E1. Geometric and Spatial Reasoning
Specific Expectations
Geometric Reasoning
E1.1
describe and classify cylinders, pyramids, and prisms according to their geometric properties, including plane and rotational symmetry
 prisms:
 right prism: the angles between the base and the sides are right angles, and the lateral faces are rectangles:
 oblique prism: the angles between the base and the sides are not right angles, and the lateral faces are parallelograms:
 pyramids:
 one polygonal base with triangles as the other faces that join at a single vertex, known as the apex
 right pyramid: the apex is directly above the centre of the base
 oblique pyramid: the apex is not centred over the base – it leans over
 named for the shape of the base (e.g., hexagonbased pyramid):
 cylinders:
 two congruent and parallel bases
 can have polygons, curved figures, or a combination of the two as bases
 prisms are a special kind of cylinder that have only polygons as faces
 right cylinder: the two bases are aligned directly one above the other:
 oblique cylinder: the two bases are not aligned one above the other:
 plane symmetry:
 a threedimensional object has plane symmetry if it can be divided into two congruent halves that are mirror images of each other:
 rotational symmetry:
 A threedimensional object has rotational symmetry if it can be turned about an axis of rotation and look the same as the original after a rotation of less than 360°:
 The order of rotational symmetry is the number of times a geometric figure can be rotated and look the same before it has been rotated all the way around (full rotation). Pictured below is the top view of a cube rotated 90° around an axis of rotation. It has an order of rotational symmetry of 4 (4 × 90° = 360°):
 A geometric property is an attribute that helps define a class of objects.
 Cylinders, pyramids, and prisms represent three broad categories of threedimensional objects.
 There are many attributes that are used to distinguish and define subcategories or classes of objects, including:
 the shape of the base or bases;
 the number of bases;
 the number of edges and vertices;
 whether the object is symmetrical (e.g., whether it has rotational or plane symmetry);
 whether the faces are perpendicular to the bases.
 Threedimensional objects can have rotational symmetry (when an object can rotate around an axis and find a new spin position that matches its original position) and plane symmetry (when an object can be split along a plane to create two symmetrical parts). Generating property lists and using them to create geometric arguments builds spatial sense. Minimum property lists identify the fewest properties needed to identify a class. The following is a list of some properties for cylinders, prisms, and pyramids.
Cylinders 

Circular Cylinders NonCircular Cylinders Special Cylinders: Prisms 
Prisms 


Pyramids 

RectangleBased Pyramid 
To deepen students’ understanding of the relationships that distinguish and connect cylinders, prisms, and pyramids, have pairs of students engage in “property challenges” by drawing nets on grid paper that satisfy given criteria. Property challenges could include the following:
 Make a net for a prism with faces that are all rectangles.
 Make a net for a squarebased prism with (nonrectangular) parallelograms for two of its other faces.
 Make a net for a right prism that has a hexagon for a base.
 Make a net for a prism with rectangles and right triangles.
 Make a net for a pyramid that is made of four triangles.
 Make a net for a squarebased pyramid that has at least one scalene triangle.
 Make a net for a pyramid whose base has rotational symmetry of order 4.
 Make a net for a pyramid featuring a rectangle and a right triangle.
 Make a net for a nonprism that has squares for its two bases and trapezoids for its other faces.
 Make a net for a cylinder with a circumference of approximately 8 cm (prerequisite learning: E2.3 and E2.4).
To support students in visualizing possible solutions, provide them with a set of drawings that satisfy the clues (see below, recognizing that perspective drawings can make the shape of the bases difficult to discern). Have them identify a match before making the net. The matching process can be made more challenging by adding two or more options for the same clue or by including nonexamples.
Once students have a plan for what object they will build to satisfy the criteria, have them make a net and explain their rationale. Through conversation, draw out the impact of using, for example, parallelograms instead of rectangles to join the two bases of prisms, or scalene triangles instead of isosceles triangles for the faces of pyramids. Support students in writing property lists for each type of prism, pyramid, or cylinder they build, and have them use these lists to generate other property challenges to share with the class.
Provide students with a set of real objects and drawings that are cylinders, pyramids, and prisms. Have students sort them and describe the rationale for their sort. In the discussion, highlight aspects of their sort that are geometric properties of cylinders, of pyramids, and of prisms, respectively.
Have students predict which prisms, cylinders, and pyramids have plane symmetry, then have them check their prediction by building the object out of clay and then splitting it in two. To support students in recognizing plane symmetry, suggest that they hold a mirror or reflection tool (e.g., a mirror) to one half of the object and check whether the reflection reproduces the whole. Guide them to identify horizontal, vertical, and diagonal planes of symmetry. In largegroup conversations, have students explain what features a symmetrical object has, and how it can be recognized. As an extension, have them find all the planes of symmetry of, for example, a cube, and check the Internet to verify their work. Challenge students to find all nine planes of symmetry of a cube.
