E1. Geometric and Spatial Reasoning
Specific Expectations
Geometric Reasoning
 types of triangles:
Classifying Triangles by Side Length  
Equilateral Triangle Three equal sides 
Isosceles Triangle Two equal sides 
Scalene Triangle No equal sides 
Classifying Triangles by Angle Measures  
Acute Triangle All angles are less than 90°. 
Right Triangle One angle is 90°. 
Obtuse Triangle One angle is greater than 90°. 
 Triangles have been an important shape for mathematicians throughout history, and they continue to be significant in engineering, astronomy, navigation, and surveying.
 Geometric properties are specific attributes that define a “class” of shapes or objects. The following are geometric properties of triangles:
 All triangles have three sides and three angles.
 The combined length of any two sides of a triangle is always greater than the length of the third side.
 The interior angles of a triangle always add up (sum) to 180° (e.g., 70° + 60° + 50° = 180°).
 The exterior angles of a triangle always sum to 360° (e.g., 110° + 120° + 130° = 360°).
 The interior angle and its corresponding exterior angle always sum to 180° (e.g., 130° + 50° = 180°).
 Triangles can be classified by the number of equal side lengths or number of equal angles:
Classifying Triangles by Number of Equal Sides or Number of Equal Angles  
Equilateral Triangle: 3 Equal Sides 3 Equal Angles 
Isosceles Triangle: 2 Equal Sides 2 Equal Angles 
Scalene Triangle: No Equal Sides No Equal Angles 
 Triangles can be classified by the type of angle measures:
Classifying Triangles by Type of Angle Measures  
Acute Triangle: All Angles Are Less Than 90° 
Right Triangle: 1 Angle Is 90° 
Obtuse Triangle: 1 Angle Is Greater Than 90° 
 There are different techniques for constructing triangles depending on what is known and unknown:
 When all side lengths of a triangle are known but the angles are unknown, a ruler and compass can be used to construct the triangle. To do so, one could draw the length of one of the sides, then set the compass for the length of another side. The compass can then be put at one of the ends of the line and an arc drawn. Now the compass should be set to the length of the third side and set on the other end of the line to draw another arc. Where the two arcs intersect is the third vertex of the triangle (see the diagram below). The sides can be completed by drawing a line from the ends of the original line to the point of intersection.
 When one side length and two of the angles of a triangle are known, a protractor and a ruler can be used to construct the triangle. The unknown vertex is the point where the arms of the angles intersect:
Note
 Triangles can also be constructed using dynamic geometry applications in many ways, including by transforming points and by constructing circles.
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.
Acute  Right  Obtuse  
Scalene 


Isosceles 


Equilateral 

After they have completed the chart, use their findings to inspire additional insights or questions. For example:
 How can you describe an equilateral triangle without using the word “side”?
 Can a triangle have more than one obtuse angle? Can it have more than one right angle?
 A triangle has one very small angle and another angle that is almost a right angle. What can you tell me about the third angle?
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:
 Use a compass (or similar tool) and a ruler to construct a triangle that has side lengths of 11 centimetres, 8 centimetres, and 7 centimetres.
 Use a protractor and a ruler to construct an isosceles triangle with a side length of 9 centimetres and two 70° angles.
 Use a dynamic geometry application to construct a triangle with given side lengths or given angles. In addition to drawing and using dynamic geometry software, students can develop code to make various types of triangles and use the code to describe their properties.
E1.2
identify and construct congruent triangles, rectangles, and parallelograms
Identifying and Constructing 
Criteria 
Congruent Triangles 

