We started off our Absolute Value unit with solving absolute value equations and inequalities this year. Then, we learned how to graph absolute value functions, and I had the students do this problem:

And the students were all, “Ohh this is why we get two solutions.” So I learned my lesson to start by graphing, and then solve simultaneously. Luckily, I didn’t have to wait til next year to try this approach. Our next unit of study was Quadratics, and once they learned how to graph them in vertex form, I gave them a similar problem:

At this point, we had not solved a single quadratic equation yet. My students graphed and launched right into solving like it was no big deal at all. I thought they had finally made the connection between the solutions and the intersection of the line and the parabola.

But, then we moved onto quadratics in standard form and solving by factoring. The factoring and solving went well, but on their assessment, I gave them a “Find the Error” problem where they had to identify which work was correct and explain their reasoning. Here are some of their responses:

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Most of my students correctly selected Kristen; however, I was extremely disappointed in their explanations. I expected their explanations to be more in depth after the many discussions we had about solving for the x-intercepts. They mostly went with the procedural explanation of setting it equal to zero. I wanted them to explain WHY we set it equal to zero. I don’t know how to ask that without directly giving away which student’s work is correct in the first place. There’s also a problem with the many responses claiming Kristen is correct because she found two solutions. This tells me I need to do more examples with only one.

At this point in the year we’re moving onto exponentials, but I’ll be thinking about this problem for a while. Any advice would be greatly appreciated!

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Hi Heather,

Two things I’m wondering. The space to leave an explanation looks rather small. I think on some level, students look at that and decide how much they need to write. With this, a one-sentence answer looks appropriate.

I’m not sure students know what you’re looking for. It might help to come right out and ask, “How does setting the equation equal to zero help us find our solutions?” There’s just too much room for interpretation here. I think that’s good for regular class discussion, but not on a quiz, where you can’t give a follow-up question when their first answer seems insufficient.

What do you think?

Nathan, I think you’re right on both points. I was trying to fit it on one page so the space is smaller than it really should be. I ended up giving full credit (two pts) for most of these answers since it was obvious they didn’t know what I was looking for, which is my fault and not theirs. I scanned the answers as I collected the quizzes and almost followed up right then with “Why?” but made the split decision to wait. I think when I pass them back I’ll have that class discussion again, and ask how they could expand upon their answers. Thanks for the advice!

It feels a bit to me like there’s two questions you’re trying to get at there. The first question is distinguishing valid from invalid work (“Determine which of these two solutions is correct”) and the second is explaining why valid work is actually valid (“Explain why this particular solution is correct”). You asked the first, but expected them to answer the second. So as you mentioned above, I think a “Why” follow-up is probably in order.

A different way of framing it might have been to say ‘here are two VALID ways of doing the problem – explain why they are equally valid’. That would work with something like x^2 = 9, where you can either root both sides, or move the nine to get the zero, then factor.

Note that in the given example, they could also explain that the second one is valid by saying “when you substitute x=2 back in, you get the 8”. Still no mention of the zero principle, but would that be answering the “why” satisfactorily? (Or were you exclusively looking for mention of x-intercepts and not even the zero principle?) Anyway, enjoy exponents!

I was looking more for the mention of x-intercepts or where y = 0 than the zero principle, but I think your suggestion to provide two valid ways might just solve my dilemma! By choosing Kristen, it was easy for them to explain their choice by mentioning her process. But with two valid ways, they are really going to have to explain their understanding behind the process. I try to have them avoid using substitution as the “why” since they could have guessed and checked. Thanks for the advice!

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An increased focus on the connection between a graph and its equation would probably help. If students know that the graph of an equation is the set of points that make it true, they are more likely to understand why a certain number of intersections matches a certain number of solutions.

Like Greg said, these students are also missing the concept that you can test an answer by using its value for the variable. These two ideas are connected — a graph of two variables is all the points (x,y) that make it true, while the solution set to an equation is the set of values (x) that make it true.