Math in the garden — going beyond the common core

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:

math common core

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:

my work

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:

w-1=10

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.

w=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:

abcd

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).

11x19

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:

real solution

(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:

math common core

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 🙂

gardens

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About Steve Goldberg

I teach U.S. History at Research Triangle High School, a public charter school in Durham, NC, whose mission is to incubate, prove and scale innovative models of teaching and learning.
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5 Responses to Math in the garden — going beyond the common core

  1. sarahsmilesalot says:

    Common Core is no different than previous math curricula. Real world application has never been the concern, but number manipulation in as short a period of time has been. Therefore my first question to TLC is how much time will be given students to solve ‘David’s problem’? To include a look at famous gardens relevant to this math problem might take much more than a class period. How will students learn about the connection between this one math problem and the gardens of Europe? Will the instructor tell the class that there is a connection or will students be allowed to discover this connection. Such a discovery might take weeks.

    Question 2 – Will students answer ‘David’s problem’ individually or work with other students?

    Common Core contnues to require students to work through math problems individually. The required level of knowlege is higher/deeper for students with no change in any other aspect of how the information is learned. The designers of this new curriculum, The Hunt Institute, has created yet another educational fade that will run its traditional 10 years that will be paid for by the students (and teachers) caught in this particular pendulum swing from its predecessor No Child Left Behind.

    I hope TLC will design each math problem with every element of what students and teachers must consider in its solution.

    • Steve Goldberg says:

      Hi Sarah. Thanks for the comment!

      Great point about individual work — at TLC we will almost always be working together when we are together. When we are working on math at home, each student will typically be working on his/her own, learning the relevant concepts for where that student is. But we will also be blogging about math and reading each others’ blogs.

      As for exploring gardens, you are right — that could take a while. My vision is that before making a garden at TLC, we will do some basic exploration of famous gardens from around the world. That might take most of a week. Once we decide on a style, we will work to make that garden a reality.

      And at some point, when we are thinking about how to define the outside of the garden, I might show students that common core problem and point them to this blog post and we can discuss such concepts as area and perimeter and budgetary constraints.

  2. sarahsmilesalot says:

    I used wiki to create the set up for the garden my sixth graders designed. They could talk to each other and me as well as post research and web 2 tools. I liked having everything in one place. You are welcome to check it out if it is useful to you.

    Stimulating stuff you are doing, Steve. How was the Duke game?

  3. jplgough says:

    So…I think the square garden is the most interesting part of the problem. Interesting, huh. I wrote a blog post, Meaning to Mechanics, about using square-foot gardens to teach polynomials.

    Do you know about Square-Foot Gardens and Mel Bartholomew? Have you watched Pam Warhurst: How we can eat our landscapes from TED?

    I think an interesting problem would be to ask about urban deserts. Mel’s motto: Do Something Positive Against World Hunger. Let’s Stop Sending Food, Let’s start teaching Self Sufficiency. Pam Warhurst says we want to start a revolution. How might we (the learners at TLC) engage in design that feeds communities and teaches self sufficiency? Could we persuade local business owners to allow us to design, plant, and grow gardens to feed our community? How much would it cost? What materials do we need? What would the design look like? What would we grow? Would we interview community members about what they would eat, know how to cook, etc?

    I might mention that you are not keeping it real when you use $100 gold bricks as the border of your garden. Pea gravel, mulch, concrete or stone tiles, maybe, but not gold bricks. Could the border be green too? What plants make a good footpath border?

    What do you think?

  4. Eliza says:

    This is the best blog post I have ever read :).

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