Let's discuss Mathematics

[Samson]

Why is a straight line y=mx+c?

A quadratic is y=ax^2+bx+c, a cubic is y=ax^3+bx^2+cx+d, why is a straight line not y=ax+b? Why c, and particularly why m?

 

[Kyriakos]

Why is a straight line y=mx+c?

A quadratic is y=ax^2+bx+c, a cubic is y=ax^3+bx^2+cx+d, why is a straight line not y=ax+b? Why c, and particularly why m?

While it's not written like that usually in Greek mathbooks (I most often see the coefficient of x in a linear function be "b" or even "a"), I would imagine that the "m" is there to help with calculations using that line as tangent to a curve? Since it would be unique.
Of course it's counterintuitive to start with that (in middle school), when it will be years before you use it for the tangent to a non-linear function. Then again all sorts of secondary school math are presented heterochronically (I am sure a simpler term exists in English, but apparently this one is also there in English).

 

[q1y11]

While it's not written like that usually in Greek mathbooks (I most often see the coefficient of x in a linear function be "b" or even "a"), I would imagine that the "m" is there to help with calculations using that line as tangent to a curve? Since it would be unique.
Of course it's counterintuitive to start with that (in middle school), when it will be years before you use it for the tangent to a non-linear function. Then again all sorts of secondary school math are presented heterochronically (I am sure a simpler term exists in English, but apparently this one is also there in English).

I like how you put meaning to those equations .
I found that to be the hardest thing to do coming out of high school and entering university with applied math as a degree. People tend to not explain every single thing in the equations and you could not find the meaning anywhere... That got me stunned for a long time.

 

[Kyriakos]

I like how you put meaning to those equations .
I found that to be the hardest thing to do coming out of high school and entering university with applied math as a degree. People tend to not explain every single thing in the equations and you could not find the meaning anywhere... That got me stunned for a long time.

I didn't see meaning in them in highschool either - back then I was only looking for personal 'tricks' to do things automatically. And sometimes the "meaning" is not even discovered - but for many part of it has been. What I like is that obvious (but abstract) tautologies (or even not so obvious ones) in algebra can also acquire very non-abstract meaning elsewhere, such as in geometry. Eg the inequality ab<=[(a+b)/2]^2 implies (I suppose among other things) that the square is the orthogonal parallelogram with the smallest perimeter for a set surface area => the algebraic phrase can be written also as functions with their min/max.

 

[q1y11]

I didn't see meaning in them in highschool either - back then I was only looking for personal 'tricks' to do things automatically. And sometimes the "meaning" is not even discovered - but for many part of it has been. What I like is that obvious (but abstract) tautologies (or even not so obvious ones) in algebra can also acquire very non-abstract meaning elsewhere, such as in geometry. Eg the inequality ab<=[(a+b)/2]^2 implies (I suppose among other things) that the square is the orthogonal parallelogram with the smallest perimeter for a set surface area => the algebraic phrase can be written also as functions with their min/max.

Yes , yes I like your point. I spend 12 years in and out of the university and I finally manage to get the lowest possible grade in order to finish it haha but I was really happy that I could explain to myself one thing. That functions and interpolation can be used to find different "things" in time. So just if we have a random sets of points in a dimension let's say a normal coordinate system we can find a representation of those points as a drawn out function, a representation of time and that Pi and the golden ratio can be used as a perfect process in time.... Don't know if I'm making sense haha to you at least I was happy when I though of that got the lowest point and never gave it much though after...

 

[Samson]

What is the definition of simplest form when it comes to Algebraic Fractions?

I have googled, and most definitions seem circular to me (eg. "In mathematics, the simplest form refers to the most reduced or simplified representation of a fraction.") This site is better than most, and says "In mathematical algebra, the simplest form is the least attainable fraction of a number or a linear equation." but I do no know the strict definition of attainable.

