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 square-based prism with (non-rectangular) 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 square-based 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 non-prism 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 non-examples.

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 large-group 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

Have students build a simple three-dimensional 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’s-eye perspective” of a physical space, and discuss its features. In pairs, have them draw a simplified blueprint (top-view 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 real-world 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.

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 non-standard 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 pencil-and-paper 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).

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 x-axis, 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 y-axis 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 x-axis and the y-axis.

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 x-axis. 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 x-axis and y-axis 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.