Comments on: Getting my math(s) right.
http://ask.metafilter.com/219663/Getting-my-maths-right/
Comments on Ask MetaFilter post Getting my math(s) right.Tue, 10 Jul 2012 08:53:13 -0800Tue, 10 Jul 2012 08:59:12 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: Getting my math(s) right.
http://ask.metafilter.com/219663/Getting-my-maths-right
Maths (math) people of the US: I need your help in working out if certain British conventions would be understood or standard in the US classroom. <br /><br /> I'm working on some text that will be read by schoolkids in the US. There is a fair bit of math involved, and while I'm confident I've "translated" most of it properly there are a few things I'm not sure about:<br>
<br>
1. "The formula that describes the steps can be applied to find any triangular number, where T(n) is the sum of the numbers from one to n"<br>
<br>
Is T(n) correct in the US here?<br>
<br>
<br>
2. "Firstly, Newton developed differential calculus, a method for calculating the gradient of a curve on a graph."<br>
<br>
Should "gradient" be "slope" here?<br>
<br>
<br>
3. Vectors/coordinates. I have the sentence "To calculate the vector that describes the movement of an object between two points, like an aeroplane, the coordinates at point A are subtracted from point B" illustrated with something like:<br>
<br>
B(2, 10, 4) - A(5,0,5) = AB (-3,10,-1)<br>
<br>
<em>except that there the numbers in the parantheses are piled on top of each other rather than being separated by commas, as in the numerous examples<a href="http://www.netcomuk.co.uk/~jenolive/vect17.html"> in this link</a><br>
AND <br>
there is a right-facing arrow above "AB"</em><br>
<br>
Is this how this should be presented in the US?<br>
<br>
<br>
4. "The gradient of a straight line is calculated by dividing the change in vertical height by the change in horizontal distance. At first, the flat ocean bed has a zero gradient – there is no slope at all. Despite moving only a few millimeters each year, folds develop over time, and a slight gradient of 0.1, or 10%, builds up. Over millions of years the fold mountain continues to grow and, as the gradient gets steeper, it rises above sea level. "<br>
<br>
Again, should all of these "gradient"s be "slope"s? Or should they be "grade"s, because we're talking about mountains? Or a mix of the two?<br>
<br>
<br>
Thanks for your help!post:ask.metafilter.com,2012:site.219663Tue, 10 Jul 2012 08:53:13 -0800cincinnatus cmathmathsterminologyUKUSresolvedBy: ocherdraco
http://ask.metafilter.com/219663/Getting-my-maths-right#3173506
I would use "slope" everywhere you've used gradient.comment:ask.metafilter.com,2012:site.219663-3173506Tue, 10 Jul 2012 08:59:12 -0800ocherdracoBy: madcaptenor
http://ask.metafilter.com/219663/Getting-my-maths-right#3173508
1: I don't think there's a standard notation for triangular numbers, so you can use whatever you want.<br>
<br>
2: "gradient" should be "slope". (In a mathematical context I'd reserve "gradient" for its vector calculus meaning, which judging from the samples you've given means you probably shouldn't use it at all.) <br>
<br>
3: some people write vectors as column vectors, some as row vectors. Usually I'd use column vectors in a context where I'm going to do matrix multiplication, but in a context where all I was going to do was addition and subtraction I'd probably use row vectors for the typographical convenience.<br>
<br>
4: I'd use "grade" here, but define it first; as the <a href="http://en.wikipedia.org/wiki/Grade_(slope)">wikipedia article</a> will tell you, there are several different definitions of "grade" in this context.comment:ask.metafilter.com,2012:site.219663-3173508Tue, 10 Jul 2012 08:59:20 -0800madcaptenorBy: leahwrenn
http://ask.metafilter.com/219663/Getting-my-maths-right#3173509