They give an example, but it just leaves me less sure. They say simplify this:


And the answer is:

But I do not understand how the simplest form can have an expression that can be further factorised on the top and an expression with brackets on the bottom. Depending on the definition of simple, surely the answer has to be one of these:

 

[Kyriakos]

@Samson Yes, the form (x^2+2x)/2(x+1) is inconsistent (as you have shown with the two counter-examples where the same way was applied). But I can imagine why it was the preferred way:
If you try to establish, through algebra (ie not through visualizing the graph) where this function is positive and where it is negative, you would never leave it as three functions because then you would need to establish combinations of +.- in three things instead of 2. Of course you can do the numerator as two first, but then keep the result as one polynomial.
This may still be a couple of years away from what the kid is asked to do - or at least one.

(edit, hm, now I will check if actually this is even true - namely if you can leave them as two without missing part of the range; will get back to you on this shortly)
(see two posts below: it wasn't true)

 

[Samson]

Yes, the form (x^2+2x)/2(x+1) is inconsistent (as you have shown with the two counter-examples where the same way was applied. But I can imagine why it was the preferred way:
If you try to establish, through algebra (ie not through visualizing the graph) where this function is positive and where it is negative, you would never leave it as three functions because then you would need to establish combinations of +.- in three things instead of 2. Of course you can do the numerator as two first, but then keep the result as one polynomial.

I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.

 

[Kyriakos]

I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.

From discussions in Facebook by math teachers for those grades, it sadly seems to be inconsistent. Better tell the kid to write both/all in the exam

 

[Kyriakos]

@Samson I was wrong to assume you could have the numerator be dealt with as one polynomial in the first place. Because indeed if you do that you will miss part of the range for +-. In this case I missed the added range for the full function being positive in also (-2,-1).

Desmos creates the graph -when I treated the numerator as one, I couldn't account for the function being positive in (-2,-1).

 

[AdrienIer]

I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.

As a high school math teacher my answer is :
-The question is badly worded. "In a simple form" is better because it doesn't imply that there is only one good answer
-Therefore I'd accept all 4 versions : (x²+2x)/(2x+2) (IMO the best one), (x²+2)/(2(x+1)), x(x+2)/(2x+2) and x(x+2)/(2(x+1))

 

[CKS]

I would say that simplest form means that the numerator and denominator have no common factors besides 1. I agree with Adrienler that all 4 versions should be acceptable, but I suspect that some teachers will have the belief that one of them is "best" and mark others wrong. I think the "best" variety will vary from teacher to teacher, and so you will have to ask the child's teacher what they want. If the teacher just goes by the answer in the book, you will be out of luck, since I expect the book answers will not be consistent.

Personally I prefer things factored out, since you have to factor everything to cancel terms, and re-distributing things at the end is an extra step, but I'm a lazy physicist, not a mathematician.

 

[Kyriakos]

If we take into account the class the pupil is in (middle school?), maybe the point of writing the function like that is strictly polynomial division? (which should be right around the corner)

For example:

It would be one rather direct reason to maintain the apparent inconsistency in how the numerator and denominator are left.

 

[Kyriakos]

Why does Desmos have no solutions for absolute variable functions when they equal 0?

For example:

It doesn't have a solution for |x|=0 either.
This does go against the definition of a function of an absolute variable, afaik, since it is defined as the distance of x, starting at a, from the value (in this case distance of x, starting from a=0, from 0), so you'd expect the line x=0 as solution to |x|=0 and x=+-3 for ||x|-3|=0.

edit: I asked on reddit, and perhaps it has to do with the program's method of establishing values of the function by comparing negative to analogous positive values.

 

[Kyriakos]

A few days ago, a general exam for entrance in some type of experimental (and supposedly better) middleschool and highschool took place in the country.
I did read online the 25 questions that each of the entrance math tests had, for kids to start middleschool and highschool.

I can't be certain about the middleschool one, as I haven't read the books they use nor recall what a good math student of that age would be capable of doing. Maybe (?) it was a test with some difficulty.
But I am entirely certain that the highschool exam test was farcically easy.

