Overview

In this lesson students begin the process of solving a differential equation by estimating solutions from a slope field. Work from first semester related to slopes of tangent lines is reinforced and built upon.

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Teaching Tips

Make sure to check in early with groups during this activity. The idea of plotting a mini tangent line is not always intuitive for students. Let students spend time exploring the slope fields before jumping to pattern recognition (such as that the slope does not depend on the y-value at all, so at one particular x-value all the slopes will be parallel). 

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A few of your students may recognize the similarities between slope fields and magnetics fields from physics classes or the maps of wind vectors during a weather forecast on TV. All three contexts are providing discrete pieces of information about a continuous event: a differentiable function, the magnetic characteristics of Earth, or the flow of air across the continent.

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Slope fields allow students to visualize a family of antiderivatives, each differing by the ubiquitous +C. Slope fields are especially useful when an antiderivative is complex or unknown. Have students work intentionally and methodically when creating their first few slope fields: they should examine ordered pairs individually until they are able to see a pattern emerge and the antiderivative become apparent.

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Differential equations written only in terms of x will have identical tangent lines in a vertical row; DiffEqs written only in terms of y will have identical tangent lines in a horizontal row.

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Exam Insights

Constructing and interpreting slope fields is often presented in conjunction with integration by separation of variables. Once a student has found a specific solution through a given point, they may be directed to sketch their solution on a slope field.

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College Board has several Curriculum Modules available to calculus teachers. The module discussing Slope Fields (by Nancy Stephenson) is straight-forward and comprehensive. For more exposure to AP-style questions, familiarize students with these questions and scoring rubrics: 2008 AB 5a, 2006 AB 5a, 2005 AB 6ab, and 2004 AB 6ab. For multiple choice practice, see 2003 BC 14. 

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Student Misconceptions

When the value of a derivative is undefined, students should refrain from making a vertical tangent line on their slope field. Lin McMullin’s blog, Teaching Calculus, has several excellent examples where the vertical tangent mark is obviously absent.

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If asked to sketch a particular or specific solution to a DiffEq through an ordered pair, encourage students to plot the point first, then sketch the solution through that point. This may prevent someone from sketching graphs above and below an asymptote and presenting a solution curve that is not even a function.