E1.3 Use scale drawings to calculate actual lengths and areas, and reproduce scale drawings at different ratios.
Skill: Calculating Actual Lengths and Areas from Scale Drawings
Scale drawings enable something very large or very small (microscopic) to be represented on a page or a screen. They are useful for visualizing, comparing, and calculating dimensions when reading a map, following instructions, or designing a plan.
Scale drawings are similar to the actual object or space. This means that the angles in the drawing and the corresponding angles in the actual object are congruent, and their corresponding lengths are proportional. A scale on the drawing explicitly describes this proportion and can be written in words, as a fraction, as a ratio, or as a graphical representation (for example, a bar scale).
Distances, areas, volumes, and angles can all be measured indirectly by referring to the measurements in the drawing. Top, front, and side views (plan and elevation drawings) are most reliable for calculating actual dimensions.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
On a scale drawing, the ratio of the object as depicted on the drawing to the actual value of the dimensions of the object is given. For a distance of 40 km on a map with a scale of 1:1 000 000, 1 cm represents 1 000 000 cm or 10 km. To show 40 km, the distance of the line must be 4 times longer, or 4 cm.
For a length of 0.2 mm with a scale of 1:\(\frac{1}{100}\), 1 cm represents 0.01 cm, which is equivalent to 0.1 mm. A distance that is twice as long (0.2 mm), is represented by a line that is 2 times longer, or 2 cm.
Skill: Reproducing a Drawing at a Different Scale
When reproducing a drawing at a different scale, a student should make a larger drawing if the scale is smaller, and a smaller drawing if the scale is larger.
For example, to go from a 1:20 scale to a 1:40 scale, the dimensions of the drawing will be twice as small, because the scale ratio has doubled.
Grid paper is useful for drawing to scale. Choosing the scale of the grid paper is the first step in designing a scale drawing, depending on whether one is making a smaller-scale (reduction scale) or a larger-scale (enlargement scale) drawing.
In the case of a smaller-scale drawing, the grid of the actual object will be larger than the grid of the scale drawing, since we want to reduce the image of the real object (for example, a distance of 10 km between two cities is represented by a length of 1 cm on a road map). The scale is therefore 1:1 000 000.
Inversely, in the case of a larger-scale drawing, the grid of the real object will be smaller than the grid of the scale drawing, since we want to enlarge the image of the real object (for example, a dimension of 0.1 mm on an electronic circuit is represented by a length of 1 cm on an industrial drawing). The scale is therefore 1:\(\frac{1}{100}\).
Source: adapted from The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Scale
A scale is a ratio of size that compares the actual dimensions to the dimensions of the drawing. Depending on the type of drawing, angles may or may not be represented accurately.
A scale drawing can be reproduced at different scales by finding the unit ratio if it is not provided.
Smaller scales show a large area with a small amount of detail. For example, 1:2 000 000 means 1 cm in the drawing represents 2 000 000 cm, or 20 km.
Larger scales show a small area with greater detail. For example, 1:2 means that 1 cm in the drawing represents 2 cm.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.