E2.3 Use non-standard units appropriately to estimate, measure, and compare capacity, and explain the effect that overfilling or underfilling, and gaps between units, have on accuracy.
Skill: Estimating Capacity Using Non-standard Units
In the primary grades, fundamental concepts are essential to the development of the understanding of measurable attributes. Iteration is one of the fundamental concepts.
Fundamental Concept: Iteration
The student who understands this concept realizes that it is possible to estimate the capacity of a container by visualizing the action of repeatedly placing, in an orderly fashion, a single non-standard or standard unit of measurement in such a way as to fill as much of the space inside the container as possible.
Example
”How can Celeste estimate the capacity of this cup in sugar cubes?“
We can first visualize the action of placing a sugar cube repeatedly at the bottom of the cup so as to cover it. This corresponds to the number of sugar cubes in a tier. We then visualize the action of placing sugar cubes repeatedly so as to cover the distance of the height of the cup. We estimate the number of sugar cubes by adding the number of cubes visualized in the first tier for each of the other tiers, or by multiplying the number of cubes per tier by the number of tiers [height]. It is sometimes difficult for students to develop benchmarks for this attribute since capacity can be expressed in grams (for example, the capacity of a cereal box), in millilitres or in litres (for example, the capacity of a container of juice) or in cubic centimetres (for example, capacity of a storage box). It can also be expressed in terms of the number of identical objects that a container can hold (for example, a cylindrical container that has a capacity of four tennis balls).
Source: translated from Fiche de la maternelle à la 3e année Attribut capacité, p. 2.
Benchmarks Associated with the Attribute of Capacity
Capacity refers to the amount container can hold. It is sometimes difficult for students to develop benchmarks for this attribute since the capacity of a package can be expressed in grams (for example, the capacity of a cereal box), in millilitres or in litres (for example, the capacity of a container of juice) or in cubic centimetres (for example, capacity of a storage box). It can also be expressed in terms of the number of identical objects that a package can contain (for example, a cylindrical container that has a capacity of four tennis balls). It should also be noted that, since liquids take the shape of the container in which they are placed, we cannot form a mental image of a capacity of 1 litre without taking the container into account. In elementary school, teachers can encourage students to use various items found at home to create benchmarks capacity (for example, milk carton, toy box).
Students need to complete several estimation activities in the classroom involving non-standard or standard units in order to develop a sense of measurement, so it is important that teachers plan learning experiences and formal instruction focused on specific estimation strategies.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Mesure, p. 16-17.
Skill: Measuring Capacity Using Non-standard Units
Fundamental Concept: Iterating Units
Iterating units of measurement is the careful placement of a number of standard objects as units to fill a three-dimensional space to determine a capacity.
Example 1
Two students determine the capacity of the container by successively pouring in several eye droppers with the same amount of liquid until the container is full. Students count the number of eye droppers poured in.
Example 2
A student determines how many cups of popcorn each bowl can hold by filling it cup by cup and counts the number of cups.
Iterating units of measurement quantifies the measurement of any attribute of an object, usually in terms of non-standard units. This strategy is particularly useful in helping students develop their understanding of the attributes length, area, and capacity since the measurement of each attribute is expressed in terms of concrete non-standard objects rather than more abstract standard units such as centimetres and metres.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Mesure, p. 91-92.
If units of different sizes are used to properly fill an object, each unit is counted and treated separately.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Skill: Comparing Capacities
Comparing and ordering involves comparing two objects based on the same attribute. We compare the measure of an attribute of two objects either by direct comparison or by indirect comparison.
From a very young age, children compare the measurement of an attribute of two objects by direct comparison (for example, comparing directly by pouring the contents of one container into another container.) They then communicate the result descriptively rather than quantitatively (for example, ”My glass can hold more milk than yours.“).
When it is difficult or impossible to compare two objects directly on the same attribute, an indirect comparison can be made by comparing the measure of the attribute for each of the objects to a third measure (for example, comparing indirectly by pouring the contents of the two containers, one after the other, into a third container, and emptying the third container between the two pours). The use of units allows comparison questions (for example, ”Which one has more?“) to be replaced by measurement questions (for example, ”How much does it have?“, ”How much more does it have than the other?“).
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Mesure, p. 87-89.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Fundamental Concept: Transitivity
A student who understands this concept can establish a relationship of equality or inequality between the capacity of three containers by comparing the capacity of one of the containers with that of the other two.
Example
Maya's pencil case can hold as many pencils as Melanie's. However, Melanie's case can hold fewer pencils than her friend Christian's case. Who has the pencil case with the largest capacity? Who has the smallest? (Since Maya's case has the same capacity as Melanie's case and Christian's case has a larger capacity than Melanie's case, Christian's case also has a larger capacity than Maya's case. Christian's pencil case has the larger capacity and Melanie's and Maya's cases have the smaller capacity.)
Source: translated from Fiche de la maternelle à la 3e année Attribut capacité, p. 2.
Skill: Explaining the Effect of Overfilling, Underfilling, and Spacing on Measurement Accuracy
The capacity of a container is the maximum amount of a given substance that it can hold. When the given substance completely fills the container, the capacity is equal to the interior volume of the container.
Source: translated from Curriculum de l’Ontario, Programme-cadre de mathématiques de la 1re à la 8e année, 2020, Ministère de l’Éducation de l’Ontario.
Structure Associated with Units
The concept of structure associated with units refers to the way in which these units are organized to determine the magnitude or size of a given space, whether one-, two-, or three-dimensional. It is closely related to the concept of iteration and is the basis of the juxtaposition strategy used to determine a measurement.
Source: translated from Guide d’enseignement efficace des mathématiques, de la maternelle à la 3e année, Mesure, p. 57.
Overfilling or underfilling (either of the object used for measurement, such as a spoon, or of the container being measured) does not give an accurate measurement of capacity. Also, the choice of non-standard units affects the accuracy of the capacity measurement. For example, the choice of marbles gives a less accurate measurement of the capacity of a container than sand, since marbles cannot measure the maximum capacity due to the spaces they leave between them.
Knowledge: Attribute - Capacity
The capacity of a container refers to the maximum amount it can hold.
Note: It is important to distinguish between capacity and volume. Capacity indicates the quantity that a container can hold, while volume is the space occupied by an object.
Source: translated from Fiche de la maternelle à la 3e année Attribut capacité, p. 1.