We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?

As students first started looking, I heard some of the following:

- positive x axis
- x is positive, y equals 0
- they come together on (2,0)
- (?,y*0)
- when z is on top of w, z is on the positive side on the x axis

Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?

As students looked for longer, I heard some of the following:

- The length of the line segment from the origin to Z is the x coordinate of W.
- w=((distance of z from origin),0)
- The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?

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howardat58

March 29, 2016 at 8:57 pm

This is real math. This is what engineers, scientists, statisticians do all the time (not quite all!)

Some queries:

I am assuming that the students have access to the “hand” and can move it anywhere, as shown where it is moving the point on a circle.

How do you hide the equation for z ?

I guess they knew that circles were involved somewhere

As a followup they could try this for a linear equation, say z=3x-2y but with no immediately preceding stuff. Some hints if needed on effective strategies for moving the (x,y) point.

Then some more general 2nd degree ones, Even z=xy or z=xy+x+y=1

Eventually a reality check, which is that in the real world you do not have unlimited information, and may need just sample points, or restrictions on the variation in the inputs.