Math quiz answers

[emmagrant01]

The quiz is here.



1. 1 is a prime number.
Answer: False. Prime numbers are defined to be those that have exactly two divisors: themselves and 1. So 1 only has one divisor and doesn't meet the criteria for a prime. But more importantly, the main reason that we're interested in prime numbers is that they are the building blocks of other numbers. Every integer can be written in a unique way as a product of prime numbers; so 24 = 2³ ⋅ 3. (This is called the Fundamental Theorem of Arithmetic.) But if we let 1 be a prime, we would lose that uniqueness -- so we could write 24 = 1² ⋅ 2³ ⋅ 3 = 1⁴⁵ ⋅ 2³ ⋅ 3, and so on. (More on this topic here.)

2. A square is a:
Answer: all of them. A square is a quadrilateral (a closed four-sided figure), a rectangle, a rhombus (an equilateral parallelogram), a parallelogram (opposite sides are parallel), and a regular polygon (a closed figure with all sides and angles equal).

3. Circumference is a measure of:
Answer: perimeter; specifically the perimeter of a circle.

4. If x squared is 9, then x is:
Answer: 3 or -3. In the equation x² = 9, what we are looking for are all the numbers that when substituted for x would make the equation true. 3² = 9 and (-3)² = 9, so both answers are correct.

5. If the square root of x is 9, x is:
Answer: 81. This is the only integer whose square root is 9. -81 doesn't have a real square root, and its complex square root is 9i.

6. 1/3 is
Answer: a rational number. Any number that can be written as a ratio (fraction) of two integers is rational. And contrary to what many of my students seem to think, 1/3 ≠ 0.3. Those two numbers are close, but not equal.

7. The number 1 is equal to the number 0.9999999999...
Answer: true! Just like we can write 1/3 = 0.3333..., we can write 1 = 0.99999... There are several interesting ways to prove this fact, but I'll direct anyone interested to a good explanation here.

8. If you draw a large triangle on a globe connecting three cities, what can say say about the sum of the angles inside the triangle?
Answer: The sum is greater than 180 degrees. To demonstrate this to yourself, draw three dots on an orange and then draw straight lines on the surface of the orange between them. You can probably see it by looking, but you can measure the angles to check. More info is here.

9. 0.555555555... is:
Answer: a rational number, also equal to 5/9. Decimal numbers that either terminate or repeat infinitely are rational numbers. I won't go into the technical details of why, but if you want to know more, go here (and scroll halfway down until you see "Rational Numbers").

10. A trapezoid is mathematically defined as:
Answer: both! This is a weird case where one mathematical definition wasn't agreed upon, and so both are considered valid. They have different consequences, so you have to specify which one you're using. For example, under the first definition, a parallelogram is not a trapezoid, but under the second, it is a trapezoid.

11. Which of the following situations represents 3 divided by 1/2?
Answer: I have 3 pizzas, and I want to give 1/2 pizza to each of my friends. How many friends can I feed? People often mistranslate the problem as "divide three into two halves", but the question really is "how many halves are in 3?" There are 6 halves in 3, 2 in each whole.

12. If a number is divisible by 24, is it divisible by 8?
Answer: Yes. If the number is divisible by 24, then it is divisible by the factors of 24, so it is also divisible by 2, 3, 4, 6, 8, and 12.

So, how'd you do?

Comments

[wordplay.livejournal.com]

I was really thrown by the trapezoid question, and I just don't like the "both" answer because, you know - it has been defined as both, but it can't actually be defined as both simultaneously. Totally missed the 1=0.9999999etc thing. Otherwise... it could have been worse? *laughs*

I would do one of these for lingustics, except that, OH YEAH, we don't do any real linguistics education in the US curriculum. So proud.

[emmagrant01.livejournal.com]

It's actually interesting to see what other folks' expertise is, you know? I know a hell of a lot about something very small, and not much outside of that. But such is academia. :-P

The trapezoid thing is funny to me, because it seems perfectly reasonable to me that mathematicians couldn't decide which was better, and so accepted both. I was once asked which was right by someone from the state education agency (which designs the high-stakes tests kids take here), and they were unhappy when I told them both were accepted. Heh.

