This problem provides students the opportunity to explore fractions in a practical context as well as identify and explain patterns and justify their ideas. Solvers are shown a sequence of five squares shaded light blue and dark blue and are asked to find what fraction of the total area of each square is covered by light blue. They are also asked to work out what the next two squares would look like if they followed the pattern. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, and ideas for extension and support.

In addition to the purely geometric and trigonometric aspects of the task, this problem asks students to model phenomena on the surface of the earth.

In this activity, students explore the relationship among angles formed by a transversal and a system of two lines. In particular, they consider what happens when the two lines are parallel versus when they are not.

This is an instructional task that gives students a chance to reason about lines of symmetry and discover that a circle has an an infinite number of lines of symmetry. Even though the concept of an infinite number of lines is fairly abstract, fourth graders can understand infinity in an informal way.

This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. It is both interesting and important that these types of quadrilaterals can be distinguished by their lines of symmetry.

This task is intended for instruction, providing the students with a chance to experiment with physical models of triangles, gaining spatial intuition by executing reflections.

This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below.

This instructional task gives students a hands-on experience with composing and decomposing geometric figures.

In this activity students apply their knowledge of triangle properties. It asks students to sort three types of triangles based on angle and side comparisons, while ignoring the size and orientation differences. Students can work online with an interactive Flash applet, or on a printed sheet (included). The Teachers' Notes page includes suggestions for implementation, discussion questions, and ideas for extension.

Use this Venn diagram to get students thinking about the different types of symmetry as they use the mathematical practices.

This goal of this task is to give students familiarity using the formula for the area of a circle while also addressing measurement error.

This classroom task gives students the opportunity to prove a surprising fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles. The solution provided (among other possibilities) uses the SAS trial congruence theorem, and the fact that opposite sides of parallelograms are congruent.

The purpose of this task is to engage students in an open-ended modeling task that uses similarity of right triangles, and also requires the use of technology.

This Math Fun Fact focuses on using calculus, or sector-triangle formulas from geometry, to compute corresponding areas.
Math Fun Facts were developed as warm-up activities. They are mathematical tidbits meant to arouse curiosity and fascination with the subject. Fun Facts give students a glimpse that mathematics is full of interesting ideas, patterns, and new modes of thinking.

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Do you like to paint? Watch this step by step video as artist Kristin Farr demonstrates how to paint your very own "Magic Hecksagon," which is a colorful, geometric design inspired by folk art. She uses a plethora of different colors to bring a sense of motion to her work. Watch and learn more in the interview with Kristin Farr: http://youtu.be/OX1r-3-VK-0

In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in P-12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes. The Geometry curriculum is available below.

In this activity students are asked to relate the numbers 1- 20 to rectangular shapes. Learners use unit squares or cubes to sort numbers by their 'shapes,' either squares, rectangles or sticks (rectangles of unit width). Ideas for implementation, extension and support are included.

Open Middle tasks provide opportunities for student to approach a mathematical task using different strategies and representations. They can be used as a warm-up/closing activity, as a formative assessment, or to facilitate discourse and discussion and get insite into student thinking and problem solving. These tasks provide a great opportunity for student to engage with the Standards for Mathematical Practice.