In this activity, students investigate the possible sizes and orientations of squares in a 4 by 4 grid. They are to place as many counters as they can on the grid and avoid making the four corners of any square. Solutions require working systematically. Ideas for implementation, extension and support are included along with printable worksheets of grids (.doc)

This interactive Flash game, played on a dot grid, offers an opportunity to practice visualizing squares and angles and also encourages students to use systematic strategies. The object is to select the four corners of a square in some size and orientation. The student can play against the computer or with another student. The dimensions of the grid can be adjusted. The student can opt to see the outline of a square once formed. The Teachers' Notes page offers suggestions for implementation, discussion questions, ideas for extension and support, and links to related activities.

This problem asks students to visualize a square drawn on a clock face and estimate and calculate its area. The problem can be solved without use of the Pythagorean Theorem. Ideas for implementation, extension and support are included along with printable sheets of clock faces.

In this investigation students work systematically and keep organized records as they explore forming triangles from unit sticks. Learners look for patterns and trends in the number of triangles possible with a given integer perimeter. Ideas for implementation, extension and support are included along with a printable sheet of the problem.

This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches. Teachers should of course feel free to provide additional scaffolding to encourage solutions or thinking in one particular direction. We include three solutions which fall into two general approaches, one based on reference to previously-derived results (e.g., the Pythagorean Theorem), and another conducted in terms of the geometry of rigid transformations.

The construction of the tangent line to a circle from a point outside of the circle requires knowledge of a couple of facts about circles and triangles. First, students must know, for part (a), that a triangle inscribed in a circle with one side a diameter is a right triangle. This material is presented in the tasks ''Right triangles inscribed in circles I.'' For part (b) students must know that the tangent line to a circle at a point is characterized by meeting the radius of the circle at that point in a right angle: more about this can be found in ''Tangent lines and the radius of a circle.''

ile patterns will be familiar with students both from working with geometry tiles and from the many tiles they encounter in the world. Here one of the most important examples of a tiling, with regular hexagons, is studied in detail. This provides students an opportunity to use what they know about the sum of the angles in a triangle and also the sum of angles which make a line.

This task aims at explaining why four regular octagons can be placed around a central square, applying student knowledge of triangles and sums of angles in both triangles and more general polygons.

In this activity learners explore the connections of digital time displays with numeric and geometric properties. Students look for times that have bilateral or rotational symmetry, or have a certain digital sum, etc. Ideas for implementation, extension and support are included.

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints. The modelling process is a challenging one, and will likely elicit a variety of attempts from the students.

This task gives students a chance to explore several issues relating to rigid motions of the plane and triangle congruence. As an instructional task, it can help students build up their understanding of the relationship between rigid motions and congruence.

This activity has students experiment with turning through angles in a conceptual, qualitative way. It can be used prior to studying angles quantitatively. Students are given descriptions of physical phenomena to emulate on either the included Flash applet or a hands-on manipulative. They rotate colored disks in a circle to mimic turns and angles. The Teachers' Notes page includes suggestions for implementation, discussion questions and instructions for making the manipulative.

In this activity students interact with rotations of a two-dimensional image. It is intended to be a bridge between the physical turning of objects and the more abstract rotations of a 2-D image. Students are asked how many quarter turns it will take to achieve a resultant image. There is an interactive Java applet available for the manipulations. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension, and printable sheets.

In this video segment from Cyberchase, Digit must a make a straight line between the two points and then follow the path created.

Jackie and Matt are looking at the same object along two different lines, in this video segment from Cyberchase.

This task combines two skills from domain G-C: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment (G-C.2), and computing lengths of circular arcs given the radii and central angles (G-C.5). It also requires students to create additional structure within the given problem, producing and solving a right triangle to compute the required central angles (G-SRT.8).

This problem gives children an opportunity to explore patterns in a practical context and to generalize the results with a rule. Students investigate how many blocks would be needed to build an up-and-down staircase with any number of steps up. An interactivity in the hints shows the blocks transformed into a square pattern. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere. Students who are given this task must be familiar with the formula for the volume of a cylinder, the formula for the volume of a cone, and CavalieriŐs principle.

The author of this one-page article describes the pedagogical benefits of using math games in the classroom. She analyzes the game context in terms of the kinds of learning that takes place, the behaviors that students exhibit, and the social-emotional experience of students. She concludes by recommending strategies for effective introduction and management of games. [While the author's observations were in the context of geometry games, they generalize to most topics.]

This video is meant to be a fun, hands-on session that gets students to think hard about how machines work. It teaches them the connection between the geometry that they study and the kinematics that engineers use -- explaining that kinematics is simply geometry in motion. In this lesson, geometry will be used in a way that students are not used to. Materials necessary for the hands-on activities include two options: pegboard, nails/screws and a small saw; or colored construction paper, thumbtacks and scissors. Some in-class activities for the breaks between the video segments include: exploring the role of geometry in a slider-crank mechanism; determining at which point to locate a joint or bearing in a mechanism; recognizing useful mechanisms in the students' communities that employ the same guided motion they have been studying.