E1. Geometric and Spatial Reasoning
Specific Expectations
Geometric Reasoning
E1.1
identify geometric properties of tessellating shapes and identify the transformations that occur in the tessellations
 tessellations:
 A tessellation uses tiles to cover an area without gaps or overlaps. The angles where tiles meet must add to 360°.
 The area of the base determines how many cubes can be placed on the base, which forms a single unit – a layer of cubes. The height of the prism determines how many layers of cubes it takes to fill the volume. Therefore, the formula for finding the volume of a rectangular prism is area of the base × height.
 Tessellating tiles are composed of one or more shapes and fit together in a repeating pattern. They are often used to create artistic designs, including wallpaper, quilts, rugs, and mosaics.
 Complex tessellating tiles can be designed by decomposing shapes and rearranging the parts using combinations of translations, reflections, and rotations (see also E1.4).
 If a shape can be transformed through a series of rotations, reflections and translations (i.e., by being turned, flipped, or slid), and still look the same, the shape is symmetric.
 There are different types of symmetries. For example, there is reflective symmetry, rotational symmetry, and translational symmetry.
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.
A regular penatagon cannont tile on its own. 
A regular pentagon will tile when combined with rhombuses with the same side lengths as the pentagon and an acute angle of 36°: 
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.
E1.2
make objects and models using appropriate scales, given their top, front, and side views or their perspective views
 constructing a model (as illustrated in the last example below) from a top view (plan), and front and side views (elevations):
 constructing a model from an isometric or cabinet projection:
 the resulting threedimensional model looks like this:
 Twodimensional drawings, if they are accurately constructed and include enough information, can be used to reproduce actualsized objects or scaled models in three dimensions (see also E1.3).
 Twodimensional drawings can 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.
 Twodimensional drawings are read and interpreted when navigating a map, following assembly instructions, or building an object from a plan.
 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.
 A perspective drawing shows three views (top, front, side) in one illustration. It is preferred for illustrations; however, angles are distorted and backside elements may be hidden. Isometric grids (also called triangular grids) are used to draw different perspectives, including isometric and cabinet projections.
Note
 Cabinet projections are so named because of their early use in the furniture industry. Isometric means “equal measure”, and isometric projections use the same scale on both axes.
 A scale is a ratio that compares actual dimensions to the dimensions in the drawing. A scale ensures that the intended lengths and proportions can be reproduced. Depending on the type of drawing, angles may or may not be represented accurately.
 Top, front, and side views of an object or space (plan and elevation drawings) use the actual angle sizes in the drawing and show all lengths in a common scale. For example, a 60° angle in the drawing is 60° in real life, and a scale of 1:100 means that 1 cm on the drawing equals 100 cm in real life (or 1 mm on the drawing equals 100 mm in real life, and so on).
 Isometric projections, drawn on a triangular grid, show all lengths in the same scale, including the lines that show depth. However, angles are distorted to create the appearance of perspective. Therefore, for example, a 90° angle in real life appears as a 60° in an isometric drawing.
 Cabinet projections also distort angles but use two scales to create perspective. The “depth” scale is half that of the “base and height” scale. So, for a scale of 1:100 a cabinet projection of a 1 cm cube would have a base and height of 1 cm, but a depth of 0.5 cm.
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 threedimensional 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 twodimensional 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.
E1.3
use scale drawings to calculate actual lengths and areas, and reproduce scale drawings at different ratios
 a scale drawing at different ratios:

 Scale drawings enable something very large or very small (microscopic) to be represented on a page or a screen. They are useful for visualizing, comparing, and calculating dimensions and are used, for example, when reading a map, following instructions, or designing a plan.
 Scale drawings are similar to the actual object or space. This means that the angles in the drawing and the corresponding angles in the actual object are congruent, and their corresponding lengths are proportional. A scale on the drawing explicitly describes this proportion and can be written in words, as a fraction, as a ratio, or as a graphical representation (e.g., a bar scale).
 Distances, areas, volumes, and angles can all be measured indirectly by referring to the measurements in the drawing. Top, front, and side views (plan and elevation drawings) are most reliable for calculating actual dimensions (see E1.2).
 A scale drawing can be reproduced at different scales by finding the unit ratio if it is not provided.
