E1. Geometric and Spatial Reasoning
Specific Expectations
Geometric Reasoning
E1.1
create lists of the geometric properties of various types of quadrilaterals, including the properties of the diagonals, rotational symmetry, and line symmetry
- types of quadrilaterals:
| Square | Rhombus |
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 (special rhombus: square) |
| Rectangle | Parallelogram |
 (special rectangle: square) |
 (special parallelograms: rectangle, rhombus, square) |
| Kite | Dart |
 (special kites: rhombus, square, dart) |
 |
| Trapezoids | |
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An isosceles triangle with one pair of parallel sides and another pair of sides of equal length. |
 |
| (special trapezoids: parallelogram, rectangle, rhombus, square) | |
- using diagonals of a quadrilateral:
| Rectangle | Square |
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| Parallelogram | Rhombus |
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| Trapezoid | Kite |
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- using line symmetry:
- using paper folding, reflection tools, dot paper, or technology to determine the number of lines of symmetry in a shape; for example:
- rotational symmetry:
- A shape has rotational symmetry if it still looks the same after it has been rotated less than 360°:
- When a square is rotated about its centre, its position matches its original position after a
rotation, a
rotation, and a
rotation; therefore, a square has rotational symmetry of order 4 because its position matches the original position four times during a complete rotation:
- A geometric property is an attribute that helps define an entire class of shapes.
- Quadrilaterals are polygons with four sides and four interior angles that add up to 360°. These are defining geometric properties of quadrilaterals. If a polygon has one of these attributes, it will automatically have the other and will be a quadrilateral.
- There are many different sub-categories, or classes, of quadrilaterals, and they are defined by their geometric properties. Certain attributes are particularly relevant for defining the geometric properties of shapes:
- angles:
- the number of right angles;
- the number of reflex angles;
- sides:
- the number of equal sides;
- whether the equal sides are adjacent or opposite;
- the number of parallel sides;
- symmetries:
- the number of lines of symmetry;
- the order of rotational symmetry;
- diagonals:
- whether they are of equal length;
- whether they intersect at right angles;
- whether they intersect at their midpoint.
- angles:
Note
- Quadrilaterals can be sorted and defined by their geometric properties. Analysing geometric properties is an important part of geometric reasoning. The goal is not to memorize these property lists but to generate and use property lists to create spatial arguments.
| Quadrilateral | Sample Properties | Example |
| Kite |
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| Dart |
A kite with:
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| Trapezoid |
Note Some definitions of trapezoid specify only one pair of parallel sides. The Ontario mathematics curriculum uses an inclusive definition: any quadrilateral with at least one pair of parallel sides is a trapezoid. |
|
| Parallelogram |
a trapezoid with:
|
 |
| Rectangle |
a parallelogram with:
|
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| Rhombus |
a parallelogram with:
|
 |
| Square |
a rectangle with:
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 |
- Relationships exist among the properties of quadrilaterals. For example, a square is a special type of rectangle, which is a special type of parallelogram.
- Minimum property lists identify the fewest properties guaranteed to identify the class (e.g., if a quadrilateral has four lines of symmetry, it must be a square).
To build an understanding of properties that distinguish one quadrilateral from another, have students sort and analyse a wide range of quadrilaterals (squares, rectangles, parallelograms, rhombuses, kites, darts, concave and convex regular and irregular quadrilaterals). As they sort these quadrilaterals, support them to identify attributes that can be used to differentiate classes of shapes (e.g., angles, sides, diagonals, line symmetry, rotational symmetry). Have them describe and record various distinguishing properties, for example, “These are all squares because they have four congruent sides, four right angles, and four lines of symmetry.”
To move deeper into creating property lists to define classes, provide groups of students with already-sorted sets of quadrilaterals: squares, rectangles, parallelograms, rhombuses, kites, darts, trapezoids. Give each group the name of each set (e.g., “These are all parallelograms”), and have them write detailed property lists to describe the class (e.g., “So what is a parallelogram?”).
To recognize relationships between different classes of quadrilaterals, have students compare property lists for various shapes and identify the common properties:
- for all squares and rectangles
- for all squares, rectangles, and parallelograms
- for all squares, rhombuses, parallelograms, and kites
Students should use these common properties to explain why all squares are rectangles, and all rectangles are parallelograms. They should also identify the properties that distinguish squares from rectangles, and rectangles from parallelograms (e.g., all squares are rectangles but not all rectangles are squares; all rectangles are parallelograms, but not all parallelograms are rectangles).
