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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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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Tough Questions

As you teach this week, look for a question a student (or colleague!) asks that you don’t immediately know how to answer.  What did you do?

1. If you came up with a good answer at the time, what thought process got you there?
2. If you came up with a good answer after class ended, what prompted the realization?  Will you communicate your better response to the student (or other original asker)?  If so, how?
3. If you have yet to come up with a good answer, we want to help!

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