Two- and Three-Dimensional Geometry

Essential content for elementary teachers

Develop an understanding of basic geometric concepts including: point, line, plane, space, line segment, betweenness, ray, angle, vertex, parallelism, perpendicularity, congruency, similarity, simple closed curve, Pythagorean relationship.
Identify types of angles including acute, right, obtuse, straight, reflex, vertical, supplementary, complementary, corresponding, alternate interior, and alternate exterior.
Recognize and define common geometric shapes.
1. Two-dimensional geometric shapes
  1. Triangles: be able to classify by sides (equilateral, scalene, isosceles) and classify by angle (right, acute, obtuse)
  2. Quadrilaterals (trapezoid, parallelogram, rectangle, square, rhombus, kite): identify characteristics and relationships among these shapes
  3. Polygons, regular polygons
  4. Circle
2. Three-dimensional geometric shapes
  1. Polyhedra (prisms, pyramids), regular polyhedra (Platonic solids): connecting polyhedra to polygons, nets
  2. Cylinder, cone, sphere

Essential content for students K-3

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.8
1. Two-dimensional geometric shapes
  1. Recognize, name, build, draw, compare, and sort shapes.
  2. Describe attributes and parts of shapes: circle, rectangle, square, triangle, parallelogram (sides and vertices); locate interior (inside) and exterior outside) angles.
  3. Compare shapes made with line segments (polygons) and identify congruent and similar geometric shapes.
  4. Identify right angles in polygons.
  5. Investigate and predict the results of putting together and taking apart shapes.
2. Three-dimensional geometric shapes
  1. Recognize, name, build, draw, compare, and sort shapes: sphere (ball), cone, cylinder (can), pyramid, prism (box), cube.
  2. Describe attributes and parts of shapes: identify faces, edges, vertices (corners).
  3. Sort using similar attributes (curved surfaces, flat surfaces).
  4. Investigate and predict the results of putting together and taking apart shapes.
Develop vocabulary and concepts related to two- and three-dimensional geometric shapes.
1. Two-dimensional shapes: angle, circle, congruency, line segment, parallelogram, polygon, rectangle, similarity, square, triangle
2. Three-dimensional shapes: cone, cube, cylinder, edge, face, prism, pyramid, sphere, vertices

Essential content for students grades 4-5

Maintain and expand on concepts introduced in primary grades.
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.9
1. Two-dimensional geometric shapes
  1. Identify, compare, and analyze attributes of shapes, and develop vocabulary to describe the attributes.
    1. Angles (right, acute, obtuse, straight)
    2. Circles (diameter, radius, center, arc, circumference)
    3. Lines (parallel, intersecting, perpendicular)
    4. Line segments
    5. Polygons (vertex, side, diagonal, perimeter); classification by number of sides (quadrilaterals, pentagon, hexagon)
  2. Classify shapes according to their properties.
    1. Triangles (classify by angles and sides)
    2. Quadrilaterals (square, rectangle, parallelogram, rhombus, trapezoid, kite)
  3. Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.
  4. Explore and identify congruence and similarity.
  5. Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.
2. Three-dimensional geometric shapes
  1. Identify shapes (cylinder, cone, sphere, pyramid, prism).
  2. Apply terms (face, edge, vertex).
  3. Classify shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids.
  4. Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.

See Appendix A for a list of sample state standards for mathematics.

Instructional activities and lesson plans for two- and three-dimensional geometry

The activities and lesson plans listed here (by grade level) are described more extensively in Instructional Activities and Lesson Plans, where they are listed alphabetically.

