I just spent about 20 minutes looking over some common core sample 8th grade math problems, posted on the web by the state of New York.
In general, they seem reasonable, but this problem bothered me:
The first part of this problem that bothered me is that it’s irrelevant that David had a square garden. He might just as well have had a circular garden, or a pentagon-shaped garden. In fact, David might be making his first garden — the part about the square is irrelevant information, and I suspect it’s in the problem to distract/trick people.
Square gardens aside, let’s try to get into this problem — let’s assume that for some reason, David wanted a garden where the length is twice the width, minus three feet. It’s a little odd, but we can work with that.
Here’s how I set this problem up:
The length (l) is 2 times the width, minus three.
When you add the sides all up (that’s the perimeter), you get 6w-6, and we’re told that for some odd reason, David wants the perimeter to be 60 feet.
That leads us to the equation 6w-6 = 60 , and when we divide both sides by 6, we get:
so w, the width of the garden, is 11.
When we plug 11 in for w (I used green), we get dimensions of 19 x 11, because the length is 2 times 11, minus three — which is 19. The width is still 11.
That’s an okay math problem, but I think it has the potential to be a more interesting problem if we made it more open-ended and said something like this:
David is thinking about building a 11 x 19 garden, and he wants to surround it with square gold bricks measuring 1 feet by 1 feet. These bricks cost $100 each, so David can only afford 60 of them (his budget for that part of the project is $6000).
Is David thinking about this project correctly? Explain why or why not.
And the answer — which would require words in addition to numbers, just like in the real world — would be “no — just because the perimeter is 60 does not mean that you need 60 blocks to surround it, because you also have four corner pieces to deal with.”
See the corners, A, B, C, and D below:
So it would cost David $6400 to surround the garden in the way he described.
But rather than saying “sorry, dude — no garden,” a better solution would be to present an alternative to David. Why not make the gold border part of the 11 x 19 dimensions — the garden inside would be a little smaller, but the overall look and feel would still have the magic 2w-3 x w ratio, which apparently matters to David.
We’re not told why David is all hung up on 11 x 19 as his garden dimensions. Maybe David saw some famous garden and decided that he liked its dimensions (I’m trying to draw 11 x 19 on the first image below, but it’s hard to draw on green, and my highlighting didn’t help).
If David told me that was the look he was going for — a Versailles sort of ratio — I could work with that — and we could even get all multi-disciplinary, and learn about Louis XIV and his famous Palace at Versailles.
But getting back to our friend, David, I find it unlikely that a real client in the world would present the problem the way New York’s common core problem presents it.
Can you imagine this conversation?
“Hello? Yes, my name is David, and I want to redesign my square garden.”
“Hi David. I’m a talented garden designer — I’d like to help you — tell me a little about what you have in mind.”
“Well, I want it to be a rectangle with a length that is three feet shorter than its width.”
“Um, okay — I’m not sure why you told me that it was square, but how big do you want it?”
“I want the perimeter to be 60 feet.”
“Okay, David — why do you want the perimeter to be 60 feet?”
“Because I want to see if you can solve the sort of math problems you might see in the common core. Bwah ha ha ha!!”
I made up the part about the golden bricks, but in the real world, it’s likely that David might have some sort of reason for making the perimeter 60 feet — it would likely be a budget constraint. And rather than just accept the 60 feet and solve the problem, I’d love to see students suggest an alternative solution, which might allow him to come in $400 under budget, rather than $400 over budget.
Instead of using gold bricks to surround the garden, we could make the gold bricks part of the outer dimensions. The outside of the garden (the blue line) would still be 11 x 19, but now we’d only be using 56 gold bricks:
(I drew the second set of “gold” bricks in purple above so they would not get confused with the outer gold bricks — and I like purple).
The way the folks in New York frame the problem, it’s not math — it’s calculation — and there’s no room for creativity:
When we do math at Triangle Learning Community middle school, we will use real-world problems. We will also do calculations, so we will cover the common core, but we will strive for a broader understanding of how and why math might apply to this sort of situation in the real world. We’d work in economics, too — maybe it makes sense to surround the garden with silver, rather than gold. How much does silver cost relative to gold?
There’s a good chance we will encounter this sort of math problem in the context of designing and building our own garden (that’s something we plan to do at TLC).
And when we make our garden, we will not limit our garden to rectangular dimensions “with a length that is three feet shorter than twice its width” — because nobody talks that way outside of math problems…
We’d keep it real. There’s lots of math and problem solving in the real world.
And we might learn a thing or two about famous gardens around the world in the process — those looked pretty cool — here’s another selection of a Google image search for “famous gardens” — not a bad image for the start of a Monday morning 🙂