Junior Cert - 2014 Sample Paper

ray giraffe

Regarding part (ii)

I'm a tutor and I don't know which answer to suggest to my students


Yes, it is symmetrical and it looks like a quadratic function [latex]y=ax^2+bx+c[/latex] with a negative number for [latex]a[/latex].

OR

No, it looks more like a circle than a quadratic.

OR

Something else??


What do you think?

bnt

You have a bit of graph paper, and space to draw two small charts ... I would draw quadratic and circle functions, and compare those with the picture and the data derived from it.

Ciaran

It looks vertical on the the left and right edges so I would say it can't be described by a quadratic. I don't know if that's the reasonaing that the question is looking for though.

Michael Collins

Ciaran wrote: »

It looks vertical on the the left and right edges so I would say it can't be described by a quadratic. I don't know if that's the reasonaing that the question is looking for though.

And if it really is vertical no function can describe it. (A rather intuitive application of the vertical line test!)

In this case is seems fair to limit the domain to what they provide in the chart, as in x = [-3, +3].

But...what an awful question!

The photo is also taken at a strange angle, not face on for some reason.

OK, if we accept that it is a symmetric quadratic about x = 0 (I'm not so sure about this, check the y values at x = +/-3...but it is close), then

[latex] y = -ax^2 + c [/latex]

with -a negative since the curve is concave downwards.

The max is at 3.7 or there abouts, so now

[latex] y = -ax^2 +3.7 [/latex]

So the question is do the data fit this curve? At y = 0, x = +/- 3.1, so this gives a = 0.36 or so, hence

[latex] y = -0.36x^2 + 3.7 [/latex]

Let's try this at x = 2: [latex] -0.36(2)^2 + 3.7 = 2.26 [/latex] , but it looks more like 3 on the graph. Hence it is not possible. The shape is closer to circular than parabolic.

This is not too surprising given that a quadratic can only fit 3 random points exactly, now obviously the arch isn't random in shape, but it would appear that it is definately not parabolic (quadratic).

ProjectManager

Are you sure you guys are not over thinking this??

It is a JC question and the circle wouldn't have been covered yet. The recommended time for the question is 5 minutes (for the whole question - not just part II) and the answer should be a 'yes' or 'no' and a reason to justify your choice of yes or no. - Make sure your students read the question. THats all they are asked for.

Some features of a quadratic are that there is only one turning point (max or min) and two roots (where it cuts the X axis) and the pictured shape has both of those.

So I think it is perfectly acceptable to say yes and give one or both of those reasons.

If you start trying to 'fit' it to an equation, you are going to have a problem for a number of reasons (accuracy of scale, photo taken at an angle etc).

This is a classic lesson for the students in terms of learning to answer what they are asked and not what they would like to have been asked.

Michael Collins

ProjectManager wrote: »

Are you sure you guys are not over thinking this??

It is a JC question and the circle wouldn't have been covered yet. The recommended time for the question is 5 minutes (for the whole question - not just part II) and the answer should be a 'yes' or 'no' and a reason to justify your choice of yes or no. - Make sure your students read the question. THats all they are asked for.

Some features of a quadratic are that there is only one turning point (max or min) and two roots (where it cuts the X axis) and the pictured shape has both of those.

So I think it is perfectly acceptable to say yes and give one or both of those reasons.

If you start trying to 'fit' it to an equation, you are going to have a problem for a number of reasons (accuracy of scale, photo taken at an angle etc).

This is a classic lesson for the students in terms of learning to answer what they are asked and not what they would like to have been asked.

Good point. It is Junior Cert and I suspect the answer they are looking for is what you have suggested.

But, you don't generally say things that are untrue in maths. You maybe simplify things, treat things less rigorously e.g. how limits are treated in the Leaving Cert is intuitive but not rigorous.

In this case, can you given an arbitrary shape by a quadratic? In general, with more than three points to fit, you can't.

You say it's not a fitting problem, but they ask you to write down curve co-ordinates. So if the answer is "Yeh, it looks sort of like a quadratic", they've kind of made the question too analytical already.

