## Exploring the roots of quadratic functions

In my algebra 2/trigonometry class, I wanted to spend a day with them exploring quadratic functions, their roots, factors, and how everything is related. This is taught in algebra 1, but the students always seem forget all this after geometry. It’s a perfect intro to our quadratics unit.

Part 1. Building off Dan Meyer’s approach to factoring trinomials, I first had the students find all values that make quadratic expressions equal to 0.

Most found this straightforward and doable, which was great because that was the point: accessibility. The third expression had one non-integer value as an answer, which I included on purpose to throw them off. A couple kids actually factored and used the zero-product property, which was ahead of the game…they actually remembered something from algebra 1!

Part 2. I had them use Desmos to examine the graphs of the three functions and find a relationship between the graphs and the values they found that made each equal 0. It took several minutes, but there were definitely some ah-ha! moments when they saw the connection, which was cool.

I then re-introduced them to be the term “roots” as a way of describing these magic numbers.

Part 3. I also wanted them explore the relationship between the factors and the roots. Because of time, we more or less did this together (instead of them working it out in groups). We first factored all the expressions. I then asked how the factors relate to the roots of each function. Most of the class realized that when each factor is set to 0, the roots result from these “mini” equations.

Overall, the lesson was solid. I really liked that, other than imparting the term “root,” there was no need for me to lead any part of the lesson. I simply provided resources and asked the right questions that spurred deep thinking.

Exit slips showed their understanding of the connection between the roots and factors wasn’t strong. This was probably due to the fact that the lesson was a bit rushed at that point. The next lesson focuses strictly on finding roots by means of factoring quadratic equations, so hopefully that helps. I also felt the lesson flip-flopped around the term expression and function too much. Leaving the lesson, the difference between the two could have been unclear and may cause some confusion amongst the kiddos. Another thing I would have changed is not having all trinomials…the kids could possibly generalize that all quadratic functions are trinomials, which is obviously not true. Even if they don’t go that far, a variety quadratic functions still would have been better for them to explore.

Here is the document.

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## Knowledge Audits

How can I know what my kids know?

I’ve been asking myself that question for a long time. In my Regents-obsessed school, I’m forced to ensure my students can perform well on end-of-year state exams. The typical learning flow in my class usually looks like this:

1. Student learns X.
2. Student demonstrates understanding of X.
3. Student learns Y and forgets X.
4. Student demonstrates understanding of Y and has no idea what X is.

Compile this over the course of a school year and you have students that understand nothing other than what they just learned. What does this mean for a comprehensive standardized exam? Disaster!

Sure, a lot of this has to do with pacing and students not diving deep into things they learn to make connections. That is a sad reality of too many teachers, including me. So given these constraints, how can I help kids build long-lasting understanding of things they learn and not forget everything other than what we’re doing right now?

In the past, I’ve “spiraled” homework and even put review questions on exams, but this never helped. There was no system to it and I never followed up. This year, I’m lagging both homework and exams, which does seem to be making a difference. But with the ginormous amount of standards that students are supposed to learn each year, I still feel this isn’t enough.

So, last week I began implementing Audits. These are exams that do not assess concepts from the current unit. The plan is to administer about one a month and because I lag my unit exams, I should have no trouble fitting them into the regular flow of things.

I’m choosing not to call them “Review Exams” or some other straightforward name in order to put a fresh spin on them and increase buy in. So far, so good.

The hope is to continually and systematically revisit older content to keep students actively recalling these standards. This should reinforce their learning and help to make it stick. On the teacher side of things, I get an updated snapshot of where they are and can plan accordingly. The SBG aspect is simple: the results from the Audit supersede any previous level of understanding.

• If a student has not previously earned proficiency on a standard that is assessed on an Audit, he or she can earn proficiency. This alleviates the need to retest on their own.
• If a student has previously earned proficiency on a standard, he or she must earn proficiency again or else lose credit for that standard. This would then require them to retest.

The first Audit resulted in a mix of students earning credit and losing credit for a set of standards. It was great. The proof is in the pudding. Knowledge isn’t static and my assessment practices must reflect this.

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## Internalizing feedback without seeing it

I’ve found that students all too often overlook, or simply ignore, the feedback I give them on assessments. For whatever reason they just don’t use it. This is a problem.

I value reassessment and see feedback as crucial in that process. If a student struggles on an exam, I need them to learn from their misconceptions. My feedback reflects this. I’ve always provided fairly detailed feedback, but this year I’ve stepped it up significantly. In fact, all I do now is give feedback. I provide no scores or grades on exams. This helps, but I still need to find ways for students to grow from the feedback they receive.

I have experimented with kids relearning concepts the day after an exam without seeing their actual exam. The day after the exam, I give a copy of the SBG results to each group. Each student uses the data to identify the specific concepts that they need to relearn or review. The groups are a mix of proficiency levels (based on the exam results) so if a student needs help with a particular standard, there’s someone in their group that understands it and can help them. I also give them blank copies of the exam to work on and discuss.

After about 15-20 minutes of peer tutoring, I give them their exams back. Based on their newfound understanding, at least some of their misconceptions should be alleviated. They now spend 15-20 minutes correcting their mistakes on a separate sheet of paper while directly responding to the feedback I’ve given them on the exam.

Ideally, this means that they are using feedback from their peers to understand and respond to the feedback I’ve given them. It serves as relearning/remediation before they retake the exam. What I’m missing, though, is a reflection piece that ties into the feedback as well.

A colleague conjured up a different spin on this. After an exam, he informs students which standards they didn’t earn proficiency on. (He doesn’t hand back their actual exam either.) He allows one week (more or less) of relearning/remediation on those standards – either on their own, or with you. He actually uses an online resource for this. Then, when they feel ready to retake, he returns their exam and asks them to self-assess and correct their original mistakes. If they can, he allows them to retake. If not, they continue relearning. It may not focus on feedback, but I like this.

Closing thoughts: what if I do get my students to internalize my feedback? Are they just going to be doing it to satisfy the requirements that I ask of them? When they leave my classroom, will they view feedback as a necessary component of success? Will my feedback really make a difference? How else could I get them to value it?

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