Designing Tasks for Introducing Functions and Graphs within Dynamic Interactive Environments

In this paper, we elaborate on theoretical and methodological considerations for designing a sequence of tasks for introducing middle and high school students to functions and their graphs. In particular, we present didactical activities with an artifact realized within a dynamic interactive environment and having the semiotic potential for embedding mathematical meanings of covariation of independent and dependent variables. After laying down the theoretical grounds, we formulate the design principles that emerged as the result of bringing the theory into a dialogue with the didactical aims. Finally, we present a teaching sequence, designed and implemented on the basis of the design principles and we show how students’ efforts in describing and manipulating the different graphs of functions can promote their production of specific signs that can progressively evolve towards mathematical meanings.

Review of Introduction To Property Testing by Oded Goldreich

The area of property testing is concerned with designing methods to decide whether an input object possesses a certain property or not. Usually the problem is described as a promise problem: either the input object has the property or the input object is far from possessing the property. Here, the meaning of object being far from possessing the property is based on a specified and meaningful notion of distance. The main objective of property testing is accomplishing this decision making by developing a super efficient tester. A tester that reads through the entire object can easily determine whether the property is satisfied or not. However, one wishes the tester to probe the input at very few random locations and determine whether the property is satisfied. As such, randomness is a necessary ingredient for testing and having the tester erring on few instances is a necessary price to pay for designing highly efficient methodologies. Much of the literature on property testing has focused on two types of objects: functions and graphs. Naturally they form the major portion of the book: functions are discussed from Chapters 2 to 6 and graph properties are discussed from Chapters 8 to 10. The final three chapters focus on distribution testing, probabilistically checkable proofs (PCPs) and locally testable codes, and ramifications of property testing on other related topics in Computer Science and Statistics. A separate chapter is devoted to query lower bound techniques.

2. Sines, cosines, and their relatives

‘Sines, cosines, and their relatives’ begins by defining the basic trigonometric functions—sine, cosine, and tangent—and explaining their use. These functions are geometric quantities defined using the ratios of the Opposite, Adjacent, and Hypotenuse sides of the right triangle. Less common functions are the cosecant, secant, and cotangent functions. The history of the naming of the trigonometric functions is discussed along with an explanation of even more obscure functions: the versed sine, versed cosine, exsecant, and excosecant. The versed sine was used frequently in practical applications like astronomy, navigation, and surveying. Finally, inverse trigonometric functions and graphs of trigonometric functions are considered.