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As mathematics teachers, we are familiar with the relationships among graphs, equations, pictures/diagrams, and words. This multiple-representation mindset is often lacking in our students, who sometimes fail to see the connections that we take for granted. The advantage of different representations of the same situation is that some representations provide information that may not have been readily seen in another representation. For example, a graph provides information about linearity versus non-linearity, as well as information about changes in slopes and intercepts. A table of values provides information about patterns in data. An equation provides information as a representation well suited to forecasting (extrapolation), as well as the ability to represent patterns in a concise manner. Pictures and words give us physical or written descriptions of reality that can be represented mathematically in other forms.
A major takeaway for students is that a graph tells a story and can provide information that is not readily available from other representations. This is a big idea in mathematics. We routinely use straight lines to provide information about cellphone plans or identify break-even points for a business; parabolas show us maximum area or minimum cost. But the best way for students to learn "the graph tells a story" concept is with student-generated data, followed by the "What if?" thinking, which results from changing the parameters of the original situation. This is a great opportunity for students to think critically and justify their ideas. It also allows other students to challenge the results and have students explain their thoughts.
I first saw the basic problem in The Mathematics Teacher almost 20 years ago. A student takes equal-sized sips of liquid (I like cranberry juice for this, since...