E1.2 Make objects and models using appropriate scales, given their top, front, and side views or their perspective views.
Skill: Constructing Objects and Models to Appropriate Scales, Using Their Front, Side and Top Views, or Various Perspectives
Two-dimensional drawings, if they are accurately constructed and include enough information, can be used to reproduce actual-sized objects or scaled models in three dimensions.
Two-dimensional 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.
Two-dimensional 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.
Image Top view: 5 blocks are placed to form an upside down "L". Front view: 5 blocks are placed so as to form an "L". Side view: 5 blocks are placed to form an "L" with the opening to the right.
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.
Image Isometric projection drawn on dotted paper. The solid has 7 cubes. Oblique projection of a solid with 7 cubes, drawn on dotted paper.
Source: The Ontario Curriculum, Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Scale
A scale is a ratio that compares dimensions in the drawing to the actual dimensions of what is being drawn. 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.
Source: The Ontario Curriculum, Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Front View
The front view (elevation) is a “flat drawing” without perspective. This view is used in engineering drawings to ensure the accuracy of three-dimensional reproductions.
Front view of the three-dimensional object:
Source: adapted and translated from En avant, les maths, 7e année, CM, Sens de l'espace, p. 2.
Knowledge: Side View
The side view (elevation) is a “flat drawing” without perspective. This view is used in engineering drawings to ensure the accuracy of three-dimensional reproductions.
Side view of the three-dimensional object:
Source: adapted and translated from En avant, les maths, 7e année, CM, Sens de l'espace, p. 2.
Knowledge: Top View
The top view (plan) is a “flat drawing” without perspective. This view is used in engineering drawings to ensure the accuracy of three-dimensional reproductions.
Top view of the three-dimensional object:
Source: adapted and translated from En avant, les maths, 7e année, CM, Sens de l'espace, p. 2.
Knowledge: Perspectives
Perspective Drawing
A perspective drawing shows three views (top, front and side views) within the same illustration. The two types of perspective drawing are isometric and cabinet projection.
Isometric Projection
The isometric projection represents an object in perspective in a “corner view”, so that the main edges of the object (namely, the three dimensions of the object) form equal angles. In an isometric projection, a scale is applied consistently to all dimensions of the object, for example, 1 cm = 2 cm for base, height and depth.
Cabinet Projection
The cabinet projection represents an object in perspective from a “direct view” of one of its sides, so that depth is represented by oblique lines (namely, at an angle). In a cabinet projection, the “depth” scale is half the “base and height” scale (for example, if the scale for base and height is 1:2, or 1 cm = 2 cm, then the scale for depth is 0.5:2, or 0.5 cm = 2 cm).
Source: translated from En avant, les maths!, 7e année, CM, Sens de l’espace, p. 2-3.