This is a great tool for plotting vector fields:
http://dlippman.imathas.com/g1/fullgrapher2d.html
You can also plot functions of all kinds: implicit, explicit, polar, parametric, etc. In addition, you can plot several different graphs simultaneously (say, a vector field and a set of flow lines), and you can modify each graph individually.
I found this very helpful is constructing a recent test!
My latest contribution to the New York Times Learning Network is a math lesson designed to get students thinking quantitatively about the increase in population growth around the world. Here is an excerpt.
A typical feature of population growth is that the rate of increase itself increases over time. Visually, this means that the line segments get steeper from left to right. When the slopes of each line segment are computed for each 10-year interval, students can look for a pattern in how the slopes, i.e. the rates of population growth, change. For example, students might notice that, every ten years, the slope of the line segments increase by 0.5 million people per year: this means that the rate of change of population increases by 0.5 million people per year.
Once a pattern is identified, students can then extend their graph beyond 2010 by drawing a line segment from the 2010 data point whose slope fits this pattern. By extending this new line segment so that it covers 10 years on the horizontal axis, this will create a population projection for the year 2020. By repeating the process, students can create population projections for 2030, 2040 and beyond.
Using a recent revision on world population growth by the United Nations population bureau as a starting point, students choose a country to profile. By using available population data, students create piece-wise linear graphs to model that countries population growth, and look for trends in order to make population projections.
You can find the full article here.
Here’s an introduction to the one-cut challenge using triangles, from my Fun with Folding series, suitable for students of all ages (including teachers!). This is a rich, compelling problem that touches on a lot of sophisticated ideas in geometry, but is simple enough to start playing around with right away.
The one-cut challenge is as follows: given a shape made up of connected straight line segments (i.e. a polygonal chain), can you produce the shape as a cut-out using only straight folds and a single straight cut?
A good place to start is with an equilateral triangle. This is a fairly easy problem to solve, given the inherent symmetry in the figure. Fold across any line of symmetry to produce a new figure that looks like two line segments meeting at an angle. Fold those together along their vertex, and cut!
The next step is trying this with an isosceles triangle, whose single line of symmetry still allows this approach to work.
Now the kicker: try it out on a scalene triangle! No more lines of symmetry, and all of the sudden this is a pretty challenging problem!
Happy folding!
Have more Fun With Folding!
This website provides running tallies on several world-wide statistics:
Data on Population, Energy, Economics, and Health are all constantly “updating”, brought you you by the Real Time Statistics Project.
In addition to the obvious questions one could ask, like “At what point will the world’s population grow to over 10 billion?” or “When will the earth run out of oil?”, there are interesting meta-questions like “Where do these models come from?” and “What assumptions are being made to calculate the amount of money spent on weight-loss programs?”.
Another nice resource to play around with!
This website aims to evaluate the accuracy of various weather-prediction services while providing its own forecast:
Using composite indices and statistical methods, data from sites like Accuweather and Weather.com is analyzed and rated.
This is a good resource for an interesting group or individual project in statistics: how accurate are the various services? What is a “good” prediction? How valuable is this information? How can we use statistics to evaluate these questions?
Several years ago I read this post from the Freakonomics blog: it details an informal study conducted by a man and his daughter who looked at seven months of TV weather forecasts in Kansas City and evaluated their accuracy. The entire article is interesting, but the bottom line is best summed up in a quote from someone from one of the TV stations: ““We have no idea what’s going to happen [in the weather] beyond three days out.”