A visual proof that the sine is subadditive on [0, pi].

March 14 is special because it is Pi Day. Mathematics is celebrated on that day because the date, 3-14, replicates the first three digits of pi. Pi-related songs, websites, trivia facts, and more are at the fingertips of interested teachers and students. Less celebrated, but still fairly well known, is National Metric Day, which falls on October…

This paper reports on teachers? practical knowledge [PK] about peer interactions [PI] in learning mathematics. The focus is on high school teachers who consistently engaged students PI in their teaching. Data consisted of interviews and classroom observations. Findings indicate that these teachers have PK of students? roles in PI and learning…

Using techniques that show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's identity. The proof involves evaluations of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.

Using modern technology to examine classical mathematics problems at the high school level can reduce difficult computations and encourage generalizations. When teachers combine historical context with access to technology, they challenge advanced students to think deeply, spark interest in students whose primary interest is not mathematics, and…

The number Pi (approximately 3.14159) is defined to be the ratio C/d of the circumference (C) to the diameter (d) of any given circle. In particular, Pi measures the circumference of a circle of diameter d = 1. Historically, the Greek mathematician Archimedes found good approximations for Pi by inscribing and circumscribing many-sided polygons…

One of the best known numbers in mathematics is the number denoted by the symbol [pi]. This column describes activities that teachers can utilize to encourage students to explore the use of [pi] in one of the simplest of geometric figures: the circle.

For at least 2000 years people have been trying to calculate the value of [pi], the ratio of the circumference to the diameter of a circle. People know that [pi] is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real…

Celebrating mathematics should be a yearlong event in which students in mathematics classes of all levels engage in mathematics activities and competitions that will encourage growth in mathematical knowledge, enthusiasm for the subject, and collaboration among students of different abilities and backgrounds. Pi Day and Pi Week festivities--a good…

"Lord Brouncker's continued fraction for pi" is a well-known result. In this article, we show that Brouncker found not only this one continued fraction, but an entire infinite sequence of related continued fractions for pi. These were recorded in the "Arithmetica Infinitorum" by John Wallis, but appear to have been ignored and forgotten by modern…

In my Calculus classes I encourage my students to actively reflect on course material, to work collaboratively, and to generate diverse solutions to questions. To facilitate this I use peer instruction (PI), a structured questioning process, and i-clickers, a radio frequency classroom response system enabling students to vote anonymously. This…

Using continued fraction expansions, we can approximate constants, such as pi and e, using an appropriate integer n raised to the power x[superscript 1/x], x a suitable rational. We review continued fractions and give an algorithm for producing these approximations.

In 1997, Bailey and Bannister showed that a + b greater than c + h holds for all triangles with [gamma] less than arctan (22/7)where a, b, and c are the sides of the triangle, "h" is the altitude to side "c", and [gamma] is the angle opposite c. In this paper, we show that a + b greater than c + h holds approximately 92% of the time for all…

The researcher employed two designs to address the research question for this particular study. This quasi-experimental non-equivalent control group study compared the math achievement of 92 eighth grade students who received Classroom Performance System (CPS)-based instruction using Peer Instruction (PI) to 76 eighth grade students who received…

This paper is a whimsical survey of the various explanations which might account for the biblical passage in I Kings 7:23 that describes a round object--a bronze basin called Solomon's Sea--as having diameter ten cubits and circumference thirty cubits. Can the biblical pi be any number other than 3? We offer seven different perspectives on this…