I would use slope everywhere you used gradient. <br>
<br>
1. T(n) is what I would use, but I doubt the students will necessarily have seen it. <br>
<br>
3. It depends on the age of the students. College-level students in linear algebra would definitely use the column vector notation. Calc III students would use the arrow over the name notation. I *think* calc III students use ordered triples rather than column vectors for vectors in R^3, but I don't have a book handy to check. And I don't know what physics students use at all. <br>
<br>
I find the A(1,2,3) notation to represent a vector with coordinates (1,2,3) named A to be weird. Not standard for me. <br>
<br>
In any case, it's "airplane". :)comment:ask.metafilter.com,2012:site.219663-3173509Tue, 10 Jul 2012 09:00:32 -0800leahwrennBy: bessel functions seem unnecessarily complicated
http://ask.metafilter.com/219663/Getting-my-maths-right#3173520
Seconding use "slope" instead of gradient. <br>
<br>
I also find the notation A(x,y,z) for a vector A to be confusing. You're defining AB=A-B, a notational convenience, and also defining a method for computing AB, via (-3,10,-1)=(2, 10, 4) - (5,0,5). I would not try to simultaneously inline the notation and the method. Rather, something like <br>
<br>
line1: AB = A - B<br>
line2: (-3,10,-1)=(2, 10, 4) - (5,0,5)<br>
(you should also align the equals sign, minus sign, and vectors).<br>
Maybe throw in a line 3?:<br>
line3: (AB1,AB2,AB3)=(A1, A2, A3) - (B1,B2,B3)comment:ask.metafilter.com,2012:site.219663-3173520Tue, 10 Jul 2012 09:08:39 -0800bessel functions seem unnecessarily complicatedBy: aimedwander
http://ask.metafilter.com/219663/Getting-my-maths-right#3173530
#3 - airplane :)<br>
<br>
And I agree with others about the use of "slope" instead of "gradient"<br>
<br>
in #4 if you wanted to use a word to talk about the physical ocean bed as separate from the (mathematical) slope characteristic of hte mathematical representation of it, I'd consider using "incline".<br>
<i>"The slope of a straight line is calculated by dividing the change in vertical height by the change in horizontal distance. At first, the flat ocean bed has a zero slope – there is no incline at all. Despite moving only a few millimeters each year, folds develop over time, and a slight slope of 0.1, or 10%, builds up. Over millions of years the fold mountain continues to grow and, as the incline gets steeper, it rises above sea level. "</i>comment:ask.metafilter.com,2012:site.219663-3173530Tue, 10 Jul 2012 09:19:44 -0800aimedwanderBy: muddgirl
http://ask.metafilter.com/219663/Getting-my-maths-right#3173531
In the US, T(n) is the standard notation for a function whose result varies with "n", so I think that's fine. When I was in school kids didn't see this function notation until pre-calculus or calculus, which is in the final years of high school.comment:ask.metafilter.com,2012:site.219663-3173531Tue, 10 Jul 2012 09:20:36 -0800muddgirlBy: scose
http://ask.metafilter.com/219663/Getting-my-maths-right#3173535
3. Usually the "output" variable is on the left of the equals sign, so I'd write AB = B - A.<br>
For your example, I'd do something like:<br>
<br>
<pre><br>
B = (2,10,4), A = (5,0,5)<br>
<br>
AB = B - A<br>
= (2,10,4) - (5,0,5)<br>
= (-3,10,-1)<br>
</pre><br>
<br>
4. I'd use "slope". In the USA any kid who has taken algebra knows the concept of slope, but they might not be familiar with "grade". I don't think I could have told you what a "10% grade" was until I started driving and noticed it on road signs.comment:ask.metafilter.com,2012:site.219663-3173535Tue, 10 Jul 2012 09:25:03 -0800scoseBy: Elementary Penguin
http://ask.metafilter.com/219663/Getting-my-maths-right#3173568
