E1. Geometric and Spatial Reasoning
Specific Expectations
Geometric Reasoning
E1.1
identify geometric properties of rectangles, including the number of right angles, parallel and perpendicular sides, and lines of symmetry
- geometric properties of rectangles:
- four sides, four vertices, four right angles
- opposite sides that are of equal length (congruent)
- opposite sides that are parallel
- adjacent sides that are perpendicular
- at least two lines of symmetry: horizontal; vertical; and, for squares, the two diagonals
- Geometric properties are specific attributes that define a class of shapes or objects. The geometric properties of a rectangle describe the attributes that all rectangles have, and include:
- four sides, four vertices, and four right angles;
- opposite sides that are of equal length (congruent);
- opposite sides that are parallel;
- adjacent sides that are perpendicular;
- at least two lines of symmetry – horizontal and vertical.
- Geometric properties are often related. Rectangles have four right angles, so they must also have two sets of parallel sides, and the opposite sides must be of equal length. This type of spatial reasoning is used by structural engineers and others.
- The geometric properties of a square include all the geometric properties of a rectangle; therefore, all squares are also rectangles. Squares have additional geometric properties (four equal sides, four lines of symmetry, including two diagonals); therefore, not all rectangles are squares.
- The geometric properties of a shape, and not its size or orientation, define the name of a shape. A rotated square that might look like a diamond is still a square, because it has all the geometric properties of a square.
Have students compare rectangles to other types of quadrilaterals, and ask them to describe how they are the same and different. Have them use these similarities and differences to write a geometric property list that is true for all rectangles and that distinguishes rectangles from other shapes. Support students in recognizing relationships between geometric properties. For example, ask: “Can you make a rectangle with four right angles that does not have opposite sides of equal length?” Guide them to understand that a square is a type of rectangle (a rectangle with four equal sides), and that not all rectangles are squares.
Have students use paper folding or a mirror to determine the lines of symmetry in a rectangle. Have them compare the number of lines of symmetry in a rectangle to the number of lines of symmetry in other quadrilaterals. Support students in recognizing that the number of lines of symmetry can be used to distinguish rectangles from other quadrilaterals. Guide them to notice that squares also have two lines of symmetry – like a rectangle – plus an additional two. This, in conjunction with other properties, is what makes a square a rectangle, but not all rectangles squares.
Location and Movement
E1.2
plot and read coordinates in the first quadrant of a Cartesian plane, and describe the translations that move a point from one coordinate to another
- coordinates in the first quadrant of a coordinate plane:
- 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 at the origin, (0, 0).
- The number lines on the Cartesian plane extend infinitely in all four directions and include both positive and negative numbers, which are centred by the origin, (0, 0). In the first quadrant of the Cartesian plane, the x- and y-coordinates are positive.
- Pairs of numbers (coordinates) describe the precise location of any point on the plane. The coordinates are enclosed by parentheses as an ordered pair. The first number in the pair describes the horizontal distance from the origin, and the second number describes the vertical distance from the origin. The point (1, 5) is located 1 unit to the right of the origin (along the x-axis) and 5 units above the x-axis. As a translation from (0, 0), the point (1, 5) is right 1 unit and up 5 units.
Note
- An understanding of the Cartesian plane supports work in geometry, measurement, algebra, and data, as well as practical applications such as navigation, graphic design, engineering, astronomy, and computer animation.
Have students draw the first quadrant of a Cartesian plane on a grid and use vocabulary such as perpendicular, x-axis, y-axis, horizontal, and vertical to describe the grid. Have them generate and sequence a series of clues, written as coordinates, that produce a series of points that, when connected, create a shape or image. Have them share their clues with each other and test whether the sequence and coordinate locations are accurate and produce the intended image. Alternatively, have them play strategy guessing games (e.g., a modified version of “Battleship”), where a secret object is hidden along points on a Cartesian plane. Have them pair up and take turns – one student guesses the location of the object by guessing coordinates and the other checks whether the guess is a hit or a miss.
Provide students with the first quadrant of a Cartesian plane with points already plotted. Ask them to describe the movement to get from one point to another. Have them describe positions using Cartesian coordinates and describe both movement and direction (e.g., 4 right and 5 up, or 4→ and 5↑).
E1.3
describe and perform translations and reflections on a grid, and predict the results of these transformations
- predicting the results:
- visualizing the location of an object based on experience with performing translations and reflections on a grid; for example, have students think about and discuss the following:
- Where would you place the line of reflection between rectangles D and E?
- Describe how to transform rectangle B to become rectangle C.
- Describe how to transform rectangle E to become rectangle F.
- Which rectangle is a reflection of rectangle B?
- Which rectangles are reflections of rectangle A?
- visualizing the location of an object based on experience with performing translations and reflections on a grid; for example, have students think about and discuss the following:
- translations on a grid:
- each vertex of the triangle is translated right 5 and up 3; or 5 right, 3 up; or 5→ and 3↑.
- reflections on isometric dot paper:
Reflections in a Horizontal Line | |
Reflections in a Vertical Line | |
- Transformations on a shape result in changes to its position or its size. As a shape transforms, its vertices (points on a grid) move. This explains how transformations involve location and movement.
- A translation involves distance and direction. Every point on the original shape “slides” the same distance and direction to create a translated image. This is called the translation vector. For example, on a grid, a vector could describe each point moving “5 units to the right and 2 units up”. It is a mathematical convention that the horizontal distance (x) is 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. The points on the original image are the same distance from the line of reflection as the points on the reflected image. Reflections are symmetrical.
Note
Online dynamic geometry applications enable students to see how transformations behave in real time and are recommended tools for the study of transformations and movement.
Have students draw a polygon on grid paper and label the vertices. Have them randomly select a direction (right, left, up, down) and distance (1 to 10) for a translation. Ask them to move each vertex of their shape to draw the translated image. Guide them to recognize that, although the shape’s position has changed, it has moved in its entirety the same distance and direction, resulting in a congruent shape (same shape and size). Have them visualize and predict the result of a second translation.
Give students an image from a translation and a description of the translation, which includes direction and distance (e.g., 5 right, 2 up or 5→ and 2↑). Ask them to draw the original image.
To build students’ understanding of reflections, have them draw a polygon on a grid and label the vertices (e.g., A, B, C). Have them use a reflection tool (e.g., a mirror) to reflect the shape in a line of reflection, then draw the reflected shape and label the vertices (A’, B’, C’). Guide them to recognize that, although the shapes remain congruent, the vertices have been “flipped” over the line of reflection, keeping the distance to the line of reflection the same. Using the new image, ask students to draw a new line of reflection and predict where the new image (A”, B”, C”) will be. They can then test their prediction with the reflective tool. Alternatively, this task could be done using an online dynamic geometry application.
Provide students with a polygon on grid paper and a variety of instructions for translations and reflections (e.g., 2 down, 2 left; 4→ and 5↑; reflect vertically in a line of reflection that goes through one vertex of the polygon). Have students randomly select a set of instructions and then predict where the new polygon will be located on the grid. Have them visualize and predict the result of the selected transformation and verify their prediction by carrying out the transformation.