#### Present the Unknown

A motivational technique for more advanced learners can be to make them aware of a lack or void in their knowledge of a subject. Students should be given an opportunity to discover this lack on their own.

An activity for presenting the unknown can be found in trigonometry. Students begin their study of trigonometry with the various trigonometric ratios (sine, cosine, tangent) on the right triangle. Eventually they will be led to consider angles of measure greater than 90º. To spark interest in this topic, a teacher might have students find values such as: sin 30º, cos 60º, and cos 120º.

Students familiar with the 30-60-90 triangle will likely be able to figure out the first two values fairly easily. The third value may confuse some students, since they are unfamiliar with trigonometric functions of angles that measure more than 90º. Students will realize that they can find the trigonometric functions of acute angles, but not of obtuse angles. In this sense, students might draw on their curiosity and be motivated to extend their knowledge to find the answer.

The technique of presenting the unknown can also be employed in geometry when introducing the measures of angles having vertices outside a given circle. To illustrate this, suppose students have learned the relationships between the measures of arcs of a circle and the measure of an angle (whose rays subtend these arcs) with its vertex in or on the circle, but not outside the circle.

The teacher might present the class with the examples like the following, asking for the value of x:

Because students are familiar with how to find the value for x in the first two diagrams, their confidence may grow. Then the knowledge may set in that they cannot find the measure of the angle formed by two secants intersecting outside the circle. If asked (appropriately) what they would like to learn during the ensuing lesson, they would likely ask to learn how to find the measure of an angle formed outside the circle. This shows they have been motivated.

#### Integrate Technology

Many teachers have begun employing presentation stations, interactive whiteboards, and other technologies to present lesson introductions and engaging activities like those above. These can provide or activate important background knowledge, as well as stimulate student interest. Such tools can make technology-based teaching resources such as virtual manipulatives, videos, animations, and software tools available (with oversight) to an entire class.

Studies report that students' attitudes toward learning improve when technology is used in instruction (Silvin-Kachala 1998, Kulik 1994). The illustration mentioned earlier, where the midpoints of a quadrilateral are joined to form a parallelogram, can be very dramatically demonstrated with geometric software such as Geometer's SketchpadTM.