Ever poured water into a bottle and suddenly it overflowed before you even noticed? Maybe you were daydreaming, or maybe your bottle has the shape shown in the interactive tool here. Why does that happen? Let’s investigate with the “Bottle Problem” interactive below!
This learning resource illustrates how graphs emerge from covarying quantities. One calculator shows how changing the volume of water in a bottle affects its height. The second calculator reveals how these quantities correspond on the x– and y-axes, first as segments, then as a point, and finally as a trace that forms the graph of volume versus height. In this way, the resource highlights both what a graph is made of (a trace of points) and how it is made (through covariation).
Bottle Problem
Adjust the slider, reveal the segments and point, record the point’s traces, and watch how the graph takes shape.
Use the following questions or instructions to make the most of the interactive tool above:
- Slide the volume slider left and right. What happens? Does every constant change in water volume result in a constant rate of change in water height? When is the rate of change in height the fastest? When is it the slowest?
- Click “Show segments” and move the volume slider again. What do the lengths of the red and blue segments on the x– and y-axes represent?
- Click “Show point” and move the volume slider. What does the point represent? What limits its movement?
- Record the point’s trace by clicking “Record trace,” then move the volume slider. What do the traces represent? What do you get if all possible traces of the point are recorded?
Happy exploring!