E1. Geometric and Spatial Reasoning
Specific Expectations
Geometric Reasoning
E1.1
sort and identify two-dimensional shapes by comparing number of sides, side lengths, angles, and number of lines of symmetry
- types of shapes that can be included in a sort:
Concave Shapes | Convex Shapes |
Shapes with Only Curves | Shapes with Curves and Straight Lines |
Regular Shapes (Shapes with Equal Side Lengths and Equal Angle Measures) | Irregular Shapes (Shapes with Different Side Lengths and Different Angle Measures) |
Shapes with Equal Side Lengths and Different Angles | Shapes with Different Side Lengths and Equal Angles |
Shapes with Symmetry | Shapes without Symmetry |
- comparing shapes:
- number of sides and number of vertices or angles:
- 3 sides: triangle
- 4 sides: quadrilateral
- 5 sides: pentagon
- 6 sides: hexagon
- 7 sides: heptagon
- 8 sides: octagon
- length of sides:
- Are all sides equal?
- Is at least one set of opposite sides equal?
- Are none of the sides equal?
- size of angles:
- Are all angles right angles?
- Is there one, two, or no right angle?
- Is the angle bigger or smaller than a right angle?
- parallel sides:
- Are there no, one, two, or more pairs?
- lines of symmetry:
- Yes or no?
- number of sides and number of vertices or angles:
- Two-dimensional shapes have geometric properties that allow them to be identified, compared, sorted, and classified.
- Geometric properties are attributes that are the same for an entire group of shapes. Some attributes are relevant for classifying shapes. Others are not. For example, colour and size are attributes but are not relevant for geometry since there are large rectangles, small rectangles, blue rectangles, and yellow rectangles. Having four sides is an attribute and a property because all rectangles, by definition, have four sides.
- Two-dimensional shapes can be sorted by comparing geometric attributes such as:
- the number of sides;
- the number of angles and whether the corners (vertices) are square;
- the number of equal (congruent) side lengths and how those sides are arranged;
- whether the sides are curved or straight;
- whether there are any parallel sides (sides that run side by side in the same direction and remain the same distance apart);
- the number of lines of symmetry.
- Each class of two-dimensional shapes has common properties, and these properties are unaffected by the size or orientation of the shape.
- A line of symmetry is an imaginary “mirror line” over which one half folds onto the other. Line symmetry is a geometric property of some shapes. For example, rectangles have two lines of symmetry and squares have four lines of symmetry. Some triangles have three lines of symmetry, some have one line of symmetry, and some have no lines of symmetry.
Have students cut out and sort a wide variety of two-dimensional shapes, including concave and convex shapes, shapes that have congruent (equal) side lengths but different angles, shapes with curved and straight sides, shapes with different side lengths, and shapes with different numbers of sides. (See BLM: Grade 2 E1.1 for a sample of assorted shapes). Have them identify their sorting rules, supporting them in using the correct vocabulary. Guide them in noticing that the orientation of a shape does not change it; for example, an “upside-down” triangle is still a triangle because it still has three sides and three angles. Have them re-sort the shapes using a different sorting rule to emphasize that the sort depends on the attribute being sorted for.
Have students play “Which One Doesn’t Belong?” by showing them four shapes and asking them to identify one shape that does not belong. Have them share their reasoning and their sorting rule. Emphasize that shapes have many sortable attributes and that the same group of objects can be sorted in different ways. Draw out the idea that whether something is included in the sort depends on the sorting rule.
Show students a two-dimensional shape on a geoboard. Place students in groups of four. Have the first student recreate the shape on their group’s geoboard, make one change to the shape, and then pass the geoboard to the next student in the group. Repeat until all four group members have made a change. Have students in the group identify the resulting shape and describe how it is different from the original shape and how it is the same.
Provide students with a variety of shapes. Have them sort these shapes based on whether they think they are symmetrical or not. Then ask them to visualize (predict) the line of symmetry, share their prediction verbally with peers, and verify by using their own strategy, such as paper-folding or using a reflection tool (e.g., a mirror). Have students adjust their sort as necessary.
Composing a Shape Using Pattern Blocks | Decomposing a Triangle to Make Other Shapes |
- Two-dimensional shapes can be combined to create larger shapes (composing) or broken into smaller shapes (decomposing). All shapes can be decomposed into smaller shapes. The ability to compose and decompose shapes provides a foundation for developing area formulas in later grades.
- If a two-dimensional shape is broken into smaller parts (decomposed) and reassembled in a different way (composed), the area of the shape remains the same even though the shape itself has changed. This is known as the property of conservation.
Have students cut out a two-dimensional shape from a sheet of paper. Have them fold the paper shape once or multiple times and identify the decomposed shapes created by the folds (i.e., the shapes within the shapes). Next, have them cut out the decomposed shapes and rearrange them to compose a new shape. Ask them if the area of the new shape is the same as, greater than, or less than that of the original shape. Have them share their thinking within a small group, and then discuss as a class.
Give students a collection of pattern blocks or tangram pieces, and ask them to build a composite shape with all the pieces. Give them a second collection of the same pattern blocks or tangram pieces and have them build a different composite shape using all the pieces. Ask them how the areas of their two shapes compare, and have them explain their thinking.
