IA Type I -- Stellar Number

[MaggieD]

use matrices to solve it. its really easy.

[CristinaV]

Sweet. All these posts were incredibly helpful when I tried finding the triangular numbers' patter mathematically. But I was just wondering if anybody knows of a program that I can use to draw the stellar shapes.

I know there's http://luckytoilet.wordpress.com/2010/06/30/investigations-about-star-polygons-inspired-by-someone-elses-math-homework/

But apparently we can't download anything from there anymore since December 15... bummer.

So yea I wouldn't mind using http://sites.google.com/site/ibstellarnumbers/home for the images, it's just that I'm concerned about the Use of Technology criterion. Would it be of any use if we drew a graph or something like that?

Thanks :-)

[dessskris]

Sweet. All these posts were incredibly helpful when I tried finding the triangular numbers' patter mathematically. But I was just wondering if anybody knows of a program that I can use to draw the stellar shapes.

I know there's http://luckytoilet.wordpress.com/2010/06/30/investigations-about-star-polygons-inspired-by-someone-elses-math-homework/

But apparently we can't download anything from there anymore since December 15... bummer.

So yea I wouldn't mind using http://sites.google.com/site/ibstellarnumbers/home for the images, it's just that I'm concerned about the Use of Technology criterion. Would it be of any use if we drew a graph or something like that?

Thanks :-)

That's what we are here for don't forget to rep the useful posts to thank the posters

And to answer your question,

IA criteria and additional notes

Criterion E: use of technology

Achievement level 3: The student makes full and resourceful use of a calculator or computer in a manner that significantly enhances the development of the task.

The emphasis in this criterion is on the contribution of the technology to the mathematical development of the task rather than to the presentation or communication.

...

Using a computer and/or a GDC to generate graphs or tables may not significantly contribute to the development of the task, and therefore may not merit more than level 1.

(Mathematics HL IA criteria and additional notes Nov 2006; page 7)

^ I think we can assume this is the same for SL. If it's the same, that means it is okay for you to 'steal' the stellar shapes. Well you don't steal (as long as you acknowledge the source) but whatever..

And the drawing is not a part of the task anyway, right? It does not help you in the development of the task. So it should be ok

But if you still want to draw it yourself... I don't know any software which can automatically draw stellar shapes, but you can draw them manually by putting points and connecting them together (I swear this would require hell lots of effort btw). You may want to try TI-Nspire Student Software or Autograph.

PS. I am so sorry I don't have the SL syllabus nor IA criteria hehe I even got this file from my Math HL teacher or from this site, I think.

[ashlaynicolexd]

actually it is:

Sn= Sn-1 + [(12)(n-1)]

and you want to try other stellar sequences

if p=4,

S₁= 1 S₄= 49

(+8) (+32)

S₂= 9 S₅= 81

(+16) (+40)

S₃= 25 S₆= 121

(+24)

S₇= S₆ + [(8)(7-1)]

S₇= S₆ + [(8)(6)]

S₇= 121 + 48= 169

[dessskris]

actually it is:

Sn= Sn-1 + [(12)(n-1)]

I am not doing this task but I think you cannot express Sn in terms of S(n-1)

you are supposed to express Sn in terms of n. and doesn't the above expression only work when p=6?or what? I cant remember

[jsand]

I decided to go the route of using quadratic equations, and it seemed to be going well with the 6-point star; however, the answer I keep getting for the general statement is when p=6, Sn=n^2 - 6n + 1 which is a wrong answer.

I noticed for the difference between the dot values of the 6-point star was a growing multiple of 12, specifically 12(n-1). when d=12. I'm not really sure how that's going to really help me thus far

anyways, what I was trying to do was to solve for a, b, and c for the 3 equations I got from the values of Sn. a + b + c = 1; 4a + 2b + c = 13; and 9a + 3b + c = 37. My logic was that solving for a, b, and c should give me the terms for my general quadratic statement.

It's all turning out wrong though, what should I do? I'm stumped.

[CristinaV]

IA criteria and additional notes

Criterion E: use of technology

Achievement level 3: The student makes full and resourceful use of a calculator or computer in a manner that significantly enhances the development of the task.

