Polygons Galore

A Mathematics Unit for High-Ability Learners in Grades 3-5

Dana T. Johnson, Marguerite M. Mason, Jill Adelson

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000495232
• Edition: 1
• Topic: Education
• Subtopic: Education General

Polygons Galore! is a mathematics unit for high-ability learners in grades 3-5 focusing on 2-D and 3-D components of geometry by exploring polygons and polyhedra and their properties. The van Hiele levels of geometric understanding provide conceptual underpinnings for unit activities. The unit consists of nine lessons that include student discovery of properties of polygons and polyhedra, investigations for finding areas of triangles and quadrilaterals, study of the Platonic solids, and real-world applications of polygons and polyhedra. It also includes activities related to identifying, comparing, and analyzing polygons by using properties of the polygons; constructing meanings for geometric terms; developing strategies to find areas of specific polygons; identifying and building regular and nonregular polyhedra; and recognizing geometric ideas and relationships as applied in daily life and in other disciplines, such as art.Grades 3-5

Part I
Introduction

Introduction to the Unit

DOI: 10.4324/9781003237204-2

Unit Description: Polygons Galore! investigates two-dimensional and three-dimensional components of geometry by exploring polygons and polyhedra and their properties.

Unit Rationale: Geometry is a fundamental and powerful strand of mathematics, and the foundation for spatial reasoning. Typical curriculum materials address geometry in a rote method that emphasizes recalling shapes and memorizing formulas. This unit goes beyond those methods by allowing students to:

• identify, compare, and analyze polygons by using properties of the polygons;
• construct meanings for geometric terms;
• develop strategies to find areas of specific polygons;
• identify and build regular and nonregular polyhedra;
• analyze the relationship of the numbers of vertices, faces, and edges in a polyhedron; and
• recognize geometric ideas and relationships as applied to other disciplines, such as art.

Differentiation for Gifted Learners: This unit gives students a much broader and deeper experience with geometry than do most curriculum materials. It is challenging in that it requires deep understanding of the mathematical strand of geometry rather than merely memorizing formulas and definitions or recognizing shapes. Reasons to use this unit with gifted learners include the following:

• The regular school curriculum at this level does not generally probe geometry beyond lines, area, and perimeter. This unit provides additional content experiences.
• Developing formulas for specific attributes of geometric shapes requires a much deeper understanding of the algorithms and serious thinking.
• Constructing meanings for mathematical terms mandates that the learner develop spatial thinking.
• Task demands are more strenuous than in typical curriculum materials. Students are frequently asked to complete tasks with less teacher support than would be given in a typical math class.
• Much of the work in this unit is inquiry-based. Although this approach may benefit all students, inquiry lessons are a good approach to unleash the thinking abilities of gifted students.
• The pace and density of material requires the ability to process concepts rapidly.

Links to Common Core State Standards and NCTM Standards: The Common Core State Standards for Mathematics (CCSSM) call for the study of geometry in grades K–12. As the CCSSM note: “Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades” (Common Core State Standards Initiative, 2010, p. 17). According to the CCSSM, by the fifth grade students understand that attributes belonging to a category of figures also belong to all subcategories of that category and can classify two-dimensional figures in a hierarchy based on properties. By sixth grade, students are finding the area of special quadrilaterals by composing into rectangles or decomposing into triangles and other polygons. This unit will provide the students with the means to investigate the geometric properties of figures and the relationships among various figures.

The National Council of Teachers of Mathematics’ (NCTM, 2000) Principals and Standards for School Mathematics include the following for grades 3–5: “The reasoning skills that students develop in grades 3–5 allow them to investigate geometric problems of increasing complexity and to study geometric properties” (p. 165). This unit will not only allow students to study increasingly complex geometric shapes, but also afford them the opportunity to consider the various properties of those shapes and apply them to another discipline, specifically the visual arts.

Suggested Grade Level Range: 3–5, but the unit can be adapted to both older and younger students.

Prerequisite Knowledge:

• Familiarity with basic polygons such as triangles and quadrilaterals and their attributes.
• Proficiency with tasks involving sorting and classifying.
• Proficiency with rulers and protractors and/or angle rulers.

