A Diagrammatic Construction of Indefinite Integrals: Confronting the Elusive +C

摘要

Multiple representations of mathematical concepts reveal different aspects of their properties. Constructing links between pictorial and algebraic representations allows for deeper understanding and is therefore a powerful tool to extend learning in the mathematics classroom. The interpretation of definite integrals as the area under the curve is common practice but indefinite integrals are usually approached using algebra only. This article examines a series of learning activities around the gradient function from my Year 12 Curve Sketching Summer School at UCL. Their ultimate aim is to lay down the path to constructing a diagrammatic representation of indefinite integrals. No equations are used in the activities throughout except when clearly stated. Firstly the gradient function is explored as a purely geometrical feature of a curve by using computer animations and students are then asked to sketch the corresponding gradient function of given curves. Bringing this idea to the familiar territory of quadratics and cubics, students are given a card matching activity where they need to match graphs to their corresponding gradient function and vice versa. Finally students are presented with three different quadratic graphs drawn on blank coordinate axes and requested to sketch the cubic whose gradient function corresponds to the given quadratics. The outcome is a diagrammatic construction of indefinite integrals and an insight into the elusive integration constant, +C. This task also provides the classification of cubics according to the number of stationary points. These activities are usually well received by the students with feedback along the line of 'we get the +C now.'