Provide students with a variety of regular polygons (equilateral triangle, square, pentagon, hexagon, heptagon, octagon). Ask them which shapes on their own would be good to tile a space, such as the wall or the floor. Have them list which shapes tile on their own and which ones do not. Support students in recognizing that, of the regular polygons, the equilateral triangle, the square, and the regular hexagon work because when these shapes are brought together, their vertices can fill 360° where they meet:

The other regular polygons do not work because when they are brought together, their interior angles do not add up to 360°, so a gap results. Ask students if there is a way to combine shapes that cannot tile on their own with another shape, such as illustrated below, where three pentagons must be combined with a rhombus to make 360°. Have them find different combinations and share their favourite designs with others in the class. Ensure that at the end of the activity, students recognize that a regular polygon will tile on its own if the measure of one of its interior angles divides into 360 evenly.

Have students select a shape that tessellates. Have them cut out a piece from one side of the shape and translate that piece to the opposite side of the shape and tape it in place. Then, have them place their shape on a rectangular piece of paper and translate it, each time outlining it on the paper. Have them share their designs with others in the class.

Girih tiles, invented by North African artists in the 13th century, are used throughout Islamic art and architecture and feature complex tessellating shapes and symmetrical designs. M. C. Escher (1898–1972) was inspired by the Girih tiles and created his own tessellating art designs. He described ten “systems of symmetry” that use different transformations or combinations of transformations to create a unique tessellating tile. Tessellations also occur in nature, such as in honeycombs. Have students research and analyse Girih tiles, the art of Escher, and tessellations in nature to reveal the underlying polygon or pattern core. Guide them to use the geometric language of location, movement, and transformation to describe the work. As a culminating activity, have them use these examples to inspire their own construction of other tessellating tiles, and ask them to write a description of the transformations they used. Ensure that students recognize or understand that any triangle, any square, any rectangle, and any parallelogram will tesselate.

Have students read a plan and elevation drawing (e.g., a blueprint for the school or another building, a landscape design, or instructions) and build a simplified three-dimensional scale model that preserves the dimensions and proportions. Support them in identifying the scale on the plan and choosing an appropriate scale for their model. Provide them with cardstock or some other material to build their model, and have them draw on their experience with nets and prisms to approximate objects in the plan at the chosen scale. Assure them that the goal is not to make “the perfect model” but to read and interpret scales and to understand and experience how two-dimensional drawings can be used to build a model or the actual object.

Have students draw the top, front, and side views (plan and elevation drawings) as well as perspective views (isometric and cabinet projections) of a structure made of interlocking cubes. Have them exchange drawings with a partner, then build one another’s structures from the drawings. Have them compare the original structure with the newly made structure and note any differences.

As an extension activity, have students discuss what the structure might have looked like if they had only shared their perspective drawings (i.e., any hidden backside elements would be missing), or a side view, or a top view.

Have students use the scale on a map to estimate the distance from one location to another and the area of a region. Guide a conversation about the value of scales and how they are calculated and used. Have students compare the scale on a map to the scale on a blueprint or other plan drawing. Have them use the scale drawing (plan and elevation views) to calculate measurements of the actual object or space. If the actual object or space is available, compare the predicted measurements with the actual physical measurements.

Provide students with, for example, a cereal box, and compare the picture on the front to the actual size of the object illustrated. Have them measure the dimensions of the object in the picture (the height and width) and the actual object and compare them multiplicatively as a ratio. Support students in noticing proportional relationships (see B2.8) between the dimensions of the two objects (e.g., length to length) and within each of the two objects (i.e., length to width).

Once the scale has been established, have students draw a scaled-down version of the image to fit a smaller box with given dimensions. Discuss strategies for determining the scale factor of the two boxes, and have them apply that factor to the image. To make connections to multiplying by a fraction (see B2.6), have them scale by a fraction, such as two thirds or three fourths. A ratio table is a helpful organizer to show how a dimension can be scaled up or down.

Provide students with the dimensions of a drawing, such as the one below, and ask them what the dimensions of the drawing would be if it were scaled up by a factor of 2. Then, ask them to sketch the dimensions for the same drawing scaled down by a factor of 0.5.

Provide or have students choose a drawing (e.g., a simple picture from a colouring book) with a 1 cm × 1 cm grid superimposed. Ask them to recreate the drawing on grid paper with different-sized squares, including both smaller and larger squares. Have them share their drawings with others in the class.

Have students draw a shape anywhere on the coordinate plane. Ask them to identify the coordinates for each vertex. Then, ask them to translate this shape using two or more horizontal moves and two or more vertical moves, drawing the translated image each time. Have them compare the coordinates of the vertices of the final shape with those of the original shape. Ask them to determine the single move horizontally and the single move vertically that could be used to translate the original shape to its final location on the coordinate plane.

Have students draw a shape anywhere on the coordinate plane. Ask them to reflect the original shape in the x-axis and compare the coordinates of the vertices of the reflected shape with those of the original shape. Then, ask them to reflect the original shape in the y-axis and compare the coordinates of the vertices. Have them perform a combination of reflections in the x-axis and the y-axis and determine whether the order of reflecting affects the final outcome of the shape’s position on the coordinate plane.

Have students draw any shape in the first quadrant of the coordinate plane so that one of the vertices is the origin (0, 0). Have them rotate the shape counterclockwise 90° and compare the coordinates of the vertices of the rotated shape with those of the original shape. Have them rotate the shape another 90° counterclockwise and again compare the coordinates of the vertices. Have them rotate the shape another 90° counterclockwise and compare the coordinates of the vertices. Once they have completed all of the rotations, have them discuss with a partner what they notice about the changes to their coordinates after each rotation.

Have students draw a triangle in the first quadrant with one vertex at the origin (0, 0). Have them dilate the triangle by:

• multiplying each of the coordinates of the vertices by a value greater than one
• multiplying each of the coordinates of the vertices by a value between zero and one

After each dilation, have students compare the coordinates of the vertices of the original triangle with those of the dilated triangle.

Have students use a dynamic or interactive geometry environment to draw a simple shape (shape A) on a coordinate plane and label its coordinates. This task could also be done using block-based or text-based coding applications that include coordinate grids.

• Have students translate shape A into each of the other quadrants of the coordinate plane. Discuss the strategies they used to determine the appropriate vector (distance and direction), and support them in using positive and negative integers to describe the movement. Have them analyse the relationship between the coordinates of the original and the image, and the movement specified by the vector. Support them in generalizing the pattern symbolically. For example, for the translation (+5, −2): (x, y) → (x + 5, y − 2).
• 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 them in noting how the coordinates of the original changed after each reflection. Discuss ways to generalize this pattern symbolically. For example, for an x-axis reflection: (x, y) → (x, −y). For a y-axis reflection: (x, y) → (−x, y).
• 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°, and after each rotation, discuss the patterns that emerge. Support students in generalizing the pattern symbolically. For example, for a counterclockwise rotation of 90° around the origin: (x, y) → (−y, x).
• Have students dilate shape A by different scale factors, including by negative scale factors and factors with an absolute value less than one. Support students in noticing the impact of using different scale factors and in describing how the coordinates change. Discuss ways to generalize this pattern symbolically. For example, for a scale factor of 2.5: (x, y) → (2.5x, 2.5y). For a scale factor of −3: (x, y) → (−3x, −3y).

Note that the aim of this task (and the expectation) is not memorization of the mapping rules, but rather that students recognize that coordinates behave in particular ways when they are transformed in the coordinate system; these patterns can help with predicting the result of a transformation. Support students in recognizing these patterns and developing familiarity with the actions of different transformations so that they can begin to visualize and predict the results of each change.