B1.5 Use place value when describing and representing multi-digit numbers in a variety of ways, including with base ten materials.
Skill: Using Place Value to Describe and Represent Numbers
The symbolic form of a number represents either its name, a quantity of objects, or a position in an ordered set. The value of each of the digits that make up the number depends on the position it occupies in the number (for example, the digit 1 in a 3-digit number can mean 1, 10 or 100 depending on its position).
To understand the concept of a number, connections must be made between the symbol (for example, 84), the word (for example, eighty-four), the quantity (for example, 84 objects), or the position (for example, the eighty-fourth chair in an auditorium). Numbers are also sometimes used as a simple code, without reference to quantity or position (for example, 4 in a phone number or on a soccer jersey). Adults, who have long understood that the meaning of numbers depends on the context in which they are used, are often unaware of the difficulty children may have in understanding these differences.
To understand the symbolic representation of a number in the base-10 numbering system, students need to recognize that each grouping of 10 is considered an entity called a ten (for example, in the number 21, the digit 2 represents 2 tens). They should also recognize that the same digit (for example, the digit 3) represents a different value depending on its position in the number (for example, in the number 435, the 3 represents 30 while in the number 367, it represents 300) and that the digit 0 in the positional writing of a number accounts for an empty space (for example, in the number 307, the 0 indicates an absence in the tens position).
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 73.
The concept of quantity is involved in understanding the concept of the place value of the digits that make up a number. This value increases successively by a factor of 10 when the digits are read from right to left and decreases by a factor of 10 when they are read from left to right.
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 44.
To better grasp the concept of place value, students need to understand the relationships between numbers and the anchor points 5 and 10. By grouping interlocking cubes or centicubes, students discover the relationships between ones and tens, and tens and hundreds. To develop an understanding of analogous relationships beyond hundreds, the use of base ten materials is preferred. This material includes ones (unit cubes), tens (rods), hundreds (flats), and thousands (thousand cubes). Ten unit cubes can be combined to form a rod, 10 rods can be combined to form a flat, and 10 flats can be combined to form a thousand cube. It is important to note that all of these manipulatives help students develop their understanding of the concept of place value as long as they have the opportunity to use them in well-structured activities. It is not helpful for teachers to simply model the use of the materials (for example, to add two 2-digit numbers) without giving students the opportunity to manipulate them themselves and develop the concept on their own. This would be like having them learn an algorithm by heart without understanding it.
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 63.