I had a great experience with an Algebra I student yesterday. He was solving a system of equations through a word problem. He had the equations set up but didn't know how to use elimination or substitution to solve them.
The system looked like this:
d + q = 200
.10d + .25q = 94
When I asked him to try to solve it, he started coming up with numbers that add to 200, and then used them in the second equation to see if the numbers worked. For example, he tried 96 and 104, which he found didn't work in the second equation; it gave an answer of 35.6, which is too small. He changed the numbers so they would get solutions closer to 94, meaning he made q larger because he noticed that it is multiplied by .25 in the equation, which weights the final answer more heavily than d. I let him do this for a little while (perhaps not long enough). I told him his solution path is absolutely legitimate and will work. However, it's going to take him a while to solve that way, so I wanted to show him an easier way. Then I take him through how to solve using substitution because I think it will make the most sense with the way he was originally thinking about the problem.
I was amazed that this student was actually thinking about the problem. He was the only one in the class who I could say actually understood what the systems mean. He may not be able to do the standard procedure we call substitution, or the one called elimination, but he gets what systems of equations are about.
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