In this case the teacher is assisting students in making graphs, utilizing distance and time based on real-life scenarios, and students create the necessary graphs through discussion with each other, using these graphs to calculate Riemann Sums and to develop the Trapezoidal Rule.

In this case the teacher is facilitating an activity involving functions and derivatives and students are sorting and matching card sets of graphs and descriptions of functions and derivatives, using their knowledge of algebraic relationships to make the match.

In this case the teacher is facilitating students identifying functions and their appropriate matching graph and students are doing an activity where they are discussing the graphs and hypothesizing on possible matches, justifying their choices.

In this case the teacher is guiding students in an exploration of polar curves and students are making and testing conjectures in collaborative groups to generalize the patterns they are seeing and analyzing the characteristics of graphs, given the equations in polar form.

This case shows a teacher leading a discussion about a specific AP test question for calculus and students working through the problem using mathematical discourse.

This case shows a teacher actively engaging her students in a discussion in determining derivative functions and definite integrals based on real-world problems.

In this case the teacher has assigned nine different graphs and has asked the students to identify which graphs have absolute, relative maximum, or minimum values and the students are discussing the various characteristics that warrant maximum or minimum values.

This case shows a teacher facilitating a whole-class discussion about 4 graphs and students sharing unique ideas and facts about the graphs before interpreting problems independently.

This case shows a teacher facilitating small groups of students as they match function graphs to derivative graphs and students exploring and demonstrating their knowledge of rates of change graphically.

This case shows a teacher facilitating a student discussion about open-ended calculus problems on motion and students brainstorming ideas and problem solving to better understand the concepts.

This case shows a teacher teaching about arithmetic sequences and linear equations and students developing their reasoning and mathematical confidence as they propose ideas and then support or dispute them.

This case shows a teacher managing group work while students are engaged in a calculus activity finding limits graphically, numerically, and algebraically.

This case shows a teacher facilitating a discussion on Riemann sums and students developing their knowledge of definite integrals as the limit of a Riemann sum through mathematical discourse.

This case shows a teacher demonstrating and analyzing motion through graphing and students learning how to apply derivatives to simple motion problems.

In this case the teacher is helping students become more familiar with the concept of area underneath a curve and its representation by an integral and students are using kinematic equations to solve a variety of real-life problems, engaging in productive mathematical discussion.

This case shows a teacher facilitating small group work and students engaging in mathematical discourse as they explore the concept of linear programming and apply it to real-world problems.

This case shows a teacher capitalizing on students' prior knowledge and students synthesizing information to solve systems of equations using matrices.

In this case the teacher is helping students gain a better understanding of the physical look and feel of 3-D objects and their volumes and students are using abstract integrals to create different models, analyzing them and gaining multiple perspectives from their visualizations.

In this case the teacher is assisting students in discovering for themselves the various features of the graphs of rational functions and students are working together, communicating ideas, sharing thoughts, and supporting their reasoning with evidence.