## Prompting Generalizations of Limits at Infinity

After my PreCalc students completed a rational functions packet (thanks largely to CME) I wanted to move directly into the study of limits. I started with the chart below which is pretty much exactly what JackieB shared. The difference was in the follow up; I didn’t want to scaffold their conclusions as much as she did. Which resulted in most students not understanding what I wanted for generalizations. Next time I need to reword that section, should I ask students what the possible values are for the limits first and then ask them to generalize how to know what the limit will be?  I also debated mixing up the order of the problems (as they are now the first three go to infinity, the next three go to a non-zero number and the remaining three go to zero) but hoped leaving them in order would highlight the pattern.  I try to avoid the “guess what the teacher’s thinking” game but also want to encourage them to make their own observations.  Solutions to my dilemma?

Original Post on Drawing on Math with a complete description of the unit.

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## Interpreting Graphs

I find graphical solutions to be the quickest method to approach many arduous tasks when a graphing utility is involved. Solve a system? Graph and locate the intersection. Find real roots? Graph and look for intercepts. Identify asymptotes? Graph and see where that weird squiggle is.

My honors Pre-Calculus students on the other hand, are not convinced. They thought I was torturing them when I asked them to graph to find real roots. I knew they studied all the other methods of solving polynomials last year and this year I wanted to focus on the more interesting task of complex roots. Several students announced they would rather do synthetic division than pull roots off a graph. I consider that torture! But even with graphing calculators they had such a hard time estimating the values of intercepts, and I promised to only give integer solutions so they knew to round.

We have since moved on to rational functions, yet the same issues are still cropping up. On the test I’m grading, one question is: “Describe the graph of $f(x)=\dfrac{2x^2-8}{x^2+8x+12}$ Include domain, range, holes and asymptotes (vertical and horizontal).” It is certainly a question that involves synthesizing a number of ideas, but they have graphing calculators, are good at factoring and we did plenty of problems just like this. I shouldn’t be getting multiple vertical asymptotes, domains like -6<x<6 or ranges of y<1. I don't know what to do.

How do you help students pull important information out of a graph?

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## Bloodbath

Cross Posted from Continuous Everywhere but Differentiable Nowhere.

So today I had this experience where this precalculus test I gave was a bit of a bloodbath. Not for everyone, but for more than usual. In a way where I cringe, cry out to the high heavens, and scream:

## WHYYYYYYYYYYYYYYYYYYYYYY?

The reason is because I felt pretty proud of the way I have been introducing the material. You see, in precalculus this year, the kids are coming up with everything on their own. I don’t give them anything.* And thus far they’ve been doing well with this. And during this unit, even though I didn’t quite have the same amount of time to create everything to my best ability (I relied a lot on the textbook for this stuff), I felt pretty confident about my kids’s understanding.

So I have to wonder: Where did I go wrong as a teacher? What was different about this unit than the others?

First off, this unit was some pretty heavy stuff. We were deriving and applying the trig formulas, and then we were solving more complicated trig equations (they had done basic trig equations previously). All in all, we took a total of 8 days to do this. I should also note that this is an advanced class, and they have been doing a lot of collaborative work this year.

FYI: these were the trig formulas we derived and applied… this is the “trig formula family tree” I made for them.

And for this unit, I led class in a pretty routine way. Each day I had a packet for the kids to work on. They would work on the warm ups with their groups (which were designed to activate prior things they knew but forgot, and have kids make some connections on their own). After 5-10 minutes, we would all talk through the warmup problems together.

Then I would let each group work on their own. I would walk around and facilitate, nudge, question, and answer questions. On some packets, I would have special places where I told kids to “draw in a heart, and call me over when you get to the heart.” (But to be honest, overall, I think I was throwing myself into the groups less than I usually do this unit, as I’ve been trying to let go.)

Then class would end. Most groups were where they should have been… close to done with the packet, and ready to start working on the book problems. These book problems varied in difficulty from the routine “can you do something simple?” to the “okay, apply this in a moderately deep way.” For this unit, I did assign more nightly work than I normally assign, because I knew that to get good at this stuff involves a lot of practice. (I don’t think that is true for everything in math, btdubs.)

