B1.6 Count to 10 by halves, thirds, fourths, fifths, sixths, eighths, and tenths, with and without the use of tools.
Skill: Count to 10 in Increments of One-Half, One-Third, One-Fourth, One-Fifth, One-Sixth, One-Eighth, and One-Tenth With or Without the Use of Tools
To count by a fractional amount is to count by a unit fraction. For example, when counting by the unit fraction one third, the sequence is: 1 one third, 2 one thirds, 3 one thirds, and so on. Activities that require counting by unit fractions can reinforce that the numerator is actually counting units. A fractional count equivalent to the unit fraction makes one whole. For example, when a whole is divided into two equal parts, each unit is called one half; when it is divided into three equal parts, each unit is called one third; when it is divided into four equal parts. When there are four or more equal parts, the suffix th is added to the number of units to obtain fourths, fifths, twentieths, and so on. A student who has mastered the concept of counting by a unit fraction does not need to recreate the image of the "pie" to understand, for example, that six sixths (\frac{6}{6}\) form a whole. Thus, in order to identify the fraction shown below, his reasoning might be, "I count the pieces (1, 2, 3, 4, 5, and 6) and conclude that they are sixths. Since five (5) pieces are shaded, I can state that 5 one sixths or five sixths (\(\frac{5}{6}\)) of the shape are shaded."
This process also leads them to recognize that a fraction can be the sum of unit fractions. Counting of unit fractions is implicitly the addition of unit fractions.
Example
Students who have counted parts of a whole by counting unit fractions can explain that the denominator represents what is being counted (for example, sixths) and the numerator represents the count of units.
It is important for students to understand that the more the whole is divided, the smaller its units.
Example
The fourths of a whole are larger than the tenths of the same whole.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 33-35.
Fractions can describe quantities larger than a whole. For example, 5 thirds means that there is 1 whole (or 3 one thirds) and 2 additional one thirds.
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.
Knowledge: Unit Fractions
Any fraction that has a numerator of 1; for example, \(\frac{1}{2}\), \(\frac{1}{3}\) or \(\frac{1}{4}\)). Every fraction can be decomposed into unit fractions; for example, \(\frac{3}{4}\) is 3 one-fourth units, or \(\frac{3}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\)).
Source: Ontario Curriculum, Mathematics Curriculum, Grades 1-8 , 2020, Ontario Ministry of Education.