When designing a mathematics capstone course for preservice teachers entitled 'Mathematical Patterns and Structures through Inquiry,” I wanted to include contexts in which mathematics provides the underlying structure for various visual and performing arts. The book *Mathematical Quilts: No Sewing Required* (Venters and Ellison 1999, p. 78) looks at a number of ways that mathematical concepts—such as the golden ratio, logarithmic spirals, Pythagorean triples, and Penrose tiles—are useful to know when designing various quilt patterns.

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### Jeffrey J. Wanko

### Jeffrey J. Wanko

The use of trapezoids to explore a number of mathematical concepts, including similarity, representation, and the Pythagorean theorem. Preservice teachers develop hypotheses about isosceles trapezoids which are investigated. Tiling with pattern blocks and the development of the formula for area are also examined.

### Jeffrey J. Wanko

Engage your students with four types of language-independent logic puzzles.

### Jeffrey J. Wanko

The search for prime numbers has long held a great fascination for mathematicians and for mathematics enthusiasts. Whether as a mathematical recreation or as a serious study within number theory, this quest has resulted in some profound mathematical advances and in a few surprising results that held some unforeseen applications and connections to other areas. For example, the problems of finding perfect numbers and constructible regular polygons have both been simplified through the search for prime numbers (Bell 1937, Clawson 1996).

### Jeffrey J. Wanko

In a traditional mathematics curriculum, middle school students' experiences with functions in algebra are usually limited to linear relationships. However, a number of realworld experiences (e.g., compound interest, population growth, and the amount of a drug remaining in a subject's bloodstream) involve exponential functions. In this article, I describe two problems involving exponential functions and some basic probability concepts that were used in an eighthgrade class, which led to greater student understanding about the general form of an exponential function. I also describe how students used graphing calculators to help them make sense of the values that are used in the general form and how they relate to the context of the problem.

### Jeffrey J. Wanko

Students can improve their deductive reasoning and communication skills by working on number puzzles.

### Jeffrey J. Wanko

A number of experiments modeled in mathematics classes involve repeating an action until a specific condition is met. One might be interested in seeing how long it takes until a sum of 12 is rolled on two fair dice or until a specific name is drawn from a hat. These problems are easy to model by using physical materials and running the experiment a few times. But to see what happens over the long run, statistical modeling software is a useful alternative.

### Jeffrey J. Wanko

Hashiwokakero, also known as Bridges, is one of several types of Japanese logic puzzles that help students establish proof–readiness skills.

### Jeffrey J. Wanko

Teaching problem solving to today's students requires teachers to be aware of the ways their students may use the internet as both a resource and as a tool for solving problems. In this article, I describe some of my own experiences in teaching problem solving to preservice teachers and how the existence of the internet has affected the ways in which I design and pose problems to my students.

### Jeffrey J. Wanko

A modern retelling of the historic debate between Georg Cantor and Leopold Kronecker helps explain the concept of the cardinality of different infinities.