Sunday, March 23, 2014

Why we can't find Malaysian Flight 370

This entry is with full respect to those individuals and their families who have been missing since the flight's disappearance.

Our textbooks like to give “Real World” problems. Too many times these are at the end of the section or under a section called “Modeling” and they are so canned, they are unrealistic - thus defeating their purpose. This pedagogical paradigm that our textbooks frame assumes that we should teach skills before content but this is like teaching vocabulary without context and in the world of vocabulary learning, context is everything (think spelling bee... Can you use this in a sentence?) We need to be always teaching context for math just like any other language always uses context and the Malaysian Flight 370 disappearance is a perfect example. Many people are wondering – why haven't we found the floating stuff that we see with the satellite?

What a perfect related rates problem.

If I wasn't on spring break right now, I'd ask my calculus class this exact question. Here's the information:

-7 days ago the Australian government spotted a big object floating in a location 1,400 miles south of Perth
-The current in the South Indian Ocean flows at 1m/s.
-We can imagine that the search area becomes a sector area (assuming its still floating) – we can make up some reasonable angle such as it can move 15 degrees in either direction from the center line of the current.

Now ask the question again given this information and see what happens in a calculus class. A follow up question might be: what is the rate at which the search area is changing every day/hour/minute/second. I would love to see students problem solving (making diagrams, asking questions, etc) and seeing why its so hard to find debris in the ocean.

I found this information on so why not integrate literacy into the lesson by having students read for information (your English/Social Studies/Science teachers will love you because they always assume that students are learning math the way they did 25 years ago). As of yesterday, CNN also states “ the current search area is 2.97 million (Thanks to Dan Meyer for checking me on the details) square miles” - that's roughly the size of the continental US. Can you imagine 15 planes and 30 ships trying to find something 100 feet long somewhere in the continental US?

A great follow-up discussion would be how to maximize the efficiency of searching, hoping that they find the objects soon would be even more remarkable.

(Want to make this an Trig/Precalc lesson? Ask students about how the sector area and how height changes the distance which a person can see on the ocean – how many ships are necessary and of what height to cover the search area?)

For more problems like this check out Stu Swartz's "Ripped From the Headlines" at

Sunday, March 9, 2014

Horizontal Asymptotes & End Behavior

It's been a while since we've updated this blog, but we're back!

This is a lesson that we did way back in October (better late than never!) in my honors precalculus class. In this short activity students investigate what happens to rational functions toward infinity and negative infinity. The activity is fairly simple, but it helps kids improve number sense a whole lot more than memorizing "if the degree of the numerator=the degree of the denominator"....  

The "Quick Solve" was fairly straightforward for the students. It was easy for them to see that as x gets extremely large the "plus 4x" starts to matter less and less. As x approaches infinity the "minus 3" barely even matters at all. Once they move on to the rational functions they can start thinking about how the numerator and denominator grow in relation to one another, with just looking at the terms that has the most impact on the fraction.

I liked that this activity got the students away from trying to memorize rules to find horizontal asymptotes of rational functions. Once the students completed the table on page 1 we discussed as a class what these functions look like in the coordinate plane as you approach positive and negative infinity, ignoring what's happening between the extremes. This got us into the whole discussion of end behavior (they have already been working with limit notation) and the kids started to think about the "ends" of these functions when there was no horizontal asymptote.

All in all, I think this activity did its job - there's no real practical applications embedded here, but it makes the students just think about numbers in general; something I've been really trying to push with my students this year.