In this case the teacher is discussing what a derivative is and what it means to analyze a graph of a function and students are brainstorming a minimum of four ways a derivative can be used and how to use the graph of the derivative to describe functions.

This case shows a teacher in a discussion with students about derivatives and students giving information and answering questions about derivative properties.

This case shows a teacher working with students on applying derivatives to a group of individual graphs and students trying to determine the relationship among the graphs and the first and second derivatives.

In this case the teacher is assisting the class in identifying and correcting errors in their application of u-substitution and students are integrating definite and non-definite integrals, using generalized strategies for implementing the substitution process.

This case shows a teacher working with small groups on making connections between integral and differential calculus both analytically and graphically and students working collaboratively to make predictions based on the patterns in given functions.

In this case the teacher is developing the ideas of Euler's method by using slope fields to gather information about solutions to differential equations and students are engaged in an activity using a single tangent line approximation and the slope field to draw an approximate particular solution to the differential equation.

In this case the teacher is facilitating small groups of students bridging together previous lessons about properties of functions and their relationship to derivatives and the students are using manipulatives collaboratively to deepen their conceptual understanding in multiple ways: visually, numerically, verbally and graphically.

This case shows a teacher working effectively with students on how to compound interest and students predicting which variables would have the greatest effect on an investment, manipulating one variable at a time to see which of four investments would yield the greatest return.

This case shows a teacher facilitating group work on a real-life problem in calculus and students working collaboratively to connect the surface area and volume of a sphere using integration to a real-world application.

This case shows a teacher facilitating a discussion based on data the students had collected and students seeing similarities and differences in the data, determining visual representations of Riemann and Trapezoidal sums, and drawing conclusions together as a class.

In this case the teacher is facilitating small groups who are reviewing function and limit properties by connecting and manipulating graph and function cards.

In this case the teaching is facilitating students creating and analyzing sets of parametric equations to model projectile motion and students are using prior knowledge of physics and calculus to generate equations that model motion while engaged in mathematical discourse.

In this case the teacher is facilitating an activity where students use what they know about finding average rates of change and instantaneous rates of change to deepen their understanding of derivatives and students are using technology and group discussion to make and test conjectures, discussing their errors, and coming to correct conclusions.

In this case the teacher is helping students make connections about how related rates problems change over time and students are brainstorming common themes in the problems, shapes that may be used, and using a computer application to see change as time passes.

In this case the teacher is guiding students in discovering what information the first and second derivatives of a function give about the function itself and students are discovering what characteristics of a function would be important for graphing that function, making the connection between local extrema on the function and the places where the derivative is zero or undefined.

This case shows a teacher working with students on a calculus activity about rates and students engaged in a hands-on activity determining rates and rates of change.

This case shows a teacher working with small groups and students discovering how to find the area under a curve using integration, manipulatives, and a graphing calculator while also communicating mathematically.

This case shows a teacher leading a discussion on derivatives of compositions of functions and students applying previous knowledge to fit derivatives of basic functions into the framework of the Chain Rule.

In this case the teacher is helping to expand students' knowledge of graphs related to derivatives and students are exploring this relationship and analyzing how to obtain the graph of a function from first and second derivative graphs.

In this case the teacher is facilitating a whole class discussion about derivatives and the students are learning to extend continuity to vector functions, and are introduced to the geometry of space curves to begin their understanding of derivations of the derivative of a vector function.