Thought for the day: Creative Math Prompts offer a means of introducing important concepts by having students discover and reason about ideas before you explain them.
Concepts: definition of a function, vertical line test for a function (possibly piecewise-linear functions, line graphs, trends)
Examples of noticing and wondering
I notice three graphs that are made of four or five line segments each.
I notice that the horizontal axis is labeled "time," and the vertical axis is not labeled.
I notice that each graph looks the same on the left and right sides, but different in the middle.
I notice that for the third segments from the left, one has a positive slope, one has a vertical (undefined) slope, and one has a negative slope.
I notice that the vertical segment could represent a sudden change in zero time. (It would probably make more sense not even to draw the segment).
I notice that the negative slope and zero slope segments should not both be in the third graph, because it would mean that there are two different measurements of the same thing at the same time.
i wonder what the vertical axis could represent.
I wonder what the units and scales on the graphs could be.
I wonder if I could create stories for the graphs.
I wonder if there is a quick way to recognize graphs (like the bottom two) that don't make sense.
I wonder if there is a way to make the bottom two graphs make sense as they are.
I wonder if I could figure out equations for the different parts of the graphs.
Creating something new
Students may create meanings for the graphs by choosing a quantity for the vertical axis, possibly along with units and scales for one or both axes. They may invent stories or situations to go along with their creations. They may also modify the given graphs (or create a brand new set of graphs) and then go through the same process. Encourage them to share and compare their creations!
The three graphs lead students to recognize graphs that do not make sense because they show more than one output for some inputs. By analyzing the graphs carefully, students may discover that this situation occurs when one or more points on the graph lie vertically above or below each other. This discussion can both (1) motivate the definition of a function, and (2) help students discover the vertical line test for recognizing functions and non-functions from their graphical representations.
Some students may believe (erroneously) at first that in order to change the bottom graph so that it makes sense, they must change the third segment, because it appears to represent "going backwards in time." If so, challenge them to accomplish this without changing the third segment. Others may investigate ways to make sense of the bottom two graphs by replacing the "time" variable by some other quantity for which there can be more than one output for a given input.
Students who choose to include units and scales in their examples and stories may be interested in trying to create equations for each of the segments. This could lead to a discussion of symbolic representations of piecewise functions.