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Exploring Sequences Related to Medial Triangles

Sequences are an important topic—not only in mathematics but also in the mathematics curriculum. Exploring sequences related to medial triangles offers learners opportunities to solve problems involving sequences within geometric contexts. Sequences related to medial triangles are also a rich source of beautiful and interesting geometric patterns that can be discovered with the use of dynamic geometry software (DGS) such as GeoGebra or Web Sketchpad. In addition, exploring sequences with DGS allows learners to formulate and test conjectures, an important mathematical process. In this article, I describe how my students used GeoGebra and geometric reasoning to visualize, discover, formulate,

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Triangle Center Technology

The paper discusses technology that can help students master four triangle centers -- circumcenter, incenter, orthocenter, and centroid. The technologies are a collection of web-based apps and dynamic geometry software. Through use of these technologies, multiple examples can be considered, which can lead students to generalizations about triangle centers.

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Graphing Inequalities, Connecting Meaning

tudents often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and my students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood “if its mental representation is part of a network of representations” and that the “degree of understanding is determined by the number and strength of the connections” (p. 67). I therefore developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities.

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Investigating Home Primes and Their Families

The process of prime factor splicing to generate home primes raises opportunity for conjecture and exploration.

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The Stable Pairing Problem

The Gale-Shapley algorithm can be used to match partners in a variety of contexts, such as marriage and hospital residencies.

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Are All Infinities Created Equal?

This historical—and classroom friendly—approach to the concept of infinity uses Cantor's proofs of cardinality.

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Delving Deeper: Transforming Shapes Physically and Analytically

Constructing formulas “from scratch” for calculating geometric measurements of shapes—for example, the area of a triangle—involves reasoning deductively and drawing connections between different methods (Usnick, Lamphere, and Bright 1992). Visual and manipulative models also play a role in helping students understand the underlying mathematics implicit in measurement and make sense of the numbers and operations in formulas.

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Tapering Timbers: Finding the Volume of Conical Frustums

Many everyday objects–paper cups, muffins, and flowerpots–are examples of conical frustums. Shouldn't the volume of such figures have a central place in the geometry curriculum?

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Delving Deeper: Boundaries in Higher Dimensions and Their Kinship with Combinatorics

This is a description of a collaborative investigation by mathematics teachers into the numbers of dimensional boundaries for n > 2. Functions are fit to the patterns observed, and a relationship to Pascal's triangle is noted.

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The Amazing Mathematical Race

Participants race across a university campus, completing challenging mathematical tasks that correspond to NCTM's Standards.