• Geometric shapes exist in two dimensions (pictures or drawings) and in three dimensions (objects).
• Three-dimensional objects and two-dimensional shapes can be sorted by identifying and paying attention to similarities and ignoring differences.
• Shapes and objects have more than one, and often many, attributes, so they can be sorted in more than one way. Sorting rules indicate which attribute to sort for and are used to determine what belongs and what does not belong in a group.
• Attributes are characteristics or features of an object or shape (e.g., length, area, colour, texture, ability to roll). Attributes can be used to describe, compare, sort, and measure.
• Geometric properties are specific attributes that are the same for an entire “class” of shapes or objects. So, for example, a group of shapes might all be red (attribute), but in order for them all to be squares, they must have four equal sides and four right angles (the geometric properties of a square). Geometric properties are used to identify two-dimensional shapes and three-dimensional objects.

Note

• Sorting by attributes is used in counting, measurement, and geometry. When a student counts “this” and not “that”, they have sorted; when they measure length, they focus on one attribute and not another; when they say that this shape is a triangle and not a square, their sorting has led them to identify the shape.

• Each face of a three-dimensional object is a two-dimensional shape. Often, a shape is identified by the number of sides it has. Common shapes on faces of three-dimensional objects are triangles, rectangles, pentagons, hexagons, and octagons.
• While the number of sides often determines a shape’s name, this does not mean, for example, that all triangles look the same even though they all have three sides. Triangles can be oriented differently and have different side lengths, and yet still be triangles.

Note

• Constructing three-dimensional objects helps build understanding of attributes and properties of two-dimensional shapes and three-dimensional objects.

• If two shapes or objects match in every way, they are congruent. Shapes with matching halves have congruent halves.
• Congruent halves can be superimposed onto one another through a series of slides (translations), flips (reflections), or turns (rotations). This means that congruent halves are also symmetrical.
• Both three-dimensional objects and two-dimensional shapes can have matching, congruent, symmetrical halves.

• Positional language often includes direction and distance to describe the location of one object in relation to another.
• Words and phrases such as above, below, to the left, to the right, behind, and in front describe the position of one object in relation to another. Numbers can describe the distance of one object from another.

• Movement encompasses distance and direction.
• Words or phrases such as above, below, to the left, to the right, behind, or in front of describe the direction of one object in relation to another. Numbers can describe the distance of one object from another.
• A combination of words and numbers can describe a path to move from one location to another. The order of the steps taken on this path is often important.