The emphasis in this criterion is on the contribution of the technology to the mathematical development of the task rather than to the presentation or communication.

...

Using a computer and/or a GDC to generate graphs or tables may not significantly contribute to the development of the task, and therefore may not merit more than level 1.

(Mathematics HL IA criteria and additional notes Nov 2006; page 7)

^ I think we can assume this is the same for SL. If it's the same, that means it is okay for you to 'steal' the stellar shapes. Well you don't steal (as long as you acknowledge the source) but whatever..

And the drawing is not a part of the task anyway, right? It does not help you in the development of the task. So it should be ok

But if you still want to draw it yourself... I don't know any software which can automatically draw stellar shapes, but you can draw them manually by putting points and connecting them together (I swear this would require hell lots of effort btw). You may want to try TI-Nspire Student Software or Autograph.

PS. I am so sorry I don't have the SL syllabus nor IA criteria hehe I even got this file from my Math HL teacher or from this site, I think.

Yes, for use of technology it is the same criteria as in SL. But I simply don't see how we can use technology to solve this IA. I thought we could just use a program to draw the stellar figures and that that would count as part of use of technology, but apparently that would not help in the development of the task, only on the presentation... and the criteria clearly says that that doesn't count. Only something that helps you solve the IA would be considered as appropriate use of technology.

So does anybody want to clarify what "technology" we can use that will meet the criteria? Because I seriously have no idea.

[dessskris]

I decided to go the route of using quadratic equations, and it seemed to be going well with the 6-point star; however, the answer I keep getting for the general statement is when p=6, Sn=n^2 - 6n + 1 which is a wrong answer.

I noticed for the difference between the dot values of the 6-point star was a growing multiple of 12, specifically 12(n-1). when d=12. I'm not really sure how that's going to really help me thus far

anyways, what I was trying to do was to solve for a, b, and c for the 3 equations I got from the values of Sn. a + b + c = 1; 4a + 2b + c = 13; and 9a + 3b + c = 37. My logic was that solving for a, b, and c should give me the terms for my general quadratic statement.

It's all turning out wrong though, what should I do? I'm stumped.

Hey I am not doing this task but I guess I could just help...perhaps?

I decided to go the route of using quadratic equations, and it seemed to be going well with the 6-point star; however, the answer I keep getting for the general statement is when p=6, Sn=n^2 - 6n + 1 which is a wrong answer.

Did you mean using quadratic eqn to get the general statement (GS)? Well, what did you do to get it? I actually found the GS and it is very similar to yours. Hint: you are missing a coefficient

I noticed for the difference between the dot values of the 6-point star was a growing multiple of 12, specifically 12(n-1). when d=12. I'm not really sure how that's going to really help me thus far

Good that you notice that! Haha I didn't notice

So you know that:

S1=1

S2=1+12

S3=1+12+2x12

S4=1+12+2x12+3x12

Let U1=0

Let U2=12

Let U3=2x12

Let U4=3x12

And Sn=1+U1+U2+...+U(n-1)+Un

Can you find Sn in terms of n, then?

anyways, what I was trying to do was to solve for a, b, and c for the 3 equations I got from the values of Sn. a + b + c = 1; 4a + 2b + c = 13; and 9a + 3b + c = 37. My logic was that solving for a, b, and c should give me the terms for my general quadratic statement.

How did you get those equations? yeah you are correct, solving for a, b and c gives you the coefficients for your GS.

I don't know if you learn this in SL but you can solve it using matrices, do RREF. If you do learn it in SL, use your GDC to do RREF and you'll get the answer. Otherwise, solve it using other methods.

[dessskris]

Yes, for use of technology it is the same criteria as in SL. But I simply don't see how we can use technology to solve this IA. I thought we could just use a program to draw the stellar figures and that that would count as part of use of technology, but apparently that would not help in the development of the task, only on the presentation... and the criteria clearly says that that doesn't count. Only something that helps you solve the IA would be considered as appropriate use of technology.

So does anybody want to clarify what "technology" we can use that will meet the criteria? Because I seriously have no idea.

