An interactive applet and associated web page that demonstrate the exterior angles of a triangle. The applet shows a triangle where the user can drag any vertex to reshape it. The exterior angles are shown and a running calculation shows that no matter how you change the triangle, the exterior angles always add up to 360 degrees An exterior angle is equal to the sum of the opposite interior angles 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 incircle of a triangle - the largest circle the will fit inside a triangle. The applet shows a triangle where the user can drag the vertices around. The incircle is continuously adjusted as the user drags the vertices. By experimentation the user can observe that the three sides of the triangle are always tangents to the incircle. The page text explains the definition of the incircle and how to draw it. It describes how the incenter is the center of the incircle of the triangle with a link to that page http://www.mathopenref.com/triangleincircle.html 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 concept of triangle inequality. The applet shows a triangle where the vertices can be dragged to reshape the triangle It shows that no matter what you do, the longest side is always shorter than the sum of the other two. 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 the interior angles of a triangle. The applet shows a triangle where the user can drag the vertices to reshape it. As it is being changed, the angles are continually updated, along with an expression showing that they always add to 180 degrees 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 demonstrates the midsegment (midline) of a triangle. A triangle is shown that the user can reshape by dragging any vertex. As it is being dragged, the midline changes accordingly. From this can be seen that the midsegment is always parallel to the base and half it's length. 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 that in similar triangles, the ratio of their areas is the square of the ratio of the sides. As you drag one triangle to resize it, it remains similar to another and the ratios of sides and areas is calculated as you drag. One can be seen to be the square of the other at all times. A slight 'snap-to' effect allows easy selection of integer ratios (2:4 etc). 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.

An interactive applet and associated web page that demonstrate that in similar triangles, ten items are always in the same ratio to each other (perimeter, three sides, three altitudes, three medians). As you drag one triangle to resize it, it remains similar to another and the ratios of sides and medians is calculated as you drag. They can be seen to be the same at all times. 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 illustrate the triangles that can be drawn inside a polygon. The applet has a pentagon with the triangles drawn. The user can change the number of sides, and switch between regular / irregular. The vertices are draggable. The text on the page discusses the uses of these triangles. 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 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).

A web page and interactive applet illustrating the properties of an undecagon (11 sided polygon). The applet shows an undecagon where the user can drag any vertex to reshape it. User can see that the interior and exterior angles are constant in a regular undecagon, but vary in an irregular version. Controls allow the display or hiding of the diagonals, and triangles within the undecagon. The web page lists the properties of the undecagon 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.

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.

An interactive applet and associated web page that demonstrate vertical angles. The applet shows two intersecting line segments that create two pairs of vertical angles. As you drag the endpoints of the line segments, the angles change and show that the vertical pairs are always congruent. 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 show the definition of a vertical line in coordinate geometry. The applet has two points that the user can drag which define a line. The line flagged when it is vertical (slope=undefined). The grid, details and coordinates can be turned on and off. The applet can be printed exactly as it appears on the screen to make handouts. The web page has a discussion on how to test for vertical 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.

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Ms. Whicker uses similar figures, 3D views and cross-sectional views to have her students construct figures using manipulatives.

This short video and interactive assessment activity is designed to teach second graders about visually evaluating parallel lines cut by a traversal.

An interactive applet and associated web page that demonstrate the volume of a cube. A cube is shown where the edge length be changed by dragging. The volume is continuously recalculated as you drag, and a unit cube grid is superimposed on the cube to illustrate the volume graphically. The calculations can be turned off for class discussion. The web page has the formula for the volume calculation, and a discussion about the subtle distinction between the volume of a cube and the volume inside the cube. The web page also has links to other pages defining the various properties of an ellipse and to some ellipse constructions. 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 volume of a cylinder. A cylinder is shown where the height and radius can both be changed by dragging. The volume is continuously recalculated as you drag. The calculations can be turned off for class discussion. The web page has the formula for the volume calculation, and a discussion about the subtle distinction between the volume of a cylinder and the volume inside the cylinder. The web page also has links to other pages defining the various properties of an ellipse and to some ellipse constructions. 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.

Students apply their knowledge of scale and geometry to design wearables that would help people in their daily lives, perhaps for medical reasons or convenience. Like engineers, student teams follow the steps of the design process, to research the wearable technology field (watching online videos and conducting online research), brainstorm a need that supports some aspect of human life, imagine their own unique designs, and then sketch prototypes (using Paint®). They compare the drawn prototype size to its intended real-life, manufactured size, determining estimated length and width dimensions, determining the scale factor, and the resulting difference in areas. After considering real-world safety concerns relevant to wearables (news article) and getting preliminary user feedback (peer critique), they adjust their drawn designs for improvement. To conclude, they recap their work in short class presentations.

The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.