Reply To: How do you define solutions?

[Kyle Pearce]

So happy that you articulated that you’re referencing a solution to a system of equations. I was initially thinking “the solution to a problem” (in general).

I’m definitely of the thinking that students need experiences to build their expertise and flexibility with a concept and emerging strategies. 
I wonder if bulding the understanding of “solution” in the area of systems of equations could be built off of context.

In other words, let’s build on what we know “solution” to mean for all of the math problems we’ve encountered throughout our experience.

Problems like:

• There are 2 apples. Sonya buys 3 more apples. How many apples are there in total?
• There are 3 boxes with 12 donuts in each. How many donuts are there?
• There are 84 donuts total in 7 identical boxes. How many donuts are in 3 boxes?
• and many more…

In the above, students could probably articulate that the “solution” is the answer to the question.

When we start solving systems of equations, our definition of “solution” becomes more precise. It isn’t the answer to ANY problem, but more specific problems like this one:

• In my class, you can buy-out of a detention for $105 plus $6.25 per detention, while in Jon’s class he charges $21.50 per detention. How many detentions would you need to get so that the cost to buy-out is the same from both teachers?

I wonder if you were to do a bunch of exploration (sounds like you already have) of systems of equations and then give students different systems with different question “types” and basically see if they can land on the really important information that helps them answer all of those problems… Could they land on the really important “part” is where the equations are equivalent. Revealing the point of intersection “unlocks” the problem to help you solve pretty much any question that could be asked.

Any thoughts on this?