Part of Wordless Wednesday…
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Tag: coordinates
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Cuisenaire Rods Go Pythagorean!
I’ve been playing with the lovely Cuisenaire Rods for a few weeks now. I made the (fortunate) mistake of creating this flower in a past article about coordinate pairs.
The mistake was that I would eventually have to come up with the coordinates for this thing. Fortunate because it gives the MathFourTicians out there something else to teach with the rods!
The center of the flower is the place to start.
Since all the “petals” are attached to the center, that’s probably the best place to begin.
I converted to something I could see.
Since everything is tiny (1 cm), I went to a bigger setup. And some of the coordinates were easy to pick out. So I put those in the big grid, too:
And then I started to do some work…
Next I considered what I was really dealing with: a square. And each side was 1 cm.
According to those crazy Pythagoreans, the diagonal measures :
So half the diagonal is :
I went back to the big grid.
When I put this information on the big grid, it looks like this:
Doing some arithmetic and geometry, I get:
Using my very cool TI-30 calculator, I get that .
I can accurately name the points!
Knowing that each corner pokes out roughly 0.2, I can calculate the coordinates:
From this I can create the ordered quadruples as described here for the petals of the flower. But at this point I’m pretty much needing a break. So I’ll leave that for next time.
Whatcha think? Fun? Share your thoughts in the comments and on Twitter:
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Cuisenaire Rod Art Inspires Thought and Decision
As with all good mathematics, I just spent hours pondering how to set up my super cool Cuisenaire Rod quadtuple.
It was a great experience. Made me wonder why I was being so selfish in keeping it to myself.
This is a perfect thing for kids to do!
The Setup
Seems simple enough. Make a picture on 1cm square graph paper with Cuisenaire Rods. Then write down the coordinates of the rods. Voila! You have the coordinate representation of your picture. Furthermore, you’ve practiced coordinates!
Cool, right?
The rub is that you have to correctly identify the orientation of the rods – are they up-and-down or side-to-side? Or worse – at an angle!
So now we have four things to identify: size/color, angle of rotation, x coordinate, and y coordinate.
Since the rods have width, we can’t just say “put it at (3,2).” What part of the rod goes at (3,2)? One of the four corners? The midpoint of one of the sides?
I just picked one.
My official Cuisenaire Rod Ordered Quadtuple (CROQ) looks like this:
(size of rod, angle of rod, x-value of anchor, y-value of anchor)
The size of the rod is easy.
The angle of the rod is clear, but we have to define what is zero (up-and-down or side-to-side) and how we rotate (always clockwise, counter-clockwise, either way, etc.).
I picked up-and-down to be zero degrees. It is the way I naturally look at the rods. Like this:
See the bright green dot? That’s the anchor point. It stays there (in your mind) and is the bit that you’ll put on the (x, y) point.
Which means that as you change the orientation (or rotate your rod), it sticks around. Like this:
This rod is (5, 90, 0, 0) because it is 5cm long, rotated 90 degrees (clockwise) and the anchor is at (0, 0).
You can do interesting angles too!
Check out the 45 degree one:
This one has the 4-tuple (6, 45, 0, 0). Notice that the rotation is always clockwise.
This works, but isn’t the best way.
When we graph (typically), zero degrees is horizontal. And sticking out to the right of the origin.
Perhaps the best way to represent the rods on the plane is to use this type of set up. But it wasn’t the natural choice for me.
Which is right?
It doesn’t really matter. Unless you’re running a contest (which I think ETA Cuisenaire is putting together).
Try this at home. It simulates what researching mathematicians do everyday:
- Let your child make a piece of artwork with the rods on graph paper. Go ahead and spring for another home set of rods and let him glue it. You can frame it and put it on the wall later.
- Ask him to do this: “Using only numbers, letters and punctuation, write the instructions to recreate your picture.”
- Take a piece of graph paper and recreate their picture yourself, based on what he’s written.
- Compare yours and his. Are they the same? If not, what happened? If so, were there any challenges for you while redoing it?
- Discuss it and let him think about this as long as he wants (minutes, days, weeks, years). Then ask him if he can think of a different way to describe it so it is easier to recreate.
- Repeat as many times as is enjoyable. After a while – sometimes years – he’ll create something similar (probably better) than what I have above.
But, really, which is right?
This is exactly what mathematics is all about. Not which is right, but which feels best. And which one will we agree on.
When ETA Cuisenaire puts together their contest, they’ll have the rules clearly defined. “Right” will only be according to how they set it up. When you enter, they’ll recreate your art based on their rules.
I like my rules. So much so, I’m going to use them next week to give the coordinates for the image at the top of this article. Wanna compare? Do it and post your CROQs in the comments.
(Notice there’s a whole-lotta math goin’ on in that image – not just coordinates!)
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3 Places to See the Coordinate Plane in Action
I was browsing through the book Family Math and found a nifty game called Hurkle. It goes a little something like this:
- Someone picks a place on the coordinate plane (in secret) for Hurkle to live.
- Others guess Hurkle’s home by giving coordinates and pointing out those coordinates.
- When they are wrong, they are given a hint of which way to go (from their guess) to find Hurkle.
The coordinate “space” is around too.
This made me think of the conversation I had with a neighbor the other day while walking our children to the park. When Scotty beams someone in Star Trek, he needs coordinates. But those coordinates must be measured from some origin (0,0,0) in the universe to make any sense. So where’s the origin?
A student of mine long ago found that the origin was Earth. My neighbor argued it should be Vulcan – apparently they started the United Federation of Planets.
And we can find coordinates at work in board games.
With thoughts of these two in my mind, I wonder what ever happened to the game of Battleship. It’s a great game of coordinates along with logic. Once you hit a ship, you have to go in each direction to determine how big the ship is and in what direction.
I’m excited about looking around my world today for more coordinate systems. Where do you see them in your world? And how do you apply them in your teaching? Leave your thoughts and ideas in the comments.
