Lesson 3: 7.3 Sketching Slope Fields
Duration of Days: 1
Lesson Objective
Students will be able to construct a slope field for a given first-order differential equation by calculating the slope at various points (x, y) in the plane.
Students will be able to estimate the value of a solution at a specific point by using a tangent line derived from the slope field.
How can a differential equation dy/dx be used to describe the "flow" of a family of functions?
What does a horizontal line segment in a slope field tell us about the derivative at that specific point?
What happens to the slope field when the differential equation depends only on x, only on y?
How does a slope field represent the "infinite" nature of general solutions (the +C)?
Slope Field (or Direction Field)
Linear Approximation
Tangent Line
Family of Functions
Solution Curve
Equilibrium Solution
Grid Points
Segment
MP4: Model with mathematics.
Using slope fields to model the behavior of a differential equation visually.
MP5: Use appropriate tools strategically. Determining when a slope field is the best tool for understanding a solution's behavior.
This lesson introduces slope fields as a graphical tool. Students learn that since a differential equation dy/dx = f(x, y) gives the slope of a solution at any point, they can draw tiny line segments at various grid points to visualize the overall "shape" of the solution curves. The lesson covers the manual calculation of these slopes and the interpretation of patterns—such as identifying where the derivative is zero or undefined.
Purpose: To provide a visual bridge between the algebraic differential equation and the actual function that solves it. It allows students to "see" the behavior of solutions (growth, decay, oscillations) without finding an explicit formula.
DOK Level 2 (Skill/Concept): Calculating slopes and drawing segments for 6–12 points on a provided grid.
DOK Level 3 (Strategic Thinking): Matching a given slope field to its corresponding differential equation by identifying patterns
Support (Scaffolding):The "Slope Table": Provide a T-chart with three columns: (x, y), dy/dx Calculation, and "Visual Slope".
Interactive Applets: Use tools like GeoGebra or Desmos to let students see how changing a differential equation instantly shifts the entire slope field.
Isocline Analysis: Ask students to identify "isoclines"—paths along which all slopes are the same.
Backwards Design: Give students a finished slope field and ask them to write a possible differential equation that would produce that specific visual pattern.
AP College Board