Then at the start of the next class, I would have one set of my handwritten solutions per each table (that way, three kids have to share, and thus talk!). I would give kids 5-10 minutes to compare their answers, talk with their groups to figure out things they were doing wrong, and then we would come together as a class and I would field questions that groups couldn’t answer. Then we started a new packet, and the process continued like that for most days. [We did have a bit where we did a paper folding activity, which was pretty cool.]

To see what these packets look like, I combined all of them here so you can scroll through them. I highlighted some of the problems/questions which I thought were good at getting at something hard/interesting/conceptual

As you can see, these aren’t really great. Not bad either, though. [1]

So where did things go wrong?

When I look through the tests, here are some things I noticed as a trend:

• Kids struggled with some of the basic “apply the formula” questions
• Kids had trouble figuring out which sign to use when using the half angle formulas (e.g. $\cos(\beta/2)=\pm\sqrt{\frac{\cos\beta+1}{2}}$) [2]
• Kids really nailed the conceptual explanation part of “how many solutions does this trig equation (e.g. $\cos(24\theta)=-1$) have on the interval $\latex 0\leq\theta<2\pi$?” question
• Kids struggled with remembering that when you take the square root of both sides, you get two solutions (so $\sin^2(\theta)=1/2$ is really two equations to solve)
• Kids did a pretty good job of deriving the trig formulas
• Even though kids did a pretty good job on the “how many solutions does this trig equation have?” they didn’t find all the solutions to the basic trig equations given.

As far as I can tell, here were the contributing factors (in no particular order):

(a) Lots and lots of sickness. I still have 5 kids who haven’t taken it (out of 19).

(b) I thought I was getting formative feedback when I gave regular little mini non-graded “do you remember the trig formulas we’ve derived” at the start of some classes…

And honestly, I felt proud that I have been making a conscious effort to collect this formative feedback. But now I see it wasn’t the right formative feedback.

(c) I usually get a good amount of formative feedback in Precalculus. Mainly I do it by collecting of the nightly work, marking it up, and handing it back. Thus I usually know what students are understanding and what they are not, and they also know what they understand and what they don’t. However, because I was swamped, I didn’t really do that. Maybe once in eight days? So each day, kids got to compare their own work to my solutions, which I thought would at least give THEM feedback… But I never got to see what kinds of mistakes they were making, or where they were getting tripped up, not in detail and not in a big-picture way. So I didn’t build these things into the lessons… which is important because…

(d) This material is hard. Harder than some of the previous units/ideas. That’s because this unit required conceptual understanding, juggling a lot of memorized formulas, a bunch of intuition (as to how to start solving the trig equations), and a lot of “fact” information (like where in the unit circle is $\sin(\theta)=-1/2$?). It’s just pulling a lot of stuff together.

(e) I should have spent more time reminding them of the trig equations they had previously solved. I assumed that they remembered all of that stuff we did weeks ago and could apply it. I jumped in too fast.

(f) The test was a bit too long. The kicker is, I thought it was too long, so I cut some stuff out. I was trying to be conscious of that. Well, the road to hell…

So there we are. Surprisingly, typing this out has made me feel a lot better. I feel like I now have a better grasp on why something I thought was going pretty well was actually not going as well as I thought. I also have some concrete ideas on what to do next year. The main takeaways for me are: go slower, bring in more visual understanding for trig equations, don’t mess around with the harder stuff, get a lot of formative feedback on the basic types of problems, and make the assessment shorter than my intuition tells me.

*Okay, to be fair, I have given them two things — one which we proved later, the other which I never proved. (The former was the sum of angles formula for sine and cosine, the latter was Heron’s formula.)

[1] It was a stressful time when I was doing this unit, and so I just didn’t have time to come up with anything better. But still, I think they get at good stuff. Even if there needs to be A LOT MORE GRAPHING next year. We did a lot of graphing when we did basic trig equations. We should have done graphing here too.

