**ERIC Number:**EJ1044754

**Record Type:**Journal

**Publication Date:**2014-Oct

**Pages:**4

**Abstractor:**ERIC

**Reference Count:**4

**ISBN:**N/A

**ISSN:**ISSN-0025-5769

Activities for Students: Filling a Square with a Curve

Martin, David R.

Mathematics Teacher, v108 n3 p218-221 Oct 2014

Finding patterns and making conjectures are important thinking skills for students at all levels of mathematics education. Both the Common Core State Standards for Mathematics and the National Council of Teachers of Mathematics speak to the importance of these thought processes. NCTM suggests that students should be able to recognize reasoning and proof as fundamental aspects of mathematics, make and investigate mathematical conjectures, develop and evaluate mathematical arguments and proofs, and select and use various types of reasoning and methods of proof. CCSS states that students should "make conjectures and build a logical progression of statements to explore the truth of their conjectures" (CCSSI 2010, p. 6). The activity presented in this article makes such reasoning accessible to high school students with some previous study of trigonometry. In this activity, students use technology to graph parametric equations, make conjectures based on their graphs, modify their conjectures on the basis of evidence, and find counterexamples or investigate methods of proof. Students will be left with some open-ended questions to engage them in investigations beyond the classroom. Students learn to plot parametric curves, including those involving sine and cosine. After exploring several of these curves, students will conjecture about parameters that will make a curve appear to fill in the area of a square. Students deal with the idea of spacefilling curves from a naive perspective; it is important to note that none of the curves encountered in this activity actually fill space.

Descriptors: Mathematics Instruction, Academic Standards, Mathematical Logic, Validity, Mathematics Skills, High School Students, Trigonometry, Secondary School Mathematics, Equations (Mathematics), Graphs, Mathematical Concepts

National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail: orders@nctm.org; Web site: http://www.nctm.org/publications/

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**High Schools; Secondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A