Also, they don't ask "is it possible to approximate", they ask "is it possible to represent", which in maths would generally mean characterise completely.

So, while I definately agree that your answer is most likely what they are after, I think the question is poorly designed to elicit that response.

Also, I feel it necessary to say that I have absolutely no axe to grind when it comes to Project Maths. I don't know enough about the overall syllabus anymore to comment on it. But I have seen a few questions about which I'd have reservations. This is one.

ZorbaTehZ

Whoever set this question needs to be fired - absolutely woeful.

pianoperson

No, it can't be quadratic as the 2nd difference (of table) isn't constant

platypus

pianoperson wrote: »

No, it can't be quadratic as the 2nd difference (of table) isn't constant

This

Being able to identify quadratic patterns is on the syllabus. If you list the points the 2nd difference is not constant therefore the arch is not described by a quadratic function. I don't see any problem with the question

ray giraffe

bnt wrote: »

You have a bit of graph paper, and space to draw two small charts ... I would draw quadratic and circle functions, and
compare those with the picture and the data derived from it.

Sounds good.

Ciaran wrote: »

It looks vertical on the the left and right edges so I would say it can't be described by a quadratic.

That was one of my first thoughts too - but that idea may be beyond Junior Cert!

Michael Collins wrote: »

And if it really is vertical no function can describe it. (A rather intuitive application of the vertical line test!)

In this case is seems fair to limit the domain to what they provide in the chart, as in x = [-3, +3].

But...what an awful question!

The photo is also taken at a strange angle, not face on for some reason.

OK, if we accept that it is a symmetric quadratic about x = 0 (I'm not so sure about this, check the y values at x = +/-3...but it is close), then

[latex] y = -ax^2 + c [/latex]

with -a negative since the curve is concave downwards.

The max is at 3.7 or there abouts, so now

[latex] y = -ax^2 +3.7 [/latex]

So the question is do the data fit this curve? At y = 0, x = +/- 3.1, so this gives a = 0.36 or so, hence

[latex] y = -0.36x^2 + 3.7 [/latex]

Let's try this at x = 2: [latex] -0.36(2)^2 + 3.7 = 2.26 [/latex] , but it looks more like 3 on the graph. Hence it is not possible. The shape is closer to circular than parabolic.

This is not too surprising given that a quadratic can only fit 3 random points exactly, now obviously the arch isn't random in shape, but it would appear that it is definately not parabolic (quadratic).

Very nice work. I agree that this question is pretty awful - maths should be concrete at Junior Cert and not so open ended!

ProjectManager wrote: »

Are you sure you guys are not over thinking this??

It is a JC question and the circle wouldn't have been covered yet. The recommended time for the question is 5 minutes (for the whole question - not just part II) and the answer should be a 'yes' or 'no' and a reason to justify your choice of yes or no. - Make sure your students read the question. THats all they are asked for.

Some features of a quadratic are that there is only one turning point (max or min) and two roots (where it cuts the X axis) and the pictured shape has both of those.

So I think it is perfectly acceptable to say yes and give one or both of those reasons.

If you start trying to 'fit' it to an equation, you are going to have a problem for a number of reasons (accuracy of scale, photo taken at an angle etc).

This is a classic lesson for the students in terms of learning to answer what they are asked and not what they would like to have been asked.

I think you make very good points. The question is supposed to be quick to answer, nothing complicated in a couple of minutes.

The issue that worries me is - what is the question trying to accomplish? Teaching students that any graph with one maximum and 2 roots can be "represented by a quadratic"??

Michael Collins wrote: »

Good point. It is Junior Cert and I suspect the answer they are looking for is what you have suggested.

But, you don't generally say things that are untrue in maths. You maybe simplify things, treat things less rigorously e.g. how limits are treated in the Leaving Cert is intuitive but not rigorous.

In this case, can you given an arbitrary shape by a quadratic? In general, with more than three points to fit, you can't.

You say it's not a fitting problem, but they ask you to write down curve co-ordinates. So if the answer is "Yeh, it looks sort of like a quadratic", they've kind of made the question too analytical already.

Also, they don't ask "is it possible to approximate", they ask "is it possible to represent", which in maths would generally mean characterise completely.

