Question about math aimed at those who studied it in college

[Grammar Stage Parent]

Our daughter recently wrote a standardized test for Grade 4 which the public schools in our area use. The math section consisted of two questions. The first was this:

 

"The class is building a rectangular pen for puppies. They have 24 sections of fence. Each section is 1 m long. Draw ALL the possible rectangular shapes that can be made and label their sides."

 

This problem was accompanied by a drawing of several kids holding rectangular pen pieces, with wires connected to iron borders.

 

She has done well in Singapore Math, but it didn't prepare her - or me - for a question of this nature. A friend who did graduate work in mathematics glanced at it and said that this was a college-level question.

 

Does anyone have anything to say about the suitability of this question? I'm not having a crisis of faith with Singapore. I just want to know if this test question was improper at the Grade 4 level, and to what degree.

______________________________________________

Daughter: 9; Singapore Primary Mathematics 3B; Story of the World Level 3; Writing Strands 3; Spelling Workout Level D; Science experiment books recommended in WTM; WWE3

 

Son: 6: First Language Lessons Level 2; Singapore Primary Mathematics 1B; general handwriting practice and reading practice; SOTW 1; WWE 1; Science experiment books recommended in WTM

[Jen500]

It seems like a suitable 4th grade question to me.

[Grammar Stage Parent]

Could you elaborate on what standard of suitability you are using?

[OlgaLA]

I am pretty sure I saw a problem like this in one of the elementary level workbooks, although I don't remember which one. It is certainly not a college level problem.

 

ETA: I have a rising third grader, so it was at her level or below.

[Xander]

I'd consider that easy for 4th grade.

A better question for that level would be one that asked them to find the configuration which gave the largest area for a rectangular pen.

[Jen500]

Could you elaborate on what standard of suitability you are using?

 

Knowing the definition of a rectangle, knowing how to figure out the perimeter of a rectangle, and finding the side lengths of a rectangle given the perimeter.

[Dana]

They've been using questions like this for well over a decade. NCTM (National Council for Teaching of Mathematics) has questions like this as illustrations in their material for illustrations of questions that are open-ended.

 

It would only be a college-level question if you were trying to find the maximum area enclosed by the field - and were limited to using calculus (optimization problems / max/min problems) - and had no limitation on size of fence segments.

 

So although it's a slightly more challenging problem - especially if you haven't seen it before and it came as a surprise - it's not out of line.

 

The idea behind it is to test a student's knowledge of the concept of perimeter but without saying "Draw all possible rectangles that have a perimeter of 24 m."

[skueppers]

Seems entirely appropriate to me, if a bit high on busywork. (Draw them ALL? Really?)

 

I would definitely not consider this a college-level question, and remember answering questions like it when I was in elementary school.

[Snickerdoodle]

If you are interested, you can read some of Andrei Toom's articles about word problems.

 

http://www.de.ufpe.br/~toom/my_articles/engeduc/index.htm

 

In particular I'm thinking of #8

Edited August 17, 2010 by Snickerdoodle

[Guest Cheryl in SoCal]

It seems appropriate to me, and would be easy to figure out by finding the factors of 24.

[Grammar Stage Parent]

Hmmm, I appreciate the responses, but since we've been deep in Singapore for the last four years, it's hard for me to get a wider perspective on this. I'm going to consult some other math texts.

 

Thanks to everyone who responded.

[Jen500]

Do you use CWP or the IP books? I think they're very helpful for problem solving.

[EKS]

It seems appropriate to me, and would be easy to figure out by finding the factors of 24.

 

Actually, I don't think that's what they're asking. It's not an area problem. It's a perimeter problem. You would have rectangles of the following dimensions:

 

1x11

2x10

3x9

4x8

5x7

 

It's actually kind of an interesting problem, certainly within the reach of a 4th grader.

[Guest Cheryl in SoCal]

Actually, I don't think that's what they're asking. It's not an area problem. It's a perimeter problem. You would have rectangles of the following dimensions:

 

1x11

2x10

3x9

4x8

5x7

 

It's actually kind of an interesting problem, certainly within the reach of a 4th grader.

