Questions and Dialogue

At the beginning of our unit on systems of equations we discussed the question, What is a solution?
We returned to the question as I introduced solving linear inequalities and systems of linear inequalities.  Students participated readily ... less pause than the first time around ...

What is a solution?

Student A: anything that satisfies the equation ...
Me:  could someone add to that thought?
Student B:  it's the numbers that solve the equation ...
Me:  Hmmmm, I don't think we can use the root word in the definition ... any other thoughts?
Student C:  it fits in the equation
Me:  OK ... it satisfies, it solves, it fits ... any other thought?
Student D:  it makes the equation true
Me:  Yes ... to all of the above!
Me:  Now, if I give you an equation with 2 variables, how might you represent the solution visually?
Student E:  could you restate that question?
Me:  (I write on the board) y = 2x + 3; How might I represent the solution to this equation visually?
Student F:  (2, 7)
Me:  OK, that point is true.  How else might I represent the solution visually?
(pause)
Me:  OK ... are there other solutions besides (2, 7)?
Several students:  yes ... many of them ... infinite number
Me:  Yes!  So how might I represent those infinite solutions visually?
Student G:  a line
Me:  Yes!  So how or why does a line represent the infinite solutions to that equation?
Student H:  a line continues in both directions forever
Student I:  a line is made of infinite number of points
Me:  Yes!

We then extended our conversation to include inequalities ... talking about why we shade and what that tells us.

I want to get better at asking questions ... I want to ask better questions.  I also want to develop strong math discourse in our class!

These are the "essential" questions suggested by our district curriculum document:

1. Describe the solution to a system of equations with 3 variables.
2. How do you determine the variables for a system of equations?
3. How do you write a system of equations from information given in a problem situation?
4. How do you know if your solution to a system of equations is reasonable?
5. How do you connect the solution to a system of equations to the context of a problem 
6. situation?
7. How do you decide which method to use to solve a particular system of equations?
8. How do you solve systems with more than two variables?

These questions hit the surface and skim across the topic.  I want something deeper, richer for my students.