# Introduction

Up next: an assessment on rational functions from Sam Shah. As you read through the assessment and consider adding your comments to the discussion, keep these questions in mind:

- What do you like (and why)?
- What would you do differently (and why)?
- What questions do you have for Sam?

# Submission Type

“Share an assessment you *don’t* hate…”

# Assessment

# Author’s Commentary

“I have a lot of not-so-good assessments, ones that I would be embarrassed for any of y’all to see. But I’m not sharing those because I know they aren’t so good, and I know I can come up with ways to improve them. (Mainly that will involve improving how I teach/introduce the material, which means that the assessments themselves will be better because my kids will have done more and better thinking, so I can ask more and better questions.) So I’m going to share a strong assessment that I just gave, on rational functions.

I am teaching an advanced precalculus class for the first time this year. And so they get things pretty quickly, and are a strong bunch. You can see how I introduced/taught rational functions here in this blogpost. This was the assessment I came up with to cover the unit.

I don’t have too much to really say about individual questions.

The goal of this unit was to be highly conceptual, and the goal of the questions was to uncover misunderstandings.

For example, when I asked the vertical asymptote question (#3), only about a 1/3 of the kids got full credit. I was looking for something like: “when x gets close to zero, you have a fraction that is 1/(small number) which results in a big number. So for example, 1/0.001 is 1000. The closer the x value is to zero, the larger the output! That is why the function approaches infinity as x gets close to zero!”

However, what was clear to me in hindsight was that we talked about that idea briefly, but then we spent a long time talking about why horizontal asymptotes appear (or don’t appear). And so kids would say things about not dividing by zero, and then almost in a non-sequitor switch their explanation to something like “for large positive and negative x-values, the output is close to 0.” So I was able to see where their misunderstandings were through this question (and also where I need to shore up my own teaching next year).

Overall, though, I have to say I really am proud of this assessment. Not only that, but my kids did quite well on it — I think I had a B+ average! That being said, I’d love ideas on places to improve it.”

Sam

Thanks for sharing! I’ll share my thoughts here sequentially

#1 – I would expect most students to work backwards here and think of the given quotient graph as y = 1 for all x other than -2 and sort of unfactor from there. Is that the approach you discussed or expected from them? I like the emphasis on this sort of backward thinking.

#2 – Now looking at problem 2 it’s clear that this is a skill you must have really emphasized. By the way, why choose the no calculator route for this assessment?

#3 – Do you think that they might have been more detailed in their written response if they had more room to write? I think that students sometimes gauge the detail they provide based on the space I give them. For awhile I always had students write their work on separate paper and I may switch back to doing that to encourage them to take any room necessary to stretch out and think.

#5 – I like the clause of using an Algebra II student as their audience. In class I will often ask them to explain to their annoying younger sibling.

#6 – Had they explored oblique asymptotes yet?

#9 – I might have split this into three questions (a) when is 0 < h(x) (b) when is 0 = h(x) (c) When is h(x) < 0

#10 – Part b the height of the hole is a funny phrase. I assume you mean to ask them what the y coordinate of the hole is.

How long did this quiz take? I could see my Precalc Honors kids slaving over question #8 in particular.

Again, thanks for sharing!

I really liked Q1. It is not a kind of question I’ve asked before. It certainly isn’t a problem that could be solve using wolframalpha.

#1 really is a neat way to introduce this topic. Like #2 as well. Nice bonus. I completely agree with making this a no calculator test.

I have found that 1/small is big & 1/big is small discussion really takes some time for some students – not sure if the issue is me or poor number sense.

Did you use limit notation for any of this? As x– > 2+, y — > ____? That is another way to test how well they understand what is causing the vertical asymptotes – sometimes kids that do understand just stink at writing explanations.

You may want to vary the form a bit when asking about vertical or horizontal asymptotes: y = (3x – 1)/(1 – x) + 3 or y = (x-1)/x + 1/x-1 + 3 for example. Otherwise, they sometimes latch onto “patterns” that give the right answer as long as we use the same form.

Hihi! Thank you mrdardy, dan, and l hodge!

I’ll work backwards.

l hodge:

I agree that the 1/small and 1/big takes time for my kids too! It is something I have (not on purpose, but it came up a lot) been emphasizing a lot all year. But it seems so crucial!

As for limits, I decided to take them out of the curriculum this year (even though it would have been easy to add them in) because I figured if they knew the concepts, then the WORDS/NOTATION going along with it would be cake. And they will get that next year in AP Calculus.

I really love how you threw a wrench in the works with your y = (3x – 1)/(1 – x) + 3 or y = (x-1)/x + 1/x-1 + 3 so this doesn’t become a game of patterns. I am DEFINITELY going to be using that next year.

dan:

Thanks! I was really excited when I thought of it!

mrdardy:

#1: For #1, a few kids put a horizontal line at y=1 (or y=2 or y=3) for both f and g, but then put a hole in one of them. Most drew y=x+2 for both f and g. They discovered holes mainly with lines. I wonder if I say “if you can come up with f and g so that they aren’t lines, you get a sticker!” or something if it might spark some creative/independent thinking in more kids.

#2: Yes and no. They really quickly figured out how to come up with the equations for graphs, without me doing much. As for the no-calculator, I was going for deep conceptual understanding, and they don’t need a calculator for that on this assessment, I don’t think. What would be the benefits of allowing a calculator?

#3: Yes. Absolutely. I kept on going back and forth as to whether to give more room or not, but in hindsight, you were right, I should have.

#6: We ran out of time. I didn’t want to introduce it if I didn’t have the 20 minutes for us to really talk about why they appear and make it meaningful, so I made the executive decision (after asking the AP Calc teacher) to just cut it out and focus on the key parts. We’re (the AP Calc teacher and me) both excited to see if the deeper conceptual understanding will translate into a different level of understanding next year or not.

#9: Porque?

#10: I have found that when I ask “where is the hole” I am often being unclear with kids — in both calculus and precalculus. Sometimes all I want is the x-coordinate, and less often I want the y-coordinate. So I told them I would specify needing to know the “height of the hole” when I wanted the y-coordinate.

This assessment took a whole 50 minute period.

Thanks again all for your thoughts!

Sam

Regarding #9 as a Calc teacher for many many years now, I find that I am always trying to emphasize the idea that we are often interested in finding roots when we actually say that we are looking for sign changes. I think that breaking up the inequality that also includes the equals part might set them up later to really focus on the importance of that zero as the barricade between positives and negatives. A small point for sure.

Thanks for sharing a terrific assessment (I’ll be ‘borrowing’ from this next year) and for the detailed responses.