Getting my math(s) right.
July 10, 2012 8:53 AM Subscribe
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.
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:
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"
Is T(n) correct in the US here?
2. "Firstly, Newton developed differential calculus, a method for calculating the gradient of a curve on a graph."
Should "gradient" be "slope" here?
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:
B(2, 10, 4) - A(5,0,5) = AB (-3,10,-1)
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 in this link
AND
there is a right-facing arrow above "AB"
Is this how this should be presented in the US?
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. "
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?
Thanks for your help!
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:
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"
Is T(n) correct in the US here?
2. "Firstly, Newton developed differential calculus, a method for calculating the gradient of a curve on a graph."
Should "gradient" be "slope" here?
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:
B(2, 10, 4) - A(5,0,5) = AB (-3,10,-1)
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 in this link
AND
there is a right-facing arrow above "AB"
Is this how this should be presented in the US?
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. "
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?
Thanks for your help!
Best answer: 1: I don't think there's a standard notation for triangular numbers, so you can use whatever you want.
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.)
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.
4: I'd use "grade" here, but define it first; as the wikipedia article will tell you, there are several different definitions of "grade" in this context.
posted by madcaptenor at 8:59 AM on July 10, 2012
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.)
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.
4: I'd use "grade" here, but define it first; as the wikipedia article will tell you, there are several different definitions of "grade" in this context.
posted by madcaptenor at 8:59 AM on July 10, 2012
Best answer: I would use slope everywhere you used gradient.
1. T(n) is what I would use, but I doubt the students will necessarily have seen it.
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.
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.
In any case, it's "airplane". :)
posted by leahwrenn at 9:00 AM on July 10, 2012
1. T(n) is what I would use, but I doubt the students will necessarily have seen it.
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.
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.
In any case, it's "airplane". :)
posted by leahwrenn at 9:00 AM on July 10, 2012
Best answer: Seconding use "slope" instead of gradient.
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
line1: AB = A - B
line2: (-3,10,-1)=(2, 10, 4) - (5,0,5)
(you should also align the equals sign, minus sign, and vectors).
Maybe throw in a line 3?:
line3: (AB1,AB2,AB3)=(A1, A2, A3) - (B1,B2,B3)
posted by bessel functions seem unnecessarily complicated at 9:08 AM on July 10, 2012
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
line1: AB = A - B
line2: (-3,10,-1)=(2, 10, 4) - (5,0,5)
(you should also align the equals sign, minus sign, and vectors).
Maybe throw in a line 3?:
line3: (AB1,AB2,AB3)=(A1, A2, A3) - (B1,B2,B3)
posted by bessel functions seem unnecessarily complicated at 9:08 AM on July 10, 2012
Best answer: #3 - airplane :)
And I agree with others about the use of "slope" instead of "gradient"
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".
"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. "
posted by aimedwander at 9:19 AM on July 10, 2012
And I agree with others about the use of "slope" instead of "gradient"
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".
"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. "
posted by aimedwander at 9:19 AM on July 10, 2012
Best answer: 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.
posted by muddgirl at 9:20 AM on July 10, 2012
posted by muddgirl at 9:20 AM on July 10, 2012
Best answer: 3. Usually the "output" variable is on the left of the equals sign, so I'd write AB = B - A.
For your example, I'd do something like:
posted by scose at 9:25 AM on July 10, 2012
For your example, I'd do something like:
B = (2,10,4), A = (5,0,5) AB = B - A = (2,10,4) - (5,0,5) = (-3,10,-1)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.
posted by scose at 9:25 AM on July 10, 2012
Best answer: nthing "slope" and "airplane". T(n) is good for triangular numbers, but you might also consider Tn. In either case, high school students in the US will probably never have heard of triangular numbers.
posted by Elementary Penguin at 10:04 AM on July 10, 2012 [1 favorite]
posted by Elementary Penguin at 10:04 AM on July 10, 2012 [1 favorite]
Best answer: 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.
1) This is fine, although a lot of people use subscript n to denote the individual terms in a series.
2) In the U.S., slope refers to the magnitude of the gradient vector. The directional aspect of the slope (i.e., the gradient) isn't usually addressed until multivariate calculus.
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.
4) The gradient at a point is perpendicular to the level surface through that point. The magnitude corresponds to the steepness of the surface.
posted by dsword at 10:40 AM on July 10, 2012
1) This is fine, although a lot of people use subscript n to denote the individual terms in a series.
2) In the U.S., slope refers to the magnitude of the gradient vector. The directional aspect of the slope (i.e., the gradient) isn't usually addressed until multivariate calculus.
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.
4) The gradient at a point is perpendicular to the level surface through that point. The magnitude corresponds to the steepness of the surface.
posted by dsword at 10:40 AM on July 10, 2012
Best answer: 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".
posted by Rash at 11:06 AM on July 10, 2012
posted by Rash at 11:06 AM on July 10, 2012
Response by poster: 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.
posted by cincinnatus c at 12:33 PM on July 10, 2012
posted by cincinnatus c at 12:33 PM on July 10, 2012
I would use "grade" instead of gradient, because I live in the real world.
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.
posted by oneirodynia at 1:33 PM on July 10, 2012 [1 favorite]
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.
posted by oneirodynia at 1:33 PM on July 10, 2012 [1 favorite]
"First" instead of "Firstly".
posted by Harvey Kilobit at 1:38 PM on July 10, 2012 [3 favorites]
posted by Harvey Kilobit at 1:38 PM on July 10, 2012 [3 favorites]
I would not use the following phrase:
At first, the flat ocean bed has a zero gradient – there is no slope at all.
"No slope" generally refers to an undefined slope, as in a vertical line.
posted by Earl the Polliwog at 3:10 PM on July 10, 2012
At first, the flat ocean bed has a zero gradient – there is no slope at all.
"No slope" generally refers to an undefined slope, as in a vertical line.
posted by Earl the Polliwog at 3:10 PM on July 10, 2012
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.
posted by chairface at 5:22 PM on July 10, 2012
posted by chairface at 5:22 PM on July 10, 2012
This thread is closed to new comments.
posted by ocherdraco at 8:59 AM on July 10, 2012 [3 favorites]