[tm-nicholas.livejournal.com]

Terribly.

I always hated math. Must be why I'm an accounting major. :-x

[emmagrant01.livejournal.com]

My mom's degree is in accounting, and I don't understand half of what she says when she's in accounting mode. :-P

[phaballa.livejournal.com]

Wow, I got like, 2 of those right. I am dumb at math! But! I know so many funny stories about Calvin Coolidge!

[emmagrant01.livejournal.com]

Isn't it fun to see what narrow little area you know a lot in compared to others' little areas of expertise? Just goes to show how smart we all are when we work together. ;-)

[nimroddess.livejournal.com]

I had no idea about rational vs. irrational numbers, so I got those two parts wrong (at least I was consistent!) and I said 0.5555555... was 0.56 because it can be rounded to that and you used the phrase "is" rather than "is equal to" and so I basically created a trick question for myself by being too anal retentive. And I didn't know trapezoid had two definitions so I just picked one. But everything else I got spot on :)

[emmagrant01.livejournal.com]

I actually tried to keep the language more colloquial than I would at work, because I didn't want people to have to sift through the technical stuff. But it's actually really hard to talk about mathematical ideas without that very specific language. :-P

[thalia]

That's really interesting about 0.99999... -- I didn't know that. I guess I would have said the limit as the number of digits approaches infinity would be 1, but it wouldn't actually get up to 1. Dr. Math's explanation was good, though.

(I'm ignoring the trapezoid question. [g])

[emmagrant01.livejournal.com]

I teach a course for teachers and that's the problem that causes the most discussion every semester. People understand logically why it's true, but somehow they want 1 to be special and not subject to being represented in multiple ways like other numbers. Sometimes pointing out that 1 = 2/2 = 3/3 = 4/4 = ... helps, but there are always one or two who just say, "No, I don't believe it." ;-)

The trapezoid one is a rarity in school-level mathematics. But it doesn't really matter which definition you choose, which is why there is no consensus.

[letmypidgeonsgo.livejournal.com]

technically, i got 5 (& a half) right. i totally didn't think that pizza question through, & only realised the answer when i went to comment about it

:(

(psst, i re-stole this idea & did a history one!)

[emmagrant01.livejournal.com]

Oooh! Let me see how much I suck at history! ;-)

[jedirita.livejournal.com]

Woo-hoo! I got more than half right!

[emmagrant01.livejournal.com]

Go you! :-D

[jedirita.livejournal.com]

Have I mentioned how sexy you are when you talk math?

[emmagrant01.livejournal.com]

Hee! I do love it so! ;-)

[littleroo27.livejournal.com]

A) I got three right; I suck
B) What the hell middle school did you go to? I learned some of this stuff my sophomore and junior year in high school! Though obviously not well enough to remember, lol.

[emmagrant01.livejournal.com]

B) I had a shitty math education, so I didn't learn that stuff when I was a kid. But if you take a look at state and national standards for middle school math in the US, this is the sort of thing that's in there. The emphasis is now on moving abstract ideas, algebra, and geometry down into elementary school, so that kids see it throughout school and not just in high school, if ever.

[acromantular.livejournal.com]

OK, you got me on the trapezoids- I didn't know it was under argument! I definitely remember that a parallelogram can be a trapezoid, so went with the "at least" answer. Tricksy!

Otherwise, I'm a rock-solid middle school math nerd. Hooray for nerd pride!

[emmagrant01.livejournal.com]

That was the one "tricky" one, yeah. I figured the only people who would know that one would be teachers. :-P

Yay nerds! (I need a nerd icon.)

[bibliokat.livejournal.com]

Half right and half wrong! Not as well as I thought I'd do but not horrible either. (Although really, I should have known about 1 not being prime *facepalm*)

I'm off to fail the American History one!

[emmagrant01.livejournal.com]

I failed that one pretty badly. :-P

[zaph.livejournal.com]

10. A trapezoid is mathematically defined as:
Answer: both!

Aw, crap, really? That's the only one I missed, too.

