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Developing the vertex formula meaningfully. (English)
Math. Teach. (Reston) 108, No. 6, 429-433 (2015).
Summary: As teachers working with students in entry-level algebra classes, the authors realized that their instruction was a major factor in how their students viewed mathematics. They often presented students with abstract formulas that seemed to appear out of thin air. One instance occurred while they were teaching students to graph quadratic equations. The algebra textbook in their school district introduced the concept by giving students a formula for the $x$-coordinate of the vertex as $x = -b/(2a)$. (Note: The vertex form $y = a(x - h)^2 + k$ is not introduced in the textbook and district standards until advanced algebra.) For a number of years, they provided students with the vertex formula, and the students successfully graphed by substituting values into the formula. Yet when asked where the formula came from or how it connected to the defining characteristics of quadratics, students did not know. They were performing procedures using this “magical” formula but did not understand how the formula developed. In this article, Nebesniak and Burgoa present a method for graphing quadratic equations that connects students’ prior knowledge of graphing linear equations with intercepts and calculating mean. This atypical approach is effective only for a quadratic that has real roots. However, they use this method because the goal is to help students reason how to graph a quadratic with understanding rather than memorize the vertex formula. They believe that this initial approach to graphing is beneficial, despite the limitations, because students are making sense of the concept by connecting it to prior knowledge. (ERIC)
Classification: I23
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