[2] The kicker is that I said in class that figuring out the correct sign is the most important thing about applying that formula. Multiple times. But me saying it until I’m blue in the face the same as them totally understanding it. Next year I need to build in some warm up questions like: if $\alpha=200^o$, what quadrant is $\alpha/2$ in? Draw a picture. If $\beta$ is in the fourth quadrant, explain in words and with pictures why $\beta/2$ is in the second quadrant..

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## Isn’t negative infinity small?

Studying limits again, this time a pair was finding a limit that they said got very small, but then wrote negative infinity instead of zero. When I acknowledged the difficulty of this question, one student mocked me “It’s not like I asked you a touchy subject!”  I laughed and explained that negative infinity has large absolute value, so it is a large negative number.  His partner protested “but negative numbers are small!” so I countered with “If you owe someone 100 dollars is that a lot?” They both concurred that it is, so maybe my explanation was valid.  The original student wanted a non-economic example though, and he proposed toothpicks (this kid cracks me up all the time).  Then I was stuck talking about the non-existence of negative toothpicks, unless we return to the concept of owing someone.  The entire conversation was amusing and I think I was moderately convincing this time.

How do you differentiate between small numbers and negative numbers?  We didn’t even include “less than” in our discussion…

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## Is infinity a number?

A student asked me “Is infinity a number?” while I was introducing limits.  My immediate response was “infinity is a concept” and I started talking about $\infty$ vs. $2\infty$.  I didn’t really know where I was going with that though, so I let the students debate for a bit and finally related the whole conversation back to the problem at hand.  The discussion was interesting, but other than acknowledging the difficulty of comprehending infinity, I didn’t contribute much (which is usually good, but I certainly didn’t provide clarity).

How do you define infinity?

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## Midterm Analysis: Inverse Trigonometry

Another area of struggle on my PreCalc midterm is inverse trig functions:

Students had a completed unit circle (radians, degrees and coordinates – they earned them on the exam by completing one on their own the previous week) but could not use a calculator for this section. What types of mistakes do you think students would make?

Feel free to check out the exact mistakes my students made at the Math Mistakes blog.

What I noticed:

While inverse trig was a struggle all year, I was concerned how many students over-thought $sin^{-1}(sin(\dfrac{5\pi}{6}))$. When my co-worker and I wrote the midterm we thought that question would be a gimme, but student after student evaluated $sin(\dfrac{5\pi}{6})$ (to varying degrees of success) and then found the arcsin. It didn’t occur to me to include many of the correct answers I saw (π/6 and even 17π/6!) on my answer key but luckily I was paying attention when I saw the first one and added solutions to my key as I found them. While 17π/6 isn’t a mistake, I am curious why students didn’t notice that the sine and arcsine cancel.

For the next problem I anticipated students would struggle since inverse tangent isn’t easy to read directly off of the unit circle. Not only do they have to recall that tangent is $\dfrac{sine}{cosine}$, but they also needed to be able to simplify a fraction within a fraction. The red flag was students who didn’t recognize that there was an intermediate step of simplification; instead they expected there to be an angle where sine=1 and cosine=$\sqrt{3}$ (apparently I didn’t get a picture of that mistake, sorry). The mistake I did photograph was an interesting one though. The student recognized that the -1 meant inverse, however they misattributed it to the inverse of the ratio rather than the inverse of tangent. That shouldn’t be a mistake I see on the midterm though. Especially after explaining countless times that for tangent the input is an angle and the output is a ratio, while for arctan the input is a ratio and the output is an angle.

On the final problem I saw good attempts and students who couldn’t solve it generally forgot that triangle trig was an option. I think those issues were more indicative of a lack of studying than a misconception. Or maybe an inflexibility in their approach? I’ve certainly seen students start a problem only to find that they can’t solve it. At this point they call for help assuming they made a mistake somewhere. Many times their work is accurate and what they actually need is to consider a different method. They could do with a good dose of productive struggle.

