Whacky math homework problems

[Paul C. Anagnostopoulos]

My son is studying volumes of solids. All his homework last week and this week is on that subject. In the middle of this evening's homework is this gem:

Two birthday gifts are ready to be wrapped. The magic kit box measures 10.5" x 8" x 2". The model ship box measures 16" x 7" x 1.5". Which box will use more 30" wide wrapping paper?

WTF? My first reaction was that the person writing the problem does not realize that the volume of the boxes is irrelevant. Then I figured I was being unfair. But then what? Calculate the surface area of the six sides of each box and tile them optimally on the 30" paper?

We ended up just placing the length of the box on the 30" paper and rolling the box around until it was covered.

Who makes up this stuff?

~~ Paul

 

[neutrino_cannon]

Cut the 30' wrapping paper in small squares and then glue those to the outside of the gifts.

Then calculate the surface area, 242 for the magic kit, 293 for the model ship.

There you go.

Alternatively,


the magic kit is magic because it takes up no space in the box.


The model ship is made of the same magic no-thickness paper that appears in all math questions, so it doesn't take up any space either.






And by cover the box in wrapping paper, they really mean to fill it with water.







This couldn't be some sort of review could it?

 

[Paul C. Anagnostopoulos]

We went with the magic kit, because it uses more paper if you roll it along the length of the paper and cut off the excess.

The problem is bogus, I tell you!

~~ Paul

 

[Tom]

Well, you can cut the paper, so strictly speaking the piece of data regarding the 30" side is irrelevant. The least amount of paper will be used if you cut it to exactly cover the box.

However, the problem doesn't stipulate which method "will" be used to wrap the gifts, so you cannot say which gift "will use" less paper.

The problem is ill-posed.

 

[Paul C. Anagnostopoulos]

Yes, good point, Tom. You can simply cut the exact shape needed to cover a box. I was thinking too practically when I mentioned tiling, trying to end up without a bunch of useless scraps. Does "use more" include wastage or not?

I suspect that it's a simple surface area problem tossed in the middle of a bunch of volume problems. But I'm still suspicious.

~~ Paul

 

[T'ai Chi]

Two birthday gifts are ready to be wrapped. The magic kit box measures 10.5" x 8" x 2". The model ship box measures 16" x 7" x 1.5". Which box will use more 30" wide wrapping paper?

I guess at first glance I'd calculate the suface area of each box. For a rectangular prism with dimensions a x b x c, the suface area is S = 2ab+2bc+2ac = 2(ab+bc+ac).

S(magic kit box) = 2(10.5"*8" + 8"*2" + 10.5"*2") = 2(84" + 16" + 21") = 242"

S(model ship box) = 2(16"*7" + 7"*1.5"+16"*1.5") = 2(112"+ 10.5" + 24") = 293".

Since S(model ship box) is greater than S(magic kit box), S(model ship box)/30 is greater than S(magic kit box)/30, so the model ship box will use more of the 30"-wide paper.

?

 

[Paul C. Anagnostopoulos]

Yes, that's probably the answer they want. However, when you wrap a package, you don't do it optimally. And the problem says "... use more 30" wide paper," so you have to wonder whether that dimension is a red herring.

~~ Paul

 

[jimlintott]

Hey it's a gift wrapping problem.

30 inches will wrap around the magic box with 5 inches overlap. 10.5*2+2*2=25 because it is 8 inches wide a nine or ten inch strip will do. The other box is 16 inches long so 30 inches won't cover. You will need about a 17 inch strip to wrap it.

What this has to do with volume? I don't know. How many gallons of water will the boxes hold? Makes the metric system look pretty good, eh!

 

[Zep]

Why don't you just PAINT the boxes. Save the wrapping paper for Christmas...

 

[T'ai Chi]

Yes, that's probably the answer they want. However, when you wrap a package, you don't do it optimally. And the problem says "... use more 30" wide paper," so you have to wonder whether that dimension is a red herring.

~~ Paul

I tend to think so. I think that the width of the paper could be anything to try and throw off the student. I'd think the question reduces to 'wrapping which box uses the most material?', unless it is trickier than I am thinking (which it could be).

 

[Paul C. Anagnostopoulos]

Okay, here's the next one:

Construct a rectangular prism with volume 36. The height is 4 units. How many can you construct?

Say what? How much time do they want the kids to spend on this problem?

~~ Paul

 

[Skeptoid]

Assuming whole number dimensions, 2. 4x1x9 and 4x3x3.

 

[Brian the Snail]

That's another pretty badly worded question. I read it as wanting kids to actually physically construct prisms out of cardboard or something. I have this mental image of an 80-year bending over some card and a plot of glue, saying "Well that's 4,326,948, and I guess I can't do anymore because of my arthritis. So now I've answered that question..."

 

[Soapy Sam]

Paul- What age group is this aimed at? The present wrapping problem is in reality a very hard one, as many frustrating hours on Christmas Eves have shown me.

Unless this is at least 15-17 year old stuff, either
1)I'm even dumber than I thought. (Which is feasible).
or
2)The questions are indeed badly expressed.