Length of Lessons: Some of the lessons can be done in 45 minutes whereas others may take two 45-minute sessions. These lessons can be customized by lengthening or omitting various activities.

Timing: This unit could be completed within 2 weeks, spread out throughout the semester, or done in a pull-out enrichment class. Not all lessons need to be included and the order of the later lessons may be determined by the teacher. It is recommended that Lessons 1, 2, 3, and 6 be included. The most serious and substantive treatment of the material will include all of the lessons and some of the extensions.

Extensions: Suggestions for extension activities are included within the lessons. Extensions can be done by groups or individually. You might also keep a piece of chart paper in your classroom and encourage students to generate questions for further study. Individuals or student groups can be assigned questions and report to the class on findings of additional extension activities. The extensions often require students to function somewhat independently. However, you may choose to assign an extension to less able students by writing a more scaffolded version of the task. The extensions will provide some opportunities to offer greater engagement to individuals who need extra challenges.

Assessment

DOI: 10.4324/9781003237204-3

Each lesson has suggested assessments, but teachers will find many more ways to determine student understanding.

Math Journals: If students maintain a math journal, they can be asked to solve a single problem in the journal and explain their reasoning. A good technique to give students an audience for their writing is to suggest that they are to write a postcard to a friend who has asked for help in solving the problem.

Preassessment: This is not a readiness test. You should administer it before you start the unit. It assumes that students know something about geometry. It is intended to give the teacher a baseline indicator of what students know before they begin work on the unit. Typically, they should not do well on the preassessment. (However, if they do know the material, be sure to assign appropriate extensions from the lessons.)

Postassessment: This is included at the end of the unit. You should administer it after you complete the unit. You are welcome to add any questions to both the pre- and postassessments.

Unit Glossary

DOI: 10.4324/9781003237204-4

Acute triangle: A triangle that has three acute angles (less than 90°).

Adjacent sides: Two sides that are side-by-side and share a common vertex.

Angle: The figure formed by two rays that share a common endpoint.

Archimedean solids: Solids made of two or more different regular polygons; semi-regular solids.

Area: The amount of surface of a region or shape (measured in square units).

Base of a triangle: Any one of the three sides of the triangle; usually the one drawn at the bottom, parallel to the floor.

Bisect: To cut into two equal parts.

Circle: A closed plane (two-dimensional) figure with all points of the figure equidistant (the same distance) from the center.

Congruent: Having the same size and shape.

Convex: Curving or bending outward (a convex polygon has no angles larger than 180 degrees).

Concave: Curving or bending inward (a concave polygon has at least one angle larger than 180 degrees).

Cube: A three-dimensional shape with six congruent square faces (it has 12 congruent edges and eight vertices or corners).

Cylinder: A three-dimensional shape with bases that are parallel, congruent circles.

Dart: A concave quadrilateral.

Decagon: A polygon with 10 sides.

Diagonal: A segment that joins two non consecutive vertices in a polygon or polyhedron.

Dodecagon: A polygon with 12 sides.

Dodecahedron: A polyhedron with 12 pentagonal faces.

Edge: A line segment where two faces of a three-dimensional figure meet.

Equilateral triangle: A triangle with three congruent sides and angles (each angle measures 60°); note that an equilateral triangle is a special case of an isosceles triangle.

Face: A flat side of a three-dimensional figure (a face will be a polygon).

Geometry: The branch of mathematics that deals with position, size, and shape of figures.

Height of a triangle: The perpendicular distance of the vertex (that is opposite to the base side) from the line containing the base.

Heptagon: A polygon with seven sides.

Hexagon: A polygon with six sides.

Hexahedron: A polyhedron with six faces (if the faces are squares, it is a cube).

Icosahedron: A polyhedron with 20 triangular faces.

Isosceles triangle: A triangle with at least two equal sides and angles (note that an equ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Part I: Introduction
• Part II: Background for Teachers
• Part III: Lesson Plans
• References
• About the Authors
• Common Core State Standards Alignment