I gave my Advanced Algebra 2 students the following question today to open class:
Write a system of equations with two equations and two variables with a solution of (2,3). Be prepared to justify your answer in two ways.
We have done solving systems of equations by graphing and substitution. We started elimination yesterday. I had hoped that they would come up with equations by working backwards - for example, 2 + 3 = 5, so x + y = 5 is one equation they could use in the system. I had hoped that once they got the equations, they would use either substitution or elimination to check the solution was (2,3).
What I got was blank stares. They had no idea where to start. I ask them to give me one equation and we'll put together a system. Then the student I have from the ED (emotionally disturbed) unit - who is bright mathematically - raises his hand and gives me an equation. Then he gives me a second one. And, right or wrong, I take the time to show them that (2,3) is a solution to it, first by substituting the values in to check and secondly by using substitution since it made the most sense for the equations he had given me.
I asked them to give me two more equations, and after a few moments, I had two more equations, one each from two different students. Demonstrated that (2,3) was the solution again. I asked them for two more equations and got two more equations, this time a little quicker from my students.
After I was done checking the third system of equations, I did talk with them a little about how I wanted them to understand what the solution to the system meant.
I'm still behind on More Good Questions and I guess I need to read some more to have a better idea of how to handle open questions with my classes. I decided to try this with my Advanced Algebra 2 class since they are a little more willing to think and work at stuff than my regular kids. I'm not totally sure what I wanted to get out this, but I had hoped for better responses from them. Will have to read more and think more before bringing in the next open question.
I actually kind of dig this question. My solution strategies weren't at all what yours were. Here are mine...
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I know that any system of the form ax+by=0, cx+dy=0 will have intersection point (0,0). So I make one up. Say it's 2x+3y=0, 4x+5y=0. Then I translate everything up 3 and over 2, so now it's 2(x-2)+3(y-3)=0, 4(x-2)+5(y-3)=0. Distribute, reorganize, voila.
(2)
I tend to think of linear equations in y=mx+b form. So I know I need two different lines of this form that go through (2,3). I choose my two slopes (say, -1 and 1), solve for y-intercepts. And again voila.
As I thought that through, I wondered whether a way to herd those blank looks along might be this...Have each kid write an equation for A line that goes through (2,3). Then they have to find someone who has a different line. Together they make a system. Kids verify this and correct any errors they discover in the process. Repeat once more with a new point, and maybe now they're ready for the larger task of writing their own system with a given solution.
Maybe.
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