Continuing from last week, we have another potential Pythagorean’s Theorem Project/Problem. This one was sent in by Steve.
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AUNT BITTY’S GARDENS
Launch: My Aunt Bitty has a business creating “designer gardens”. These are beautiful little triangular gardens that fit into a particular space–usually the corner of a yard. You tell her your space, and she tells you just how to put in the rocks and plants to make it beautiful.
She is making models of different sized gardens that she wants to sell–and she has a problem.
(At this point, the most fun thing to do would to bring in Aunt Bitty as a guest lecturer. For a male teacher, dressing up in a false wig/skirt would be a hoot.)
Aunt Bitty:
“So here’s what I do. I find out the size of the garden–and then I go into my back yard, and experiment planting different gardens of that size until I can get it looking just right! People tend to like corner gardens, so all my designs are nice right triangles with two little legs around the corner and a nice long hypotenuse.”
Aunt Bitty draws a diagram.
“To plan each garden, I hired a group of high school kids down the street to make a balsa wood frame just that size. Then, I put the frame down in my back yard, and design the garden inside the frame.”
“It worked great for a while. Here’s the frame they made with legs of 3 meters and 4 meters, and a hypotenuse of 5 meters.”
(Show frame)
Here’s the frame they made with legs of 2.5 meters and 6 meters, and a hypoteneuse of 6.5 meters.
(Show frame.)
But then I needed a frame for a garden with legs 3 meters and 4 meters and a hypoteneuse of 6.5 meters–and look what they gave me!
(Show frame.)
That corner there is just not a nice right angle! So I fired them.
I’m looking for a group who can make frames for me–and my nephew said his class could help me out. He’s got the materials (glue and balsa wood and all that) here. I’d like each group to make me three frames. I’ll tell you the size of the frames I need, and then you-all just go ahead and make ’em. Make sure that each frame is a nice right triangle with a real corner. Make sure that angle between the two legs is a right triangle. I don’t want any more frames looking like this.”
Facilitator Notes: Each group should be given two sets of triangle dimensions that will work, and one that won’t. Teacher discretion about whether each group gets its own three unique triangles, or whether there is overlap between triangles. It will take some experimenting to find appropriate size triangles but probably won’t be too hard, since you can multiply a pythagorean triple by any rational number to get three numbers that will work (e.g. the triple 5/12/13 can become 1.25, 3, 3.25 (dividing by 4) or it can be 3, 7.2, 7.8 (multiplying by 0.6) etc.).
PART 2: Web search and Presentation.
After students have discovered that some sets of numbers will make a right triangle, and others won’t.
Is there a mathematical formula you can use when Aunt Bitty gives you three lengths, to check whether or not you can make a right triangular frame? Your job is to:
1. Find the formula
2. Prepare a power-point presentation or a hands-on activity to convince Aunt Bitty that your formula will always work! (She is a sceptical old aunt, so you will work hard to be clear and to convince her that it will ALWAYS work.)
The teacher can provide web resources with various proofs of the pythagorean theorem. My favorite for visual clarity is the Proof by Rugs from IMP –but there are many approaches.
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My comments:
- I’m a huge fan of this problem. In particular I’m a huge fan of the “skeptic” role played by Aunt Bitty. Steven and I got into a good email discussion about this problem. He said a friend told him there are three levels of mathematical proof: convince yourself, convince a friend, convince an enemy (played by Aunt Bitty in this case). I like that framework of proof a lot.
- Speaking of which, Steve and I agree that some sort of hands-on activity with various wood (or other material) lengths is in order.
- Hopefully students will be able to see the relationship in right triangles and conditions for making an acute or obtuse triangle (i.e. when A²+B²<C² or A²+B²>C²). With a little prodding I’m sure they could.
- I also like having students develop their own hands-on activity to convince Aunt Bitty, or, say, a younger sibling. Granted the hands-on activities will all probably end up being similar, but wouldn’t it be great if students had to come up with step-by-step instructions and/or their own worksheets?
- I wonder if there’s “too much” up front information. Could Aunt Bitty roll out different aspects of the project in stages? For example, provide that first diagram, and perhaps and area of garden space and have students come up with the dimensions at first. Then examine student results, followed by the incorrect diagram of the obtuse triangle. At that point, there may be a “need to know” regarding the possibility of a right triangle with particular dimensions. Here’s an example of what I mean.
Artifact
Aunt Bitty makes these gardens in the corners of yards.
She’s been commissioned to create corner gardens with areas of 6 m², 30 m², and 60 m². Whoever gives her the dimensions first earns a hard candy and two bits.
If you want you can tell students that she only deals with whole numbers because she hates calculators or something.
At this point, you let students struggle with this problem a bit before you start introducing concepts like hypotenuse. You can use graph paper, Geometer’s Sketchpad, or some other resource if you so desire.
- I think this problem offers a myriad of paths to extensions. Creating a stone path around each garden brings perimeter into consideration. Constructing a similar garden for your school would bring in some hands-on community service. Any other ideas?
- The IMP Pythagorean’s Theorem proof that Steve pointed me to had a bunch of other nice ready-to-go Pythagorean Theorem activities. Be sure to check it out if you’re still looking for inspiration.
What do you guys think? Is this something you could use in your classrooms?
emergent math 