kitchen table math, the sequel: Between

Wednesday, September 12, 2012


Pick a number between 1 and 10.

OK, 10.


Is there an official mathematical definition of "between" somewhere when it comes to integers versus reals? I just got caught on a SAT question when it asked for the number of integers between two points of a sequence. I included the end points. Of course, the choices included both with and without the ends. Also, I got caught on another problem when I selected the wrong choice even though I had the right answer. It's kind of a variation of mis-bubbling.


Daniel Ethier said...

Between always excludes the endpoints unless the word inclusive is in there somewhere ... at least that's how I always interpret it.

Glen said...

There may be some "official" definition of "between" used by the college board, but it seriously annoys me that they would make the correctness of an answer hinge on knowing it.

I certainly consider "between" to be ambiguous, as would most other STEM professionals, I would guess. That's why we don't use "between" in a specification; we use "in the range such-and-such."

A range is specified as something like [0,1), which means from zero (including zero) to one (excluding one). The '[' or ']' means "include this endpoint" and the '(' or ')' means "exclude this endpoint." (You also specify the domain, such as "integers" or "real numbers," unless it is understood.) You would say something like, "Python's random.random() function returns a floating-point number in the range [0,1)," or "Rolling n dice together produces an integer in the range [n, 6n]."

Glen said...

And, Daniel, do you really interpret it that way? Would you normally say that rolling a die gives you a random number between zero and seven? If I say, "Pick a number between one and ten," do you interpret one as excluded?

I'm not saying you're wrong, though, just that it is ambiguous enough that even if there is some official definition recognized by the College Board as the default value of an implicit {inclusive | exclusive}, I don't consider it important enough to lose points over. (Of course, I don't expect my opinion to carry much weight with the College Board.)

Jen said...

Yes, between two numbers excludes those two numbers on the SAT. I think of it as a physical thing (or a number line) -- any number the tiniest bit bigger than 1 and any number the tiniest bit less than 10 count -- but one and ten do not.

They will say inclusive, or including if they mean for you to include them.

"Would you normally say that rolling a die gives you a random number between zero and seven?"

No, I would say that rolling a die gives me a number from 1 to 6.

SteveH said...

I took a quick look in the SAT Blue Book, but I didn't see any definition. It's hard to do a search on it because "between" is a common word.

This is the question (from Drill #2 in PWN the SAT Math Guide - I will give a review of the book when I'm done. I'm doing the drills right now).


1,2,2,3,3,3,4, ...

In the sequence above, there is one 1, followed by two 2s, three 3s, four 4s. and so on. How many terms fall between the first occurrence of 80 and the first occurrence of 85?

(A) 243
(B) 330
(C) 409
(D) 411
(E) 495

The answer is 'C'. I remember coming up with 410 as the total number, but wondering whether to include the ends or not. I chose 'D' after mis-remembering what I read somewhere.

The worst case senario is if this is your third error on the whole test. You lose 1.25 points for each mistake on bubble questions. After your first error, your 54 - 1.25 = 52.75 raw score rounds up to 53 and your SAT score is still 800 (from Blue Book sample test 1 scoring). After your second mistake, your raw score is 52.75 - 1.25 = 51.5, which gets rounded up to 52, giving you a SAT score of 770. You can see what's coming. With your third error, your raw score 51.5 - 1.25 = 50.25 gets rounded down to 50, giving you a SAT score of 730. With that third error, you SAT score drops 40 points. Looking at a number of test scorings, the drop from 2 to 3 errors is always about 40 points. The difference between 1 error to 3 errors is 70 points. I don't know why the scoring table is not refined to a resolution of 1/4 points, at least above 700.

After doing two (20 question each) drills in the book, I came away with three errors; a mis-selection of a choice, a misunderstanding of a definition, and selecting an answer for the radius when they asked for diameter - even after I knew very well to look for those tricks!

I am very methodical about math and programming. Speed is anathema to me. Above a high 600 SAT score level, it's really about speed and preparation.

Anonymous said...

Do the real SATs use "between" in the ambiguous way? or just SAT prep tests?

I agree that most STEM professionals are leery of using "between", preferring to use "<" and "≤" or the open/closed interval notation with ")" and "]".

Off-by-one errors (where some uses "<" instead of "≤") are extremely common in computer programming, so the problem is a conceptual one, not just a problem with the vagueness of English.

Shannon Severance said...

I looked at a number of online freely accessible math encyclopedias and dictionaries for the definition of "between"

Wolfram MathWorld has three definitions for between. The first two are from geometry and are exclusive. The third confuses me, but looks to be inclusive in the front half, but exclusive in the back half.

MathWorld also has an entry for "strictly between" that looks to be an open interval.

But please don't take my word for it, the definitions involve partially ordered sets, not something I have a lot of experience.

SteveH said...

"Do the real SATs use "between" in the ambiguous way? or just SAT prep tests?"

