E2.2 Solve problems that involve converting larger metric units into smaller ones, and describe the base ten relationships among metric units.
Skill: Solving Problems Associated with Converting Large Metric Units to Smaller Metric Units
In the metric system, conversions are based on an understanding of the relative magnitude or size of metric units and the multiplicative relationships of the positional value system.
Since both the positional value and the metric system are based on a tens system, a metric conversion simply requires moving the digits a number of positions to the left or right of the decimal point. The number of moves depends on the relative size of the units to be converted. If, for example, 1 kilometre equals 1000 metres, then 28.5 km equals 28,500 m, because the digits move three positions to the left.
There is an inverse relationship between the size of a unit and the number of units: the smaller the unit used, the greater the number of units required. It is important to remember this principle when estimating whether a conversion will increase or decrease the number of units.
Note: Although this learning content focuses on converting large units to smaller units, students should understand that a conversion can also be done in the opposite direction using decimal numbers. It is appropriate to expose grade 5 students to decimal measurements.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Skill: Describing the Base Ten Relationships between Metric Units
Inverse Relationship
The number of units required to determine the measurement of an attribute is inversely proportional to the size of the unit used. In other words:
- the smaller the unit used, the greater the number of units required to determine the measurement of the attribute;
- the larger the unit used, the smaller the number of units required to determine the measurement of the attribute.
If, for example, the duration of the same activity is measured once in minutes and a second time in seconds, there will be more seconds than minutes because the second is a smaller unit than the minute. Although the concept of an inverse relationship may seem obvious in this type of situation, it poses a problem for many students who are more accustomed to situations involving direct relationships (for example, the greater the distance to be travelled by car, the greater the duration of the trip). In order to help them understand this concept, teachers should present them with a variety of concrete measurement situations that encourage them to make this connection.
Relationships between the Units of Length, Mass and Capacity
The fact that the various standard units associated with the attributes length, mass , and capacity are part of a decimal system of units is used to establish equivalence relationships between these units; for example, since the gram (g) is 10 times larger than the decigram (dg) and 10 times smaller than the decagram (dag), the following equivalence relationships can be established:
1 g = 10 dg, 1 g = 0.1 dag
Students need to explore several learning situations using concrete materials to develop a good understanding of these equivalence relationships. In order to move easily from one unit to another, they also need to have a good understanding of the concept of inverse relationships.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Mesure, p. 64.