Tag Archives: mathematical practice

A Small Change: Solving Equations with Logarithms

In my Math 10 class, did my lesson today involving solving exponential equations that cannot be solved using knowledge of integral powers. My start was the same as it has been for that lesson over many years:

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I have students start with an iterative guess-and-check method since it's something that will pretty much always work. This was no big deal to the students. When one student said her TI calculator gave the exact answer, I asked if she really thought that was the exact answer. She said no, but I used Python to rub it in a bit.

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This was another opportunity to show the difference between exact and approximate answers - always something I try to teach implicitly whenever it comes up. As with many of the Common Core Standards for Mathematical Practice, I think this (MP6 - Attend to Precision) is always an idea that comes with context.

The big shift in this lesson came when we started solving the equation algebraically. I always do a bit of hand-waving at this point saying 'isn't it great that these logarithm properties let us do this?', while getting a class full of students giving me just enough of a sarcastic head nod to make me feel bad about it.

Instead, I made reference to the process of switching back and forth from logarithmic and exponential form.

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The students are pretty skilled at doing this. I wrote it up in the notes myself because most students wrote it faster than I could get anyone to explain the process.

The key here was that when I asked students to calculate these values on the calculator, nobody could do it. One found the LOGBASE command on their TI, but for the most part, this stayed as an abstract number. It made sense to them that they ended up with 'x =' in the end, but that didn't make a big difference in terms of being able to talk about what that meant. They did a couple of these on their own.

Only then did I show them the logarithm property trick that lets us get the answer in a different form:
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I admittedly connected some dots here, but I didn't do so in a formal way of introducing change of base. A couple of them figured out that this was a form that they could calculate using the common logarithm button on their calculators.

I'm not emphasizing log properties this year outside of what they allow us to do in solving equations. This is something that we will devote more time to next year in IB Mathematics year 1 class. I will mention this change of base property as a nice tool to use for confirming graphical and iterative solutions, but probably won't assess them knowing how to apply change of base directly.

Any time I can get rid of hand-waving and showing mathematics as a list of tricks to be memorized, it's a win.

Before a Break: CCSS Math, Bogram Problems, and Peer Feedback

I spent the day in a room full of my colleagues as part of our school's official transition to using the Common Core standards for mathematics. While I've kept up to date with the development of CCSS and the roll-out from here in China, it was helpful to have some in-person explanation of the details from some experts who have been part of it in the US. Our guests were Dr. Patrick Callahan from the Illustrative Mathematics group and Jessica Balli, who is currently teaching and consulting in Northern California.

The presentation focused on three key areas. The first focused on modeling and Fermi problems. I've written previously about my experiences with the modeling cycle as part of the mathematical practice standards, so this element of the presentation was mainly review. Needless to say, however, SMP4 (Model with mathematics) is my favorite, so I love anything that generates conversation about it.

That said, one element of Jessica's modeling practice struck me by surprise, particularly given my enthusiasm for Dan Meyer's three-act framework. She writes about the details on her blog here, so go there for the long form. When she begins her school year with modeling activities, she leaves out Act 3.. Why?

Here's Jessica talking about the end of the modeling task:

Before excusing them for the day, I had a student raise their hand and ask, "So, what's the answer?" With all eyes on me, a quick shrug of my shoulders communicated to them that that was not my priority, and I was sticking to it (and, oh, by the way, I have no idea what time it will be fully charged). Some students left irritated, but overall, I think the students understood that this was not going to be a typical math class.
Mission accomplished.

Her whole goal is to break students of the 'answer-getting' mentality and focus on process. This is something we all try to do, but perhaps pay it more lip-service than we think by filling that need for Act 3. Something to consider for the future.

The other two elements, also mostly based in Jessica's teaching, went even further in developing other student skills.

I had never head of Bongard problems before Jessica introduced us to them. This involves looking at well defined sets of six examples and non-examples, and then writing a rule that describes each one.

Here's an example: Bongard Problem, #1:

You can find the rest of Bongard's original problems here.

In Jessica's class, students share their written rules with classmates, get feedback, and then revise their rules based only on that feedback. Before today's session, if I were to do this, I would eventually get the class together and write an example rule with the whole class as an example. I'm probably doing my students the disservice by taking that short-cut, however, because Jessica doesn't do this. She relies on students to do the work of piecing together a solid rule that works in the end. She has a nicely scaffolded template to help students with this process, and spends a solid amount of time helping students understand what good feedback looks like. Though she helps them with vocabulary from time to time, she leaves it to the students to help each other.

Dr. Callahan also pointed out the importance of explicitly requiring students to write down their rules, not just talk about them. In his words, this forces students to focus on clarity to communicate that understanding.

You can check out Jessica's post about how she uses these problems here:
Building Definitions, Bongard Style

The final piece took the idea of peer feedback to the next level with another template for helping students workshop their explanations of process. This should not be a series of sentences about procedure, but instead mathematical reasoning. The full post deserves a read to find out the details, because it sounds engaging and effective:

"Where Do I Put P?" An Introduction to Peer Feedback

I want to focus on one highlight of the post that notes the student centered nature of this process:

I returned the papers to their original authors to read through the feedback and revise their arguments. Because I only had one paper per pair receive feedback, I had students work as pairs to brainstorm the best way to revise the original argument. Then, as individuals, students filled in the last part of the template on their own paper. Even if their argument did not receive any feedback, I thought that students had seen enough examples that would help them revise what they had originally written.

I've written about this fact before, but I have trouble staying out of student conversations. Making this written might be an effective way for me to provide verbal mathematical details (as Jessica said she needs to do periodically) but otherwise keep the focus on students going through the revision process themselves.

Overall, it was a great set of activities to get us thinking about SMP3 (Construct viable arguments and critique the reasoning of others) and attending to precision of ideas through use of mathematics. I'm glad to have a few days of rest ahead to let this all sink in before planning the last couple of months of the school year.