A Little Inquiry For Teachers

My summer begins Friday. One of my goals over the summer months is get my thoughts down about the mathematics behind the applet below. It is stunningly simple, but it is the details I want to make clear. So, I present it to you as a sort of summer inquiry activity for teachers. As you move the green points around and as you drag the purple-ish circle around and change its radius, what happens?

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Follow Up: After sharing this with some of my students after one of their exams, they expressed a desire to be be able to place points and draw lines, so I updated the applet to reflect their wishes.

I was pleasantly surprised over how much time my students spent with this applet and the quality of their observations.




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Can't Quite Get My Head Around The Numbers

A lot of BIG numbers is this story.

• A month's worth of leaking oil could fill enough gallon milk jugs to stretch more than 11,300 miles.
• At worst, it's enough to fill 102 school gymnasiums to the ceiling with oil.
• The Gulf has five billion drops of water for every drop of oil.

Random or Real?

A WCYDWT moment from Mind Your Decisions centering on the online game ARORA. Can you tell the difference between random data an real data?

This Changes My Game

I have been trying to create moodle questions that get my students to construct something in GeoGebra, get some sort of information from the construction, and answer the question. I have been struggling with the idea that any calculated moodle questions I come up with is ultimately reduced to an algorithm. Nevertheless, if my students are unaware of the algebra behind the algorithm, and need to think of what to construct, then perhaps that is O.K.

One question I have used recently is this:

A circle has a center (17,15) and a radius 59. Write the equation of the circle. Then, find the larger of the two y-values when x=33.

By algorithm, the better algebra students will easily be able to solve this circle equation for the two y-values, but the students in this particular class are not my better algebra students. What I hope would happen is for them to graph the circle in GeoGebra (meaning they must write the equation correctly...#1 goal),

find a good window setting (which is essentially scrolling out until you can see the entire circle...do they know what they are looking for? Goal #2),

graph the line x = 33 (which is difficult to do on my calculator of choice, the Nspire, or other graphing calculators),

constructing the two intersection points, and taking the larger of the two y-values (recognizing that this is the desired answer to the question),

To answer the question correctly, there is still a lot of intuitive mathematical understanding that is being displayed, especially when you ask them why the answer must be 71.79.

I have been encouraging my students to find different web-based resources, lately, and one of my students found a web-based calculator web2.0calc.Imagine my surprise when one of my students did this to solve this problem:

producing this result

Not a CAS, but a pretty nifty numerical solver. And intuitive! When I asked the student what made him try this, he told me that he just wrote the equation, and substituted 33 in for x like the question asked him to do. There were two answers for y, so he used the larger of the two. Intuitive.

THIS changes my game.

Go ahead and try this for yourself on the calculator below.

Function Family Guy (Solution)

Here is one.

and with the grid and axis hidden

With the "model" for a comparison

What I Wish I Did First

Though things have worked out fine, I wish I had given this Magic Point first.


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Function Family Guy

I discussed my approach to function families in an earlier post. The instructions for each of these graphing activities were the same:

Write an equation for each of the following exactly as you would enter them into your Nspire, sketching the graph of each as you go. Only when you have checked your equations and graph with me should you graph them on the Nspire.

Nothing fancy, but it was fun. Kids whined about the numbers (too many decimals), and argued about the finished result (there are two famous football-shaped headed cartoon characters). We did all of these on the Nspire (hmm...might be my first Nspire-related post), but we also used GeoGebra. There are some differences in how you enter the equations which make the Nspire a little better for this activity.

For example, to enter the equation of a parabola from vertex A to B in the graph below, it would look like this on the Nspire

But like this in GeoGebra

I think this might be one area where the Nspire is superior, but even as I write this sentence, I think of how graphing circles and and other equations is so much easier in GeoGebra. You do not need to isolate y in order to graph a function in GeoGebra. You can graph non-functions like x = 4. Ellipses and Hyperbolas are peices of cake (or of cones...you choose).

For example, graphing circles in GeoGebra look like this

 but look like this in the Nspire

where you must graph the circle in two parts, top and bottom. Of course, I hope that by now, my students will graph a circle on the Nspire like this,

using function notation and the algebraic representation of geometric transformations, reflecting the top half of the circle in f1 over the x-axis, then translating the reflected circle up six units.

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1. A parabola with vertex A, from A to B.
2. A parabola with vertex A, from A to C.
3. A parabola with vertex D, from D to F.
4. A CUBIC with inflection point D, from D to E.
5. The LEFT HALF of a circle from B to E.
6. The RIGHT HALF of a circle from C to F.
7. A line segment from (1,1) to (1.5, 0.5).
8. A line segment from (1.5, 0.5) to (1,0).
9. The LEFT HALF of a circle, centered at (1,-1) and radius of 0.5.
10. A circle centered at (-3,1) with a radius of 1.25.
11. A circle centered at (4, 0.5) with a radius of 1.25.
12. A parabola from the vertex (3,2) and stopping at (4, 2.25).
13. A parabola from the vertex (-2, 2.5) and stopping at (-3, 2.75).
14. A parabola with vertex (4,0), opening downward, from (3, -0.25) to (5, -0.25).
15. A parabola with vertex (-3,0.5), opening downward, from (-4, 0.25) to (-2, 0.25).

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1. Circle centered at (1,3) with radius 0.5.
2. Square root from vertex (1.5, 3) to (0.5, 3.25).
3. Square root from vertex (0.5, 3) to (-0.5, 3.25).
4. Square root from vertex (0.5, 3) to (-2, 2.75).
5. A cubic from inflection point (-2, 2.75) to (-3,2).
6. A parabola with vertex (1, 3.75) from (0, 3.5) to (2, 3.5).
7. A circle with center (-3,1) and radius 1.
8. A cubic from inflection point (-2, -0.75) to (-3,0).
9. A square root from vertex (0.25, -1) to (-2, -0.75).
10. A parabola from vertex (0,-1.25) to (-1, -0.75).
11. A parabola with vertex (2.25, 3.75) from (1, 3.5) to (3, 2).
12. The bottom half of a circle centered at (2.25, 2) with radius 0.75.
13. A parabola with vertex (-0.25, 3.75) from (-1, 2.75) to (0, 3.5).
14. A square root from vertex (2.75, -1) to (2.5, 1.25).
15. A parabola with vertex (1, -1.5) from (-1,-1) to (3, -1).

The Magic Point

I provided this applet to my students a couple of weeks ago. It fit well with where we were at the time, exploring matrix transformations and all. I left it on my class Moodle for a week before we began talking about it in depth. My only question to the students was "What does point D have to do with anything?"

I pose the same question to you.

Or perhaps you will share a different question to ask.


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