Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
Reason with shapes and their attributes.
2.G.1. Recognize and draw shapes having specified attributes, such as a
given number of angles or a given number of equal faces. Identify
triangles, quadrilaterals, pentagons, hexagons, and cubes.
3.G.1. Understand that shapes in different categories (e.g., rhombuses,
rectangles, and others) may share attributes (e.g., having four
sides), and that the shared attributes can define a larger category
(e.g., quadrilaterals). Recognize rhombuses, rectangles, and
squares as examples of quadrilaterals, and draw examples of
quadrilaterals that do not belong to any of these subcategories.
Van Hiele Levels of
Geometric Thought
Level 0: Visualization
See geometric shapes as a whole; does not focus on their particular
A student would identify a square but would be unable to articulate
that it has four congruent sides with right angles.
Teacher Activity
Reinforce this level by encouraging students to group shapes
according to their similarities
Shape Sort
van Hiele
This is a level 0 activity because students are
operating on shapes they see in front of them.
These shapes may “change” or have different
properties as they are rearranged or rotated.
The object of this activity is to begin to see that there are
likenesses and differences in shapes.
van Hiele Levels of
Geometric Thought
Level 1: Analysis
Recognize that each shape has different properties; identify the
shape by that property.
A student is able to identify that a parallelogram has two pairs of
parallel sides, and that if a quadrilateral has two pairs of parallel
sides it is identified as a parallelogram.
The products of thought at level 1 are the properties of
Property Lists for
Work in groups of 3 or 4.
List as many properties as you can that are applicable to all
the shapes on their sheet.
•Use an index card to check right angles.
•Use rulers to compare side lengths and draw straight lines.
•Look for lines of symmetry.
•Use tracing paper for angle congruence.
•Use the words “at least” to describe how many of
Does the property apply to all the shapes in the category?
Math Talk Videos
van Hiele Levels of
Geometric Thought
Level 2: Informal Deduction
See the interrelationships between figures
Given the definition of a rectangle as a quadrilateral with right
angles, a student could identify a square as a rectangle.
Teacher Activity
Create hierarchies (i.e. organizational charts of the relationships) or
Venn diagrams of quadrilaterals to show how the attributes of one
shape imply or are related to the attributes of others.
Minimal Defining Lists
Work in your group to find “Minimal Defining Lists”.
A Minimal Defining List (MDL) is a subset of the properties for a shape
that is defining or minimal.
Defining means that any shape that has all the properties in the MDL
must be that shape.
Minimal means that if any single property is removed from the list it is
no longer defining.
Find two or three MDLs for your shape.
A proposed list can be challenged as either not defining or not minimal
by giving one counterexample.
Quad Squad
AIMS Activity – See Directions
(quadrilateral with
exactly one pair of
parallel sides)
isosceles trapezoid
(trapezoid with
equal parallel sides)
(quadrilateral with
2 pair of parallel
(parallelogram with
right angles)
kite (quadrilateral
with 2 pair of equal
adjacent sides)
(parallelogram with
equal sides)
square (rectangle
with equal sides or
rhombus with
congruent angles)