Requests for feedback:

I am continually surprised by how much difficulty students have doing anything backwards. In geometry we will study angles formed by parallel lines until they can fill in all 8 angles without a second thought. But then when I give them angles and ask if the lines are parallel, they look at me like I’m asking them if unicorns have purple stripes. The same thing happened here, students are great at finding values of sine, cosine and tangent if I give them the angle. But once I switch it up and ask for the angle instead, they start giving nonsensical answers. What is it about working backwards that students find so challenging? How do you help students recognize the structure?

p.s. Do unicorns have purple stripes?

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## Logs and Decibels

Cross Posted from Hilbert’s Hotel

On Wednesday I started a lesson that I thought would be more engaging than “do these problems and keep going”… and the students complained MORE than usual!  Ugh, what a depressing day.

Well, now it is 2 days later, and I’ve had some time to reflect on what happened.  Let me explain what I did, and how I think I could have improved, although I would appreciate feedback as to more ways for me to improve.

The lesson was about Logarithms and came after my students had learned about how to solve for logarithms, and how to add, subtract, and multiply the logarithmic function.  I had actually created the lesson back in graduate school 3 years ago and was excited to actually get to use it (last year I forgot about this lesson).  In involves having students use the equation for decibels, which I provide below, and finding sound pressure (p) when given decibels (dB) and visa versa.

$dB = 10 \times \log ( \frac{p^2}{(2 \times 10^{-5})^2})$

I actually started the lesson by giving the students three areas of their lives where logarithms (and logarithmic scales) are used: earthquakes, magnitude of stars (you can see the stars really well out here in NM), and measuring sound volume.  I then allowed them to choose one of those three areas, and had them briefly research in class the following three points/questions:

1. Find one equation involving logarithms for the situation, and explain the variables in the equation.
2. Compare a few of the items on a scale and explain “how many times more” one of these things is than the other using the equation (e.g. “how many times brighter is Venus than the North Star and where do they fall on the scale?”)
3. Explain why logarithms (and a logarithmic scale) is used in this situation.

The students researched and presented, and all was smooth sailing up to this point.  Many of them were getting the point I was trying to make: that if you didn’t have a logarithmic scale, the numbers would be ridiculous to use and would be difficult to compare.  Instead, because scientists use a logarithmic scale, many people don’t understand when these are reported in the news (comparing a magnitude 4 earthquake to a magnitude 8 earthquake… the second one isn’t twice as big–it’s 10^4 times bigger!).

At this point, I decided to give “my presentation” on decibels (this was the least selected topic, and, in my opinion, one of the most interesting).

So as you can see above, I gave them the equations and showed a cool scale comparing decibels.  After this, I gave them the following worksheet (just pages 1 and 2) to fill out.

Note: I would NOT recommend stealing this worksheet without heavy editing!

Now note that this was my advanced Precalculus class, so I was trying to let them figure out the majority of this on their own–like Dan Meyer says “Be less helpful.”

Well, not only were students hopelessly lost, but they started complaining about the activity, which is rare for that class.  In my eyes, they were complaining about an activity which drew the math closer to the “real world” and gave purpose to what we were doing.  As one of the students commented “you guys always complain and ask ‘why are we doing this’ and this is why, and now you’re asking ‘why are we doing this?’ again??”

Well, I left the classroom rather discouraged and frustrated because I had expected a “better than average” lesson at worst, and was hoping that the students would really get into it, but instead I found a classroom of frustrated, tired, bored, and unengaged (is that being redundant?) students.

So what went wrong?

I want to start by think about something I read on a few people’s blogs, who had failed lessons and reflected on them.  The common theme between these was that the lesson was a great idea because it applied the content to the math very well, but it lacked a hook.