 

I'm super tired and misunderstood the question:001_huh: Guess it's a good thing I wasn't taking the test:lol:

[nmoira]

Knowing the definition of a rectangle, knowing how to figure out the perimeter of a rectangle, and finding the side lengths of a rectangle given the perimeter.

:iagree:

 

All you need start with is, "The width could be 1 unit." Even if the pattern doesn't become apparent, the child can still figure out each corresponding length by calculation.

 

P = 2W + 2L

 

W.....L

=====

1......11

2......10

3......9

4......8

5......7

6......6

[nmoira]

Do you use CWP or the IP books? I think they're very helpful for problem solving.

:iagree:

 

It's just an agreeing-with-Jen kind of day. :D

[Kuovonne]

"The class is building a rectangular pen for puppies. They have 24 sections of fence. Each section is 1 m long. Draw ALL the possible rectangular shapes that can be made and label their sides."

 

All you need start with is, "The width could be 1 unit." Even if the pattern doesn't become apparent, the child can still figure out each corresponding length by calculation.

 

P = 2W + 2L

 

W.....L

=====

1......11

2......10

3......9

4......8

5......7

6......6

 

I'd agree with this answer, except the problem doesn't say that you must use all 24 sections of fence.

[nmoira]

I'd agree with this answer, except the problem doesn't say that you must use all 24 sections of fence.

Snap! :D

 

P = 2W + 2L

 

W.....L

=====

1......11, 10, 9, 8, 7, 6

2......10, 9, 8, 7, 6

3......9, 8, 7, 6

4......8, 7, 6

5......7, 6

6......6

[Greeny]

I am in the last year of my BSN and have taken many college level math classes. This is certainly not college level. It seems more like beginning algebra, maybe 7th grade?

[Kuovonne]

Snap! :D

 

P = 2W + 2L

 

W.....L

=====

1......11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

2......10, 9, 8, 7, 6, 5, 4, 3, 2

3......9, 8, 7, 6, 5, 4, 3

4......8, 7, 6, 5, 4

5......7, 6, 5

6......6

 

You missed a few. I included them above.

That's way too many to draw for a test.

[nmoira]

You missed a few. I included them above.

That's way too many to draw for a test.

More than a few...

 

Grammar Stage Parent: Did the test specify whether all fence sections had to be used?

[EKS]

I'm super tired and misunderstood the question:001_huh: Guess it's a good thing I wasn't taking the test:lol:

 

Actually, I interpreted it to be an area/factor problem at first as well. So I completely understand!

[Guest Cheryl in SoCal]

Actually, I interpreted it to be an area/factor problem at first as well. So I completely understand!

You're very gracious:001_smile: I'm thinking maybe I need a nap, but I don't think that's going to happen. I swear, small children are born knowing how to use sleep deprivation to turn their parents into babbling idiots:lol:

[Sebastian (a lady)]

Keep in mind that a standardized test is going to have questions that are both easier and harder than "on grade level" so that they can get a more accurate measure of each student's abilities. The analogy that I use when I proctor is that you don't measure someone's height with a ruler and just say that they are bigger than 12 inches. You use a measuring device that will expand to beyond their height. So while it may or may not be a fourth grade question, it is reasonable for this to be on a test that a fourth grader takes. (If you can see the difference.)

 

On the question itself. It is only asking for rectangles, not every shape that could be made. Each rectangle is made up of four sides where opposite sides are equal length and the angles are 90deg. So you are looking for rectangles that can be made up of 24 units.

 

I would start by dividing it up in half since the opposite sides are equal. That leaves me with 12 units to make up length and width. I would get the following combinations.

1x11

2x10

2x9

4x8

5x7

6x6 (this makes a square, which is a special rectangle)

 

I would probably stop there, assuming that the test designers would agree that a 2x10 pen is the same as a 10x2 pen. That would give six different rectangles to draw and label.

