Why Are There Two Possible Values of x in Similar Shapes Geometry Problems?

[Steve4Physics]

I think the best approach for a student at this level is to draw two diagrams corresponding to the two possible solutions.

One diagram has BE and CD in parallel; the other doesn't . The OP needs to draw both diagrams (roughly to scale). If they can't draw them, they can't understand and answer the question properly. (That's why I gave a big hint in Post #16.)

Minor edits.

 

[DaveC426913]

I stand by malicious.

OK. I disagree, and I'll explain why.

Malice, at its core, involves intent to harm.

Drawing a diagram that is not representative of the final solution is an important tool to teach the student to be aware of exactly the kind of pitfall we are all witnessing in this thread. Geometry is not just about the student getting the answer by rote; it is also about the student understanding what is being asked.

If the opening question had been drawn as you expect, the student would have learned nothing about real-world problems that are not always obvious. They would go out into the world seeing two lines that look to their naive eye to be parallel, but in fact are not. They would have not learned rigour in geometry. And geometry is all about rigour.

The student must learn to see when such things as parallelism are not true - and notably, that similar triangles need not be un-mirrored images of each other. They're not going to learn that if every question's mathematically rigorous answer can be short-circuited by enabling eyeball estimation.

It's not malicious - not harm - it is, instead, a critical part of the lesson.

The OP's struggle is a perfect example. Because of this particular question, they will never make this mistake again.

 

[Mark44]

OK. I disagree, and I'll explain why.

Malice, at its core, involves intent to harm.

Or, in the case of the drawing in this thread, the harm arises by the effort of the problem poser to show how clever he is, and conversely, how stupid the poor student who is the target.

Drawing a diagram that is not representative of the final solution is an important tool to teach the student to be aware of exactly the kind of pitfall we are all witnessing in this thread. Geometry is not just about the student getting the answer by rote; it is also about the student understanding what is being asked.

Baloney. It would be one thing if the inaccuracy of the drawing were inadvertent. But as pointed out by @pbuk in post #26, the drawing is wildly inaccurate, which is not an accident.

 

[renormalize]

Baloney. It would be one thing if the inaccuracy of the drawing were inadvertent. But as pointed out by @pbuk in post #26, the drawing is wildly inaccurate, which is not an accident.

Would you find the problem more palatable if its author had added an explicit warning such as:
"Note: illustration is not necessarily drawn to scale."?

 

[DaveC426913]

Or, in the case of the drawing in this thread, the harm arises by the effort of the problem poser to show how clever he is, and conversely, how stupid the poor student who is the target.

You are second-guessing the intent of the poser. Why look for malice when there is a perfectly valid educational basis for the set up of the problem?

How does a student learn rigour of they can just eyeball the answer? How would they exercise their analytical muscle if the work is done for them?

In fact: why did we bother letting this thread get 34 posts in, if not to encourage the OP to think it through himself. Isn't that equally "malicious"? Aren't we all guilty of that which what you accuse the problem poser?

Can we accuse you of showing you how clever you are, and how stupid the OP is - when you didn't tell them the twist in the problem right off the bat? (Or did you, too, have your reasons for not spoon-feeding the OP?)

Baloney. It would be one thing if the inaccuracy of the drawing were inadvertent. But as pointed out by @pbuk in post #26, the drawing is wildly inaccurate, which is not an accident.

Nobody ever suggested it was an accident. It is a deliberate, tried-and-true technique to teach the student a critical geometry analysis skill.

Learning is more than rote number-crunching; learning starts with problem-solving: what is being asked. In geometry, more than many other subjects, that is a matter of deducing and what is given, and - notably - what is not given.

 

[Mark44]

Would you find the problem more palatable if its author had added an explicit warning such as:
"Note: illustration is not necessarily drawn to scale."?

Yes, I would not object so strenuously.

You are second-guessing the intent of the poser. Why look for malice when there is a perfectly valid educational basis for the set up of the problem?

How about sadism?

Nobody ever suggested it was an accident. It is a deliberate, tried-and-true technique to teach the student a critical geometry analysis skill.

I agree that it probably was not accidental, but posing a problem that is deliberately misleading is NOT a tried-and-true pedagogical technique.