So I started looking for ways to improve that and I want to pause and quote a post from Dan Meyer’s blog:

Both of these things interested me, but the line from there to a classroom modeling task forces me to ask myself:

1. What question would lead to that interesting knowledge?
2. Is there some way I can provoke that question visually?
3. Could a student guess at that question?
4. What information would a student need to answer that question?
5. What mathematical tools would a student need to answer that question?
6. Is there some way to confirm the answer visually?

So the next time you see something that’s simultaneously a) interesting to you and b) mathematical, try running through those questions above and see how they’d play out. In the meantime, you can check out my specific answers above.

Those are all very, very important questions if you are designing a lesson for students to be engaged and interested because you think the topic is interesting.  As Evan Weinberg said:

I need to be a lot more aware of the level of my own excitement around activity in comparison to that of the students.

So how can I improve the hook in this particular lesson?

Well, for starters, I could not start by giving them the equation.  Could they find the equation above simply given a list of sounds, their respective sound pressures and decibel levels?  Possibly and possibly not, but now we’ll never know.  Either way, they would have a much more intuitive grasp of what the equation means.  Instead, I found myself explaining the equation and the parts of the equation several times over–I probably wasted more time than I would have if they had struggled with it!

This lesson also enlightened me to a difference between good struggling and bad struggling, which I previously did not know existed.  Students struggling to find “the right place to plug in the number you gave them into the equation you gave them” is not good struggling.  Students working to find a relationship between numbers and working to have a rigorous conversation about it is better.  Students struggling to see what that relationship means and how they can use that relationship for future problems is even better.

I thought I had I created a pretty neat extension activity, if I can just find the right hook and present it better (this class is still working their way through this worksheet) I might have a chance to redeem this activity for the students and for myself.  As for answering Dan’s questions above, no, I do not think there is an easy way to verify this visually.  If the students were elite musicians, I could possibly verify our finds through experiments and having the students bring their instruments into class.  Even with speakers, I’m not sure students (nor I) could hear the differences in decibels acutely enough to know whether our calculations were correct.

I will also not discount the effect of students’ lives into the situation.  I spoke with one particularly frustrated student later and found out that this student was having a “bad 3 days outside of class”, for which she and I apologized to each other for our respective lack of thoughtfulness.  You can never discount the baggage that many students bring to your room on a daily basis.  However, I will not use that as an excuse not to improve this lesson which clearly could have been better handled.

Please let me know if you have a better idea on this lesson or if you have a better idea for a lesson plan using logarithms (I suppose I haven’t had time to search many sources for one of these).

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## Midterm Analysis: Inverse Functions

After attending the middle school data analysis meeting, I decided to do some analysis of my own.  I started with the inverse function problem on the PreCalculus midterm since it was the only topic students did well on during the semester but struggled with on the exam.

What types of mistakes do you think students would make?

Have a hypothesis?  Okay, now you can view the exact mistakes my students made at the Math Mistakes blog.

Back?  Great!  So here’s what I noticed:

Many students made small errors in basic algebra that may have been due to fatigue (this was question 23 of 25 and most students had to stay past the scheduled 90 minutes to finish the exam).

It was a “check if these are inverses” question but many students attempted to check by finding the inverse themselves, rather than composing the functions. When their solution didn’t look exactly like the given inverse they said “not inverses” and moved on. In fact, the inverse many students found was equivalent to the one given (they found $\dfrac{1}{x} + 3$).

Next time I teach/assess this:

In class/homework I need to give more examples of equivalent functions that don’t look equivalent so students are compelled to check more carefully.

I wonder if giving the functions in reverse order would change the results for the better (so students who found the inverse themselves would start with $\dfrac{1+3x}{x}$ and work from there).  I’m inclined to think I would see more algebra mistakes but fewer students who found the inverse correctly and didn’t see that their solution was equivalent to the given one.

My co-worker and I found it difficult to come up with a problem that requires students to solve the inverse since many of our students could immediately “see” the inverse without having to write anything down.  I will need to spend more time finding good problems with interesting inverses that aren’t too complex.

Requests for feedback:

Did the mistakes my students made match the ones you expected to see?  What would you do to remediate this year or prevent such mistakes next year?

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