logic) education in this country. Kids in the
US are on par with other advanced countries until middle school, and then the US students steadily fall behind.
I do know from many foreign friends that algebra is typically taught starting in the 7th grade for normal students.
So my son, being advanced in Math as early as he could, by HS graduation had only completed in math what the average technically oriented student would have in other countries. That seems to be a disservice to the students.
What does this have to do with finance? There are only two ways to attain greater wealth when we are at full employment.
1) Take it from someone else, or
2) Use technology to increase productivity, and grow the total economy.
I believe that it is the latter that keeps us having a growing GDP, and where the increased earning come that
allow an indexed equity strategy to work well.
Edited to correct ling spacing.
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Re: A math question ...
They are congruent. Are they similar as well? Yes, if you go with "All congruent figures are similar, but not all similar figures are congruent". Congruent figures are subsets of similar figures. However, some textbooks state that congruent figures are dissimilar. Ultimately, as others have pointed out, this is a matter of semantics. As Karl Popper stated, what matters is substance, not semantics, and that exposing the argument as semantic can put the matter to rest.
The correct answer is whatever definition the textbook uses, but let's not fool ourselves that we are taking a substantive position.
The correct answer is whatever definition the textbook uses, but let's not fool ourselves that we are taking a substantive position.
Re: A math question ...
I'd vote for CONGRUENT AND SIMILAR.
There are definitions floating around that seem to state that similar figures are NOT congruent. I believe they should say that similar figures are not NECESSARILY congruent. Congruent figures are similar, but similar figures may or may not be congruent depending on their sizes. At least that's the way I learned it - way back when.
There's an explanation of this on Khan Academy:
https://www.khanacademy.org/math/test-p ... -triangles
There are definitions floating around that seem to state that similar figures are NOT congruent. I believe they should say that similar figures are not NECESSARILY congruent. Congruent figures are similar, but similar figures may or may not be congruent depending on their sizes. At least that's the way I learned it - way back when.
There's an explanation of this on Khan Academy:
https://www.khanacademy.org/math/test-p ... -triangles
Re: A math question ...
Congruent is just a special case of similar.
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Re: A math question ...
This is a very common problem with testing. Many questions are trying to see if students remember specific definitions of concepts, but the terms used may not be ubiquitous, in fact may only be made up or applied to that particular unit in the textbook, or surprisingly often, used inconsistently or slightly differently than most practitioners would commonly understand them. As a result, the test is testing a particular test taking skill and short term memorization more than it is testing the concept. You can write better questions defining the terms in the question, but they are rarely done that way. Many test writers do not understand the concepts all that well anyway, so falling back on vocabulary and "tricks" is all too common. And of course there is the long term problem that while some concepts may be useful knowledge, the specific unit-level terminology used is probably next to useless long term, so mostly pointless to test.This is more an English Language test than a math test, imo...
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Re: A math question ...
Yes.downshiftme wrote:This is a very common problem with testing. Many questions are trying to see if students remember specific definitions of concepts, but the terms used may not be ubiquitous, in fact may only be made up or applied to that particular unit in the textbook, or surprisingly often, used inconsistently or slightly differently than most practitioners would commonly understand them. As a result, the test is testing a particular test taking skill and short term memorization more than it is testing the concept. You can write better questions defining the terms in the question, but they are rarely done that way. Many test writers do not understand the concepts all that well anyway, so falling back on vocabulary and "tricks" is all too common. And of course there is the long term problem that while some concepts may be useful knowledge, the specific unit-level terminology used is probably next to useless long term, so mostly pointless to test.This is more an English Language test than a math test, imo...
My particular horror story is that after I changed schools I got a zero on a math test because among other things I used into, onto and 1-1 correspondence in an essay instead of injection, surjection and bijection. The teacher expounded that if I wanted her good opinion I had to do things her way. I explained that her good opinion was not worth a plugged nickel.
Re: A math question ...
Let's put in new curriculum and standards and assure our youngsters come out uneducated. Rant over. I love non-super-imposable mirror images. You learn about this in organic chemistry and they are called enantiomers. It is also referred to as being "chiral" or "having chirality". This is absolutely crucial in medical and biologic systems where a receptor will recognize one enantiomer and completely disregard the other.