This is made worse by the fact that there were also two questions there which you could answer wrongly due to reading issues (general spectrum issues). In fact the only question I got wrong was one of those, because I was too careless to notice the trap and was content with the first four digits being what I had calculated (it's a multiple choice test, so you can very easily make this mistake if you aren't careful - although it's not like I bothered to reread my answers) but of course a later option had the correct fourth and fifth decimal parts.
So in theory, due to the test being so easy, the mark to get in (which will have to be dependent on what the others kids scored as there is a fixed number of people to be accepted), won't help at all kids who are excellent in math (and if they were careless, they may even do worse than the average!) (assuming a 23/25 or 24/25 score would lead to failure). I don't consider myself excellent even on that level of math, but I have a good sense of what it would imply and it's simply not to play a role in this test.

I should also note, however, that math was just one of the subjects of the exam. Still, with so low a level, it will be pushed to the side.

 

[Harv]

Knowing nothing about test design, here is my general thought:
If the test is well designed, then the average score should be 15/25 or 60%. This is based the average between guessing at all 25, which is 5 correct, and getting all 25 correct. Next is to guess at the standard deviation. I believe 2.5 is too small, based on the number of questions, 25, and the number of options, 5. I will work with 5.0 instead, for the sake of easy numbers to work with.

I believe the test would be attempting to distinguish the top 15%. So your passing grade to get in would be 20/25.

Distinguishing the top 2% on a 25x5 multiple choice test would be difficult, if they are also trying to make a distribution curve for the general population. If we used 15.0 mean and 4.0 standard deviation, then we would be looking for a score of 23.0. If we correctly answered 21 of 25 and guessed at the last 4, then it comes down to luck if we pass or not.
40.96 - 0/4 for a total of 21
40.96 - 1/4 for a total of 22
15.36 - 2/4 for a total of 23
2.56 - 3/4 for a total of 24
0.16 - 4/4 for a total of 25
As we can see, we already have an 18.08% to get into the top 2%. Also note that we have a 2.72% of getting a score of 24 or 25 and the 2.72% figure for being lucky exceeds the proportion we expect for +2 standard deviations.

You mentioned reading / spectrum issues. I am not sure which spectrum you are referring to and it is not my business. Maybe it is another 5 option multiple choice test. I brought this up to suggest the test question might be poorly designed if some people on a spectrum misread the question. As an example, I will put myself on a blindness spectrum (a common age related issue) and we can say I answer this 7 x6 =56 incorrectly. I thought of this as an issue, because in a recent Hearts of Iron game I was playing, I was having trouble distinguishing between a 6 and an 8 on the displayed date. So if the text is not formatted well, it might be hard for me to read.

I suspect on this forum, there are people who understand test design better.

 

[Kyriakos]

Knowing nothing about test design, here is my general thought:
If the test is well designed, then the average score should be 15/25 or 60%. This is based the average between guessing at all 25, which is 5 correct, and getting all 25 correct. Next is to guess at the standard deviation. I believe 2.5 is too small, based on the number of questions, 25, and the number of options, 5. I will work with 5.0 instead, for the sake of easy numbers to work with.

I believe the test would be attempting to distinguish the top 15%. So your passing grade to get in would be 20/25.

Distinguishing the top 2% on a 25x5 multiple choice test would be difficult, if they are also trying to make a distribution curve for the general population. If we used 15.0 mean and 4.0 standard deviation, then we would be looking for a score of 23.0. If we correctly answered 21 of 25 and guessed at the last 4, then it comes down to luck if we pass or not.
40.96 - 0/4 for a total of 21
40.96 - 1/4 for a total of 22
15.36 - 2/4 for a total of 23
2.56 - 3/4 for a total of 24
0.16 - 4/4 for a total of 25
As we can see, we already have an 18.08% to get into the top 2%. Also note that we have a 2.72% of getting a score of 24 or 25 and the 2.72% figure for being lucky exceeds the proportion we expect for +2 standard deviations.