Learning is more than rote number-crunching; learning starts with problem-solving: what is being asked. In geometry, more than many other subjects, that is a matter of deducing and what is given, and - notably - what is not given.

I maintain that what was given was deliberately misleading. As I said before, the intent appears to be to convince the student how clever the teacher is, and less so about problem-solving techniques.

@DaveC426913 I taught mathematics at the college level for about 25 years. I don't recall that any of my peers or myself posing any problems as willfully misleading as this one. What's your experience?

 

[fresh_42]

I think it is important to teach not making hidden assumptions, and being led to false conclusions only because pictures are easier to read than text. This is not only important in mathematics, but also important for any kind of investigation. Maybe a small hint like "e.g." at the picture label would have been helpful, but that is a matter of taste, and of how easy an exercise is intended to be.

 

[TensorCalculus]

Honestly, it is good that the OP gets exposed to this sort of question in homework. Whether questions like this are a good thing or not (though I think I find myself leaning more towards @DaveC426913 and @fresh_42 's side - even if the question is written to try and show how smart the question writer is, it still forces the person doing the question to not rely on the diagram and make assumptions, which is a valuable lesson) they will almost certainly come up in formal exams (GCSE, A-Level, etc) so practicing them, and struggling on them, means that they have a better chance of not making similar assumptions on the actual exam, where whether they get the question right or wrong does actually matter.

That being said, technically the diagram is correct for the first possible value of x (x=2), just not the second, less obvious one (which has probably been done on purpose). To the OP (I hope I am not giving too much away with this): we have already established the maybe BE and CD are not parallel. What happens when we get rid of this assumption? Can you draw a diagram where this assumption is not the case? If BE and CD aren't parallel, then $\angle ABE \not = \angle ACD$ and the same can be said for $\angle ABE$ and $\angle ADC$
but for triangles to be similar, then they must have 3 equal angles. How can BE and CD be parallel, yet the triangles still be similar? There is one other possibility...

 

[DaveC426913]

I taught mathematics at the college level for about 25 years.

I was afraid of that ...

 

[hutchphd]

Malicious is an unusual choice of words. It's deliberate, but it's for a purpose.

And that purpose seems to me to be malicious.
Is this a restatement of the parental admonition to "do as I say, not as I do": "do as I say not as I draw?"
I am not seeing this as interesting........am I still not "getting it"?

 

[pbuk]

[questions like this] will almost certainly come up in formal exams (GCSE, A-Level, etc)

Edit: I have just realised that this is Q.22 from Edexcel Maths Paper 1 November 2017. I'm off to read the examiner's report.

They almost certainly will not. GCSE and A-Level exams have been developed over many years to test the ability to apply knowledge, not the ability to spot trick questions. Any (UK) exam board that let a question like this slip through the net of pre-testing would be heavily criticised within the profession.

 

[pbuk]

Nobody ever suggested it was an accident. It is a deliberate, tried-and-true technique to teach the student a critical geometry analysis skill.

No, far from being a 'tried and true technique', undermining a pupil's confidence by setting trick questions is bad practice.

From the OP's posts in this thread we can see that this poor question has taught nothing, and has destroyed any confidence they had in their knowledge of similar triangles:

I might have to revisit the similar shape units of Maths Genie GCSE Revision...

I'll give this a go after I've revisited the very basics; I fear I've got myself in a right muddle :)

Hang on in there @paulb203, the whole world is not out to get you! I'd advise you to use a different source for learning reinforcement questions in future though (where did you get this shocker from?)

 

[pbuk]

Edit: I have just realised that this is Q.22 from Edexcel Maths Paper 1 November 2017. I'm off to read the examiner's report.

They almost certainly will not. GCSE and A-Level exams have been developed over many years to test the ability to apply knowledge, not the ability to spot trick questions. Any (UK) exam board that let a question like this slip through the net of pre-testing would be heavily criticised within the profession.

So here is the examiners' report for this question:

"It was good to see a number of students having some understanding of similar shapes and awarding 2 marks for a value of x =2 correctly found wasn’t uncommon. Almost no students were then able to deduce the second assumption that could be made and as such no further marks were awarded."​

Doesn't look to me like anything to be proud of.

