Interpreting Slope and Intercept in a Fundraiser Model
Ms. Chen's algebra class collected data on a car wash fundraiser. Students worked h hours and raised d dollars, modeled by d = 12h + 40. A table shows (0, 40), (2, 64), (5, 100), and (8, 136). On a quiz, three students made these statements about the graph. Ana: The y-intercept is 40, so the car wash started with $40. Ben: The slope is 12, so each hour raises $12. Cara: The graph should not extend to negative hours because the context does not allow them. Ms. Chen wants to use the statements to discuss what slope and intercept mean in context, not just how to compute them. She also notices several students plotted h on the vertical axis.
1. Evaluate each student's statement. Which are mathematically defensible, and what refinement does each need?
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Ana's statement is close but needs precision: at h = 0, d = 40, so $40 was raised before any hours were worked. That could be a starting donation or prior sales, not necessarily cash the car wash 'started with' in a literal sense. Ben's statement is correct in context: the slope 12 means the fundraiser raises $12 per additional hour, on average, and the units are dollars per hour. Cara's statement is mathematically and contextually sound: the domain should be restricted to h >= 0, and if the fundraiser ends at 8 hours, the graph should be a set of points or a segment rather than an infinite line. For students who plotted h vertically, the axes are reversed: the slope would be 1/12 hour per dollar, and the intercept would be an hours value, not a dollar value. The teacher can ask which variable depends on which.
2. Design one question Ms. Chen could ask to probe whether students understand the intercept rather than memorize it.
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Ask: 'If the group had no $40 head start but still earned $12 per hour, what equation and graph would result, and what would the y-intercept mean?' This forces students to connect the intercept to the initial condition rather than to a memorized procedure. A follow-up: 'What does d = 40 mean when h = 0, and is $40 a plausible amount for this fundraiser?' Students should explain that the intercept is the value of d when h = 0, so it is the amount raised before any hours are counted. If students say 'it is just the number in the equation,' ask them to change the story so that the intercept is 0, then 100, and describe the graph. The teacher should accept 'starting amount' only if students can say what real quantity it represents and how the graph changes when it changes.