Examine the data below. Can you tell what situation these data model? Or, can you explain the general relationship?

x y

1 1.00

2 1.41

3 1.73

4 2.00

5 2.24

Plot these data. Can you determine a line of best fit? Does this explain the phenomena that the data model?

Here is more information to add to your investigation;

-these data model a geometric concept

Here are more data to plot:

6 2.45

7 2.64

8 2.83

9 3.00

Import these data into a spreadsheet and create a scatterplot. Investigate several lines or curves of best fit.

Name at least 3 equations you obtained through this investigation.

Many times, students will end the investigation with a linear equation. But remember, if this were a linear relationship, the rate of change would be a constant value. For instance, the rate of change between the second and first y-values is 0.41 and the rate of change between the second and third y-values is 0.32. Obviously, this are different rates of change and therefore, the data are not linear-- even though on a graph it is tempting to plot a nice straight line to model the data!

Let's look closer. Did you notice what happened when the x-values were 1, 4, and 9?

Now plot the previous and the following values on another graph. Think carefully about the scales of the axes!

x y

16 4

25 5

36 6

49 7

64 8

Has the mathematical relationship become clear? Explain it in words.

Now, what could the geometrical phenomena be? Recall relationships you have investigated and learned. Think about how the x-values relate to y-values. Explain a phenomenon that can be modeled by the data.

You see, not all data sets are linear, and all must be investigated. "Dark Mathematics" can be any data set that is not yet understood!