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.

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?

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

Elizabeth (aka @cheesemonkeysf on Twitter)

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?

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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.

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?

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?

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.