In the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), the 9–12 standards call for a shift from a curriculum dominated by memorization of isolated facts and procedures to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving. One approach that affords opportunities for achieving these objectives is the use of diagrams and drawings. The familiar saying “A picture is worth a thousand words” could well be modified for mathematics to “A picture is worth a thousand numbers.” As an example of visual approaches in algebra, this article uses diagrams to solve mixture problems.
Combinatorics, someone has said, is that branch of mathematics in which one devises ways to count without counting. In this article, we consider the general combinatorial problem of counting the number of regions into which the interior of a circle is divided by a family of lines. A general formula is developed and its use illustrated in two situations. The first of these is the “pizza problem”: What is the maximum number of pieces into which a pizza can be divided by n cuts? The second is the “circle problem”: What is the maximum number of regions into which the interior of a circle can be divided by placing n points on the circle and connecting each pair of points? In the final section, applications of the general formula to other problems are discussed
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