Congruent Rectangles 

Congruent Parallelograms 

 Congruent twodimensional shapes can be superimposed exactly onto each other. Congruent shapes have congruent angles and congruent lengths.
 If all the side lengths of two triangles are congruent, all the angles will also be congruent.
 If all the angles of two triangles are congruent, it is not necessarily true that the side lengths are congruent.
 Parallelograms, including rectangles and squares, require a combination of congruent angles and congruent side lengths to be congruent.
 Constructing congruent shapes involves measuring and using protractors and rulers. For more information on using protractors, see E2.4. For more information on using rulers, see Grade 2, E2.3.
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:
 given the measurements for all three sides (e.g., 3 cm, 4 cm, 5 cm)
 given the measurements for two sides and the angle between them (e.g., 3 cm, 45°, 8 cm)
 given the measurements for two angles and a side (e.g., 30°, 6 cm, 50°)
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:
 Can two triangles have congruent side lengths without congruent angles? Can two triangles have congruent angles and not have congruent side lengths? Explain and give examples to support your argument.
 Can two parallelograms have congruent side lengths without congruent angles? Can two parallelograms have congruent angles and not have congruent side lengths? Explain and give examples to support your argument.
 Can two rectangles have congruent side lengths and not have congruent angles? Can two rectangles have congruent angles and not have congruent side lengths? Explain and give examples to support your argument.
 What measurements are needed to construct congruent triangles? Congruent rectangles? Congruent parallelograms?
E1.3
draw top, front, and side views of objects, and match drawings with objects
 top, front, and side views of an object:
Representation of the Object  Front View  RightSide View  Top View 
Note: Students are not expected to draw a threedimensional representation of the object in Grade 5.
 Threedimensional objects can be represented in two dimensions.
 Given accurate top, front, and side views of an object, with enough information included, the object can be reproduced in three dimensions. Conventions exist (e.g., shading squares to show different heights; using lines as way to show changes in elevations) to clarify any potential ambiguities.
 Architects and builders use plan (top view) and elevation (side view) drawings to guide their construction. STEM (science, technology, engineering, and mathematics) professionals use threedimensional modelling apps to model a project before building a prototype. Visualizing objects from different perspectives is an important skill used in many occupations, including all forms of engineering.
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:
Representation of the Object  Front View  RightSide View  Top View 
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.
Location and Movement
E1.4
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
 coordinates in the first quadrant:
 Moving from A to B is a translation of 1 unit right and 3 units down.
 To add context to translated points, describe movements between locations that are relevant to students. For example, the coordinates could represent places in the school or the community.
 The Cartesian plane uses two perpendicular number lines to describe locations on a grid. The xaxis is a horizontal number line; the yaxis is a vertical number line; and these two number lines intersect at the origin, (0, 0).
 Pairs of numbers (coordinates) describe the precise location of any point on the plane. The coordinates are enclosed by parentheses as an ordered pair (x, y). The first number in the pair describes the horizontal distance from the origin, and the second number describes the vertical distance from the origin.
 The point (1, 5) is located 1 unit to the right of the origin (along the xaxis) and 5 units above the xaxis. As a translation from the origin, the point (1, 5) is right 1 unit and up 5 units.
 The x and yaxes on the Cartesian plane, like any other number line or graduated measurement tool, are continuous scales that can be infinitely subdivided into smaller increments. The numbering of the axes may occur at any interval.
 Sometimes a gridline is marked in multiples of a number and the subdivisions must be deduced (e.g., for an axis marked in multiples of 10, a coordinate of 15 is half the distance between 10 and 20):
 Sometimes the axes are labelled in wholenumber increments, and the location of a decimal coordinate must be deduced (e.g., for an axis labelled 1, 2, 3, 4, ..., a coordinate of 1.5 is plotted five tenths or one half of the distance between 1 and 2):
 Sometimes not every gridline is labelled, and the value of the unlabelled grid line must be deduced (e.g., when every fifth line is labelled 10, 20, 30, 40, …):
 The number lines on the Cartesian plane extend infinitely in all directions and include both positive and negative numbers, which are centred by the origin, (0, 0). In the first quadrant of the Cartesian plane, the x and ycoordinates are both positive.
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.
E1.5
describe and perform translations, reflections, and rotations up to 180° on a grid, and predict the results of these transformations
 translations on a grid:
Horizontal Translation  
5 units to the left:  3 units to the right: 
Vertical Translation  
5 units up:  6 units down: 
 showing a translation with a vector:
 reflections on a grid:
Horizontal Line of Reflection  
Vertical Line of Reflection  
Diagonal Line of Reflection  
 rotations:
 90° counterclockwise rotation about a point of rotation at the vertex of a triangle:
 90° clockwise rotation about a point of rotation inside the shape:
 90° clockwise rotation about a point of rotation outside the shape:
 Transformations on a shape result in changes to its position and sometimes orientation or its size (or both). As a shape transforms, its vertices (points on a grid) move. The transformation describes the results of the movement. This explains how transformations involve location and movement.
 A translation involves distance and direction. Every point on the original shape “slides” the same distance and direction to create a translated image. This combination of distance and direction is called the translation vector. For example, on a grid, a vector could describe each point moving “5 units to the right and 2 units up”. It is a mathematical convention that the horizontal distance (x) is given first, followed by the vertical distance (y).
 A reflection involves a line of reflection that acts like a mirror. Every point on the original shape is “flipped” across the line of reflection to create a reflected image. Every point on the original image is the same distance from the line of reflection as the corresponding point on the reflected image. Reflections are symmetrical.
 A rotation involves a point (also known as the centre) of rotation and an angle of rotation. Every point on the original shape turns around the point of rotation by the same specified angle. Any point on the original is the same distance to the point of rotation as the corresponding point on the rotated image.
Note
 At this grade level, students can express the translation vector using arrows; for example, ().
 Dynamic geometry applications are recommended for visualizing and understanding how transformations, and especially rotations, behave.
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:
 the original shape and its image are congruent (i.e., the side lengths and the angles are the same)
 the angle of rotation is constant (e.g., ∠ADA’ = ∠BDB’ = ∠CDC’= 90°)
 the distance from any point on the original to the point of rotation is the same as the corresponding distance in the image (e.g., the length of AD = the length of A’D)
Repeat this process for reflections and translations, and write lists of the properties of each.