Oh well. At least I can still prove that 1=2. :)

[emmagrant01.livejournal.com]

If you only missed one, you're way above average! ;-)

[ashe-frost.livejournal.com]

What of this I even heard about in middle school:

1
2
3
6
7
9
11
12


The rest of it was in high school. And we did learn about powers (or whatever it's called when things are squared or cubed or whatever) until I was in Chemistry in my senior year.

No one ever told me what a trapezoid was. Or a parallelogram. We learned about angles inside of triangles in 10th grade. But I think I was in the hallway during that test.

I was never told what a prime number was in school. I learned it from the film House of Cards when I was a kid where these Autistic children communicate speaking only in prime numbers and it was explained. The 1 not being I prime number came up in reading later in life.

Most of the negative and positive, though we did focus a little on that in middle school.. We learned in 9th and 10th grade, though most of 9th grade was oragami and most of 10th grade was a fairly competent first year teacher trying to teach high school Algebra to kids who didn't understand middle school math.

Oh, it's so sad. I want better teachers in crappy places for kids whose parents don't care whether or not they're educated and don't understand why it's important for their teachers not to be horrible. Not all of them, obviously.. It just. It feels like that a lot of times. It sems like when I go to my nephew's school there's always people there picking up their kingergartener's and they want to know what happened and there's so much responsibility and the teacher's send things home and I know these kids have to be helped to learn to read..

But when I was working with 7th graders on their writing portfolios.. Their faces were dirty and their clothes were wrinkled and didn't match and some of them were walking down the highway and what? Did their parents just give up hope.

[emmagrant01.livejournal.com]

It really is sad, absolutely. I've seen a lot of that sort of thing, and it's hard to deal with day in and day out (which is why I'm not a high school teacher any more). It's hard to pinpoint exactly when and why kids check out of school, and it's very complicated. :-(

[liesbyomission.livejournal.com]

Ack, the prime number and trapezoid questions got me. I distinctly remember being really amazed when my 7th grade math teacher proved that 0.999999... is 1, so I'll never forget that one. :D Oh, and the 180 degree one got me too...I think I missed the whole "on a sphere" concept. x.x

[emmagrant01.livejournal.com]

The 1 = 0.99999... one usually does make an impact on people. I think people who've seen that one tend to remember why it works.

I was trying to avoid using the word "sphere" in that one item, thinking that the image of drawing a triangle on a globe would make more sense, but apparently not. :-P

[rubymiene.livejournal.com]

If you connect three cities on the surface of a global, then it's not a triangle anymore...

[emmagrant01.livejournal.com]

It is -- a triangle is just a closed three-sided figure. You can draw three dots on a sphere and then connect them with straight lines, which gives you a triangle. More information on geometry on a sphere here (http://math.rice.edu/~pcmi/sphere/gos3.html#2). :-)

[ms-mindfunk.livejournal.com]

I have not done any math in literally almost 20 years. The last class I took was as a high school junior, and I remember basically nothing.

Gawd, I'm gonna flunk out of college, aren't I? Maybe I should rethink the whole idea...

Btw, I couldn't log in to apres l'ecole a couple of days ago. Even when I reset my password, it gave me trouble. It may be fine now, so don't worry about it unless I can't get in tomorrow night after work - I'll let you know. Anyway, I thought you might like to know that the links with the password reset message still go to the squatter site.

[emmagrant01.livejournal.com]

the links with the password reset message still go to the squatter site.

Do you mean after you reset your password, it gives you a bad link, or something else?

[xntrick.livejournal.com]

Well let's see...I got 7 totally correct, 4 completely incorrect, and 1 only partially correct. ;)

In junior high I was a year ahead of all the other kids, math-wise. But only because I went in already ace at algebra. If I got back into it, I imagine I'd still find it entertaining. But definitions and such (especially with geometry) never interested me. :D

I actually was awarded one of those "Bank of America Achievement in Mathematics" things in my senior year of high school. Though that might just be because I somehow had a grasp of things in Math Analysis.
I don't even know what "Math Analysis" is come to think of it...

[alternateyou.livejournal.com]

Hmmm

didnt do as bad as i thought... but could have done a lot better!

