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How do you define solutions?
Posted by Mary Manske on November 25, 2019 at 11:30 pmI don’t feel like my students understand what a solution really means–they can parrot stems back, but the understanding isn’t there. I want to be able to focus on this more, but I need to be able to articulate better what a solution is, both for myself and for my team. I have a coworker who follows up all of my suggestions with “so I can just tell them to identify where the numbers are the same in both tables” and I can’t seem to make myself clearer. Help!
Mary Manske replied 3 years, 1 month ago 3 Members · 6 Replies 
6 Replies

Hi @MaryManske!
Thanks for sharing your current problem of practice!
Can you share some examples of what students are typically sharing as solutions that you feel aren’t hitting the mark?

Part of the problem is that I’m struggling to ask a question that will elicit that understanding. A few years ago when we started Systems, I used one of the Desmos activities in their collection (one of the earlier ones). It gets at what a system is and what a solution to a system is. I felt my students *could* tell me what a solution is, but clearly the questions I was asking weren’t going deep enough (and I’m not sure what to ask instead).
I also want to be able to reflect this understanding on an assessment, which is where we really hit a roadblock. This is last years back and forth (over systems, but its still solutions):
I want them to understand what a solution is.
So we tell them the solution is where the lines cross or the pair that they get.
But they still don’t know what it means…what about a table? How do they connect their solution to a table? What does that mean?
So if I tell them the solution is where the y values are the same in the table, would that work?
..You probably get the idea. I want them to understand what they’re finding and what it means when they find a solution, but I really need to get my team onboard first (thankfully, they are much clearer thinkers than me).
I’m not sure how to ask students in a way that avoids them parroting back an answer in an assessment type situation but I’m not too concerned with that because I feel pretty comfortable formatively assessing this. But I can’t explain it well enough to my team for us to build in more time for this and to make sure we are addressing it throughout the year.
That probably still was not a good enough answer to what a solution is–I’m trying but I’m not the best at being clear.

Hey @marymanske
This totally makes sense. I’ve experienced this same issue teaching this same concept. That’s why I teach this lesson to start any linear system topics: http://slamdunkmath.blogspot.com/2013/11/solvingsystemswithmanipulatives.html It’s from Alex Overwijk.
I find it helps with students understanding what they are actually finding when solving a linear system.
After that lesson we move to writing equations and then graphing them on Desmos to see that the same solutions can be found by the intersection.
This lesson is also a go to for me: https://teacher.desmos.com/activitybuilder/custom/56d139907e51c4ed1014b51f


I like that start–we’ve done similar and I use Racing Cars as well! I used both of the two intro activities in the Systems bundle (Systems of two linear equations and Solutions to Systems of Equations)–I felt like my students were getting closer to identifying what a solution actually is, but struggling to connect that across multiple representations. We can identify that its the price of each with a problem in context (with or without manipulatives) and we can find where the y values are the same in a table or the lines cross on a graph. We can usually then restate that matching part in context as a solution–but how all of that fits together is unclear for most of them.
My big issue though is how to better define that distinction. I think there’s a big difference between making that link or connection & being able to see how it carries through many representations and having done problems with solutions from tables/graphs/words/whatever and be able to identify a solution in any of those contexts. (Or maybe there isn’t and I should get over it.) Its pretty conceptual though so its hard to discuss with my team so we can get some more clarity (that usually works) because of it turning into “so we’ll give them tables!” “We’ll make sure they answer in a sentence!” “If I can teach them to say ____, would that work?”
I know this needs to be addressed starting now (or yesterday) and it seems like there is something about solutions to one variable or linear equations that is an useful connection, I just can’t quite get it.

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 buyout 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 buyout 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?


This is really helpful. I’m trying to use solutions to systems as a specific place in which the lack of conceptual understanding becomes clear, but I do think it goes back to the earlier contexts you describe (and you’re getting exactly what I’m going for with the idea of “solution”). While I think they can say thats the answer, I’m still not sure they even fully associate that with being the solution. I’m wrapping up one variable equations right now and yesterday we looked at no/infinite solutions. I circled back around to the first problem I’d opened with (x=6) and asked if any other numbers would work. They were overall not sure, even my strong students. We opened equations with a double number line, we’ve worked with them conceptually, some of them have worked with manipulatives–but they still seem to be associating that 6 with the result of an algorithm rather than as a value with a relationship to that specific equation (that no other variable has).
I’m wondering where I can start doing the exploration you mentioned before we get to systems–what are the places in other topics, in tasks, in a warm up, that I can continue to look at the idea of a solution. I don’t think it lies in word problems–we’ve talked about answering in a sentence, making sure its reasonable so I don’t know that there is a ton more to get out of there. And I suspect that it is when it is abstract that it falls apart. Something around solutions to one variable equations? Pulling out specific solutions to a linear equation (I semitried this last year but I’m not sure I helped much)?
I’d love to hear further thoughts but even just the post above is hugely helpful. My team is really thoughtful and I know they can help design better instruction around this and introduce it earlier–I just can’t seem to get them to get what I mean 100%. We end up going in circles with me going, but that isn’t quite it. I think your post above could help them to see what I’m hoping for the students to understand.