So, while I definately agree that your answer is most likely what they are after, I think the question is poorly designed to elicit that response.

Totally with you on that. "Possible to represent" is a very precise idea. A curve is either a parabola or not.
There is no natural reason that ideal bridge's arch should be a parabola.

(As an interesting aside, an ideal compression arch would be a catenary [latex] y = a\cosh \left( \frac{x}{a} \right) [/latex] )

If a student goes on to do Leaving Cert and a textbook mentions that it is possible to "represent a projectile (in a vaccuum) by a quadratic function" they should not be thinking "oh they just mean it has one maximum and two roots" - they should be able to trust that that it's exactly the same!

ZorbaTehZ wrote: »

Whoever set this question needs to be fired - absolutely woeful.

My thoughts exactly

pianoperson wrote: »

No, it can't be quadratic as the 2nd difference (of table) isn't constant

I didn't think of that! Good point!

platypus wrote: »

This

Being able to identify quadratic patterns is on the syllabus. If you list the points the 2nd difference is not constant therefore the arch is not described by a quadratic function. I don't see any problem with the question

The idea of a "quadratic sequence" certainly is on the Junior Cert syllabus. If they had spelled this out a bit more: "Are the y values making a quadratic sequence?", that would be a nice idea for a question.

But surely if you are to ask such a question at JC then the given sequence should be either exactly quadratic or way off - not "roughly quadratic" as it is.

platypus

ray giraffe wrote: »

Sounds good.



That was one of my first thoughts too - but that idea may be beyond Junior Cert!



Very nice work. I agree that this question is pretty awful - maths should be concrete at Junior Cert and not so open ended!



I think you make very good points. The question is supposed to be quick to answer, nothing complicated in a couple of minutes.

The issue that worries me is - what is the question trying to accomplish? Teaching students that any graph with one maximum and 2 roots can be "represented by a quadratic"??



Totally with you on that. "Possible to represent" is a very precise idea. A curve is either a parabola or not.
There is no natural reason that ideal bridge's arch should be a parabola.

(As an interesting aside, an ideal compression arch would be a catenary [latex] y = \cosh(x) [/latex] )

If a student goes on to do Leaving Cert and a textbook mentions that it is possible to "represent a projectile (in a vaccuum) by a quadratic function" they should not be thinking "oh they just mean it has one maximum and two roots" - they should be able to trust that that it's exactly the same!



My thoughts exactly



I didn't think of that! Good point!



The idea of a "quadratic sequence" certainly is on the Junior Cert syllabus. If they had spelled this out a bit more: "Are the y values making a quadratic sequence?", that would be a nice idea for a question.

But surely if you are to ask such a question at JC then the given sequence should be either exactly quadratic or way off - not "roughly quadratic" as it is.

The question is not "pretty awful". Identifying quadratic patterns is on the syllabus, knowing the links between tables, graphs and functions is also a key aspect of the course. The question is scaffolded also in that they are asked to fill in a table. Once they do this it's simple, if 2nd difference constant it's quadratic otherwise it's not. That's pretty concrete in my eyes

I have no idea why you think this question isn't appropriate for JCHL.

MathsManiac

I agree with platypus (and, by implication, pianoperson!) You're directed clearly towards producing a set of points that cannot be fitted with a quadratic, by virtue of the fact that the second differences are not constant (and, I would say, not close enough to constant for the variation to be due to measurement error). This technique of looking at first and second differences in patterns to see if they are linear or quadratic is one that is well covered by the syllabus and at inservice, as is the link between patterns, tables, and graphs.

I would imagine that a candidate who went further and said something like "It can be approximated by a quadratic but not very closely." would also be get the marks.

I doubt that ProjectManager's answer would be regarded as acceptable for full marks, but I think it would deserve partial credit.

Would it be possible for a "yes" answer to get full credit with a really good justification? It's a bit hard to see how, given the data you'll end up with in the table, but perhaps it could if the reason given displayed a clear recognition that any such representation can be at best only an approximation. Or maybe "Yes, because the second differences, while different, are close enough to constant that their variation might be due to inaccuracies in my measurements" would be regarded as acceptable.

