Quadrilateral Activities Leading to Level 2 Understanding

Definitions Part 1

In your table groups, write a definition for each of the following on chart paper

1. A parallelogram is a .....

2. A rhombus is a ......

3. A rectangle is a ......

4. A square is a .....

5. A trapezoid is a ....

6.A kite is a .....

Definitions Part 2

In your table groups, refine your definitions

1. A parallelogram is a quadrilateral with….

2. A rhombus is a parallelogram with …

3. A rectangle is a parallelogram with…

4. A square is a parallelogram with ….

5. A square is a rectangle with….

Venn Diagrams

Construct a Venn diagram showing the relationship between

  1. Squares and rectangles
  2. Rectangles and parallelograms
  3. Squares, rectangles and parallelograms
  4. Trapezoids and rectangles

Three Dimensional Activities

Sorting 3-D Shapes

  1. Sorting 3-D shapes by edges and vertices
  2. Sorting 3-D shapes by faces and surfaces

Venn Diagrams

  1. Two non-overlapping groups
  2. Two overlapping groups

Constructing 3-D Polyhedra with Straws

Revisiting and Refining Definitions

Slicing Polyhedra

Slicing Polyhedra

An interesting connection between two and three dimensions is found in slicing solids in different ways.

  1. Pyramids
  2. Look at the model of a square pyramid
  3. Now visualize a plane slicing through the pyramid. The places where the plane passes through the pyramid form a shape in the plane. Which shapes in the plane can be made this way, by slicing the pyramid? Use your model to help you, but also try to visualize each case without using your model.
  4. Use modeling clay to make a pyramid. Slice through your pyramid with dental floss, as if you were slicing the pyramid with a plane. The place where the dental floss cuts through the pyramid should make a plane shape. What plane shape did you get? Put the pyramid back together and try slicing it a different way. Now what shape did you get? Record as many different shapes as you can.
  1. Repeat the above steps for a cube.
  1. Repeat the above steps for a rectangular (non-cube) prism
  1. Repeat the above steps for a cylinder.
  1. Repeat the above steps for a sphere.

Clarifying Your Geometry Objectives

In the early grades, when you can expect your students to be level 0 thinkers, you want to be sure that their thinking is increasing in its sophistication and is moving toward level 1. Here are a few suggestions for things to look for:

  • Child attends to a variety of characteristics of shapes in sorting and building activities.
  • Child uses language that is descriptive of geometric shapes.
  • Child shows evidence of geometric reasoning in solving puzzles, exploring shapes, creating designs, and analyzing shapes.
  • Child recognizes shapes in the environment.
  • Child solves spatial problems.

Each of these statements can be assessed as indicative of either a level 0 thinker or a level 1 thinker. For example, at level 0, the type of characteristics that students are likely to pay attention to are not properties of general classes of shapes ( “pointy”, “fat”, “has five sides”, “goes up”, etc.) Properties such as “parallel” or “symmetrical” may be used by level 0 thinker as well as those at level 1. The distinction is found in what the properties are attributed to. At level 0, students are restricted in thought to the shapes they are currently working with, while at level 1, students attribute properties to classes of shapes (all rectangles or all cylinders). Language, reasoning, shape recognition and spatial problem solving can all be assessed as being appropriate for level 0 or level 1. By thinking in this manner, teachers can begin to get a sense of the geometric growth of their students beyond the specific content knowledge that may have been just developed.

At the upper elementary and middle grades, teachers can begin to think in terms of students being at level 1 or level 2 in their geometric thought. Before grades 6 or 7, very few students will have achieved level 2 thinking, but teachers need to be aware of progress in that direction. The following general indicators are more indicative of level 2 thinkers than level 1:

  • Child shows improvement in spatial visualization skills.
  • Child has an inclination to make and test conjectures in geometric situations.
  • Child makes use of logical explanations in geometric problem solving.
  • Child justifies conclusions in geometric contexts.
  • Child assesses the validity of logical arguments in geometric situations.

Most of these indicators or objectives include elements of reasoning and logical sophistication that are not generally present in level 1 thinkers. A level 1 thinker is using inductive reasoning to discover relationships in shapes, whereas a level 2 thinker is, with guidance, beginning to develop arguments that explain why a particular relationship exists.

Van de Walle, 2004 p. 383-384