 

[CurtC]

I'm confused as to why you think this is a bad problem. Is it because it's about surface areas, when he's studying volumes? It seems to me that putting this in the midst of the volume section is more likely to make your kid think, instead of just blindly multiplying the dimensions to get the volume.

For the answer, you could do it either of two ways. The first pass would be to just see which has more surface area. If I were the teacher, I would accept that technique, but give extra credit if the student actually planned how it could be arranged on the paper to use less of a 30" roll.

The magic kit would take 10 inches of the roll, if you used the 30" width to wrap around the 10.5+2+10.5+2 way, then you'd need 8+2*(2"/2) for the length.

The model ship would take 17.5 inches of the roll.

If I were a math teacher, this is exactly the kind of stuff I would do.

 

[Paul C. Anagnostopoulos]

Lucas is in 5th grade.

Curt, I have no particular objection to tossing a surface area problem in the middle of volume problems, although he hasn't brought home a single 3D surface area problem yet.

My problem is the 30" number. I have seen problems with superfluous numbers before, that's fine. But in this case you could consider the number superfluous and just calculate the surface areas, or you could consider the number salient and do what you suggest (or possibly try to optimally tile separate pieces of paper).

We rolled the boxes lengthwise along the 30" paper and used up 20" and 17", respectively. So the magic kit requires more paper, even though it has the smaller surface area.

This kind of trickiness is unnecessary, I think.

~~ Paul

 

[ceptimus]

This reminds me of Richard Feynman's rant about questions in some math books he was asked to review:

Finally I come to a book that says, "Mathematics is used in science in many ways. We will give you an example from astronomy, which is the science of stars." I turn the page, and it says, "Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees . . ." -- so far, so good. It continues: "Green stars have a temperature of seven thousand degrees, blue stars have a temperature of ten thousand degrees, and violet stars have a temperature of . . . (some big number)." There are no green or violet stars, but the figures for the others are roughly correct. It's vaguely right -- but already, trouble! That's the way everything was: Everything was written by somebody who didn't know what the hell he was talking about, so it was a little bit wrong, always! And how we are going to teach well by using books written by people who don't quite understand what they're talking about, I cannot understand. I don't know why, but the books are lousy; UNIVERSALLY LOUSY!

Anyway, I'm happy with this book, because it's the first example of applying arithmetic to science. I'm a bit unhappy when I read about the stars' temperatures, but I'm not very unhappy because it's more or less right -- it's just an example of error. Then comes the list of problems. It says, "John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?" -- and I would explode in horror.

My wife would talk about the volcano downstairs. That's only an example: it was perpetually like that. Perpetual absurdity! There's no purpose whatsoever in adding the temperature of two stars. Nobody ever does that except, maybe, to then take the average temperature of the stars, but not to find out the total temperature of all the stars! It was awful! All it was was a game to get you to add, and they didn't understand what they were talking about. It was like reading sentences with a few typographical errors, and then suddenly a whole sentence is written backwards. The mathematics was like that. Just hopeless!

 

[patnray]

I have no problem with throwing in superfluous numbers as a way to teach students to select the relevant info when solving a problem. But the problem is stated ambiguously and can be solved for least surface area, least linear amount off the roll, or most efficient tiling.

I agree with Curt C's analysis (in which the paper is oriented differently on the two packages) for the second approach, though in practice you need an extra half inch for folding and overlap. The width of the roll is relevant in this method.

 

[T'ai Chi]

Okay, here's the next one:

Say what? How much time do they want the kids to spend on this problem?

~~ Paul

Construct a rectangular prism with volume 36. The height is 4 units. How many can you construct?

The formula for volume of a rectangular prism with dimensions a x b x c is:

V(rect. prism) = abc.

Without loss of generality, let a be the height, so a = 4. So,

V(rect. prism) = 4bc.

It is given that 4bc = 36, or bc = 9.

So you just have to list various (b,c) combinations such that bc = 9 as desired. I would also assume whole numbers, but maybe your son could list fractions in addition for some extra credit?

 

[JesFine]

It seems to me that no matter how poorly worded the question is, if you word your answer appropriately, the teacher has to accept your answer. If you are really concerned about correctness, write every answer you can think of and explain your rationale for each. I can't imagine the teacher not giving credit even if you only gave one answer.

Insert amusing anecdote here: When I was in 7th grade, we had just learned that the way we count is base-10, and you can just as easily count in base-2, base-6, etc. Anyway, we had a pop quiz with ten numbers and their corresponding base, and were asked to convert them to base-10. The test looked something like this --
1) 101 - base 2
2) 16 - base 5
3) 47 - base 4
etc...

Anyway, being the smartass I am, I answered the first one and put "Not possible" for the rest. Of course, I was right, but I was also the only one who answered the questions this way. Everyone else converted, for example, 16 in base-5 to 11 in base-10. I got full credit since I could back up my answers, and so did the other kids, <s>even though they were dumber than me</s> since they didn't really do anything "wrong".