Again, I am not doing this task, so I am not so sure on which part you can use calculator in. But from the questions that I see, you can get the use of technology mark by:

- drawing an appropriate table in Ms. Word or Ms. Excel for this part: "Organize the data so that you can recognize and describe any patterns"

- using your GDC or graphing software to draw stuffs where necessary

- using your calculator to solve matrices (if you use matrices to find the general statement)

- drawing an appropriate graph of Sn against n when p=6 or p=7 if you do a graphical method to find the general statement

[CristinaV]

Yes, for use of technology it is the same criteria as in SL. But I simply don't see how we can use technology to solve this IA. I thought we could just use a program to draw the stellar figures and that that would count as part of use of technology, but apparently that would not help in the development of the task, only on the presentation... and the criteria clearly says that that doesn't count. Only something that helps you solve the IA would be considered as appropriate use of technology.

So does anybody want to clarify what "technology" we can use that will meet the criteria? Because I seriously have no idea.

Again, I am not doing this task, so I am not so sure on which part you can use calculator in. But from the questions that I see, you can get the use of technology mark by:

- drawing an appropriate table in Ms. Word or Ms. Excel for this part: "Organize the data so that you can recognize and describe any patterns"

- using your GDC or graphing software to draw stuffs where necessary

- using your calculator to solve matrices (if you use matrices to find the general statement)

- drawing an appropriate graph of Sn against n when p=6 or p=7 if you do a graphical method to find the general statement

Hmm ok, I've more than solved mine, but I pretty much just used the organizing data to recognize patterns part. Do you think we should show all the other methods to get more points? As in, show how we solved it with a graph, with matrices, with quadratic equations etc.?

[dessskris]

Hmm ok, I've more than solved mine, but I pretty much just used the organizing data to recognize patterns part. Do you think we should show all the other methods to get more points? As in, show how we solved it with a graph, with matrices, with quadratic equations etc.?

Nah one method will do. But you can do that to Test the validity of the general statement, maybe? It's an SL IA anyway, you don't need to prove your general statements Lucky you! I envy you Math SL students mwahaha. And they may not like it if you write too much of redundant things btw

[fhaheen]

I just recently finished my type 1 and it was also stellar number. I used mathematical induction was a useful tool in proving Pn^2+Pn+1 and i'd recommend it to gain high grades since mathematical induction is not in the SL syllabus but using it will get you more marks. (As told by my maths teacher ofcourse)

[ashlaynicolexd]

actually it is:

Sn= Sn-1 + [(12)(n-1)]

I am not doing this task but I think you cannot express Sn in terms of S(n-1)

you are supposed to express Sn in terms of n. and doesn't the above expression only work when p=6?or what? I cant remember

well actually it should be in the U of n format

U of n= U of n-1 + [ (2 p) (n -1)]

p being like 6 in the 6 stellar numbers

example. with 6 stellar numbers

U of 2 = U of 1 + [ 2(6) (1) ]

U of 2 = 1 + 12= 13

and it works with all the other ones as well

[dessskris]

actually it is:

Sn= Sn-1 + [(12)(n-1)]

I am not doing this task but I think you cannot express Sn in terms of S(n-1)

you are supposed to express Sn in terms of n. and doesn't the above expression only work when p=6?or what? I cant remember

well actually it should be in the U of n format

U of n= U of n-1 + [ (2 p) (n -1)]

p being like 6 in the 6 stellar numbers

example. with 6 stellar numbers

U of 2 = U of 1 + [ 2(6) (1) ]

U of 2 = 1 + 12= 13

and it works with all the other ones as well

Well, yes it is correct. But if you want to find U of 5, then you need to find U of 4 first! How do we find U of 4, find U of 3 first! How do we find U of 3, find U of 2 first! And so on and so forth.. So I would actually suggest finding Sn in terms of n, instead of Sn in terms of S(n-1).

[jjjjoseph]

I need help on this too. What does the part "Hence, produce a general statement, in terms of p and n, that generates the sequence of p-stellar numbers for any value of p at Sn

my first math portfolio is due in three days. we were given the one about the stellar numbers. I am about halfway through, but i am stuck on finding a formula for the stellar numbers. for the 6-stellar number, i found the pattern is you add the previous multiple of 12 to the previous stellar number. but how do you put this in an equation?

example:

table:

1 1

2 13

3 37

to find 37, you go 2(12)+ 13.