Have students write property challenges for each other, such as:
- Make a four-sided shape that has one pair of parallel sides that are not of equal length (e.g., a non-isosceles trapezoid).
- Make a shape that has two diagonals that are equal and that cross at right angles (e.g., a square).
- Make a shape that has two pairs of equal sides with none parallel (e.g., a kite or a dart).
Classmates visualize the properties and the spatial relationships, they verbalize a quadrilateral and make a prediction, and they verify it by constructing it on a geoboard.
As an extension, challenge students to find the minimum number of clues needed to guarantee a correct identification. Or have them pick a property (e.g., sides and angles; symmetry; diagonals) and define a quadrilateral using only that property. If the quadrilateral cannot be guessed by that property alone, have them add another property to the list. This type of geometric reasoning stimulates deduction and induction skills.
E1.2
construct three-dimensional objects when given their top, front, and side views
- top, front, and side views of an object:
- two possible objects:
| Top View | Front View | Right-Side View |
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 |
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- Three-dimensional objects can be represented in two dimensions.
- Given accurate top, front, and side views of an object, with sufficient information included, an object can be reproduced in three dimensions. Conventions exist (e.g., shading squares to show different heights) to clarify any ambiguities.
- Architects and builders use plan (top-view) and elevation (side-view) to guide their construction. Visualizing objects from different perspectives is an important skill used in many occupations, including all forms of engineering. STEM (science, technology, engineering, and mathematics) professionals use three-dimensional modelling apps to model a project before building a prototype. Three-dimensional objects can be represented in two dimensions.
- Given accurate top, front, and side views of an object, with sufficient information included, an object can be reproduced in three dimensions. Conventions exist (e.g., shading squares to show different heights) to clarify any ambiguities.
Have students build a structure out of interlocking cubes based on its top, front, and side views. Students compare what they built with others who were given the same clues and discuss any differences. As they analyse the various structures and compare them to the drawings, support them to troubleshoot possible errors. Have them determine if all, none, or some of the structures fit the specifications and explain their reasoning.
- Two possible objects:
| Top View | Front View | Right-Side View |
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To increase the possibility of more than one correct answer and, thereby, enrich the troubleshooting process and reasoning, provide students with only a top and a front view; alternatively, provide them with one view at a time, and use each subsequent view as a way for students to gradually refine their structure.
Location and Movement
E1.3
plot and read coordinates in all four quadrants of a Cartesian plane, and describe the translations that move a point from one coordinate to another
- The Cartesian plane uses two perpendicular number lines to describe locations on a grid. The x-axis is a horizontal number line; the y-axis is a vertical number line; and these two number lines intersect perpendicularly at the origin, (0, 0), forming four quadrants.
- 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 and the direction from the origin. The second number describes the vertical distance and the direction from the origin. For example, the point (4, 2) is located four units to the right of the origin and two units up; the point (4, −2) is located four units to the right of the origin and two units down; the point (−4, 2) is located four units to the left of the origin and two units up; the point (−4, −2) is located four units to the left of the origin and two units down.
Have students draw the four quadrants of a coordinate plane on a grid or graph paper. Have them use a scale of 1 and place the positive integers to the right of the origin on the x-axis and above the origin on the y-axis and the negative integers to the left of the origin on the x-axis and below the origin on the y-axis. Have them plot various points as a horizontal movement to the right or left of the origin and then a vertical movement up or down. For example, to plot the point (−3, 4), move 3 to the left from the origin and then up 4. Next, have them draw an image on the grid using the point that they have just plotted. Then, have them write down the coordinates for their image and exchange them with a partner to re-draw the image.
Have students play strategic guessing games with a partner. Behind a screen, each should plot secret objects, such as a treasure chest, along points in all four quadrants of a coordinate plane. They will take turns guessing the location of their partner’s hidden objects, using positive and negative coordinates to identify locations. Each partner will need a blank coordinate plane to keep track of their guesses. The game ends when one person has found all the objects hidden by their partner.
Provide students with the four quadrants of a coordinate plane with points plotted on it. Ask them to describe the movement to get from one point to the next. Guide students to use appropriate signs to indicate the movements; for example, 3→ or right 3 and 3↑ or up 3.
