A Concept Map for Mathematical Functions

Analyzing functions.
Analyzing functions.

This year I’m trying teaching pre-Calculus (and it should work for some parts of algebra as well) based on this concept map to use as a general way of looking at functions. Each different type of function can by analyzed by adapting the map. So linear functions should look like this:

Adapting the general concept map for linear functions.
Adapting the general concept map for linear functions.

You’ll note the bringing water to a boil lab at the bottom left. It’s an adaptation of the melting snow lab my middle schoolers did. For the study of linear equations we’ll define the function using piecewise defined functions.

The rising temperatures in the middle of the graph can be modeled with a straight line. Graph by A.F.
The relationship between temperature and time on a hotplate. The different parts of the graph can be defined by a piecewise function. Graph by A.F.

How to Give Effectively

Over the weekend I ran into this article from Planet Money/This American Life about a charity that just gives money directly to people. It should help to prepare my students for the Heifer International Ranch trip next month. It’s interesting to hear what people spend the money on; things like metal roofs and cows. Heifer gives cows and training instead of just cash. The article compares and contrasts the two approaches.

Preparing for Rabbit Dissection

References

Linn (2000) has a good overview of general rabbit biology and the internal organs in Rabbits: Biology (pdf).

For lots of detail on how to dissect a rabbit try Bensley’s Practical Anatomy of the Rabbit (free on Google Books)

AnimalLearn.org has a good list of free and for fee online dissections of various organisms. No rabbits as of this moment though.

Teaching Organic Farming

Teaching Organic Farming and Gardening
UCSC’s Teaching Organic Farming and Gardening

One of these days I’d like to put in a garden at school. Or maybe a few gardens. An indoor hydroponic system would be nice for the winter months, as would a greenhouse. However, the easiest thing to start with might be putting in some raised beds. To this end, the University of Santa Cruz’s Center for Agroecology and Sustainable Food Systems has detailed information in their Teaching Organic Farming & Gardening: Resources for Instructors manual.

↬ Mr. Sansone.

Understanding the Extinction of the Dinosaurs (and the Survival of Mammals)

This neat paper (Robertson et al., 2013) in the Journal of Geophysical Research makes an interesting attempt to explain the pattern of extinctions that occurred at the end of the Cretaceous: why most of the dinosaurs died out, and why ocean organisms were more severely affected than freshwater organisms by the long winter after the asteroid impact.

The flow chart explains:

Diagram of contrasts between freshwater and marine environments for factors potentially causing extinction in aquatic environments after the Chicxulub impact. (Image and caption from Robertson et al., 2013).
Diagram of contrasts between freshwater and marine environments for factors potentially causing extinction in aquatic environments after the Chicxulub impact. (Image and caption from Robertson et al., 2013).

They also include an interesting figure showing how long an organism might survive based on how large it is, which I may be able to use in pre-Calculus when we’re discussing log scales and linearizing equations.

Allometric relationship between body size and time to death by starvation for multicellular poikilotherms in the absence of food (red, drawn from the equation of Peters [1983, p. 42]). Names of various types of organisms are shown as an indication of body size. Image and caption from Robertson et al., 2013.
Allometric relationship between body size and time to death by starvation for multicellular poikilotherms in the absence of food (red, drawn from the equation of Peters [1983, p. 42]). Names of various types of organisms are shown as an indication of body size. (Image and caption from Robertson et al., 2013.)

The article is written well enough that an interested high school biology student should be able to decipher (and present) it.

Crayfish: Charismatic Mesofauna

This year, the creek is teeming with crayfish, especially compared to last year during the drought when the creek dried up and the crustaceans were hard to find. I had five students out collecting organisms on Wednesday, and they came back with ten crayfish ranging in size from a couple centimeters long, to one that was about 12 centimeters from claws to tail.

Crayfish in a jar.
Crayfish in a jar. Seen through stereoscope. Magnification unknown.

I was just looking at one of the pictures I took and realized that I did not know what species it belonged to. I’ll be having students do reports on individual species for biology next year, and I’d be very surprised if someone did not choose crayfish. They’re so many of them and, as my students from Wednesday will attest, they’re just so charismatic. While I’ve not looked into it much myself, the Crayfish & Lobster Taxonomy Browser seems a decent place to start researching.