 Small scales show a larger area with a small amount of detail. For example, 1:2 000 000 means 1 cm in the drawing represents 2 000 000 cm, or 20 km.
 Larger scales show a larger area with greater detail. For example, 1:2 means that 1 cm in the drawing represents 2 cm.
 Grids are helpful for making scaled drawings. Scaling the grid is one way to produce a scaled drawing (e.g., overlaying a 2 cm grid on the original and using a 1 cm grid to produce the scaled version). Another way is to use the same grid and apply a scale to the drawing (e.g., 10 cm on the original grid becomes 1 cm on the scaled version). The first strategy may be used for smallerscale drawings; however, the second strategy is necessary for largescale representations.
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 scaleddown 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 differentsized squares, including both smaller and larger squares. Have them share their drawings with others in the class.
Location and Movement
E1.4
describe and perform translations, reflections, rotations, and dilations on a Cartesian plane, and predict the results of these transformations
 translation on a Cartesian (or coordinate) plane:
 △ABC is translated (4, −3):
 reflection on a coordinate plane:
 △ABC is reflected in the xaxis:
 △ABC is reflected in the yaxis:
 rotations on a coordinate plane:
 △ABC is rotated 90° counterclockwise:
 △A’B’C’ is rotated 90° counterclockwise:
 △A”B”C” is rotated 90° counterclockwise:
 dilation:
 scale factor of 2, with (0, 0) as the point of dilation:
 When shapes are transformed on a Cartesian plane, the coordinates of the original vertices are transformed to create corresponding coordinates known as image points. Each of the transformations can be defined using a mapping rule in which each point is transformed using that rule.
 Mapping rule for translations:
 (x, y) (x + a, y + b). If a is positive, then the xvalue of the image point is ‘a’ units to the right of the original point. If a is negative, then the xvalue of the image point is ‘a’ units to the left of the original point. If b is positive, then the yvalue of the image point is ‘b’ units up from the original point. If b is negative, then the yvalue of the image point is ‘b’ units down from the original point. For example, (x, y) (x − 5, y − 2); each image point is 5 units left of and 2 units down from the original point.
 Mapping rules for reflections:
 a shape reflected in the xaxis has mapping rule (x, y) (x, −y). For example, the vertex of the shape originally at (2, 3) is now at (2, −3).
 a shape reflected in the yaxis has a mapping rule (x, y) (−x, y). For example, the vertex of the shape originally at (2, 3) is now at (−2, 3).
 Mapping rules for rotations about the origin:
 a shape rotated 90° counterclockwise has mapping rule (x, y) (−y, x);
 a shape rotated 180° counterclockwise has mapping rule (x, y) (−x, −y);
 a shape rotated 270° counterclockwise has mapping rule (x, y) (y, −x).
 Mapping rules for dilations:
 (x, y) (ax, ay). For example, for (x, y) (2x, 2y), the coordinates of the image points are double those of the original points. For example, the image point for (−3, 4) is (−6, 8).
 Mapping rule for translations:
Note
 Transformations on the Cartesian Plane involve points being relocated to another position.
 Translations “slide” a point, segment, or shape by a given distance and direction (vector).
 Reflections “flip” a point, segment, or shape across a reflection line to create its opposite.
 Rotations “turn” a point, segment, or shape around a point of rotation by a given angle.
 Dilations (or dilatations) enlarge or shrink a distance by a given scale factor. Scale factors with an absolute value greater than 1 enlarge the distance, and those with an absolute value of less than 1 reduce the distance. Negative scale factors dilate the shape and rotate it 180°.
 Translations, reflections, and rotations all produce congruent images:
 Lines map to lines of the same length.
 Angles map to angles of the same measure.
 Parallel lines map to parallel lines.
 Dilations (or dilatations) produce scaled images that are similar:
 Lines map to line lengths at a constant scale factor.
 Angles map to angles of the same measure.
 Parallel lines map to parallel lines.
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 xaxis 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 yaxis and compare the coordinates of the vertices. Have them perform a combination of reflections in the xaxis and the yaxis 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 blockbased or textbased 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 xaxis and yaxis 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 xaxis reflection: (x, y) → (x, −y). For a yaxis 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.