I think there was a reason I flunked out of doing maths in higher education, if I cant get that stuff right. hehe

[bolboreta.livejournal.com]

I got 7 right, 3 wrong and 2 partially right (I missed the rhombus ['cause the Spanish ones don't have "right angles"] and one of the trapezoid's definitions).

Apparently I don't give a damn about: prime numbers, whether 1 = 0.9999... (it made me think of limits, and I did think it was "virtually" the same thing, but it didn't seem right to say it was), and angles (geometry in general isn't my thing)

I also basically "guessed right" the rational numbers stuff. Math theory has never been my thing, and the fact that my two last years of math were "Maths applied to Social Sciences" probably didn't help. We basically did almost the same stuff than the "hard sciences" people (whether you were in one group or the other depended on your other subjects) were doing in math , but without all the theorems (and I've seen my sisters' notes, most of them were stupid!) and long explanations of everything, just "this is how this works and this is how you use it."

Sitting a year on a "Senior Algebra" class in the States was hell, the teachers knew I knew more than they did, and they were fucking terrified. I didn't learn a single thing, but it really was frustrating. The things that some of the kids didn't know made no sense (like going from "x²+x" to "x(x+1)", whatever that's called in English, the formula for second grade equations, etc.), and the stuff that they made us do was stupid too. I think I almost screamed at the teacher when she didn't let me solve 3 variable equation systems with a matrix (as I'd learnt to do) and forced me to muddle through the strategies I used for two variables, which usually resulted in me getting lost (I'm disorganized *sheepish smile*). It was almost just as pointless as the English teacher insisting I had to write my exams "in cursive" *grumbles* that's for primary school kids too.

I should study math again, at least pick up my high school notes because somebody mentioned optimization problems not too long ago and I thought "I used to know how that worked" *is sad*

[daphne486.livejournal.com]

I did quite well - I'm surprised by that.

[brumeux77.livejournal.com]

Only one wrong--but not the spherical geometry question I was hazy on. It was the trapezoid. We definitely never learned that a parallelogram could be called a trapezoid. Our trapezoids had only two parallel sides. Guess things have changed in the intervening *mumbles a large number* years.

[dreambastion.livejournal.com]

I sucked - but then, I knew I would... lol Twas fun though! :)

[oconel]

Well, I still disagree about 1=0.9999 because I can always find the diference between them, no matter how small 0.0001 is (add as many 9 as you want, I can add as many 0 before the 1). The link proves nothing, as you can also prove that 1=2:

a=b

a²=a·b

a²-b² = a·b-b²

(a+b)(a-b) = b(a-b)

(a+b) = b

since a=b:

2b = b if b=1 means 2=1

[emmagrant01.livejournal.com]

Actually, you can't legitimately prove that 1 = 2. That's a common "trick" shown to children to demonstrate that you have to be very careful when proving mathematical statements. In that middle of that proof, you divide by zero, which we all know is impossible, and so the proof doesn't actually prove anything. And yes, you divided by zero above in the 5th line: if a=b, then a-b=0, and when you "canceled" a-b from both sides, you divided by zero. It's not a mathematically correct proof.

The issue with 0.9999... = 1 is that you can't find a number between the two. The 9s go on forever, and the moment you assume you can find a point at which to stop (as you suggest above) you aren't talking about 0.99999... any more.

It's exactly the same as saying that 1/3 = 0.333333... This is an artifact of our base ten system for writing numbers. Those are just two ways of writing the same number. And by the way, there are infinitely many ways to represent the number 1: 2/2, 3/3, 4/4...

I use this problem in my courses a lot and many of my students struggle with it -- so I do understand that it's difficult to wrap your mind around it.

[hearts-n-roses.livejournal.com]

I didn't do too bad actually. I knew the 1=0.9999... thing because we were forced to prove it in first year Math but I couldn't prove it now because I don't remember...lol.

[freckles42.livejournal.com]

Here's how I did:

1. CORRECT
2. CORRECT (5/5)
3. CORRECT
4. CORRECT (2/2)
5. CORRECT
6. CORRECT
7. CORRECT
8. wrong. :(
9. CORRECT (2/2)
10. wrong (only guessed one)
11. CORRECT
12. CORRECT

Total: 10/12.

Not bad at all, if you ask me. :D