Kachima is making triangular bandanas for the dogs and cats in her pet club. The base of the bandana is the length of the collar with 4 inches added to each end to tie it on. The height is 1/2 of the collar length. If Kachima’s dog has a collar length of 12 inches, how much fabric does she need in square inches? If Kachima makes a bandana for her friend’s cat with a 6-inch collar, how much fabric does Kachima need in square inches?

I’m not a pet owner so somebody please set me straight: is pet apparel a productive context for mathematical inquiry? Does PETA know about this?

**Previously**: Unnatural Currents

**Featured Comment**:

Kari:

Yes, they make triangular bandanas for dogs, single ply. Usually cut with zig-zag scissors. My dog comes home from every stay at the kennel looking like a boy scout. every. time. This is still a terrible problem though.

## 15 Comments

## Breedeen

May 23, 2012 - 8:49 pm -Also, do people really make triangular bandanas?

I mean, I could be wrong about this, but don’t people usually just take a square bandana and fold it in half to get the triangle? Maybe I’ve been doing this wrong for my whole life…egads!

## Marshall Thompson

May 23, 2012 - 9:07 pm -I actually am shocked by how much money and time people spend obsessing about their dogs and cats. I have learned over time that yes, people do think about these sorts of things!

Now this is clearly a stretch, but how much of a stretch is it? Dogs come in many different sizes, and dog necks come in many different radii. This I think could be surprisingly fertile ground…

…I wonder how much more cloth I’d need for a St. Bernard vs a Chihuahua? Or for that matter how much edible meat per breed?

## JenW

May 23, 2012 - 9:19 pm -No, it’s not a particularly good pet-related question *or* sewing-related question as you don’t typically buy fabric by the square-inch. And, it would depend on the relative position of those square inches.

I’d say a slightly better math-related-to-sewing question might be amended to include something like “Kachima’s mother is a quilter and has fabric in ‘fat quarters’. Can Kachima make both a dog bandana and a cat bandana from one fat quarter?” or “How many bandanas for her dog can she make from one fat quarter, with a standard hem with a 5/8″ seam-allowance?”

## Kari

May 23, 2012 - 9:26 pm -Yes, they make triangular bandanas for dogs, single ply. Usually cut with zig-zag scissors. My dog comes home from every stay at the kennel looking like a boy scout. every. time.

This is still a terrible problem though.

## Tim

May 23, 2012 - 10:18 pm -Answer: 270 square inches

Assumption: The bolt of fabric is 45 inches wide

Assumption: The fabric store will sell you a partial yard

Assumption: The triangles are isoceles triangles

Assumption: The cuts of the fabric from the store are exact

When you buy fabric, it comes on a bolt (the thing it’s wrapped around). Some bolts have 45 inch wide fabric, some have 60. You decide how much length you want of the 45 or 60 inch bolt, it is cut off, and you buy it. So, for a 45 inch bolt of fabric, if you buy 1 yard of fabric, the bolt is cut at 1 yard in length, and you end up with a piece of fabric that is roughly 36 inches (1 yard) by 45 inches.

But you have to be careful in deciding how much fabric to buy. If you take the 61 square inches that will actually end up as part of the bandanas and only purchase that much fabric, you’ll end up with a piece of fabric that is 45 inches wide and just under 2 inches long. That’s not very practical for making bandanas.

So you have to figure out how to lay out the cuts you need on the fabric. For this project, you’d probably lay the long end of the triangles against the 45 inch edge. The bandana in this example with the largest height has a height of 6 inches, so you end up needing 6 inches of the 45 inch fabric. 6 x 45 = 270.

## Michael

May 24, 2012 - 7:19 am -Perhaps the question writers are working from their own personal bias and experience. Maybe one of them own or at least go to a pet club.

That might be like a basket ball player or fan making quadratic equation problems or someone who rides a train making a “two trains leave the station . . . ” problem.

Does trying to force an applied context of a math concept help students learn that concept initially, or do they need to have ample practice of that concept context free, before attempting an applied problem?

## Peter Price

May 24, 2012 - 10:33 am -Is this question better than a context-free practice exercise? Or worse? Sure, a person actually making a triangular bandanna would work out the amount of cloth differently. But does that really matter?

If students know that math exercises aren’t meant to be taken as literal examples of how math is used in real life, does a picture of a cute dog with a bandanna make this problem more interesting, or would students really rather practice context-free plug-and-pray exercises?

My feeling is that some students wouldn’t care, they dislike math anyway. Some would enjoy seeing a picture of a dog in the middle of math class. Some would be happy to play the game, and who cares if it’s realistic. And a small few would say “hang on, do pet owners really calculate the area of a triangle when making a bandanna?”

Perhaps we care more than the students whether the math they use for practice is realistic or not?

## Timfc

May 24, 2012 - 10:38 am -A bit of response to #7:

There’s decent evidence that the answer to the question:

Does trying to force an applied context of a math concept help students learn that concept initially, or do they need to have ample practice of that concept context free, before attempting an applied problem?

>>>

Is: eh, it depends. But, I’d say that decades of schooling of thousands of kids suggest that practice->applied problems only works for a very small portion of the population and most people end up hating math instead.

There’s a set of studies about “candy sellers” who, when operating in the context of selling candy are quite sophisticated in terms of their arithmetic and fluency. When you give them the same numbers and same operation in a “naked math” problem, they can’t do it.

Similarly, there are numerous studies showing that students might have procedural fluency but can’t do a word problem that can be solved using the procedure.

diSessa and Wagner talk about transfer-in-context and mean that students see different elements of items as the ‘structure’ of the problem than math teachers. This means that learning to do math is more than just teaching procedures, it’s teaching what the “mathematical structure” is and how to recognize it. It’s reinforcing the structure idea, much more so than the procedure.

## Johanna

May 24, 2012 - 10:59 am -I think so-called decontextualized problems do have a context: it’s just the school math context. A good contextualized problem helps students connect their context to the school math context, so it works on a whole other (very crucial) skill on top of learning (and motivating) the mathematics inherent in the situation. Of course the best problems pedagogically are the ones in which the school math provides resources to understand/resolve the problem situation AND the student’s experience with the problem context provides resources to understand/make-sense-of the math.

## Elizabeth

May 25, 2012 - 8:10 pm -OK, look, people. We are in a serious recession. We need to think outside the box. Whatever people can do to boost revenue — and taxes — is apparently fair game. This seems to include buying bandanas at the dollar store, cutting them down the diagonal (cf. Breedeen #2), and selling them for a whopping marginal profit.

McGraw Hill is just trying to do their part. That is all.

Elizabeth (aka @cheesemonkeysf on Twitter)

## Fitz

May 26, 2012 - 4:27 pm -We have a school service learning fundraiser where students make these same triangular bandanas for dogs and sell them (the money goes to the no-kill pet shelter). They make three different sizes. I don’t know that the problem by itself does much, but the math of pet bandana sales is real at our school.

## Dan Meyer

May 26, 2012 - 6:39 pm -Fitz:I’m sure people — students, even — make bandanas for different-sized dogs. But are they really measuring the circumference of the dog’s neck to calculate the material?

## Fitz

May 29, 2012 - 4:13 pm -I’ll ask. I’m not sure (I’m in the science dept, not math). I doubt it. But I still think it is a productive context for math inquiry, it’s just that this word problem is trying to squeeze circle math out of a triangle problem. The kids can just cut a triangle that they like for the dog and, after making sure it fits the right size kind of dog, they use their triangle, rectangle, money, and profit math. Not circle math.