Abstract
Given a 5-peg × 5-peg geoboard, how many different lengths can be made by stretching a rubber band to form an oblique (nonhorizontal, nonvertical) segment between any two pegs? This question launches an investigation that requires students to make connections to the Pythagorean theorem, congruence, and combinations. It also leads to the question, “Can the results be generalized to an n-peg × n-peg geoboard?” (Note that we use “n-peg × n-peg” to refer to the dimensions of the geoboard to prevent confusion with the length of the side of the geoboard, which would be n − 1.) With its use of visual representation and potential mathematical ideas that can be explored from simple to complex, this task makes meaningful mathematics accessible to learners with varied levels of prior knowledge. In addition, mathematics vocabulary such as oblique and congruent are introduced in a meaningful context, making the lesson ideal for developing academic language.
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