I cannot find anything similar (please excuse the pun) in more recent papers.

 

[DaveC426913]

No, far from being a 'tried and true technique', undermining a pupil's confidence by setting trick questions is bad practice.

It's not a "trick question". Students are taught to not trust diagrams.

An angle that looks about 90 degrees - is not 90 degrees unless it is labeled so.
Two lines that look sortta parallel - are not parallel unless they are labeled so.

This is how one teaches geometry.

Students are taught that similar triangles can be mirrors of each other.

The OP didn't get it because he has not yet learned the lesson that this particular problem is designed to challenge him with.

From the OP's posts in this thread we can see that this poor question has taught nothing, and has destroyed any confidence they had in their knowledge of similar triangles:

We should spoon feed them only answers that don't challenge them?

Would you be comfortable driving on a bridge made by a student who was fed only the easy answers?

Would you drive on a bridge built by a student that did not know similar triangles can be mirrors of each other?

 

[DaveC426913]

So here is the examiners' report for this question:

"It was good to see a number of students having some understanding of similar shapes and awarding 2 marks for a value of x =2 correctly found wasn’t uncommon. Almost no students were then able to deduce the second assumption that could be made and as such no further marks were awarded."​

Doesn't look to me like anything to be proud of.

All the more reason why this kind of question needs to be taught.

Or:

"We found that understanding stress fractures in airplane structures was too hard, and many students were unable to deduce the correct answer. In light of that, we have decided to simply not give them hard problems. We will just hand out a cheat sheet of all the formulae they'll need for their aviation engineering degrees and give every single one of them a degree."

 

[DaveC426913]

posing a problem that is deliberately misleading is NOT a tried-and-true pedagogical technique.

How exactly is the problem "deliberately misleading"?

Are you asserting that

- that in a given geometric diagram, two lines that look pretty parallel can be assumed to be parallel?

- that, in this diagram, I can assume AD and BC are parallel simply because they look parallel?

- that this diagram is "maliciously misleading"?

 

[mathwonk]

I am also a retired college math teacher (over 40 years teaching). So I am on the side of Mark44 in this matter. I do not disagree with anything anyone else has said. I just know from decades of experience that it is hard enough to teach even the most basic facts, that one does not ever have time to waste tricking 98% of people with questions like this.

When ever you pose a test question, you should ask yourself what it is testing. Then ask yourself whether it furthers your announced classroom goals to test that property. And ask yourself also if it is consistent with your grading policy whether getting it right should count towards a grade.

However I also know that it is useless to discuss teaching philosophy in the abstract, as in this thread, as no one ever agrees. People only grasp what one intends by his teaching philosophy when they see him carry it out in class. But I could not leave Mark out on a limb here, given my actual opinion, as well as my respect for those who disagree.

yes one does want to encourage caution in students who ignore unstated assumptions. It would be easy to do that here in a more helpful way I think. One could simply ask if there is a second interpretation of the question that yields a second answer, for extra credit, and remind students that it is necessary to specify the actual correspondence of vertices when invoking a similarity.

The goal is for most of the students to actually learn something, not for the question to be phrased so as to fool almost all of them.
I believe negative reinforcement is not actually the most highly recommended method of effective learning.

Apologies if I have misinterpreted the many cogent arguments that have been suggested. I admit that, even after 40 years, I myself never mastered the art of teaching.

peace.

 

[Mark44]

How exactly is the problem "deliberately misleading"?

You're kidding, right?
Here's from post #26, as to why the figure in the problem is deliberately misleading.

The first thing to notice is that the line segment DE that is 3 cm long is drawn almost the same length as AD which is 12 cm: the diagram has been deliberately drawn to mislead you.

It's not a "trick question".

See @pbuk's opinion on this (copied above) as well as @mathwonk's (copied below).

The goal is for most of the students to actually learn something, not for the question to be phrased so as to fool almost all of them.
I believe negative reinforcement is not actually the most highly recommended method of effective learning.

Students are taught to not trust diagrams.

???
Are they also taught not to trust definitions and theorems? This seems ridiculous.

 

[A.T.]

... the harm arises by the effort of the problem poser to show how clever he is,...

