Jennifer Wexler

The human unit circle

Students become the unit circle. This lesson reinforces the concept of the link between a right triangle, the angles of the unit circle, and the graphs of the sine and cosine functions.

Bridging algebra and calculus

Algebra students learn to distinguish between an average and an instantaneous rate of change as they begin to develop an intuitive understanding of a concept that is usually first introduced in calculus.

A graphic organizer for polynomial functions

bibLiography Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washing‐ ton, DC: National Governors Associa‐ tion Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/assets/ CCSSI_Math%20Standards.pdf. National Council of Teachers of Math‐ ematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM. we can start from any vertex of the trapezoid and fill in the information for all vertices. For example, if we are given that the real and only zeros of the function are 2 and –3, we can create the graphic organizer in figure 1, beginning with the vertex at the upper right. If we start with g(x) = 12x – 19x + 5, students can travel counterclockwise from the vertex at lower left. Next, we focus on polynomials of degree greater than 2. We move from the general to the factored form, using factoring by grouping, the rational root theorem, and technology and discussing the appropriateness of each method. The factor and remainder theorems along with synthetic division allow us to move from the top vertices of the trapezoid to the bottom and vice versa. When presented with a problem, stu‐ dents consider the organizer, locate the vertex that corresponds to the given infor‐ mation, determine which vertex corre‐ sponds to the required answer, and find a path to that answer. We work around the trapezoid, using only rational numbers at first; the result is some factored forms consisting of linear and quadratic factors. We extend our work to include ir‐ rational zeros, allowing some factored forms to consist of all linear factors. When appropriate, we include complex zeros as well. Working with different number sets allows for differentiated instruction. When mathematical procedures are complicated, the link to conceptual understanding can get lost, even when working with something as concrete as the roots of a polynomial function. Since creating and using this graphic organizer, my teaching of polynomial functions has become more coherent. The graphic organizer gives students a vehicle for A Graphic Organizer for Polynomial Functions Donna M. Young