E1. Geometric and Spatial Reasoning:
describe and represent shape, location, and movement by applying geometric properties and spatial relationships in order to navigate the world around them
create lists of the geometric properties of various types of quadrilaterals, including the properties of the diagonals, rotational symmetry, and line symmetry
An isosceles triangle with one pair of parallel sides and another pair of sides of equal length.
A kite with:
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.
a trapezoid with:
a parallelogram with:
a kite with:
a rectangle with:
a rhombus with:
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:
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:
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.
construct three-dimensional objects when given their top, front, and side views
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.
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.
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
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.
describe and perform combinations of translations, reflections, and rotations up to 360° on a grid, and predict the results of these 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.