E1.4
describe and perform combinations of translations, reflections, and rotations up to 360° on a grid, and predict the results of these transformations
- combination of translation and reflection:
- the triangle is first translated 5 units to the right and then reflected in a horizontal line below the shape:
- combination of rotation, reflection, and translation:
- the triangle is reflected in a vertical line, then rotated 180° clockwise about an exterior point, and then translated 3 units down:
- rotating a triangle 90° counterclockwise about a vertex (three times):
- rotating a triangle 90° counterclockwise about a point outside the triangle (three times):
- rotating a triangle about a centre point inside the triangle:
- Transformations on a shape result in changes to its position or its size. 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.
- Transformations can be combined or composed. Sometimes a single transformation can be created by combining multiple transformations.
- 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 that each point moving “5 units right and 2 units up”. It is a mathematical convention that the horizontal distance (x) be 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 centre of rotation and an angle of rotation. Every point on the original shape turns around the centre of rotation by the same specified angle. Any point on the original is the same distance to the centre of rotation as the corresponding point on the reflected image.
- Because a rotation is a turn, and 360° produces a full turn, a counterclockwise rotation of 270° produces the same result as a clockwise rotation of 90°. Convention has it that a positive angle describes a counterclockwise turn and a negative angle describes a clockwise turn, based on the numbering system of the Cartesian plane (see E1.3).
Note
- At this grade level, students can express the translation vector using arrows; for example, (
).
- Dynamic geometry applications are recommended to support students to understand how transformations behave, either as a single transformation, or a combination of transformations.
Have students draw a triangle on grid paper and label the vertices A, B, and C. Have them perform a variety of combinations of translations and reflections (e.g., right 10 and then reflected horizontally), using tracing paper as a tool, to draw new triangles. Ask them to 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. Have them verify that the original triangle and the translated triangles are congruent. Ask them to determine whether they will get the same result if they do the transformations in a different order. Repeat with other types of polygons.
Have students draw a triangle on grid paper and label the vertices D, E, and F. Have them rotate the triangle around one of its vertices by 90°, 180°, 270°, and 360°, both clockwise and counterclockwise, using tracing paper as a tool. Ask them to label the rotated triangles using the symbols for prime (D’, E’, F’), double prime (D”, E”, F”), triple prime (D”’, E”’, F”’), and so on, to distinguish among the triangles. Ask them to compare the triangles and describe what is the same and what is different about them.
Have students draw a triangle on grid paper and label the vertices J, K, and L. Have them rotate the triangle by 90°, 180°, 270° and 360°, both clockwise and counterclockwise, around a point outside the triangle, using tracing paper as a tool. Ask them to label the rotated triangles using the symbols for prime (J’, K’, L’), double prime (J”, K”, L”), triple prime (J”’, K”’, L”’), and so on, to distinguish among the triangles. Ask them to compare the triangles and describe what is the same and what is different about them.
Have students draw a triangle on grid paper and label the vertices P, Q, and R. Have them rotate the triangle about a point inside the triangle by 90°, 180°, 270°, and 360°, both clockwise and counterclockwise, using tracing paper as a tool. Ask them to label the rotated triangles using the symbols for prime (P’, Q’, R’), double prime (P”, Q”, R”), triple prime (P”’, Q”’, R”’), and so on, to distinguish among the triangles. Ask them to compare the triangles and describe what is the same and what is different about them.
Have students use a dynamic geometry application to construct a polygon, rotate it 270°, and measure the angles and distances involved. Ask them to move a vertex on the polygon or the point of rotation, and guide them to notice the impact of the move (i.e., what changed and what stayed the same). Have them compare the impacts of clockwise and counterclockwise rotations (negative or positive angle) and discuss why a rotation of 270° clockwise is the same as a rotation of 90° counterclockwise. Have them perform successive transformations, and have others in the class identify the transformations involved.
Have students perform a combination of a translation, a reflection, and a rotation (up to 360°) on a grid and mentally note the transformation. Have them share their original shape and image with a partner and ask them to identify the transformation and explain their reasoning. Using the same process, have students perform two successive transformations and again have a partner identify and explain the transformations involved.
Challenge students to predict two transformations that, when combined, create a single transformation, and have them test their predictions using a dynamic geometry application.