 

A common tripping point would be forgetting that you are doing a perimeter and starting to factor 24, which is what you would do if you were working with area (say if the problem were how many rectangles could you make with 24 carpet squares).

 

I don't think this is a college level question as long as you catch that the pen has to be a rectangle.

[Sebastian (a lady)]

Snap! :D

 

P = 2W + 2L

 

W.....L

=====

1......11, 10, 9, 8, 7, 6

2......10, 9, 8, 7, 6

3......9, 8, 7, 6

4......8, 7, 6

5......7, 6

6......6

 

Ah, the problem of poorly written and unintentionally vague questions. I think they are probably looking for the simpler answer that uses all of the fencing. They also didn't observe that Johnny has a handsaw and can cut down some of the sections to smaller pieces.

 

Reminds me of the science question that asked "Which of the following would you expect to be the hottest planet?" with answers "Mars, Mercury, Venus, Neptune". The desired answer was Mercury, because it was closer to the sun. My son couldn't get past the fact that he knew that Venus was hotter because of its atmosphere. And of course he expected that to be true, because he knew enough about Venus. :lol:

[Carrie12345]

My 8yo is finishing up Calvert Math 3, and *should* be able to do the problem based on what she's been taught about rectangles and perimeter.

 

HOWEVER, being the kind of person she is (and much of our family is), she might be distracted by the "real life" application/extraneous information. If it were in the context of lines alone, or even cubes, she could do it. Visualizing a puppy pen might side track her (for more than one reason, lol).

 

She definitely wouldn't use any sort of formula though. It would be more trial and error.

[linders]

Well, the problem only requires a knowledge of how to caluclate perimeter and basic addition skills. We've had similar in RS math since 3rd grade where they ask for various combinations of things. Maybe she was just thrown by the presentation?

Edited August 18, 2010 by linders

[Grammar Stage Parent]

More than a few...

 

Grammar Stage Parent: Did the test specify whether all fence sections had to be used?

 

No, the test didn't. I agree that the presentation threw her and me off. Singapore's problems are presented in a very formulaic manner, and I still can't find where in their cycle questions of this nature are discussed.

 

Interesting that someone else said that the question isn't college-level, but might be grade 7 algebra level ...

[EKS]

6x6 (this makes a square, which is a special rectangle)

 

 

 

I missed this one when I did the problem. So I guess I would have failed the test. :tongue_smilie:

[nmoira]

 

So I guess I would have failed the test. :tongue_smilie:

You're not the only one. :)

[nmoira]

No, the test didn't. I agree that the presentation threw her and me off. Singapore's problems are presented in a very formulaic manner, and I still can't find where in their cycle questions of this nature are discussed.

Ideally the question should have required the student to list or perhaps sketch (as opposed to draw) the possible rectangles that could be made with all 24 pieces.

 

Both the Singapore IP and MEP have exercises like this. Sometimes they can be head scratchers until you start talking/reasoning your way through it: "Hmmmm... the shortest width can be 1 unit," etc. It's not a difficult concept -- I suspect few students in Singapore 4 would have trouble with "If a rectangle with a perimeter of 24 units has a width of 1 unit, what is its length?" -- however, it's easy to be thrown when confronted with a novel presentation... as you can see from many of our answers here. :tongue_smilie:

 

How did your daughter react to the question? How did she start to attack it?

[Perry]

It seems like a suitable 4th grade question to me.

:iagree:

 

But I would expect many 4th graders wouldn't get it right. That's okay, a test should have questions that differ in difficulty.

 

It isn't a college level question.

[Ali in OR]

For working on this type of problem solving, I think I have pretty much decided to join Drexel University's Math Forum for their Problem of the Week program. I have seen PoWs implemented in a top-notch PS middle school, and they are great for learning how to approach problems, try different methods, and learn how to communicate your answer. This problem looks like a PoW problem to me more than an assessment problem.

 

http://mathforum.org/pows/