The formula I got was -------

Hate to troll here, but that is just rude. Most of us have tried, toiled and worked on this for long enough that it's alright for us to just have the formula; we are obligated to explain how we obtained it at the end, so we have to have some conception of the math behind it or the teacher will dock plenty of points. It is the teacher's job, not yours, to keep people accountable for learning correctly; moreover, "correct" learning is just "institutional" learning, and I find IB to be useful, but also to be the farthest extension of passion-sapping institutional BS. I did it to get into a great college, and I hope to never, ever, have to study anything related to mathematics again; unfortunately, I am must learn this **** to graduate. It won't do me a damn bit of good later on—I think everyone but the engineers acknowledges that—but we still have to toil through it, scraping for the A's, in pursuit of the best colleges possible. Is it really that much of a sin to give those of us who try the answer? (And understand I am replying to the admin not you lol, thanks for the help)

Edited January 5, 2011 by jjjjoseph

[dessskris]

Hate to troll here, but that is just rude. Most of us have tried, toiled and worked on this for long enough that it's alright for us to just have the formula; we are obligated to explain how we obtained it at the end, so we have to have some conception of the math behind it or the teacher will dock plenty of points. It is the teacher's job, not yours, to keep people accountable for learning correctly; moreover, "correct" learning is just "institutional" learning, and I find IB to be useful, but also to be the farthest extension of passion-sapping institutional BS. I did it to get into a great college, and I hope to never, ever, have to study anything related to mathematics again; unfortunately, I am must learn this **** to graduate. It won't do me a damn bit of good later on—I think everyone but the engineers acknowledges that—but we still have to toil through it, scraping for the A's, in pursuit of the best colleges possible. Is it really that much of a sin to give those of us who try the answer? (And understand I am replying to the admin not you lol, thanks for the help)

I appreciate your opinion..but I hope you have actually read the Math Forum Rules about the IA.

I know some people are very struggling to get the formula, and it is not that easy to get it. I also realise that not everybody can see such patterns/trends in the series. However IBO still states that students are not allowed to gain direct help, whether oral or written, in writing their Math IA. Well sometimes we break this rule actually but we are still not supposed to do that. And the formula is worth a mark, I believe. You can still BS the proof or explanation anyway and if the teacher/examiner is too lenient, they may still give you the mark for correct formula but wrong explanation.

You can always ask here if you want. Like whenever you need any hint or clue. I have got the formula btw so I think I can help in this IA. Now, I tell you what, if anybody wants to know the correct formula, PM me and I'll give it to you. I would be happier if you can give me your formula first (anything you have got...or if you haven't got anything yet then fine) so I can just say whether it is correct or not and how you will get to the correct formula. Seriously just PM me now if you want the formula :/ anybody... I don't mind.

[ronali]

Hello! I've figured out all of the formulas and expressions and things for this (much thanks to everyone who's posted here!), but I've still got a couple questions.

How are we supposed to "test the validity of the general statement?" Do we just give 'em a bunch of tables full of data for different stars? And for each of the straight-forward-just-math questions, do we need to include a lot of extra explanation/what should that be like?

It would be wonderful if someone could help me tomorrow today (I should probably go to bed) as it's due Friday.

Thanks again. You guys are rad.

[Julie]

Hello! I've figured out all of the formulas and expressions and things for this (much thanks to everyone who's posted here!), but I've still got a couple questions.

How are we supposed to "test the validity of the general statement?" Do we just give 'em a bunch of tables full of data for different stars? And for each of the straight-forward-just-math questions, do we need to include a lot of extra explanation/what should that be like?

It would be wonderful if someone could help me tomorrow today (I should probably go to bed) as it's due Friday.

Thanks again. You guys are rad.

