An interactive applet and associated web page that demonstrate opposite rays. The applet shows two rays with a common endpoint and can be dragged with the mouse. If the rays are in a line, the message 'Opposite rays' appears. A detent makes it easier to get them perfectly aligned. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet demonstrating the concept of the orthocenter of a triangle. The applet shows a triangle where the user can drag the vertices around. The orthocenter and the altitude lines that generate it are continuously moved as the user drags the vertices. By experimentation the user can observe that the orthocenter can be inside or outside the triangle. The page text explains the definition of the orthocenter and how to find it. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

This challenging problem and brainteaser gives students an opportunity to compose and decompose polygons to make rectangles.

This short video and interactive assessment activity is designed to give fourth graders an overview of ellipses, trapezoids, rhombi, and polygons.

The purpose of this task is to provide students an opportunity to use mathematics addressed in different standards in the same problem.

This high level task is an example of applying geometric methods to solve design problems and satisfy physical constraints. This task is accessible to all students. In this task, a typographic grid system serves as the background for a standard paper clip.

A parabola is a u-shaped curve that arises not only in the field of mathematics, but also in many other fields such as physics and engineering.

Solar Dishes such as the one shown use a parabolic shape to focus the incoming light into a single collector.

Students act as civil engineers developing safe railways as a way to strengthen their understanding of parallel and intersecting lines. Using pieces of yarn to visually represent line segments, students lay down "train tracks" on a carpeted floor, and make guesses as to whether these segments are arranged in parallel or non-parallel fashion. Students then test their tracks by running two LEGO® MINDSTORMS® NXT robots to observe the consequences of their track designs, and make safety improvements. Robots on intersecting courses face imminent collision, while robots on parallel courses travel safely.

An interactive applet and associated web page that demonstrate the concept of parallel lines. The applet shows two lines, one of which has draggable endpoints. As the line is changed by dragging them, the other line changes to remain parallel to it. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the properties of parallel lines in coordinate geometry. The applet presents two lines, each defined by a pair of draggable points. If they are parallel the slopes are the same and the applet adds the parallel arrow marks to them. The slope of each line is continuously recalculated and the case of vertical lines (with undefined slope) is handled correctly. The is a software 'detent' that snaps the lines to parallel if they are close to it. The grid, axis pointers and coordinates can be turned on and off. The slope calculation can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web page has a full description of the method for determining if lines are parallel, a worked example and has links to other pages relating to coordinate geometry. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet showing the definition and properties of a parallelogram. The applet shows a parallelogram where the user can drag any vertex. The other points then move in such a way that the figure remains a parallelogram at all times. A control to hide the details allows a classroom discussion where students can try to infer what the properties are as it is reshaped by the discussion leader. Text on the page has the formal definition and properties of the parallelogram with links to related pages. A companion page is http://www.mathopenref.com/parallelogramarea.html showing the ways to calculate the area of a parallelogram Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

This short video and interactive assessment activity is designed to teach third graders about fraction of a whole.

A web page and interactive applet illustrating the properties of a pentagon (5 sided polygon). The applet shows a pentagon where the user can drag any vertex to reshape it. User can see that the interior and exterior angles are constant in a regular pentagon, but vary in an irregular version. Controls allow the display or hiding of the diagonals, and triangles within the pentagon. The web page lists the properties of a pentagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles. Links to pages with generalized properties of all polygons. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a kite. A kite is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a parallelogram. A parallelogram is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a polygon. A polygon is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a rectangle. A rectangle is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a rhombus. A rhombus is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://www.mathopenref.com.

A web page and interactive applet show how to compute the perimeter of a square. A square is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://www.mathopenref.com.