Experience First

In this lesson students again reason about solutions to an equation, but this time from a geometric perspective. Learners must come up with new designs for cans within a given constraint (volume stays constant). Students may not be used to the open-endedness of the second question, so encourage students to choose any dimensions, as long as one can is wider and another taller. Ask students to consider what dimensions would be reasonable for a can and why. For students that are struggling to grasp some of the relationships between radius and height, ask students to predict what will happen to the other dimension once they adjust one of them (a wider can requires a shorter height!). Ask students if they can just choose any two dimensions for the radius and height to invite conversation about dimensions that will “offset” each other. 

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As the activity progresses, students will be asked to be more concrete about how the two dimensions offset each other. Before jumping to an analytical proof, have struggling students test out certain values (like doubling the radius and halving the height). 

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Formalize Later

Students may begin noticing that doubling the radius quadruples the volume, solely by observation, but use the debrief to justify or prove why this happens. Using color coding in the formulas so that students can clearly see the new volume in relation to the old volume is very helpful. To check for understanding, ask students to discuss what would happen if the radius is tripled or if the height is octupled to see if students can articulate how when changing the height, the volume changes proportionally, but if changing the radius, the volume changes by the square of the scale factor. 

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This type of reasoning about the relationship between variables in an equation is a huge skill tested on the SAT. Students should feel comfortable isolating any variable in an equation and explaining how this new version of the equation highlights a particular variable.

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To extend the learning, have students create their own “Which is a better deal?” question.