Ironically, it can also suggest that the problem poser wasn't clever enough to come up with a problem that is interesting by itself, and instead had to resort to stuffing a boring problem with misleading queues.

As for the pedagogical value, it makes perfect sense and does no harm to give students this as exercise that is not graded, to teach them to watch out for misleading pictures. And that includes not fooling yourself with your own exemplary diagrams that you draw when solving a problem.

But I would not put that into a graded test.

See also:

 

[fresh_42]

The exercise mentions explicitly two possible values of x. What more hint do you need to think about your hidden assumptions? It is literally an invitation to reflect on the situation and re-read the problem statement.

A criminal investigator won't be given such hints in his daily work. Where should he learn not to trust the obvious? The real world is complex, and we far too often fall for the simplest explanation, disregarding its complexity. Calling this question malicious, despite it actually stating that there is more than one solution, is ridiculous. It calls for a reflection of the problem statement, and that seems to be the goal to be learnt, not the interception theorems.

 

[DaveC426913]

How exactly is the problem "deliberately misleading"?

Are you asserting that

- that in a given geometric diagram, two lines that look pretty parallel can be assumed to be parallel?

- that, in this diagram, I can assume AD and BC are parallel simply because they look parallel?
View attachment 363325
- that this diagram is "maliciously misleading"?

I just know from decades of experience that it is hard enough to teach even the most basic facts, that one does not ever have time to waste tricking 98% of people with questions like this.

So, in this diagram, if AD and BC are not parallel, you would consider this "tricking" people, yes?

 

[DaveC426913]

You're kidding, right?
Here's from post #26, as to why the figure in the problem is deliberately misleading.

And I am taking the gist of your "deliberately misleading" criteria and applying to another case:

By your logic, it seems to me, this digram is misleading, because AD and BC appear parallel, though it will turn out they are not.

Are they also taught not to trust definitions and theorems? This seems ridiculous.

I am speechless. That is a straight up straw man. And an outrageous one at that.


An angle in a diagram is not 90 degrees unless it is labeled (or proven) 90 degrees.
Two lines in a diagram are not parallel unless they are labeled (or proven) parallel.

Agree or disagree?


BE and CD cannot be assumed to be parallel in this diagram. Agree or disagree?

 

[TensorCalculus]

Edit: I have just realised that this is Q.22 from Edexcel Maths Paper 1 November 2017. I'm off to read the examiner's report.

They almost certainly will not. GCSE and A-Level exams have been developed over many years to test the ability to apply knowledge, not the ability to spot trick questions. Any (UK) exam board that let a question like this slip through the net of pre-testing would be heavily criticised within the profession.

Yeah, they do come up... Our school gives up problems like these often to practice for maths GCSE, and I have seen problems like this on the A level as well. More so for things like Olympiad/maths challenge papers, from UKMT, but also in GCSE. Or maybe our school is just being careful and therefore trying to expose us to everything, who knows. But I have heard from older students a lot about deceptive A level maths questions.

As for the pedagogical value, it makes perfect sense and does no harm to give students this as exercise that is not graded, to teach them to watch out for misleading pictures. And that includes not fooling yourself with your own exemplary diagrams that you draw when solving a problem.

But I would not put that into a graded test.

My opinion exactly.


I agree that the question is deliberately misleading, but that doesn't mean that it's malicious. Plus, in the case where x = 2, the two lines are parallel, and the diagram is correct (at least, when worrying about which lines look parallel and which don't). If the lines clearly weren't parallel, then you could argue that it is misleading for the case where x = 2. (But I do understand that x = 2 is the more obvious solution and therefore a diagram where the lines weren't parallel might have been a better option)

 

[Mark44]

@DaveC426913 you must be very fond of that quadrilateral drawing, as you've posted it three times. I assume it's a quadrilateral inasmuch as I can plainly see four sides. Or maybe that's just an assumption on my part?

For the problem from post #1, @pbuk explained why the figure is deliberately misleading; namely, that the two segments of side AD -- AE and ED -- are drawn roughly the same length but labeled with one of them being four times as long.

