In this undergraduate level seminar series topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found (MP.7).

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

This task provides a concrete geometric setting in which to study rigid transformations of the plane. It is important for students to be able to visualize and execute these transformations and for this purpose it would be beneficial to have manipulatives and it will important that the students be able to label the vertices of the hexagon with which they are working.

In this web-based application from Illuminations students must sort shapes based on the categories of the Venn diagram. Users can choose the categories from a drop down menu. The application includes instructions and exploration steps.

This interactive tool allows a user to create many geometric shapes. Squares, triangles, rhombi, trapezoids and hexagons can be created, colored, enlarged, shrunk, rotated, reflected, sliced, and glued together.

In this math lesson, learners read the poem "Shapes" from "A Light in the Attic" by Shel Silverstein. Then, learners create their own illustration of the poem. In this lesson, learners explore geometric figures and positional words.

In this math lesson, learners read the poem "Shapes" from "A Light in the Attic" by Shel Silverstein. Then, learners create their own illustration of the poem. In this lesson, learners explore geometric figures and positional words.

Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result is a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle, which is crucial for many further developments in the subject.

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever _AXB is a right angle.

Total solar eclipses are quite rare, so much so that they make the news when they do occur. This task explores some of the reasons why. Solving the problem is a good application of similar triangles

In this lesson, students measure the sides of many squares and their diagonals, then consider the ratio of diagonal length to side length. They can note that in all cases the ratio hovers near 1.4 or the square root of 2. The very complete lesson plan contains handouts, questions for discussion, and problems for applying the new learning.

In this math game, learners play a one-pile game between two players. In this game, the maximum number of tokens that can be removed on each turn remains constant throughout the game. Learners represent the positions as the vertices of a directed graph and the moves as the edges of the graph. Learners also discover that solving a game means finding a partition of the vertices into two sets such that three important properties are satisfied. This lesson guide includes questions for learners, assessment options, extensions, and reflection questions.

This is the second i-Math in a four-part series of i-Maths entitled Symmetries and Their Properties. In this second i-Math you will investigate reflection, mirror, or bilateral symmetry. Mirrors can be used to create reflection symmetry. Many objects in nature, such as butterflies, the human body, and many types of leaves have bilateral symmetry. Objects we use every day, such as spoons, chairs, and cars also have bilateral symmetry. In this i-Math, you will learn about the mathematical properties of mirror symmetry and have a chance to create designs with mirror symmetry.

We all encounter symmetry in our everyday lives, in both natural and man-made structures. The mathematical concepts surrounding symmetry can be a bit more difficult to grasp. This unit explains such concepts as direct and indirect symmetries, Cayley table

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.''

Students use this Flash tool to build tessellations, discovering which shapes tile the plane with copies of themselves or together with other shapes. Regular polygons with 3-12 sides of the same length are provided; students may arrange, rotate, group, ungroup, paint, copy, and erase them.

Students construct "a tetrahedron and describe the linear, area and volume using non-traditional units of measure. Four tetrahedra are combined to form a similar tetrahedron whose linear dimensions are twice the original tetrahedron. The area and volume relationships between the first and second tetrahedra are explored, and generalizations for the relationships are developed." (from NCTM Illuminations)