Re: A math question ...
Hard for me... I didn't do well in 3rd grade though.Call_Me_Op wrote:Sounds hard for 3rd grade.
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Re: A math question ...
Ignoring previous posts because I want to work it out for myself...
In math, special cases and such can be important, so when it comes to the question of whether "congruent" objects are "similar," it isn't necessarily as much a question of pure arbitrary authority (is Australia a continent or an island? Is Pluto a planet? Is it correct English to say "He took his laptop to the office today?") as it might be in other fields. I don't know the answer but I would certainly look up the definitions before answering. If I were going to take a geometry course, I would memorize any definitions that weren't intuitive, and I would use the glossary of my class textbook as my authority for the definitions.
But for now I will just use m-w.com.
congruent: "superposable so as to be coincident throughout." "to lay (as a geometric figure) upon another so as to make all like parts coincide." Not good enough, are you allowed to move something out of the plane to superpose it? I think so, but the definition isn't quite good enough.
But I remember from geometry that two triangles are considered congruent if SSS = SSS, which implies that mirror flips are allowed.
I'm pretty sure that that is straight out of Euclid's Elements although not phrased the same way.
similar: "not differing in shape but only in size or position." Also not clear enough. As I read that, literally, two triangles must differ either in size or in shape to be "similar," which I suspect is not correct.
Wikipedia is much better, I think I trust what it says, and it says "In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other." "Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection."
According to these definitions, I think the correct answer would be 1, they are congruent and similar.
In math, special cases and such can be important, so when it comes to the question of whether "congruent" objects are "similar," it isn't necessarily as much a question of pure arbitrary authority (is Australia a continent or an island? Is Pluto a planet? Is it correct English to say "He took his laptop to the office today?") as it might be in other fields. I don't know the answer but I would certainly look up the definitions before answering. If I were going to take a geometry course, I would memorize any definitions that weren't intuitive, and I would use the glossary of my class textbook as my authority for the definitions.
But for now I will just use m-w.com.
congruent: "superposable so as to be coincident throughout." "to lay (as a geometric figure) upon another so as to make all like parts coincide." Not good enough, are you allowed to move something out of the plane to superpose it? I think so, but the definition isn't quite good enough.
But I remember from geometry that two triangles are considered congruent if SSS = SSS, which implies that mirror flips are allowed.
I'm pretty sure that that is straight out of Euclid's Elements although not phrased the same way.
similar: "not differing in shape but only in size or position." Also not clear enough. As I read that, literally, two triangles must differ either in size or in shape to be "similar," which I suspect is not correct.
Wikipedia is much better, I think I trust what it says, and it says "In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other." "Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection."
According to these definitions, I think the correct answer would be 1, they are congruent and similar.
Last edited by nisiprius on Mon Nov 18, 2013 1:02 pm, edited 2 times in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: A math question ...
So what was the correct answer - OP?
Re: A math question ...
This discussion illustrates the issue with multiple choice questions. Word problems, geometrical proofs, long algebraic transformations are all much more useful.
Victoria
Victoria
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Winner of the 2015 Boglehead Contest. |
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Re: A math question ...
You might have to wait a few more days.So what was the correct answer - OP?
I will refrain from posting my point of view until a couple of days (at least)!
Re: A math question ...
Not without definitions of the words.VictoriaF wrote: Word problems ... are all much more useful.
Victoria
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Re: A math question ...
Hmmm... looking up Euclid's Elements online it does not appear as if he uses any word whose English translation is "congruent." So, that particular apparatus, along with such baggage as CPCTE, is a later pedagogical innovation.
Yep... I read here that "This is the first of the congruence propositions for triangles. Euclid did not explicitly use the concept of congruence, although it would have simplified his exposition a bit." And I read that
Yep... I read here that "This is the first of the congruence propositions for triangles. Euclid did not explicitly use the concept of congruence, although it would have simplified his exposition a bit." And I read that
It is not entirely clear what is meant by "superposing a triangle on a triangle" means. It has been variously interpreted as actually moving one triangle to cover the other or as simply associating parts of one triangle with parts of the other. For the two triangles illustrated in the figure, you can actually slide one over the other in a continuous motion within the plane. Note, however, that if one triangle is the mirror image of the other, then any continuous motion would require moving one triangle outside of the plane. But the triangles don't have to be same plane to begin with, and they often are not in the same plane when this proposition is invoked in the books on solid geometry.