Have students make prisms, cylinders, and pyramids using paper nets, and have them colour or number each of the sides differently. Demonstrate how to determine whether a cube has rotational symmetry by inserting a wooden skewer through the centre and turning the cube around, noting that it looks the same in different positions, until it reaches its original position again. Ask them to predict whether the prisms, cylinders, and pyramids they made have rotational symmetry, then have them check their predictions. Next, have students determine the order of rotational symmetry of those objects that have it. As an extension, challenge them to find the 13 rotational symmetries of a cube.
Have students make prisms and pyramids with the following properties:
 a prism that has rotational symmetry and plane symmetry
 a pyramid that has rotational symmetry and plane symmetry
 a pyramid or a prism that has no rotational symmetry and no plane symmetry
E1.2
draw top, front, and side views, as well as perspective views, of objects and physical spaces, using appropriate scales
 sample drawings of physical spaces:
 floor plans of the classroom, the school, or a favourite room
 wall plans showing doors and windows
 top, front, and side views on grid paper of a structure made with cubes:
 threedimensional structure:
 plan and elevation drawings (scale: 1 cm × 1 cm grid represents the face of a cube that is 2 cm × 2 cm):
 perspective drawings (on grid paper) of a structure made with cubes:
 isometric projection, where each parallelogram represents the 2 cm × 2 cm face of a cube:
 cabinet projection, where each square and each parallelogram represents the 2 cm × 2 cm face of a cube:
 Threedimensional objects can be graphically projected in two dimensions. Twodimensional representations show how things are made, how they can be navigated, or how they can be reproduced, and can be used to represent anything from very small objects to very large spaces. They are used by designers, builders, urban planners, instruction illustrators, and others.
 Top (plan) views, and front and side (elevation) views are “flat drawings” without perspective. They are used in technical drawings to ensure a faithful reproduction in three dimensions.
 Scales are used to convey the proportions of the original (i.e., angles and relative distances). If the scale is 1 : 100, then 1 cm always represents 100 cm at full size, regardless of the view. A legend communicates the scale.
 A perspective drawing shows three views (top, front, side) in one illustration.
 Perspective views cannot show the back side, so some elements may be hidden.
 They are also better at representing straight edges than curves.
 To achieve the appearance of perspective, they may distort angles and lengths.
 They are often easier to visualize than elevation drawings and are typically preferred for illustrations.
 Two types of perspective drawings are isometric projections and cabinet projections.
Note
 Isometric projections show an object from the “corner”, with the width and depth going off at equal angles. In isometric projections, a scale is applied consistently to all dimensions (e.g., 1 cm : 2 cm for the height, width, and depth).
Have students build a simple threedimensional structure using six to ten interlocking cubes and draw its top, front, and side views (i.e., plan and elevation drawings). Ask them to include a scale for each view. Ensure that they use the same scale for all three views.
Next, have students make an isometric drawing of the same structure using isometric dot paper. Discuss what transformations need to happen to make a cabinet projection drawing. For example, in the picture below left, a rhombus is rotated to convey the perpendicular faces of a cube:
Show students a blueprint or floor plan of the school as an example of a top view or “bird’seye perspective” of a physical space, and discuss its features. In pairs, have them draw a simplified blueprint (topview floor plan) of the classroom or other physical space, either by hand or using technology. If the physical space is tiled, discuss the fact that a grid is already present and can be aligned to the square grid of the scale drawing. If this is not the case, discuss what other strategies could be used, such as using ceiling tiles or lights, superimposing a grid on their scale drawing, and so on. Have students share their blueprints with others in the class and discuss similarities and differences, including how their scales affect the size of the blueprints.
To model other realworld contexts for elevations, have students bring in examples of instructions for putting something together. Ask them to analyse the different approaches to representing three dimensions on paper and discuss how the designer or illustrator used plan and elevation drawings or perspective drawings.
Location and Movement
E1.3
perform dilations and describe the similarity between the image and the original shape
 dilations of a square from a dilation point:
 a dilation point and the original 1 × 1 square:
 dilation of the original 1 × 1 square by a scale factor of 2: the dilated square is now 2 × 2, meaning that the lengths of its sides are double those of the original:
 dilation of the original 1 × 1 square by a scale factor of 0.5: the dilated square is now 0.5 × 0.5, meaning that the lengths of its sides are half those of the original:
 A dilation (or dilatation) is a transformation that enlarges or reduces a figure by a certain scale factor. Unlike translations, reflections, and rotations, a dilation does not produce a congruent image, unless the dilation factor is 1 or −1.
 A dilated image is similar to the original. In everyday language, a similar image means it simply resembles something else. In mathematics, similar has a very precise meaning. Similar figures have the same shape (angles are congruent) and their corresponding sides are proportional. If the width of a dilated rectangle is now twice as long, so too is its length.
Note
 Dynamic geometry applications are recommended tools for understanding how transformations behave in motion. In a dynamic environment, repositioning the point of dilation has an immediate impact, as does changing the scale factor.
