**FEATURES** |

Unexpected Answers
*Harris Shultz, Janice Shultz, Richard Brown* Questions from various branches of mathematics that have answers that many teachers and students will find to be rather unexpected. |

A Direct Approach to Computing the Sine or Cosine of the Sum of Two Angles
*Timothy Gutmann* Students struggle with the formulas for computing the sines and cosines of sums of angles. My experience suggests this in part caused by the disconnect between the common, algebraic derivations of these identities and more hands-on approaches to trigonometry in which the sine and cosines functions can be experienced by students as measurements of segments within the unit circle. Here I explain a measurement-based derivation of these identities that has been helpful to my precalculus students. |

Problem Solving Can Generate New Approaches to Mathematics: The Case of Probability
*Jeremy Kahan, Terry Wyberg* A probabilistic situation that can be studied through simulation, tree diagrams, and generating functions. This example illustrates the more general theme of teaching through problem solving. |

Mathematics Examinations: Russian Experiments
*Alexander Karp* The objectives, methods and tasks of the high school final examinations in mathematics administered in St. Petersburg (Russia). |

On Inscribed and Escribed Circles of Right Triangles, Circumscribed Triangles, and the Four-Square, Three-Square Problem
*David Hansen* Some remarkable relationships between the radii of a right triangle's inscribed and escribed circles and its sides, leading to the solution of a fascinating problem in number theory. |

From Exploration to Generalization: An Introduction to Necessary and Sufficient Conditions
*Martin Bonsangue, Gerald Gannon* How a problem involving sums of integers gave students insight into the idea of necessary and sufficient conditions. |