I think that the SAT uses the word "between" to mean open ends, but I will have to search the Blue Book to find examples where you have both choices. It is true, however, that you can't rely too much on prep tests. Nothing compares to real questions on tests that have already been given.

"Off-by-one errors (where some uses "<" instead of "≤") are extremely common in computer programming, so the problem is a conceptual one, not just a problem with the vagueness of English."

Off-by-one errors may be common in programming (I've written over a million lines of code), but the issue here is not about understanding the concept, but understanding the definition.

I'm not going to argue about what the definition should or should not be, but I will argue the "gotcha" nature of the question. I am willing (my son will have to be) to play the game, but I won't be convinced that the results are meaningful.

Glen said...

Yes, thanks Shannon.

says, "If x ≤ y ≤ z, then y is said to be between x and z. If y is between x and z and y is [neither x nor z], then y is strictly between x and z."

Once again, I'm not arguing that any particular definition is the correct one, just that the term "between" is sufficiently ambiguous that the SAT should not make your score contingent on knowing their favored definition.

However, since they don't care what I think, if they do test for knowledge of their definition of between, I'm grateful for the advance warning.

Michael Weiss said...

Is there an official mathematical definition of...

You may as well stop right there, because the answer is already "No". There is no official definition of anything in math, because there is no regulatory body that has the authority to prescribe standards of usage (thank goodness!). All definitions are conventional, and almost all conventions are local.

So, for example, according to some authors 0 is a natural number; according to others it is not. According to some authors a rectangle is a trapezoid; according to others it is not. Euclid used the word "line" to refer to what we would call a "line segment". A good author makes his or her conventions explicit at the outset.

Shannon Severance said...

"So, for example, according to some authors 0 is a natural number; according to others it is not."

That explains whey I've never been able to remember if the natural numbers include zero or not.

Cal said...

Between absolutely does not include the end points. The SAT says "between 2 and 9, inclusive" or "between 2 and 9". The second means 2 and 9 are not included.

THis is not even remotely complicated, and anyone who argues otherwise is committing that most ineffectual of all acts, arguing with the test.

Anonymous said...

What was it that Mae West said?

"Between two evils, inclusive, I always choose the one I've never tried before."

Or maybe that's just how my math teacher told it...

SteveH said...

"THis is not even remotely complicated, ..."

Who said it was complicated?

Even if someone said that it was complicated, how would that be arguing with the test?

Where in the Blue Book is "between" defined?

Mike McClenathan said...

Maybe I should've put 412 or 413 as an answer choice instead of 411. In fact, I'm making a note right now to revisit this question in my next revision.

The word between occurs all the time in SAT math, and I've yet to find a place where it implies inclusivity, but it's also not something they deliberately try to nail students on, so I might be overreaching with the gotchas there.

For a few examples of "between" implying exclusivity, see page 668 #9, or page 953 #17. See page 789 for two instances of the word "inclusive" being used on the same page!

Sorry about the confusion. Glad you're enjoying the book. :)

SteveH said...

It's not confusion about your book, but a question about how much the SAT tries to catch students. I'm trying to make a list of key words to watch out for. Another one is "must". This means that the answer covers all conditions, not just some. This isn't a problem unless you stop looking at answers when you find just one that works. Works isn't must.

In problem #9 (page 669 in my book) there is no issue because all of the answers use the less than symbol.

In problem #17, one would not normally want to pick a slope at the ends; zero or 3/8.

Is there a SAT problem where they use "between" with integers?

Mike McClenathan said...

Not many, actually, and when they appear the answer choices avoid the particular trap that we're discussing in this post. I haven't found any questions with an incorrect choice that includes the boundaries. On page 716 #11 and page 830 #1, it appears the writers go out of their way to take the boundaries out of play.

See also page 583 #9, which is the question in the Blue Book that I'd say is most similar to the question we're discussing. Including the ends would require a reading error and an addition error.

In the questions I mentioned above that go to the trouble to use "inclusive," though, note that answering correctly hinges on your understanding of that word. Neither question uses "between." Both use "from __ to __" instead.

If you're compiling a list of words to look out for, I'd say "between" need not be on the top.

Gravy: It's worth noting that, as a preposition, it always requires the objective case pronoun. They'll get you once in a while with that in the grammar multiple choice. See page 839 #24 for an example of that.

Judy Tefft said...

Between means between. If someone says that they live between Denver and Colorado Springs, you know that they don't live in either of those cities. If the sign tells you to park between the yellow lines, most of us know that mean you can't park on top of one of the yellow lines. I have taught Saxon math to 3rd graders for years. It has the kids solve for mystery numbers using clues such as, "It's an even number between 6 and 10." This is not rocket science.

SteveH said...

"This is not rocket science."

Who ever said it was?

It's also not:

"Between means between."

Did you read the entire thread? The SAT is not just asking for an even number between 6 and 10.