Non-Standard Measurement

Learning experiences in measuring with non-standard units become opportunities to emphasize important concepts with students:

• Initially, students should receive a sufficient quantity of non-standard units, enough to match, cover, or fill the object being measured completely, with- out gaps or overlays. Counting the non-standard units stresses the idea that measuring involves finding the number of units that represent a measure.

• Later, students use one unit and move it repeatedly to measure. For example, students might measure the length of a table by moving a pencil in consecutive positions along the length of the table and keeping track of the number of pencil lengths that match the length of the table. This process of unit iteration is important when students learn to use a ruler to measure objects and distances that are longer than the ruler.

• Students benefit from opportunities to use more than one kind of material to measure the same object. Using a variety of materials helps students under- stand the relationship between the size of the unit and the number of units needed (e.g., a greater number of small units than large units are needed to cover a surface).

• Students also learn from experiences in using the same non-standard unit to measure different objects. For example, teachers could challenge students to find the perimeter of a book and of a large carpet, using paper clips. This activity would allow students to observe that a paper clip is an appropriate non-standard unit for finding the perimeter of a small object, such as a book, but an inappropriate unit for measuring a large object, such as a carpet.

Standard Units of Measurement

Once students have had opportunities to measure objects, using non-standard units, they begin to realize the need for standard units. For example, they might observe that the area of a book cover can be expressed as both 60 square tiles and 18 square cards. They also realize that it is difficult to compare objects if they are measured with different units (e.g., a pencil having a length of 8 paper clips and a pencil having a length of 5 erasers). Teachers should help students understand that standard units:

• provide consistent units for measuring the same attribute of different objects;

• are needed to communicate measurements effectively;

• can be used to compare the measurements of two or more objects.

When students are introduced to standard units, the use of non-standard units should not be eliminated completely. Learning activities that link non-standard units with standard units (e.g., measuring a length with both centimetre cubes and a centimetre ruler) help students make connections between concrete mate- rials and measurement tools.

Congruency

Congruence is a special relationship between two-dimensional shapes that are the same size and the same shape. An understanding of congruence develops as students in the early primary grades explore shapes and discover ones that “look exactly the same”. Students might superimpose congruent shapes to show how one fits on top of the other. By Grade 3, students should use the term “congruent” and be able to describe “congruence” by referring to matching sides and angles of shapes.

Students encounter concepts related to congruence in many areas of geometry. They observe, for example, that:

• the faces of a three-dimensional figure may be congruent (e.g., the faces of a cube are congruent squares);

• many geometric patterns, especially tiling patterns, are composed of congruent shapes;

The concept of congruence can also be applied to three-dimensional geometry. Three-dimensional figures are congruent if they are identical in form and dimensions. To verify whether two figures are congruent, students can match the figures’ faces to determine whether the parts of the figures are the same size and the same shape.