E2.3 Measure and draw lengths in centimetres and metres, using a measuring tool, and recognize the impact of starting at points other than zero.
Skill: Measuring and Tracing Lengths in Centimetres and Metres
Determine Length Using a Measuring Tool
The object is shorter than the tool:
- Zero as a starting point: the measurement is the mark on the tool that is at the end of the object being measured.
- Starting point greater than zero: the measurement is the difference between the mark on the tool and the distance between zero and the starting point.
The object is longer than the tool: move the instrument repeatedly along the length of the object or the distance between two points (without spacing or overlapping).
- All measuring tools, such as rulers and tape measures, eliminate the need to place and count units. A measuring tool repeats the unit without creating overlaps or gaps, and it is graduated (has a scale) to keep track of the units.
- A scale, including the scale of a ruler, starts at the beginning of the first unit, which corresponds to 0 because no unit has yet been measured. At the end of the first unit, the scale corresponds to 1, because the length of a full unit has been measured. The numbers on the scale continue to increase as each complete unit is counted.
- Once the 0 is aligned with the end of an object, the scale on the measuring tool keeps an accurate count of the units; however, the length can be measured from any point, as long as the scale is placed according to the starting point and the number of units accurately represents the length of the object.
Source: The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
The accuracy of the measurement depends on how well the measuring tool is used. If the tool is not used correctly, the measurement will not be accurate; it will be greater or less than the quantity being measured. To help students understand the correct way to use a given measuring tool, teachers can first model its use and identify some of the difficulties that some students may encounter. In the following, some details are presented about how to use certain tools. To use a ruler correctly, one must:
- align one end of the object with the zero point or the beginning of the graduations;
- make sure that the line of sight on the other end of the object is at a 90 degree angle to the ruler (see Parallax Error below);
image A shoe is measured, it is well placed at the end of the ruler. Snapshot look at the measurement are shown. The first one is too far to the left it is "too long". The second is well aligned. The third is too far to the right is "too short". Dotted lines represent the 3 shots. The first and the third are crossed out with a red cross
- count, on the ruler, the units that go from one end of the object to the other.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Mesure, p. 95.
Skill: Recognize the Consequences of Choosing a Starting Point Other than Zero
The distance between two ends or the length of an object remains constant, no matter where the count starts on the scale. The measurement is the number of units from the beginning of the length to the end of the length, namely between the two ends of a length.
Source: The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
Structure Associated with Units
The concept of structure associated with units refers to the way in which these units are organized to determine the size of a given space, whether one-, two-, or three-dimensional. It is closely related to the concept of iteration and is used to determine a measurement. The example below clarifies the meaning of this concept using a real-life situation.
Example
To determine the length of an object, units must be juxtaposed in a one-dimensional space, without gaps or overlaps, so that they cover the distance between two ends of the object. The teacher can check for understanding of this concept by asking students, for example, to estimate the length of a pencil and then determine the measurement using a segment of a ruler that does not start at 0.
Students who do not understand the structure of units on a ruler may have difficulty using it as a measuring tool. Many students tend to simply read the number on the ruler that corresponds to one end of the pencil (for example, 11 cm), without considering the actual number of units of length that can be counted between the two ends. In contrast, students who understand this concept can visualize a linear unit of measurement (the centimetre) laid out several times along the length of the pencil, from one end to another, and determine the number of times they see it. The resulting measurement (7 cm) then becomes meaningful.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Mesure, p. 55.
Knowledge: Measuring Tools
Linear Measuring Tools
- ruler, tape measure
- ruler or metre stick made with 1 cm interlocking cubes
Source: The Ontario Curriculum, Mathematics, Grades 1-8, Ontario Ministry of Education, 2020.
Knowledge: Standard Units (Centimetres and Metres)
Units are said to be standard because they are commonly used by a large number of people and therefore have the advantage of making the communication of the measurement clear. The standard units chosen must also reflect the attribute to be measured. For example, the centimetre or metre are used for measuring lengths.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Mesure, p. 86.