You mentioned reading / spectrum issues. I am not sure which spectrum you are referring to and it is not my business. Maybe it is another 5 option multiple choice test. I brought this up to suggest the test question might be poorly designed if some people on a spectrum misread the question. As an example, I will put myself on a blindness spectrum (a common age related issue) and we can say I answer this 7 x6 =56 incorrectly. I thought of this as an issue, because in a recent Hearts of Iron game I was playing, I was having trouble distinguishing between a 6 and an 8 on the displayed date. So if the text is not formatted well, it might be hard for me to read.

I suspect on this forum, there are people who understand test design better.

This test would allow an average-at-math student to likely get 25/25, imo. It's just not representative of the schoolbook's difficulty...

Here is an example:

(note, this is question 35, as each day there is a test on two subjects, and the previous one ended at 25 so you are 10 questions into the math one)


This is asking for one of the next pairs of drawn coordinates for this function, with x E Z (graphed x are all clearly integers). Namely, it asks for the sixth position, starting with (-3.0). But:
-it is a lame linear function, which you can just establish by looking at it (no algebra needed). With algebra: coefficient a of x is the stable gradient and is established with f(x1)-f(x0)/(x1-x0)=>coefficient of x= (2-0)/(-2+3)=2=> f(x)=2x+c=>f(-2)=2=2(-2)+c=>c=6=>f(x)=2x+6.
-you don't even need to look far for the correct answer (ie you can literally just move dots on the graph if you wish). The correct answer is Γ, the pair (2,10).

Schoolbook's problems to represent the entirety of middleschool are in no way that simple.
With this question on the test, I suspect that most students of average-to-high level would simply notice that the gradient is 2 and then multiply that by 5 (as the sixth position is n-1 positions away from the first) to arrive at the pair x=-3+5, f(x)=0+5(2), ie the pair (2,10). While series and recursion aren't explicitly taught in middleschool (the latter is in the first year of highschool, maybe the former is in middleschool in some capacity), at least notionally such an approach would be easy to put to use despite (iirc) not yet formally having covered in school the general formula for an arithmetic series. Most lower-than-average level students, on their part, would again notice that the gradient is 2, and at worst crawl one step at a time to the sixth position.

As for your question about what I meant about reading/attention issues, this problem also features such a trap(though a rather conspicuous one) : a student who is careless may forgo that they were asked about the sixth position, and choose A as the correct answer (as 0,6 are indeed a pair of valid coordinates of this function, but they aren't in the sixth position starting from -3,0). Such traps are designed in a way which has nothing to do with one's math abilities, and everything to do with spectrum issues. That said, my own careless mistake wasn't in this question, but on one which had four identical first digits in two choices ^^

 

[Harv]

That is why I was curious - You said such traps are designed in a way that has nothing to do with one's math ability, but everything to do with spectrum issues. Are you talking about ASD, ADHD, or Dyslexia or a different reading disorder?

I believe we are prone to similar issues. Or maybe a different issue, because I might read the question and choose 0,6 because I thought the question read what is next in the sequence. There is an expression that goes, it's all Greek to me - and in this case, there is extra meaning.

(One issue I am running into is trying to fix a typo or spelling error on a screen, because we are looking for a single missing or misplaced letter in a long word. A lot of it is age issued. Very likely, the brain is trying to fill in blank or fuzzy spots with what it believes is supposed to be there, making actually seeing the issue even more difficult.)

What you say implies a doubly badly designed test then, because you said it traps spectrum issues instead of mathematical understanding. If the test is too easy, which you said it was, then it would be impossible to distinguish exceptional (say top 5%) students from average. This would certainly be the case if the passing grade is 20/25 or 80%.
-------

On a general subject of Statistics, I have a question. I was reading some posts from an Axis and Allies Online Discord Forum. If you are not familiar with the game, it is what I describe as a dice fest. It's amazing for statistical analysis. In this Forum, there is a section called Dice Gods, where people post their most ridiculous (one way or the other) results.

