Exploring Students' Understanding of Linear and Quadratic Relationships in a Projectile Motion Context

TL;DR

This paper investigates how covariational reasoning helps students understand linear and quadratic relationships in a projectile motion context, focusing on height $H(t)$ as a function of time $t$. It uses a teaching experiment with a Height-Time Relationship Task implemented in Desmos to trace how two middle school students' covariational reasoning develops. Findings show that prompts comparing curved and linear representations foster a progression from gross coordination to chunky continuous covariation and strengthen the interpretation of a linear relationship as a constant rate of change. The study emphasizes the role of non-canonical graphing tasks and technology-enabled representations in promoting covariational reasoning, informing instructional design for teaching linear and quadratic relationships in dynamic contexts.

Abstract

Previous research has shown that students often struggle to develop an understanding of linear and quadratic relationships. Covariational reasoning has been identified as a way to support this development. This study aims to investigate how covariational reasoning supports students in developing understandings of linear and quadratic relationships within a projectile motion context. A teaching experiment was conducted with two middle school students who engaged in a digital task exploring the relationship between height and time. The analysis characterizes how the students' covariational reasoning evolved as they interpreted the changing quantities in the task. The findings suggest that prompts encouraging students to compare linear and quadratic relationships can foster more sophisticated forms of covariational reasoning. The discussion highlights how specific features of the task design, including the affordances of technology, the emphasis on conceiving graphs as representations of covarying quantities, and the use of non-canonical graphing tasks, can support covariational reasoning.

Figures (4)

• Figure 1: Screen showing the launch activity where students select a line segment representing the ball's height over time
• Figure 2: Screen showing the dynamic relationship between height and time using moving line segments
• Figure 3: Graph interpretation screen where students analyze the ball's height-time graph
• Figure 4: The graph that was drawn by TR on the whiteboard