Session 1 Notes: Algebraic Thinking

A B C

Notes for Session 1, Part C

Note 6

One of the goals of thinking algebraically is to develop different ways of representing real-world situations. Representing mathematical ideas in pictures, tables, graphs, and words allows us to use mathematics as a way of communicating. With Eric the Sheep, it appeared that a table and a description in words were good representations of the situation.

The problems in this section involve qualitative graphs. These are graphs that concentrate on the general features of a real-world situation. As an analogy, when learning a foreign language, we are given the rules of grammar, but also opportunities to express ourselves. Similarly, qualitative graphs allow us to interpret, transform, predict, and make logical deductions from the given mathematical data. In this way, even simple graphs can communicate a great deal of information, as illustrated in Problem C1.

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Note 7

Groups: Work in pairs on Problem C1. All of the graphing activities in this section lend themselves to interesting and thoughtful discourse, so it's important to allow time for everyone to discuss their own reasoning and justify their answers.

It may seem puzzling at first that there are no numbers on any of the graphs' axes. This may be especially true for those who are accustomed to teaching precise point-plotting techniques. Consider why the axes are not labeled with values. (In fact, if there were numbers on the axes, we would not have to think about the relative positions of the points.) The graph in Problem C1 is particularly interesting because height is not on the vertical axis. This is intentional so that, once again, we have to focus on the representation and its meaning rather than on a graph that is merely a literal picture of the situation. Consider creating another graph for Problem C2, one with the axes reversed.

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Note 8

Groups: Record answers to Problem C3 on an overhead or chalkboard. It is important to describe carefully the reasoning that was used to place the specific points on the graph.

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Note 9

The reasoning of Problem C4 (to determine what must be true of Jane, Paul, and Graham) is sophisticated, and it foreshadows the work we will be doing in later sessions on slope and rate. Session 4 notes

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Note 10

Groups: Work on Problem C5 in small groups. For each description, compare answers. Be sure to explain the reasoning for selecting the associated graphs.

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Note 11

It may be helpful to illustrate the relationship between volume and height for the bottles in Problem C9 by using a conical flask and a cylindrical bottle, and filling each with water at a steady rate. Or fill each one with cups of water and measure their heights after each cupful.

Groups: If anyone has difficulty seeing how this would translate to a graph, consider drawing a figure on an overhead or chalkboard to show the conical flask. For example, the height increases by a small amount to start with (so that bottle must be wide at the bottom) and then gradually rises by larger and larger amounts (so that bottle must be getting narrower gradually towards the top). Therefore, this is the graph of the conical flask:

conical flask graph illustration

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