Activity	Description
Describe the Shape grades K-2	Help students learn to identify geometric shapes using pattern blocks, a pattern block applet, or triangle grid paper.
Footprints in the Sand grades K-2	Use blocks of various shapes to make impressions in sand. Students identify the shapes and match the blocks to the impressions.
The Button Box grades K-3	Have students sort a box of buttons according to different attributes and describe what attributes they used.
Describing Attributes of Shapes grades K-3	Have students examine and describe the attributes of attribute blocks. Place a block where students cannot see it and have them ask "yes" or "no" questions until they can fully describe the shape.
Nature Walk grades K-3	Take students on a nature walk and have them identify various shapes in nature. Take photographs to use on a bulletin board display or have students draw what they find.
Shape People grades K-3	Have students use different polygons and circles to create a person out of shapes. Connect this activity with language arts by having each student write a story about his or her person.
Winter Shapes grades K-3	Cut various two-dimensional shapes from construction paper and have students use them to make all the winter objects that they can imagine.
Draw a Shape from Memory grades K-5	Show students a shape for a few seconds and then have them try to draw the shape from memory. Start with simpler shapes and then draw more complex ones.
Pattern Block Shapes grades 2-3	Develop the relationships between the different pattern block shapes by having students create one shape by using other shapes. Have students draw and color their shapes on triangle grid paper and include a verbal explanation of the picture and the shapes they used.
Shape Journals grades 2-4	On each page of a journal, have students name a polygon and draw it. Have them look in magazines for a picture of an object that closely resembles the polygon, cut it out, and paste it into the journal.
Writing Stories grades 1-5	Have students write creative stories using as many geometric terms as possible. Suggest a general topic or theme for their story or relate it to what you are teaching in language arts, science, or social studies.
"Who Am I?" Riddles grades 2-4	Prepare "Who Am I" riddles for students to solve, such as: "I am a polygon; I have four sides of equal length, but the four angles are not of equal measure. Who am I?" Have students make up the riddles and share them with each other.
Geoboard Polygons grades 1-4	Have students duplicate shapes on a geoboard and then break them into smaller shapes. Give students an irregular shape and have them find as many triangles, squares, or rectangles as possible.
Go-Together Rules grades 3-4	Prepare sets of rules for attributes of shapes, with blanks for students to complete, such as "All ____ have ____" or "No ____ have ____." Have students make up rules for each other to complete.
Rope Polygons grades 3-5	Have a group of three students make different triangles using rope or string. Have students determine how many students are needed for other kinds of polygons.
Shape Puzzles grades 3-5	Cut out puzzle pieces for a shape and put the pieces into an envelope with the final shape on the outside. Have students arrange the pieces into the shape shown, then make shape puzzles for others to solve.
Connect the Dots grades 3-4	Connect the Dots Have students draw polygons on square dot paper, taking turns drawing one line segment at a time. When students complete a shape, have them put their initials in the shape.
Origami grades 2-5	Use origami to explore spatial relations, shapes, and their properties. Explore concepts such as congruence and classification of triangles. Introduce mathematical terms such as diagonal and midpoint.
Build a Shape grades 3-5	Use cubes to build a shape. Have students draw the shapes on graph paper and show how they look from the front, back, side, top, and bottom.
Building Vocabulary grades 4-5	Have students make a word bank, looking for common roots in words. Have students use word roots to write definitions of words.
Creating Nets grades 4-5	Have students build shapes with wooden cubes and create nets to wrap the shapes. Use the nets to discuss surface area.
Creative Writing Activity grades 4-5	Have students read books such as Sir Cumference and the First Round Table and then write their own stories using geometry vocabulary words.
Regular Polygons grades 4-5	Have students use flexible straws or K'NEX™ materials to make regular polygons and then discuss the attributes of the shapes.
Platonic Solids grades 4-5	Discuss the Platonic solids and use nets to develop the concepts of edges, faces, and vertices. Make a chart to determine the relationship between the number of edges, vertices, and faces (Euler's Formula).
Scale Drawings grades 4-5	Use scale drawings to help students develop the concept of similarity. Have students make a scale drawing of their classroom, where one square on the graph paper represents one square floor tile.
Using Geometric Software grade 5	Use dynamic geometric software to measure the interior angles and the to Explore Triangles lengths of the sides of a triangle. Have students create different types of triangles, discover the triangle inequality property, and determine that the sum of interior angles is always 180 degrees.

Lesson Plans	Description
Geoboard Squares grades 2-3	Students create squares of different sizes on a geoboard and use a system or numeric pattern to determine the number of squares possible on a 10-by-10 geoboard.
Tiling the Plane grades 2-3	Students use pattern blocks and triangle paper to develop an understanding of tessellations while reviewing names of geometric shapes.
Rep Tiles grade 3	Students develop a deeper understanding of similarity and how perimeter changes as a result of increase in size. They create "rep tiles" using four pattern blocks ("rep-4 tiles").
Can You Name That Shape? grades 2-4	Students build and draw geometric objects using manipulatives and develop vocabulary and concepts related to two-dimensional shapes.
It's a 3-D World Out There! grades 2-4	Students use manipulatives to build, invent, and draw three-dimensional shapes.
Rep Tiles grades 4-5	Similar to the grade 3 lesson above, with added element of how area changes as a result of an increase in size.
Investigating Nets grade 5	Students explore nets for three-dimensional shapes and Polyhedra.
The Sum of the Interior Angles of a Polygon grades 4-5	Students use the Geometer's Sketchpad® to find a formula for the sum of the interior angles of a polygon.
Tiling the Plane grade 5	Similar to the grades 2-3 lesson above, and addressing more complex tessellations.

National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, (Reston, Va.: National Council of Teachers of Mathematics), p. 396
Ibid.