## E1. Geometric and Spatial Reasoning

### Specific Expectations

#### Geometric Reasoning

E1.1

sort three-dimensional objects and two-dimensional shapes according to one attribute at a time, and identify the sorting rule being used

**attributes:**- length, area, colour, texture, ability to roll, number of sides, curves, straight lines

**possible ways to sort three-dimensional objects:**

Objects with Triangular Faces |
Objects with Rectangular Faces |
Objects with Circular Faces |

**types of shapes to include in a sort:**

Regular Shapes |
Irregular Shapes |
Composite Polygons |

Shapes with Only Curves |
Shapes with Only Straight Lines |
Shapes with Curves and Straight Lines |

- 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.

Have students play “Guess My Sorting Rule”, where they sort a variety of two-dimensional shapes or three-dimensional objects into two or more categories. Ask them to trade places with a partner and to come up with a rule for their partner’s sort. Then have the pairs of students describe and discuss the rules that they came up with for one another’s sorts. For example, a student has the following set of three-dimensional objects. His rule is that all the shapes have a circular face. His partner's rule is that all the objects roll.

Give students a rule, and ask them to use a geoboard to make a shape that satisfies this rule; for example, the shape must have four sides. A sorting rule might result in more than one shape. Discuss the similarities and differences for the shapes that the students make.

Have students play “Which One Does Not Belong?”, where they each identify a sorting rule, select different shapes or objects that follow the rule and one that does not, and put them all in a “sorting circle”. Next, have students share their sort in a small group. Each of the group members shares their sort, and the others identify the shape or object that does not belong and explain why.

Any disagreements can be discussed and resolved as a class. Emphasize that shapes and objects have many sortable attributes and that the same group of shapes or objects can be sorted in different ways. Draw out the idea that whether something belongs in the sorting circle depends on the sorting rule.

E1.2

construct three-dimensional objects, and identify two-dimensional shapes contained within structures and objects

**constructing three-dimensional objects:**

Solid – Interlocking Cubes |
Skeleton – Straws and Adhesive Putty |

Cone – Paper |
Composite Shape – Real Objects |

**describing three-dimensional objects and making connections to two-dimensional shapes:**

- describing how edges, vertices, and angles make up two-dimensional shapes on three-dimensional objects:

- identifying two-dimensional shapes on three-dimensional objects:

- 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.

Have students build three-dimensional objects in solid, hollow, and skeleton form. Support them in identifying and naming the sides, vertices, and angles of two-dimensional shapes and the faces, edges, and vertices of three-dimensional objects. Ask them to draw and appropriately label a picture of the two-dimensional shapes on their three-dimensional object.

Have students play “Find My Shape”, where they search inside and outside the classroom for two-dimensional shapes that are part of three-dimensional objects. Ask them to name the shapes they find and describe the attributes they used to identify it

Have students build three-dimensional objects that meet given criteria (e.g., it can stand by itself). Ask them to look at the collection of objects that the class has built and describe what each object has in common (e.g., they all have a flat base). Discuss why a flat base is necessary for something to stand by itself. Then move on to discuss how different shapes are useful for different functions (e.g., objects that stack need a flat bottom, objects that roll need a circular face).

E1.3

construct and describe two-dimensional shapes and three-dimensional objects that have matching halves

**identifying matching halves:**- mentally and physically perform

- mentally and physically perform

Matching with Flipping or Reflecting |

Matching with Sliding or Translating |

Matching with Turning or Rotating |

**constructing matching halves:**

Folding Paper |
Using Interlocking Cubes |

Using a Reflection Tool |
Using Pattern Blocks and a Line of Symmetry |

- If two shapes or objects match in every way, they are congruent, even if their orientation is not the same. 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.

Have students play “Matching Halves Treasure Hunt”, where they find everyday shapes and objects, including objects in nature, that have matching halves and explain how they know that the two halves match (e.g., they can use a mirror to test whether the shapes or objects are symmetrical).

Provide students with paper cut-outs of shapes (e.g., the letters of the alphabet), and have them determine whether they have matching halves by folding.

Ask students to draw shapes or build objects that have matching halves and explain the strategies they used to construct the shape or object, and how they know the two halves match.

#### Location and Movement

E1.4

describe the relative locations of objects or people, using positional language

**positional language:**- right/left; e.g., the frog is to the left of the tree
- above/below
- over/under
- beside/inside/outside

**contexts:**- locations of everyday objects in the classroom
- starting position of a character or digital image before it moves on a grid

- 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.

Have students follow directions in a game, song, or poem that uses positional language.

Have students write clues to describe the location of one object relative to another. Encourage them to use distance and direction vocabulary to describe the object’s location (e.g., the turtle eggs are buried 20 steps to the right of the pond).

E1.5

give and follow directions for moving from one location to another

**directional language:**- right/left; e.g., the frog moves two hops to the right
- forward/backward
- up/down

**contexts:**- movements to locate an object
- movements of a character on a grid

- 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.

Read (or co-create with students) a story that involves characters travelling on a path. Have students draw and describe the path in the story using the vocabulary of distance and direction. They might use a large floor grid or the grid on an interactive whiteboard to organize and sequence their directions.

Have students write directions from a starting point (e.g., the classroom door) to a given object in the classroom (e.g., the pencil sharpener). Randomly select from the collection of directions, and ask students to test whether the directions lead to the object. If they don’t, ask them how the directions could be adjusted.

Have students program a sequence of steps on a grid that directs a digital image (or robot or classmate) to move through a maze from a starting position to a specific location in the maze, or to the exit. For example, for the grid below, have them both give and follow directions, such as:

- How might you get from the letter X to the dog if you had to go around all the things in the way?
- If I’m at the train and go up 5 and then left 5, what object will I land on?