One one side, the game is a dice fest and we have thousands of players making thousands of rolls. So somebody might be the lucky or unlucky recipient of a one in a million dice result. We also have an internal bias of really noticing when the dice roll against us. (I believe specifically this threshold is about 1.5 to 2.0 standard deviations and at 3.0 standard deviations, then I go downstairs and smash the dice with a hammer.) Even so, extreme results seem to happen quite often.

Do we have a method of measuring how "fair" the dice were being rolled - or if we are getting much more than usual extreme dice events? The analogy would be if this once per century flood is happing more like five times per century. It will come down to some accumulation or distribution of standard deviations.

An example might be the game's anti-aircraft. For simplicity, they get one six-sided die per plane you send in for a 1/6 chance. So I send in two and the game rolls 2 dice and they both hit, a 1/36 probability. If the dice engine is rolling funny, maybe I track 600 dice throws, expecting an average of 100 and standard deviation of.... 55 9? So if we get 155 to 210 109 to 118 in our sample, it's high, but not unusual.
Edit - Ooops! I forgot a divide by 6 error and wondered why my numbers seemed so high! It should have been sd approximately 9 and now I am puzzled.... I will try again with a smaller number like 60 and be lazy and figure it out on Excel. I get sd a little under 3 and 16 would be mean +2sd. 16+ on Excel gives me 3.3%.

Streakiness is another word I was thinking of. A lot of people thought the RNG was streaky. We might roll a bunch of misses for an empty round, then normal. Or we might roll an incredible number of hits.

Maybe I will play a competitive or non-competitive game and keep a record of dice rolls from my side to see if we can take a measure of how unusual the dice have been behaving. The most likely result is it will turn out I was on the receiving end of the aforementioned bias and I unjustifiably smashed hundreds of dice in my basement.

 

[CKS]

A few days ago, a general exam for entrance in some type of experimental (and supposedly better) middleschool and highschool took place in the country.
I did read online the 25 questions that each of the entrance math tests had, for kids to start middleschool and highschool.

I can't be certain about the middleschool one, as I haven't read the books they use nor recall what a good math student of that age would be capable of doing. Maybe (?) it was a test with some difficulty.
But I am entirely certain that the highschool exam test was farcically easy.

I'm pretty sure that the problem with the exam is that you have an entirely unrealistic view of the abilities of the students.

Many years ago I was involved in a math assessment for college students. It was not multiple choice, but it consisted of questions such as "How far can you go on half a tank of gas?" given the size of the tank and the miles the car can go on a gallon of gas and "How many points do you need to pass this class if 70% is passing and there are 850 total points?" 10 questions, one point for correctly setting up the problem, one point for correctly calculating the answer. More students got a 0 than any other grade, by quite a lot.

 

[Kyriakos]

I'm pretty sure that the problem with the exam is that you have an entirely unrealistic view of the abilities of the students.

Many years ago I was involved in a math assessment for college students. It was not multiple choice, but it consisted of questions such as "How far can you go on half a tank of gas?" given the size of the tank and the miles the car can go on a gallon of gas and "How many points do you need to pass this class if 70% is passing and there are 850 total points?" 10 questions, one point for correctly setting up the problem, one point for correctly calculating the answer. More students got a 0 than any other grade, by quite a lot.

Ok, but in my case I have recently (a bit over a year ago) deliberately read all of the official middleschool math schoolbooks for this country, so I do have a general sense of relative difficulty for those students
At least if we assume that an entrance to a "special" school would require above average ability, this test is incongruous with that aspiration.
I felt shame-by-proxy and didn't mention another question, but as you posted some examples of the same variety, I was alarmed that the stereometry question in this test (and remember that the iconic stereometry in middleschool is about conic volumes and areas etc) was about... a cube.

@Harv , I refer you to this post I wrote a little while ago, about a classic case of setting a trap to hunt for non-math issues (in this case, entirely attention-related). Post in the spoiler:

Spoiler :