Have students describe and demonstrate two-dimensional shapes and composite shapes that can be made by combining different pattern blocks or tangram puzzle pieces. For example, they might visualize and verbalize (talk about) which pattern blocks would fit together to fill an outline of a shape and then verify the different combinations that could make up that same area. Or they could use the pieces of a tangram to show various ways to make half a square. Have them compare the area of the shapes in both their “composed” form (the created shape) and their “decomposed” form (the pieces) and explain how they know that the areas are the same.
E1.3
identify congruent lengths and angles in two-dimensional shapes by mentally and physically matching them, and determine if the shapes are congruent
- identifying congruent shapes:
- mentally matching lengths and angles
- directly matching lengths and angles
- indirectly comparing lengths and angles
- congruent two-dimensional shapes:
- conventions for communicating congruent lengths and angles using markings:
- Congruent two-dimensional shapes can fit exactly on top of each other. They have the same shape and the same size.
- Checking for congruence is closely related to measurement. Side lengths and angles can be directly compared by matching them, one against the other. They can also be measured.
- Non-congruent shapes can have specific elements that are congruent. For example, two shapes could have a congruent angle or a congruent side length (i.e., those elements match), but if the other side lengths are different, or the angles between the lengths are different, then the two shapes are not considered congruent.
Note
- Visualizing congruent shapes – mentally manipulating and matching shapes to predict congruence – is a skill that can be developed through hands-on experience with shapes.
A pentomino is a shape made of five squares that share a full side or sides with one other square. Have students, in pairs, draw all the different pentominoes on 2 cm × 2 cm grid paper. Tell them that there are 12 pentominoes to find. As they draw their pentominoes, they should eliminate ones that are congruent, so they are left with only unique pentominoes. After they have completed the task, discuss the strategies they used to ensure that their pentominoes were unique.
Have students make two congruent shapes on a geoboard. Ask them to change either an angle or a side length on one of the shapes. Now ask them to compare the two shapes, identifying the angles and the side lengths that are still equal, and describe what effect the change had on the shape’s other angles or side lengths (or even the number of sides). Discuss how the two shapes are no longer congruent.
Location and Movement
E1.4
create and interpret simple maps of familiar places
- simple maps:
- a concrete map of a classroom using blocks to represent the furniture and its arrangement in the classroom
- a sketch of the layout of a classroom
- a grid that overlays a picture, where the locations of objects are marked on the intersections of the grid lines
- A three-dimensional space can be represented using a two-dimensional map by noting where objects are positioned relative to each other. A map provides a bird’s-eye view of an area.
- Words such as above, below, to the left, to the right, behind, and in front of can orient the location of one object in relation to another.
- A grid adds a structure to a map. It helps to show where one object is in relation to another and can be a guide to determining distances and pathways. The location of objects on a map grid corresponds to an actual or virtual grid overlaid on the corresponding three-dimensional space.
- Sometimes location on a grid is described by the intersection of the grid lines – this provides a precise location. Sometimes location on a grid is described by the space or region between the grid lines – this describes a more general location. It is important to be clear about which approach is used.
- Labelling a grid, with either numbers or letters, helps to describe locations on the grid more accurately.
Provide students with a simple map of the school, and have them identify the locations of places such as the main office, their classroom, washrooms, emergency exits, and the gymnasium.
Have students draw a map of the classroom on grid paper, showing the locations of the different pieces of furniture. Align the grid with the floor tiles, or create a grid on the floor by measuring with rulers, metre sticks, or measuring tapes. Have students demonstrate that objects on the map are in the same relative location as they are in the classroom. Support students in noticing how labels on a grid (e.g., A1) can make it easier to describe a location.
Have students make a concrete map of their playground showing the locations of the different features (e.g., play structures, trees, basketball court). Next, have them sketch the layout of the playground on a simple map, identifying the locations of the features on their map. Now give them an aerial photo of the playground from an online mapping site. Have them compare the two maps to find familiar locations, and discuss what they notice and what adjustments they need to make to their maps.
E1.5
describe the relative positions of several objects and the movements needed to get from one object to another
- describing relative positions:
- …is above/below the…
- …is over/under the…
- …is beside/inside/outside the…
- …is to the right/left of the…
- describing movements:
- move right/left [...] steps
- move up/down [...] steps
- A three-dimensional space can be represented on a two-dimensional map by noting where objects are positioned relative to each other. A map provides a bird’s-eye view of an area.
- Words such as above, below, to the left, to the right, behind, and in front of can orient the location of one object in relation to another (direction). Numbers describe the distance one object is from another.
- A combination of words, numbers, and units are used to describe movement from one location to another (e.g., 5 steps to the left).
- The order of these steps is often important when describing the movement needed to get from one object to another.
Provide students with a grid map, or have them use the map they developed in E1.4, Sample Task 2. Ask them to describe the relative locations of objects and write directions or code to describe the movements needed to get from one location to another on the map. They could also write directions to a mystery object and share them with a peer.
Have students describe the relative positions of several objects on a grid, and the movements needed to get from one spot to another on the grid. For example, for the grid below, ask questions such as:
- Where is the blue triangle located?
- If I start at E5 and go up 4 and then right 2, what shape will I land on?
- How might I move the rectangle at B2 to J6 without bumping into any other shapes?
Have students write code to program a small robot (or digital image or classmate) to reach a predetermined destination or to move through a maze. Ask them to read a partner’s code and predict where their partner’s robot will end up. After students have made their predictions, have them run their programs to test the predictions. Discuss what they have learned from their experience of making and testing predictions.