 

[fresh_42]

Yeah, they do come up... Our school gives up problems like these often to practice for maths GCSE, and I have seen problems like this on the A level as well. More so for things like Olympiad/maths challenge papers, from UKMT, but also in GCSE. Or maybe our school is just being careful and therefore trying to expose us to everything, who knows. But I have heard from older students a lot about deceptive A level maths questions.

I remember several anecdotes about such kinds of questions we used to tell. At least, the mathematical one of the following did happen since I knew the person who told it:

1. Physics: This flower pot at the window is warm on its room side and cold on the side facing the window. Why? (Answer: The professor turned it around before the exam.)

2. Physics: Explain the function of fuses. (The student allegedly got away with his answer that he didn't know because his mother had always forbidden him to play with them.)

3. Overheard in an elevator: "The idiot actually asked me how many times one can differentiate the constant function." complaining that once obviously wasn't the correct answer.

Admittedly, these were oral exams, but they show that unusual questions and the invitation to think outside the box really do happen.

 

[hutchphd]

It's not a "trick question". Students are taught to not trust diagrams.

My problem with this pedagogy is that my first admonition as a teacher was to "draw a rough diagram". This was engendered by my getting answers, reported to seven sig figures that were off by 8 orders of magnitude. Not good engineering.
Supplying drawings that are deliberately skewed makes a mockery of this important advice.
I prefer a good rough drawing to an 8 orders of magnitude ****-up every time. Engineers (precomputer) used to regularly do calculations on graph paper to scale. They put in a little more margin of course.
But bad pedagogy.

 

[TensorCalculus]

You will not believe the amount of times children in my year have gone into exams, not known how to mathematically solve the problem, assume the diagram is to scale and make simplifying assumptions (e.g. oh, that looks like it is half the length let me assume it is half the length) and then gotten the question right without actually applying the maths that they were supposed to in order to get the correct answer. Misleading diagrams puts a stop to this.

 

[weirdoguy]

It is a deliberate, tried-and-true technique to teach the student a critical geometry analysis skill.

@DaveC426913 are you a teacher doing that in your practice? You know, some techniques look perfect on paper, but practically they suck completely.

Besides, I have only 17 years of teaching math (and physics) under my belt as tutor/private teacher (yes, I started when I was 17), but I'm with @Mark44 on this.

Also, as a tutor I very often work with materials that school/college teachers provided, and sometimes there is pure malice on their part. There were situations where students wrote to some "higher power" about that and I was asked to be a 'reviewer'. E.g. there were two teachers (one in high-school, the other one at my own uni, not related) who did not give maximum amount of points for a perfectly good answer, because it was not clever enough! Of course they had those very clever way of reasoning prepared, but please... These people should not teach. Fortunately "higher powers" did their job, and in both cases students got their points back.

 

[DaveC426913]

I am not actually sure what you are saying here.

My problem with this pedagogy is that my first admonition as a teacher was to "draw a rough diagram".

You're telling your students to draw a rough diagram. (Not sure how that's a problem.)

This was engendered by my getting answers, reported to seven sig figures that were off by 8 orders of magnitude. Not good engineering.

You're saying, the reason you've adopted this policy is because you were getting 7 sig dig answers from students.

Supplying drawings that are deliberately skewed makes a mockery of this important advice.

I don't follow how. The drawings are a "rough diagram", similar to what you described above.

I prefer a good rough drawing to an 8 orders of magnitude ****-up every time.

OK, good. A rough drawing cannot be used to spot 90 degree angles or parallel lines.

If two lines are supposed to be parallely in a rough drawing, they will have to be indicated with '> >', right?

So, no '> >' means not parallel.

Engineers (precomputer) used to regularly do calculations on graph paper to scale. They put in a little more margin of course.
But bad pedagogy.

Right. This is bad.

Geometry is not about accurate drawings and measurements; it is about rough diagrams, properly labeled, and never making assumptions based on a diagram.

The way I read you, we are in complete agreement.

 

[DaveC426913]

You will not believe the amount of times children in my year have gone into exams, not known how to mathematically solve the problem, assume the diagram is to scale and make simplifying assumptions (e.g. oh, that looks like it is half the length let me assume it is half the length) and then gotten the question right without actually applying the maths that they were supposed to in order to get the correct answer. Misleading diagrams puts a stop to this.

Thank you. I would have thought this was plainly apparent to all.