Graphing – MathFour

Graphing Gone Groovy [Conference Notes]

Bon — Tue, 03 Feb 2015 11:50:26 +0000

The fun folks at Texas Region 4 invited me to speak at their math conference. In addition to a talk on Pinterest for Math Teachers, I talked to a group about graphing.

And since I’m a cheeseball, I called it “Grueling Graphs Gone Groovy – Getting Guys and Gals to Graph.”

Yeah, pretty bad, I know.

What’s worse – I get to do it again at the this summer!

research paper format

The talk was successful!

We managed to talk about some good stuff on graphing polynomials and rational functions. We just started on transformations when we ran out of time.

The best part was a teacher in the audience shared an awesome tip: when finding vertical asymptotes, ask, “Where can the denominator not be zero.”

I have always told students they set it equal to zero and solve for x. This results in the asymptotes. Which is true. But *poofing* away the numerator is even more weird than *poofing* away the denominator (which you do when finding x-intercepts).

So her tip turned the freakish into the (hopefully) intuitive.

Get the notes!

Here are the annotated notes for the session. (And I just recognized that I forgot to include the holes in the graph of the rational function. So weave that in if you use these. :D)

And if you’re heading to Houston in the summer for CAMT 2015, plan to come. The talk will be even better!

Share this with your PLN on Twitter, Facebook and Pinterest!

Graphing with the Lite-Brite

Bon — Tue, 21 Aug 2012 16:54:39 +0000

It’s not just for playtime anymore!

Drag out the old one, pick up a used Lite-Brite or grab a brand new Hasbro Lite-Brite.

Or… click to get the online Lite-Brite simulator (the technology these days!). And get ready – because that old classic is about to become your child’s favorite graphing lesson!

1. Draw a picture on a Lite-Brite template.

Download the Lite-Brite Classic Template if you have the classic toy. If you’re using the new one or the simulator, get the Lite-Brite Square Template.

Draw a picture on it in pencil. Here’s mine:

2. Determine the best peg placement.

Rough sketches don’t quite work with the peg placement. So draw the peg holes on the template as close to your lines as possible. Start noticing patterns.

3. Use colors!

Pick some good colors you’ll want to use when you’re graphing your picture. Unless you and your kids know the exact number of Lite-Brite pegs you have, in which colors, you’ll probably have to tweak this in the next few steps.

4. Figure out where to start

Here’s where things get very interesting. The peg holes are not lined up in what you would call a Cartesian coordinate plane. They’re not really in any kind of coordinate plane system. At least not that I know of.

The crazy structure in the grid doesn’t really matter. The fact that you can still count over and up to get where you want to go is what matters.

You and your children get to devise the way you count over and up.

5. Count and push!

Your child can follow the pattern on the template and push the pegs in the Lite-Brite. This offers practice for the skills they’ll use when graphing and working with the slope of a line. This is the precursor of calculus!

Notice that when I was graphing my drawing, I had to change the door from solid to hollow. I ran out of pegs. #ARG

Try it!

Find your Lite-Brite and download the template: classic or square. Post a link to your finished work in the comments and tell me how it went!

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Put a Coordinate Plane on Your Floor

Bon Crowder — Wed, 04 Apr 2012 15:05:57 +0000

Part of Wordless Wednesday…

Thoughts? Share them in the comments. Also, [embedtweet text=”tell your friends on twitter” tweet=”Can%20you%20make%20a%20coordinate%20plane%20on%20your%20floor%3F%20Check%20this%20one%20out!%20http%3A%2F%2Fmathfour.com%2F%3Fp%3D6087%20via%20%40mathfour”].

Cuisenaire Rod Art Inspires Thought and Decision

Bon Crowder — Sun, 31 Jul 2011 17:11:19 +0000

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!)

Cuisenaire Rods

Bon Crowder — Sun, 24 Jul 2011 11:17:06 +0000

Cuisenaire Rods are brightly colored wooden sticks. Technically, they’re “proportionally sized rectangular parallelepipeds.” (But only say that if you want to hear your 3 year old repeat something really cute!)

The “proportional” thing is important. The white ones are 1cm square, the red ones are twice as long and each color is 1cm more than the next color.

I’m anticipating many articles and videos on how to teach with these (since the possibilities with these things are virtually unlimited), so I thought I would start a running series. Here are the ideas and the links to the articles/videos:

Cuisenaire Rods – graphing and practicing coordinate pairs (see video below)
Cuisenaire Rod Art Inspires Thought and Decision – more graphing and thinking like a mathematician
Cuisenaire Rods Go Pythagorean! – figuring out the coordinates on “tilted” rods on a graph
Playing with Algebra Concepts
Adding Fractions with Cuisenaire Rods
Relations – Greater Than and Less Than
Algebra – Solving Equations with Beautiful Colors
Set Theory

Creating Coordinate Pairs with Pictures

Share your own ideas on how to use Cuisenaire Rods in the comments!

P.S. I spent my hard earned money on these at Teacher Heaven. This small set was $15 in the store, but I see you can get it for less than $13 online.

This is also shared on Works for Me Wednesday.