The more I look at the question, the more I like it! It requires you to do a bit of work to probe whether a curve that looks to the casual eye like it could conceivably be parabolic really is.

MathsManiac

ray giraffe wrote: »

.... I agree that this question is pretty awful - maths should be concrete at Junior Cert and not so open ended!

But why? JCHL students are capable of thinking. And while maths in the maths world is clean and precise, applying maths in the real world is frequently messy. Problems that you can apply maths to are often open-ended in reality.

ray giraffe wrote: »

The idea of a "quadratic sequence" certainly is on the Junior Cert syllabus. If they had spelled this out a bit more: "Are the y values making a quadratic sequence?", that would be a nice idea for a question.

Sure that'd be giving the game away entirely! Maybe at Ordinary level you could do that.

ray giraffe wrote: »

But surely if you are to ask such a question at JC then the given sequence should be either exactly quadratic or way off - not "roughly quadratic" as it is.

And that'd be no fun either! When applying maths in real life, you often need to make a call as to whether a model is a good enough fit for your purposes or not. Why not start getting used to it early?

While I appreciate the occasional frustrations of not being quite sure how much depth of thought is needed in some questions, I prefer to take these questions as a great opportunity for good discussion with students. (Oops - I should say "preferred", as I'm not teaching any more.) From other posts you've made here, you're clearly well on top of your game, and I reckon you're the kind of person who could (and does) capitalise on such opportunities.

By the way, do you ever find yourself looking at a bridge and idly wondering if it's a catenary, or a cycloid, or just plain old elliptical, or none of the above?

ray giraffe

Oops it appears someone on the internet was wrong, and it was me

I am a bit 'behind the curve' on this new syllabus and didn't realise that applying the 'second difference' to graphs is standard material.

Pianoperson, Platypus and MathsManiac, you would be excellent people to write the solutions for the educational company papers - the current one is way off the mark:

Michael Collins

ray giraffe wrote: »

Oops it appears someone on the internet was wrong, and it was me

I am a bit 'behind the curve' on this new syllabus and didn't realise that applying the 'second difference' to graphs is standard material.

Pianoperson, Platypus and MathsManiac, you would be excellent people to write the solutions for the educational company papers - the current one is way off the mark:

I'm also unfamiliar with the new syllabus and if the second differences of a quadratic is on the syllabus then the question becomes a lot better. Maybe even good! And I'll eat umble pie too!

I did do a search before posting and I couldn't see anything related to second differences or anything that might help answer this question, in either the general 2014 syllabus, nor in the 2014 syllabus for the initial 24 pilot schools.

There was a section on recognising that the rate of change of a quadratic function varies, but I couldn't see anything on the second derivative being constant.

Doesn't mean it isn't in there somewhere though!

gammy_knees

− use tables, diagrams and graphs as tools for

representing and analysing linear, quadratic

and exponential patterns and relations

(exponential relations limited to doubling

and tripling)

I lifted this from section 4.2 of the JC syllabus. This is the relevant reference. The question posted also falls into the section dealing with patterns in numbers.

MathsManiac

Now that I see the syllabus extract again, courtesy of gammy_knees, I'll concede that perhaps the business of first and second differences isn't as obvious in the text of it as I had recalled.

However, if you were to "use tables...as tools for...analysing...quadratic patterns", in any reasonably meaningful way at this level, I think you'd be doing that or something equivalent. At any rate, the resource materials on the PM website certainly deal with this learning outcome in this way a fair bit.

platypus

As well as the section gammy knees posted there is a line in section 4.4 which states as students should be able to

recognise that a distinguishing feature of quadratic relations is the way change varies

According to syllabus this and the section gammy knees posted is also required at OL

Mwalimu

Adding to what MathsManiac, platypus and gammy_knees have said, the syllabus also says (in 4.3, for HL only) that students should be able to:

- find the underlying formula algebraically from which the data is derived (linear,
quadratic relations)
To figure out what the formula is for a quadratic, the second differences give a very important clue. What does the (constant) second difference value reveal about the coefficient of x^2 in original quadratic?;)