Test the validity by taking a bunch of different values. So in essence, yes it is just a bunch of tables... The reason this is considered testing the validity is because you demonstrate that your general statement works for everything... I also explained in words below, or on the side why I did what i did, and how I did it, and what effect it had. So basically, my final sentence was: therefore, the general statement works with blablabla

Main suggestions:

1. Write everything out mathematically

2. Next to the formulas/working/etc write out in words your working/thought process (how did you get what you got)

3. Make sure to draw the stellars for the values of p and n you use, because you need visual representation...

If anyone needs any specific help with their math stellar numbers, let me know... I got a 7... I don't think me giving general info will help much. So, specifics? let me know... I came up with many possible general statements, they all work... I won't give you the direct answer though (not fair to all of us who worked our asses off), but I'll try and help you to push you towards the right answer...

Edited January 6, 2011 by juliemk

[Julie]

You can always ask here if you want. Like whenever you need any hint or clue. I have got the formula btw so I think I can help in this IA. Now, I tell you what, if anybody wants to know the correct formula, PM me and I'll give it to you. I would be happier if you can give me your formula first (anything you have got...or if you haven't got anything yet then fine) so I can just say whether it is correct or not and how you will get to the correct formula. Seriously just PM me now if you want the formula :/ anybody... I don't mind.

No offense, but this is ridiculous. I am entirely open to helping in any way possible, and want people to be lead in the correct direction and get the correct answer, but this is like cheating (what you are offering). You may be willing to give the answer, but the person who takes it is cheating, and you are initiating it. So basically, you are holding the latter while someone climbs it to break into a house... I got the correct answer, not to brag, I got a 7 on the stellar numbers, but I'm not going to give away the answer(s). I took my time to figure it out, and i am offering my time to help other figure it out. But, why should one person, who may be lazy or may not, just get the answer directly. They get a 7, I get a 7 and in the end, its my answer, my hard work, my hours/time, my mind, my brains...

So please, spare us all and just be helpful... don't give out the answer...

Edited January 6, 2011 by juliemk

[Julie]

Hate to troll here, but that is just rude. Most of us have tried, toiled and worked on this for long enough that it's alright for us to just have the formula; we are obligated to explain how we obtained it at the end, so we have to have some conception of the math behind it or the teacher will dock plenty of points. It is the teacher's job, not yours, to keep people accountable for learning correctly; moreover, "correct" learning is just "institutional" learning, and I find IB to be useful, but also to be the farthest extension of passion-sapping institutional BS. I did it to get into a great college, and I hope to never, ever, have to study anything related to mathematics again; unfortunately, I am must learn this **** to graduate. It won't do me a damn bit of good later on—I think everyone but the engineers acknowledges that—but we still have to toil through it, scraping for the A's, in pursuit of the best colleges possible. Is it really that much of a sin to give those of us who try the answer? (And understand I am replying to the admin not you lol, thanks for the help)

Sorry, I need to comment:

farthest extension of passion-sapping institutional BS

you just completely bashed the education of every person on this site and you are expecting help?

It won't do me a damn bit of good later on

Actually, it will, especially stellar numbers. In almost every job you need to develop your ability to recognize patterns, even in literature. A task like this help develop those section in the brain...

but we still have to toil through it, scraping for the A's, in pursuit of the best colleges possible.

I don't sense any scraping and toiling from you, just you simply asking for the answer and getting mad about someone pointing out the rules...

Is it really that much of a sin to give those of us who try the answer?

Read my comment to Desy below, and maybe you will realize why, yes, it is a "sin", if that is what you wish to call it... it's actually cheating according to IB regulations... It clearly states that you can not get the answer in this manner... you must work at it...

I am must learn this **** to graduate.

May I just give a sarcastic "thanks" for that comment... you are talking to some people who hate math, but you are also talking to people who love math... and it isnt ****... and it isnt just to graduate... You should be thankful people actually teach this type of math... you think someone randomly came up with computers, iPods, engineering, TV's, etc? They all have math in them, and a lot more advanced math than simple stellar numbers, obviously... so... can you withhold from the insults by any chance and just get to work on your own IB stuff? for example: the stellar numbers answer you desperately want, but are trying to get other people to tell you?

I hope that all of my quoting and changing colors, and everything else I have done will beat some sense into your brain...