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.
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)
- 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?