Last edited by nisiprius on Mon Nov 18, 2013 12:42 pm, edited 1 time in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: A math question ...
If you have a picture of the two figures it's worth a thousand words.sscritic wrote:Not without definitions of the words.VictoriaF wrote: Word problems ... are all much more useful.
Victoria
Victoria
Inventor of the Bogleheads Secret Handshake |
Winner of the 2015 Boglehead Contest. |
Every joke has a bit of a joke. ... The rest is the truth. (Marat F)
Re: A math question ...
Shouldn't that be two thousand?VictoriaF wrote: If you have a picture of the two figures it's worth a thousand words.
Victoria
Re: A math question ...
No. That would be a loophole for charging thousands of words by sticking extra figures into every picture.sscritic wrote:Shouldn't that be two thousand?VictoriaF wrote: If you have a picture of the two figures it's worth a thousand words.
Victoria
Victoria
Inventor of the Bogleheads Secret Handshake |
Winner of the 2015 Boglehead Contest. |
Every joke has a bit of a joke. ... The rest is the truth. (Marat F)
Re: A math question ...
My understanding (one of the mathematicians here can correct me) is that your instructor is right and you're wrong. You're allowing that for any predicate B without restriction, there exists a set P such that ∀x(x∈P ↔ B(x)). The problem is, you just let B be the predicate "∉P" and you've got Russell's paradox.dad2000 wrote:I once got into a heated debate with a college instructor who insisted that a set was NOT a subset of itself. By every definition I'd ever read, a set IS a subset of itself, but not a proper subset of itself. Differences like this lead to confusion and frustration for students.
BTW, I'm surprised how many people chose B -- the one choice that is false by definition and could be ruled out without even reading the question. I'm also really impressed that the kid is getting this in 3rd grade! This was 9th grade math where I grew up.
Last edited by Kulak on Mon Nov 18, 2013 1:09 pm, edited 2 times in total.
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Re: A math question ...
P.S. To those of us who think in coordinate transformation and matrices, etc. it does not feel at all right to say that a triangle is the same size if you flip it over. According to the way I would do it in a computer, to flip it over would involve a multiplication by -1 somewhere. That is to say, the matrix would have a determinant of -1.
And in calculating the area, if you go around the triangle clockwise before flipping, you ought to go around the triangle counterclockwise after flipping. That is to say, if the area of a triangle is 0.433 before flipping, there are an awful lot of situations in which it would make more sense to say that the area of a triangle is -0.433 after flipping. For example, if it were a loop of wire in a magnetic field...
And in calculating the area, if you go around the triangle clockwise before flipping, you ought to go around the triangle counterclockwise after flipping. That is to say, if the area of a triangle is 0.433 before flipping, there are an awful lot of situations in which it would make more sense to say that the area of a triangle is -0.433 after flipping. For example, if it were a loop of wire in a magnetic field...
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Re: A math question ...
Perhaps. But if they are two plots of land, adjacent but mirror images of each other, then they certainly do have the same area. No -1 involved at all.
Re: A math question ...
Which is why we probably need standards until we get well into advanced college math. With the audience of this particular class being Finite Math for business majors, I'm 100% certain that we were supposed to be grasping Naive Set Theory (where I think I would be right) as opposed to Russell's Paradox.Kulak wrote:My understanding (one of the mathematicians here can correct me) is that your instructor is right and you're wrong. You're allowing that for any predicate B without restriction, there exists a set P such that ∀x(x∈P ↔ B(x)). The problem is, you just let B be the predicate "∉P" and you've got Russell's paradox.dad2000 wrote:I once got into a heated debate with a college instructor who insisted that a set was NOT a subset of itself. By every definition I'd ever read, a set IS a subset of itself, but not a proper subset of itself. Differences like this lead to confusion and frustration for students.
Due to some bizarre graduation requirements, I had been forced to take this particular class after having completed some heavy duty math (Calculus, Linear Alg., Diff Eq, Discrete Math, etc.), and might have been the only student in the class capable of arguing this the other way.
