E1.5 Describe and perform translations, reflections, and rotations up to 180° on a grid, and predict the results of these transformations.
Skill: Describing and Performing Translations
Students can describe and perform horizontal, vertical, or oblique translations indicated with an arrow. The arrow can be placed on the figure or outside the figure. The direction of the arrow corresponds to the direction of the translation and its length corresponds to the magnitude of the displacement.

To describe and perform the translation, students must determine the magnitude of the horizontal and vertical movements represented by the arrow. For example, the arrow below defines the translation (4 left and 2 up), that is, a movement of four units left and two units up.

During a translation, all the points of the initial shape undergo the same movement, that is to say that all the points of the initial shape are moved in the same direction and are equidistant from the corresponding points of its image.

Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 29-30.
Skill: Describing and Performing Reflections
To describe and perform a reflection, it is necessary to understand that all points in the original shape and corresponding points in its image are at equal distances from the line of reflection. The reflection results in a change in the original position and orientation of a shape, and the reflected image is congruent with the initial shape. In other words, the reflected image has the same size and shape, but it will be oriented in a different direction, and will be in a different position. It is important to note that the line of reflection can be drawn in any direction relative to the original shape - horizontally, vertically or diagonally, at any angle.
In this example, points A, B and C are exactly the same distance from the line of reflection as points A', B' and C'.

Source: A Guide to Effective Teaching of Mathematics in Grades 4 to 6, p. 35.
Students perform reflections of simple and complex shapes (for example, to create friezes and tessellations) on dot paper or grid paper, using a Mira, tracing paper, or dynamic geometry software or application.

Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 31.
Skill: Describing and Performing Rotation of up to 180° on a Grid
To describe and perform a rotation, one must be able to understand and state:
- the location of the point of rotation around which a shape rotates (for example, the point of rotation can be a point on the outline of the shape, inside the shape or outside the shape);
- the measure of rotation (for example, a quarter turn or 90°, or a half turn or 180°);
- the direction of rotation (for example clockwise, counterclockwise).
Students need to understand what a quarter, half, and three-quarter turn clockwise or counterclockwise represents. It is important to introduce them to a variety of concrete activities that will help them develop this understanding.
Using the clock as a model is an effective strategy to help students develop a sense of how a full revolution can be represented using fractions. For examples, students could use a cardboard clock made with two hands attached using a brass fastener. Ask students to place both hands at 12. Then, to represent a rotation of one quarter-turn clockwise, ask students to move the minute hand to 3. The hour hand represents the initial position while the minute hand represents the position after the rotation.
Students should be supported to make the connection between the fraction of the rotation represented by the hands of the clock, the number of minutes between the two hands and as well as the number of degrees between them.

\(15 \ minutes = \frac{1}{4} \ rotation = 90^{\circ}\)
Note: On a real clock, this correspondence is not quite accurate, since the hour hand moves slightly when the minute hand rotates a quarter rotation.
Again, it is important to vary the initial direction of the hands. For example, the teacher can place the two hands of the clock at 2 and have the students rotate them clockwise one-half turn. The students can then use the fact that one-half turn of the large hand corresponds to 180° or 30 minutes to determine that the minute hand should be placed at 8.

\(30 \ minutes = \frac{1}{2} \ rotation = 180^{\circ}\)
Once students have developed their understanding of fractions of a rotation using concrete materials, they are able to represent them on circles drawn on paper.

Students should perform rotations of shapes using one of the vertices of the shape as the point of rotation and a variety of tools (for example, tracing paper, polar coordinate graph paper, square grid paper, dynamic geometry software or application). Performing the rotation according to the specified fraction will support students to better understand the impact of this transformation. Students should also perform rotations with the point of rotation on the perimeter of the shape or inside of the shape, and also with the point of rotation outside the shape.
Example
A quarter turn clockwise rotation, with the center of rotation located:
- on a vertex of the shape;
- on the perimeter of the shape or inside the shape;

- outside the shape;

Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 32-35.
Skill: Predicting the Results of Transformations
In the junior grades, students predict the outcome of a transformation and describe what will happen to the object when the transformation is completed. Through guided investigations, students will eventually be able to observe the initial orientation of an object and the outcome of a transformation, and describe the transformation that has been completed without having to see it.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 34.
Knowledge: Transformations
Transformations contribute to the development of a spatial sense and the ability to visualize the movement of objects in two- and three-dimensional space. The movement of geometric shapes can be described using various transformations, such as translation, reflection, and rotation.

Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 37.
Knowledge: Translation
A translation is defined by its size and direction (represented symbolically by coordinates or by an arrow). It represents a linear, horizontal, vertical or oblique (diagonal) movement. The initial shape and its image are congruent. The distance is constant between each point of the initial shape and each corresponding point of its image. The orientation of the image is the same as the orientation of the initial shape.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 36.
Knowledge: Reflection
A reflection is a flip performed over a line of reflection. Each point on the initial shape and the corresponding points on the image are at the same distance from the line of reflection. The initial shape and its image are congruent, but the orientation of the image is different from the orientation of the initial shape.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 36.
Knowledge: Rotation
A rotation is a transformation that moves each point of a shape around a fixed point, also called the point or centre of rotation. A rotation creates an image that is congruent with the original shape. However, the orientation of the image is different from the orientation of the initial shape.
The point of rotation can be anywhere on the plane, either outside the figure or inside.

When the point of rotation is on a vertex of the initial shape, the initial shape and its image will share this point.

Source: A Guide to Effective Teaching of Mathematics in Grades 4 to 6, p. 36.