<em>n</em>thing "slope" and "airplane". T(n) is good for triangular numbers, but you might also consider <em>T<sub>n</sub></em>. In either case, high school students in the US will probably never have heard of triangular numbers.comment:ask.metafilter.com,2012:site.219663-3173568Tue, 10 Jul 2012 10:04:44 -0800Elementary PenguinBy: dsword
http://ask.metafilter.com/219663/Getting-my-maths-right#3173606
My math training was entirely in the U.S., and the only unfamiliar term you've listed is "aeroplane." I can point to examples perfectly matching each of your examples in textbooks sitting on my shelf at home.<br>
<br>
1) This is fine, although a lot of people use subscript n to denote the individual terms in a series.<br>
<br>
2) In the U.S., slope refers to the magnitude of the <i>gradient vector</i>. The directional aspect of the slope (i.e., the gradient) isn't usually addressed until multivariate calculus.<br>
<br>
3) This notation is somewhat common in multivariate calculus textbooks. I find the notation clumsy at times, as it blurs the distinction between points and vectors. So I prefer to explain that A(x,y,z) is really shorthand for 0A(x,y,z) -- i.e. it refers to a vector pointing from the origin to the point A. Since A sits at (x,y,z), and the origin at (0,0,0), the vector has magnitude (x-0,y-0,z-0)=(x,y,z). Column vector notation, bolded latin letters and/or latin letters with little arrows over them are all more common throughout physics texts, in my experience.<br>
<br>
4) The gradient at a point is perpendicular to the level surface through that point. The magnitude corresponds to the steepness of the surface.comment:ask.metafilter.com,2012:site.219663-3173606Tue, 10 Jul 2012 10:40:24 -0800dswordBy: Rash
http://ask.metafilter.com/219663/Getting-my-maths-right#3173625
I would use "grade" instead of gradient, because I live in the real world. For some reason USA math teachers and textbooks favor that word "slope" but the sign on the roadway says "6% grade" -- most people I know forget all about "slope" once they're out of high school, and don't realize "grade" (or gradient) is what they were being taught when they were learning about "slope".comment:ask.metafilter.com,2012:site.219663-3173625Tue, 10 Jul 2012 11:06:19 -0800RashBy: cincinnatus c
http://ask.metafilter.com/219663/Getting-my-maths-right#3173717
Thanks for the answers - very helpful. As there's general consensus about most of these things, I thought it fairest to deal out best answers all round.comment:ask.metafilter.com,2012:site.219663-3173717Tue, 10 Jul 2012 12:33:54 -0800cincinnatus cBy: oneirodynia
http://ask.metafilter.com/219663/Getting-my-maths-right#3173851
<em>I would use "grade" instead of gradient, because I live in the real world.</em><br>
<br>
I live in the real world of landscape architecture, and we use "slope" when doing calculations because the word "grade" is often a verb in my profession- one would "grade a hill to a slope no greater than 3:1", for example.comment:ask.metafilter.com,2012:site.219663-3173851Tue, 10 Jul 2012 13:33:38 -0800oneirodyniaBy: Harvey Kilobit
http://ask.metafilter.com/219663/Getting-my-maths-right#3173858
"First" instead of "Firstly".comment:ask.metafilter.com,2012:site.219663-3173858Tue, 10 Jul 2012 13:38:12 -0800Harvey KilobitBy: Earl the Polliwog
http://ask.metafilter.com/219663/Getting-my-maths-right#3173975
I would not use the following phrase:<br>
<br>
<i> At first, the flat ocean bed has a zero gradient – there is no slope at all. </i><br>
<br>
"No slope" generally refers to an undefined slope, as in a vertical line.comment:ask.metafilter.com,2012:site.219663-3173975Tue, 10 Jul 2012 15:10:17 -0800Earl the PolliwogBy: chairface
http://ask.metafilter.com/219663/Getting-my-maths-right#3174186
I was going to add what Earl just said. A flat line has a slope/gradient of zero which is not the same as having no slope.comment:ask.metafilter.com,2012:site.219663-3174186Tue, 10 Jul 2012 17:22:22 -0800chairface