E1. Geometric and Spatial Reasoning:
describe and represent shape, location, and movement by applying geometric properties and spatial relationships in order to navigate the world around them
identify geometric properties of triangles, and construct different types of triangles when given side or angle measurements
Three equal sides
Two equal sides
No equal sides
All angles are less than 90°.
One angle is 90°.
One angle is greater than 90°.
3 Equal Sides
3 Equal Angles
2 Equal Sides
2 Equal Angles
No Equal Sides
No Equal Angles
All Angles Are Less Than 90°
1 Angle Is 90°
1 Angle Is Greater Than 90°
Have students analyse and sort various triangles that are scalene, isosceles, equilateral, right, obtuse, acute, and a combination of these types (e.g., isosceles right triangles, isosceles acute triangles). (A sample of triangles can be downloaded at BLM: Grade 5 E1.1 Triangle Sort.) Have them sort the triangles based on their sides, then based on their angles. Support students in understanding that triangles can be classified based on their side lengths and their angle measures. Discuss how some shapes and images look like a triangle but are not because they do not have three straight sides, or the three sides are not completely closed.
Create a worksheet similar to the one below. Ask students to sketch the appropriate triangle in each cell. If, for any of the combinations, a triangle cannot be drawn, they should write an explanation in the appropriate cell.
After they have completed the chart, use their findings to inspire additional insights or questions. For example:
Have students analyse their collection of sorted triangles to uncover other properties of triangles. For example, have them add the interior angles of a triangle and guide them to notice that, for all triangles, the interior angles add up to 180°. Have them develop a visual proof for this property by tearing off the corners of their triangles and rearranging them to form a straight angle (180°), as illustrated below:
Provide students with a variety of other triangles with missing angle measures, and ask them to apply this property to determine the missing angles. They could also construct their own triangles with missing angles and exchange them with a peer.
Have students construct triangles using various techniques. For example:
identify and construct congruent triangles, rectangles, and parallelograms
Identifying and Constructing
Provide students with a collection of triangles, rectangles, and parallelograms, some of which are congruent, and some of which are not. Ask students to predict the shapes that are congruent. Then, have them measure the side lengths and angles to confirm their predictions.
Have students construct congruent triangles when:
Have students use rulers and protractors to construct congruent triangles, rectangles, and parallelograms. As they construct these shapes, pose questions to promote spatial reasoning and develop geometric arguments about congruence, such as:
draw top, front, and side views of objects, and match drawings with objects
Note: Students are not expected to draw a three-dimensional representation of the object in Grade 5.
Have students use interlocking cubes to build different “robot sculptures”. Place a sculpture in the centre of each small group of students. Have students move around the sculpture to see it from all angles, then draw its top, front, and side views on 2 cm grid paper. For example, for the object represented below, they might draw the following:
Once done, students in each group should share the strategies they used to make their drawings. Next, collect all the drawings and redistribute them to different students. Move all of the sculptures to a common table. Have students match their new drawings to the correct sculpture on the table and explain their reasoning to a partner.
Provide students with a variety of structures along with their front, side, and top views. Ask them to match the views with the appropriate structures.
plot and read coordinates in the first quadrant of a Cartesian plane using various scales, and describe the translations that move a point from one coordinate to another
Provide students with the first quadrant of a coordinate plane that uses a scale other than 1 on at least one of the axes and has points already plotted. Have them analyse the scale to determine the coordinates of each point. For example, have them describe the location of A on the following:
Have students inspect and draw grids that use a variety of scales, and discuss strategies for determining unlabelled intervals. Have them plot points in the first quadrant of the coordinate plane and describe the locations. To apply this learning, have them play a strategy guessing game (e.g., an adaptation of “Battleship”) where a secret object is hidden along points on a coordinate plane. Students take turns guessing the location of the object by guessing coordinates and seeing if the guess is a hit or a miss.
Connect the learning about scales on the coordinate plane to scales used in number lines; graphs; and a variety of measurement tools, particularly spring scales and graduated containers.
Have students describe movement on a Cartesian plane using the language of translations (i.e., by describing distance and direction). Have them begin at the origin, (0, 0), and create different paths to move from there to another point in the first quadrant, and from that point to other points. To connect to work in the Algebra strand on coding, students might also use a Cartesian plane and code to guide a robot (or digital image, or classmate). Note that work with Cartesian planes can also be done using online geometry applications or certain educational video games. As students develop their code and directions, have them test their instructions and adjust as necessary.
Have students draw a triangle, a rectangle, or a parallelogram in the first quadrant of a Cartesian plane on a grid or graph paper using a scale of 1. Have them label each vertex with its coordinates. Next, have them draw the first quadrant of a Cartesian plane on a grid or graph paper using a scale of 2, and ask them to redraw their image on this grid using the original coordinates. Discuss how the image is similar to yet a different size than the original. Ask them to predict what would happen to their image if it was drawn on a scale of one half. Have them verify their prediction by redrawing their image using this scale.
describe and perform translations, reflections, and rotations up to 180° on a grid, and predict the results of these transformations
Have students draw a triangle on grid paper and label the vertices A, B, and C. Have them perform a variety of translations (right, left, up, down, and combinations of these) to draw new triangles. Label the new triangles using the symbols for prime (A’, B’, C’), double prime (A”, B”, C”), triple prime (A”’, B”’, C”’), and so on, to distinguish the original triangle from the new triangles. Repeat the activity for other types of polygons. Have students verify that the original polygon and the translated polygons are congruent.
Have students draw a triangle on grid paper and label the vertices J, K, L. Have them perform a variety of reflections (horizontal, vertical, diagonal, and combinations of these) to draw new triangles. Include lines of reflection on one side of the triangle and outside the triangle. Label the new triangles using the symbols for prime (J’, K’, L’), double prime (J”, K”, L”), triple prime (J”’, K”’, L”’), and so on, to distinguish the original triangle from the new triangles. Repeat the activity for other types of polygons. Have students verify that the original polygon and the reflected polygons are congruent
Have students draw a triangle on grid paper and label the vertices P, Q, and R. Have them do rotations of 90° and 180°, clockwise and counterclockwise, about a triangle vertex. Label the new triangles using the symbols for prime (P’, Q’, R’), double prime (P”, Q”, R”), triple prime (P”’, Q”’, R”’), and so on to distinguish the original triangle from the new triangles. Repeat the activity for other types of polygons. Have students verify that the original polygon and the rotated polygons are congruent.
Use a dynamic geometry application to construct a polygon such as the one below:
Have students rotate the polygon up to 180°, and measure the angles and distances involved in the rotation. Have them move a point on the original shape, or move the point of rotation, to notice how the rotation behaves. Support students in noticing that:
Repeat this process for reflections and translations, and write lists of the properties of each.