# Real Questions

Every year in precalc, two things happen. First, the exponential and logarithm unit gets squished (education jargon, I know). Second, we do the “student loan” blog post, which Steve came up with when we were first working on precalculus together. The former drives me crazy because I think that unit is more useful than some of the other things we spend time on. The latter I look forward to because it’s such a great learning experience for students.

The problem goes something like this:

Suppose you take out a \$5000 student loan every year for four years. How long would it take you to pay back the money? Please make sure to include research on government subsidized and unsubsidized loans and private bank loans.

Yeah, it’s vague. There’s no rubric, besides the standard blog rubric. The openness of the problem drives students a little crazy, which I’ve decided isn’t a bad thing. Let me explain.

The students in this class are, most likely, college bound. I don’t teach in a particularly affluent part of the state so most of my students are going to have to take out loans to pay for college. This means that the question, how long will it take to pay back loans, is painfully relevant.

Here are my main objectives for this assignment:

• I want students to understand the different types of loans and some of the verbiage they’ll encounter, if only at a surface level.
• I want students to understand the connection between exponential functions loans.
• I want students to understand how important interest rates are in total cost of a loan over time.
• I want students to realize that in this fairly common scenario, it’s not unlikely that they’ll be paying back loans for 20ish years.

I should also note that I don’t make students run these calculations by hand. I encourage them to use loan repayment calculators. I want them to understand that there is mathematics at play here, but I don’t want them to get too hung up in computations.

For many students this is an eye opening project. I’ve had students say they plan on paying it back in five years and they want to be a teacher. I had to gently explain to them that it was unlikely they’d be able to afford that. Steve has had students in tears doing this project. It has many great opportunities for students to learn about life, and also mathematics.

### Reflection

I wrote the beginning of this post when students were working on the project. I’m writing this part after grading them. Here’s a couple things I’m going to connect year to make it go better.

• Clear up the guidelines. There’s explanations in the video and a synopsis on the website. They’re both slightly different. I also plan on making them clearer.
• Listing things for students that are wrong and that students have done in the past. Things like only running one scenario, only computing a loan for \$5000, and not providing explanations for their numbers.
• I will include a question asking them to explain what they learned or gained from doing the projects.
• I will look up some useful repayment calculators as suggestions. I noticed that this can be a bit help or hinder ace depending on where google leads them.

Other than that, this activity will be used next year.

# Abstracting the Abstract – More Math GIFs and Function Talk

Creative individuals often take something that is concrete or complex and abstract it in some way that makes it more meaningful or provides a more useful perspective. When charged with the task of creating an abstract representation of my concept (functions) I was first at a loss. How do I create an abstract representation of something that is fundamentally abstract? The descriptions and media below describe the two methods I chose. For a summary of how these abstractions impact my teaching and understanding of functions scroll to the end of the media.

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Method 1

As I thought about this, I realized that when we write a function, we often only talk about a specific function. Yes, occasionally we will discuss a “family” of functions. But what does that mean and students really understand it? When we discuss a family of functions we are really talking about an infinite number of functions. What does that mean? How can we visualize such an abstract concept? Yes, sliders can help, but it can be visualized more effectively. Below are my ideas for abstracting various families of functions.

This also made me think…Could an abstraction be (at the surface level) more complex than what you’re abstracting? The GIFs show a more complex picture than just a curve with a slider (y=ax^2 with “a” as a slider). A family of functions is an infinite number of functions, right? How do we abstract an idea that is so complex? The GIFs attempt to abstract infinity (or an infinite number of curves) by suggesting what the escalation to an infinite number of curves would look like. (Click a GIF to look at a slideshow of each one individually.)

y=ax^2

y=a•e^x

y=a•sin(x)

y=(x-h)^2

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Method 2

The task set before me was to abstract my concept in two different ways. The second method I chose for abstracting the concept of functions was to pare downs functions down to their simplest…well…abstraction. Then, I sketched these abstractions and below are the images of my sketches. I decided that at their fundamental level functions have three things: an input, an operation (something that modifies the input), and an output. The images all contain these fundamental pieces.

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I think both abstractions can impact my teaching. The GIFs would be something that I could either show my students to give them a visual for the concept of a family of functions. A good project would be to have students create similar GIFs for different functions and their transformations. This solidifies the concept in a visual way and gives us something we can anchor back to throughout the rest of each unit.

The second abstraction might be useful in a different context. I like that the images provide visuals for the simplicity of functions. Often students don’t realize the vastness of functions and their stretch throughout the world. Most things with an input, operation, and output is a function. When you ask a student to give you a function they are likely to say something like y=2x+3 or f(x)=sin(x). They are unlikely to say “The number of cars in the school parking lot is a function of the time of day.” If students have the abstract concept of a function as their fundamental understanding then they might start to see function behavior in their worlds more frequently.

# Multiple Methods for a Simple Problem

For each video I have students watch I ask them, among other things, to submit one question they have after watching the video. After a student watched the “Solving Logarithmic and Exponential Equations” video, he submitted the following question.

How would you solve 8^x = 16^x ?

The following day I used this question in our WSQ chat over the video. What appeared to be a simple problem revealed some interesting solutions. I’ve provided the main types of solutions that I found.

I anticipated the second method from most students but only two of the five groups approached it that way. All methods are valid and what I really liked is that not only were they different methods but also different thought processes that led to the method/solution.

I would love to hear your feedback or observations that you have seen in your class!