Write a system of equations that has a solution of (3,-2) that you think most (all) students would think it is easiest solved using the linear combination [elimination] method.It was really fun to see their thinking and push back on them a little bit with--"Hmm...I'm not sure if I'd solve that one with linear combinations. You haven't made it appealing enough to me yet." What I love about tasks like this is some students do it exactly in the way I was hoping based on what we've already done in class and others do something more intuitive (and sometimes simpler) than what I had in mind.
We started our unit of solving linear systems with this activity and I have been really happy with the deeper understanding students have had because of the greater focus on the properties of equality. In a couple classes we even looked at doing similar operations using matrices and ideas from my college linear algebra class. (I think I went through that course hardly knowing why I could do what I did to those matrices and hated how much notebook paper it took to do it...)
Here are a couple of examples of what students said in class today when I talked with them and asked them to share their process with the class:
I started with 3x = 9 because I wanted x=3 and then I made the equation x + y = 1 because that equation was easy to have the correct solution.
3x = 9 (starts in the middle of the problem in a way)
x = 3
x + y = 1 (back to the beginning)Then I made 2x - y = 8 because the 2x would give me the 3x I needed and the 8 would make the 9 when I combine the equations.
x + y = 1
2x - y = 8 (manipulating the y's to cancel and to get his 3x = 9 situation)The y's would also make zero to make someone want to use the combinations method.
I just made up some numbers on the left side and put in the coordinates to see what they would evaluate to:
2x + 3y --> 2(3) + 3(-2) = 0 so the equation 2x + 3y = 0 has a solution of (3,-2).Then I made the next equation to have a -2x because I wanted the x's to cancel when I do combinations and the y-term could be anything. I put in the coordinate again to see what the number on the right side of the equation had to be:
-2x + (anything)y = whatever you get back
-2x + 4y --> -2(3) + 4(-2) = -14 so then the equation -2x + 4y = -14 also has a solution of (3,-2).I wasn't sure how well the activity would go but the student seemed very engaged from the start and those that I talked to had great strategies in how to go about solving the problem.
It didn't take a whole lot of planning up front and it gave the students the opportunity to be creative and come up with unique products that demonstrate their understanding of the concept in the reverse direction.I believe that in doing so students will also have a better understanding of when to use the combinations method instead of the substitution or graphing method when appropriate.