B2.8 Represent the connection between the numerator of a fraction and the repeated addition of the unit fraction with the same denominator using various tools and drawings, and standard fractional notation.
Activity 1: Counting in Unit Fractions
One person on the team chooses a unit fraction (for example, \(\frac{1}{5}\))
The 1st person says “1 one-fifth”, the 2nd person says “2 one-fifth”, the 3rd person says “3 one-fifth” and so on…
When a person gets to a whole number (for example, "5 one-fifth"), that person should stand up and say the equivalent whole number.
Be sure to count well beyond 1.
The student who is standing when all other students are sitting wins the game.
Activity 2: In Kindergarten
Example 1
In a kindergarten class, the students in your class help younger students count to 10 using objects. You decide to use small plastic animals and place them on a ten frame. The teacher circulates and asks you to analyze the items in this set as fractions.
Strategy
Using a Set Model
The frame is complete with 10 animals, which represents the whole. Each dog or cat represents \(\frac{1}{10} \) animals. There are 3 dogs. I count them in unit fractions: “1 one tenth”, “2 one tenth”, “3 one tenth”. So \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} \) of the set of animals. According to this fraction, the 3 is the numerator and it represents the number of dogs compared to the total number of animals. According to this fraction, 10 is the denominator and it represents the number of parts by which the whole is divided.
There are 7 cats. I count them in unit fractions: “1 one tenth”, “2 one tenth”, “3 one tenth”, “4 one tenth”, “5 one tenth”, “6 one tenth”, “7 one tenth”. So, \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \ frac{1}{10} + \frac{1}{10} = \frac{7}{10} \) of the set of animals. According to this fraction, the 7 is the numerator and it represents the number of cats compared to the total number of animals. According to this fraction, 10 is the denominator and it represents the number of parts by which the whole is divided.
Example 2
Kindergarten students practice colouring boxes on a rectangular strip by choosing colours of their choice. Here is a reproduction of one student sample.
How could you describe the boxes using fractions?
Strategy
Using an Area Model
There are 5 squares in all; that is 2 green squares and 3 squares that are not coloured. There are 2 green squares out of a total of 5 squares in all, so \(\frac{1}{5} + \frac{1}{5} = \frac{2}{5} \). In the fraction \(\frac{2}{5} \), the 2 is the numerator. It represents the number of green squares compared to the total number of squares. The 5 is the denominator and it represents the total number of equivalent squares.
There are 3 squares without colour out of a total of 5 squares in all, so \(\frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{3 }{5} \). In the fraction \(\frac{3}{5} \), the 3 is the numerator. It represents the number of uncoloured squares compared to the total number of squares. The 5 is the denominator and it represents the total number of equivalent squares.
Source: En avant les maths! 3e année, CM, Nombres, p. 3-5.