 Dilations have connections to the concept of onepoint perspective in Grade 7 of the Visual Arts strand of the Arts curriculum (see The Ontario Curriculum, Grades 1–8: The Arts, 2009, p. 143).
Have students use grid paper to draw physical dilations by building enlargements of each pattern block by a given scale factor. Support them in noticing that if the scale factor is 2, each length in the dilation is twice the corresponding length in the original. Have them measure the angles, side lengths, and area of the original and compare each to that of the enlargement (using nonstandard units such as pattern blocks). Have them explain what remains the same and what changes and use these observations to describe the similarity of the shapes.
Have students use a dynamic or interactive geometry application to perform, analyse, and manipulate dilations of shapes. Have them use the application to measure angles and distances, and support them in noticing which measures remain constant and which measures change. Discuss the length relationships between the original and the dilated image, and guide students to make connections to the scale factor. For example, for the accompanying image below, discuss where the “times three” factor shows itself:
Moving to pencilandpaper constructions, have students draw a triangle on grid paper and label its vertices. Have them select a point on the grid to be the point of dilation. Have them draw lines from the point of dilation, through the vertices, and beyond to draw a dilated image. Support students in recognizing that each vertex of the image will fall somewhere along these lines, and that the scale factor determines the size of the image and the distance of a vertex from the point of dilation. In the example below, the distance from the point of dilation to a vertex of the original triangle compared to the distance from the point of dilation to its corresponding vertex of the dilated triangle has a ratio of 1:3.
Have students use dynamic or interactive geometry applications or grid paper to draw and compare several dilations of the same shape, including both enlargements and reductions. Have them make connections between the side lengths of the original shape, the image, and the scale factor (with the scale factor for enlargements being greater than one and for reductions being between zero and one).
E1.4
describe and perform translations, reflections, and rotations on a Cartesian plane, and predict the results of these transformations
 translation on a Cartesian (or coordinate) plane:
 △ABC is translated 4 units to the right and 3 units down:
 △ABC is reflected in the xaxis:
 △ABC is reflected in the yaxis:
 rotation about the origin (0, 0) on a coordinate plane:
 a triangle with vertices (0, 0), (3, 0), and (0, 2) is rotated counterclockwise 90°, 180°, and 270°:
 Translations, reflections, and rotations all produce congruent images.
 Translations “slide” a shape by a given distance and direction (vector).
 Reflections “flip” a shape across a reflection line to create its opposite.
 Rotations “turn” a shape around a centre of rotation by a given angle.
 When shapes are transformed on a Cartesian plane, patterns emerge between the coordinates of the original and the corresponding coordinates of the image. These patterns are particularly evident when:
 the translation vector (distance and direction) is compared to the coordinates of the original and the translated image;
 a shape is reflected across the xaxis or the yaxis;
 a shape is rotated by 90° or 180° around the origin (0, 0).
Provide students with a shape in the first quadrant of the coordinate plane. Ask them to identify the coordinates of each vertex. Ask them to translate the original shape in different ways, compare the vertices of the translated shape with those of the original shape, and use the proper labelling conventions. For example:
 Translate the shape so that all of its vertices are in the second quadrant of the coordinate plane.
 Translate the shape so that some of its vertices are in the third quadrant and some are in the fourth quadrant.
Provide students with a shape in the first quadrant of the coordinate plane. Ask them to identify the coordinates of each vertex, then draw its reflection in the xaxis, using proper labelling conventions. Ask them to compare the vertices of the reflected shape with those of the original shape and note their observations. Then, ask them to draw the reflection of the original shape in the yaxis and make similar comparisons. Finally, have them draw their own shape in any of the other quadrants and then exchange with a partner, who reflects the new shape in both the xaxis and the yaxis.
Have students draw a triangle in the first quadrant of the coordinate plane, with one vertex at the origin (0, 0) and another vertex on the xaxis. Ask them to rotate the triangle counterclockwise 90°, then compare the vertices of the rotated triangle with those of the original triangle. Then, ask them to rotate the triangle another 90° counterclockwise (180° from its original position) and again compare the vertices. Next, ask them to rotate the triangle another 90° counterclockwise (270° from its original position) and again compare the vertices. Ask them to write a summary of their findings.
Have students use a dynamic or interactive software application to draw a simple shape (shape A) on a coordinate plane and mark its coordinates.
 Have them translate shape A into other quadrants of the coordinate plane. Discuss how they can describe the movement using a translation vector (arrow showing distance and direction, as illustrated in the examples), and support them in using positive and negative integers to describe the movement (e.g., +5, −2). Have them analyse the relationship between the coordinates of the original and the image, and the movement specified by the vector.
 Have students reflect shape A into other quadrants of the coordinate plane using the xaxis and yaxis as lines of reflection. After each reflection, support students in noting how the coordinates of the original shape change after each reflection.
 Have students rotate shape A into other quadrants of the coordinate plane using the origin (0, 0) as the point of rotation. Have them rotate shape A by 90° repeatedly, and after each rotation, discuss the patterns that emerge.