This lesson unit is intended to help teachers assess how well students are able to visualize two-dimensional cross-sections of representations of three-dimensional objects. In particular, the lesson will help you identify and help students who have difficulties recognizing and drawing two-dimensional cross-sections at different points along a plane of a representation of a three-dimensional object.

An interactive applet and associated web page that demonstrate acute angles (those less than 90 deg). The applet presents an angle (initially acute) that the user can adjust by dragging the end points of the line segments forming the angle. As it changes it shows the angle measure and a message that indicate which type of angle it is. There a software 'detents' that make it easy capture exact angles such as 90 degrees and 180 degrees The message and angle measures can be turned off to facilitate classroom discussion. The text on the page has links to other pages defining each angle type in depth. 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 three types of triangle: acute, obtuse and right. The applet shows a triangle that is initially acute (all angles less then 90 degrees) which the user can reshape by dragging any vertex. There is a message changes in real time while the triangle is being dragged that tells if the triangle is an acute, right or obtuse triangle and gives the reason why. By experimenting with the triangle student can develop an intuitive sense of the difference between these three classes of triangle. 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 the concept of adjacent arcs. The user can 3 drag points around a circle that define the endpoints of two adjacent arcs. Web page has the properties of adjacent arcs. 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 alternate interior angles that are formed where a transversal crosses two lines. The applets shows the two possible pairs of angles alternating when in animation mode. By dragging the three lines, it can be seen that the angles are congruent only when the lines are parallel. When not in animated mode, there is a button that alternates the two pairs of angles. The text on the page discusses the properties of the angle pairs both in the parallel and non-parallel cases. 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 lesson unit is intended to help teachers assess how well students are able to: work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles; Identify and understand the significance of a counter-example; Prove, and evaluate proofs in a geometric context.

The students will be able to define the radian measure of an angle.

We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).Differentiation is also used in analysis of finance and economics.

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will support you in identifying and helping students who have the following difficulties: Solving problems relating to using the measures of the interior angles of polygons; and solving problems relating to using the measures of the exterior angles of polygons.

An interactive applet and associated web page that demonstrate the area of a circle. A circle is shown with a point on the circumference that can be dragged to resize the circle. As the circle is resized, the radius and the area computation is shown changing in real time. The radius and formula can be hidden for class discussion. 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.

Images of rectangle on coordinate plane, one with easily identifiable area, other image with same rectangle rotated. Distance formula shown to calculate same area.

Definition of area of triangle, image of triangle on coordinate grid with height constructed. Area of triangle calculated using distance formula.

An interactive applet and associated web page that deals with the area of a kite, (a quadrilateral with two distinct pairs of equal adjacent sides). The applet shows a kite and the user can reshape it by dragging any vertex. The other vertices move automatically to ensure it always remains a kite. As the vertices are dragged, the area is continuously recalculated and displayed. The kite is filled with a grid of unit squares so that the students can estimate the area. The on-screen calculation can be hidden until the estimates are done. The web page lists two different ways to compute the area of a kite. 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 ways to calculate the area of a parallelogram. The user can drag the vertices of the parallelogram and the other points change automatically to ensure it remains a parallelogram. A grid inside the shape allows students to estimate the area visually, then check against the actual computed area, which is continuously recomputed and displayed. The text on the page gives three different ways to calculate the area with a formula for each. The applet uses one of the methods to compute the area in real time, so it changes as the rhombus is reshaped with the mouse. A companion page is http://www.mathopenref.com/parallelogram.html showing the definition and properties 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.

A web page and interactive applet showing the ways to calculate the area of a trapezoid. The user can drag the vertices of the trapezoid and the other points change automatically to ensure it remains a trapezoid. A grid inside the shape allows students to estimate the area visually, then check against the actual computed area. The text on the page gives three different ways to calculate the area with a formula for each. The applet uses one of the methods to compute the area in real time, so it changes as the trapezoid is reshaped with the mouse. A companion page is http://www.mathopenref.com/trapezoid.html showing the definition and properties of a trapezoid. 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 explain the area of a triangle. The applet shows a triangle that can be reshaped by dragging any vertex. As it changes, the area is continually recalculated using the 'half base times height' method. The triangle has a fixed square grid in its interior that can be used to visually estimate the area for later correlation with the calculated value. The calculation can be hidden while estimation is in progress. The text page has links to a similar page that uses Heron's Formula to compute the area. 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 task explores the real world topic of building light rails. Throughout the implementation of this task the students will learn about the cost of building railways and how to implement them within a budget. This task explores such mathematical concepts of using coordinates to find the distance between points, using coordinates to build polygons and find the area and length of sides, and writing equations of parallel lines.

This lesson unit is intended to help teahcers assess how well students solve problems involving measurement, and in particular, to identify and help students who have the following difficulties; computing measurements using formulas; decomposing compound shapes into simpler ones; using right triangles and their properties to solve real-world problems.

Definition of Cavalieri’s Principle as it applies to two and three dimensional objects, with image examples.