In this case the teacher is facilitating the learning of how to determine the length of a missing leg of a triangle through the use of real-life problems and students are discussing the problems, showing their understanding of when and how to apply the Pythagorean Theorem.

In this case the teacher is helping students acquire a thorough understanding of the relationships between the volume of cones, cylinders, and spheres through the use of manipulatives and students are engaged in a hands-on activity discovering and analyzing the relationships that exist between these different shapes.

In this case the teacher is assisting students in developing better conceptual knowledge of the Pythagorean Theorem by analyzing and explaining proofs and students are using geoboards and puzzles to prove the theorem.

In this case a teacher works with students to facilitate their analysis of the graph of a circle and students discuss the circle's equation as they make the link between algebra and graphing.

In this case the teacher is working with small groups as they work on an activity designed to help them come up with a formal definition of a circle and students are making circles with their materials, reflecting and constructing explanations while discussing what makes a true circle.

In this case the teacher is assisting students in solving problems involving quadrilaterals and students are discovering these properties by working with isosceles trapezoids using manipulatives and justifying the theorem(s) they believe fit their shapes.

In this case the teacher is using whole-class discussion to help students discover the side-angle-side conjecture in geometry and students are sharing ideas, reasoning mathematically, and making inferences.

This case shows a teacher supporting student investigations of Sierpinski's Triangle, and students using measurement tools to create precise drawings, explore number patterns, and develop an understanding of the relationship between geometry and algebra.

In this case the teacher is facilitating small groups working on area and perimeter problems and students are discussing with their partners, persevering and reasoning mathematically while working on real-life problems involving perimeter.

In this case the teacher has given the class information about Amelia Earhart's attempt to fly around the world and her subsequent fatal crash and students are striving to make sense of how many miles off course she was by using angle and distance measurements.

This case shows a teacher managing students working with dimensional objects and students measuring and determining surface areas of 3-D objects.

This case shows a teacher facilitating small groups in identifying quadrilaterals and students working collaboratively and using a manipulative to figure out the type of quadrilateral for each figure.

In this case the teacher is addressing misconceptions about surface area and volume and students are working with manipulatives on isometric dot paper creating 3-D prisms, collaborating with their group, and engaging in mathematical reasoning as they work.

This case shows a teacher helping students learn about 3-D shapes and predicting which shapes might roll the furthest and students engaged in a cooperative activity rolling the shapes, measuring and recording data.

In this case the teacher is helping students understand and prove angle relationships in circles and students are creating proofs, accurately describing and demonstrating how they proved the particular angle relationship property, applying theorems, postulates and corollaries learned previously.

In this case the teacher is facilitating small groups working collaboratively with geometry software to explore the side ratios of special right triangles and students are investigating through constructing, measuring, transforming, and animating figures to create a general rule for side ratios of 45-45-90 triangles.

In this case a teacher expertly encourages mathematical discourse while students justify their answers about polygon angles and students share their ideas on the whiteboard with their peers.

In this case the teacher is facilitating a discussion to help students set up and ultimately design formal proofs in geometry and students are making assumptions about geometric diagrams and justifying these through discourse with their peers.

In this case the teacher is engaging the students in a problem-solving activity to help them make real-world applications about perimeter and area and students are making arguments about the way perimeter and area are used in the construction of a deck.

This case shows a teacher effectively engaging students in a hands-on activity about perimeter and area while relating it to real-life experiences and students using manipulatives to model the problem given to them and to see the real-life connections.