In any event, I frequently get confused by my grade schoolers math assignments for similar reasons. Teaching methods have changed, and context and vocabulary are relevant.
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Re: A math question ...
Indeed. By the way, does anyone know why current high school geometry texts are almost fetishistic about saying "the MEASURE of the angle is 60°" instead of merely saying "the angle IS 60°?" I get it that "the angle is 60°" is not quite precision language, but what harm does it do?dad2000 wrote:Teaching methods have changed, and context and vocabulary are relevant.
What I think is a pity is that [the] high school geometry texts [I've seen] no longer use the word "axiom," and no longer describe the unprovable foundation facts as "self-evident." This may not be a loss as far as mathematics goes, but I think it is a real loss in not understanding the reference in the Declaration of Independence.
Last edited by nisiprius on Mon Nov 18, 2013 1:50 pm, edited 1 time in total.
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Re: A math question ...
Maybe I've missed something, but Russel's Paradox results from making self-referential statements.Kulak wrote:My understanding (one of the mathematicians here can correct me) is that your instructor is right and you're wrong. You're allowing that for any predicate B without restriction, there exists a set P such that ∀x(x∈P ↔ B(x)). The problem is, you just let B be the predicate "∉P" and you've got Russell's paradox.dad2000 wrote:I once got into a heated debate with a college instructor who insisted that a set was NOT a subset of itself. By every definition I'd ever read, a set IS a subset of itself, but not a proper subset of itself. Differences like this lead to confusion and frustration for students.
BTW, I'm surprised how many people chose B -- the one choice that is false by definition and could be ruled out without even reading the question. I'm also really impressed that the kid is getting this in 3rd grade! This was 9th grade math where I grew up.
(E.g. With the assumption that statements can only be true or false, consider the self-referential statement:
This statement is false.
If it is true statement, then it is false.
If it is a false statement, then it is true.
So the statement can neither be true nor false, which is the paradox.
Of course, this was before Kurt Godel's Incompleteness Theorem.
This has nothing to do with whether {1} is a subset of {1}.
In fact, if a set cannot be a subset of itself, why does the term "proper subset" exist?
Re: A math question ...
If you want my guess, the authors probably wrote it your way, and were later convinced or forced by the publisher's editors to write it the new (gramatically correct) way. I always just referred to it as a "60 degree angle".nisiprius wrote:Indeed. By the way, does anyone know why current high school geometry texts are almost fetishistic about saying "the MEASURE of the angle is 60°" instead of merely saying "the angle IS 60°?" I get it that "the angle is 60°" is not quite precision language, but what harm does it do?dad2000 wrote:Teaching methods have changed, and context and vocabulary are relevant.
Re: A math question ...
MathWizard wrote:Maybe I've missed something, but Russel's Paradox results from making self-referential statements.Kulak wrote:My understanding (one of the mathematicians here can correct me) is that your instructor is right and you're wrong. You're allowing that for any predicate B without restriction, there exists a set P such that ∀x(x∈P ↔ B(x)). The problem is, you just let B be the predicate "∉P" and you've got Russell's paradox.dad2000 wrote:I once got into a heated debate with a college instructor who insisted that a set was NOT a subset of itself. By every definition I'd ever read, a set IS a subset of itself, but not a proper subset of itself. Differences like this lead to confusion and frustration for students.
BTW, I'm surprised how many people chose B -- the one choice that is false by definition and could be ruled out without even reading the question. I'm also really impressed that the kid is getting this in 3rd grade! This was 9th grade math where I grew up.
(E.g. With the assumption that statements can only be true or false, consider the self-referential statement:
This statement is false.
If it is true statement, then it is false.
If it is a false statement, then it is true.
So the statement can neither be true nor false, which is the paradox.
Of course, this was before Kurt Godel's Incompleteness Theorem.
This has nothing to do with whether {1} is a subset of {1}.
In fact, if a set cannot be a subset of itself, why does the term "proper subset" exist?
I think the paradox shows up when you define a set S, as a set of sets that are not members of themselves. Then ask if S is a subset of (belongs to?) S.
It seems that many of us are well versed in math (as I'd expect), yet can't agree on a correct answer. How does that bode for people taking multiple choice exams?