Concept #4: Introduction to Unit Fractions and Defining General Fractions as Multiples of the Unit Fraction
Introduction to Unit Fractions
My students need to improve their facility with fraction. In a very clear way students need to see 1that number sense cannot be directly taught. Rather, it emerges from an awareness that there is a connectedness to the subtle relationships among concepts and procedures. This is exactly what my students are lacking. They rarely see the connections between subtle nuances. Doing so will lead to deeper understanding.
Whole numbers with orientation and placement have already been discussed. Hopefully by this point in the unit, my students will confident in placing and comparing integers. As mentioned before, revisits will often take place so that information will be retained throughout the unit and beyond. After the measuring activity using the unit measure, students were left with an incomplete answer to how big their object was. Instead of having a range of distances as outlined in Figure 3, we will need a more refined way to measure to make their measurements more exact. We will start to supply this by studying placement of unit fractions on the number line.
A length is 1/d if n of it makes up the unit length. Students will have to grapple with the unit of partition and the whole. That is, a unit fraction is in some sense a new unit, of which it takes d to make the original unit. This will help my students think about size. Especially, how each numerator and denominator controls the size.
As mentioned before, students are taught to see fractions in terms of an area model. This model needs to be transferred to the number line as a distance. This will erase the misunderstanding that fractions are just slices of pizza or portions of a brownie tray. The area model is important for students to understand. However, the number line is much better at putting these concepts together under one roof. To transition my students’ thinking about fractions to the number line, they will be asked to create area models of fractions with denominators 2-10. They will then crate a picture similar to Figure 11 for each different denominator. Students will have to pay particular attention to the size of their intervals on the number line because they have to be the same size. This activity will also assist my students in learning the placement of these fractions later on in the unit.
Figure 11
Defining General Fractions as Multiples of the Unit Fraction
My students have been taught at the primary level that a general fraction is the product of a whole number and the unit fraction. That is, the general fraction k/d is k times 1/d or k copies of 1/d. Furthermore, my students will see the connection between the definition and the location of each fraction between 0 and 1.
Below, in Figures 12-14, are a few examples of zooming into the unit interval from 0 to 1. Students will be expected to place all unit fraction multiples from halves to tenths. From this point, students should see that the endpoint of the whole number 1 would just be another fraction. For example, 4/4 will be the 1 in Figure 11.
We can expand k times 1/d past the unit interval by letting k become greater than d. This will allow my students to understand that a fraction still is a distance not just a partition of a whole. Using fourths as an example, we can continue 5/4, 6/4, 7/7… for as long as we want. See Figure 15 below. I want my students to see that each unit interval gets partitioned into four intervals of length 1/4, just as the unit interval did. This is highlighted in blue below.
Figure 15
Students will create many examples on their own that mimic Figure 15. They will use their own choice of a fixed denominator and expand their models past the unit interv. Students will do this multiple times in groups and alternate the participants so that students can compare models and rely on each other to make meaning for themselves.
After my students have completed all of their transitional “pizza to number line” pictures, I want them to demonstrate how equivalent fractions can be represented. Students will line up two number lines at the same time. For instance, in order for my students to see why 1/5 = 2/10, they will see that both fractions have the same position on the number line. Drawing from our definition of fractions stated above, five collections of 2(1/10) s fill the unit interval. So, 2(1/10) = 1/5. That is of course what you see if you line up the fractioned number lines from fifths and tenths. Students can also use their own artifacts similar to Figure 15 to represent equivalent fractions past 2.
Furthermore, whole numbers can be represented as a multiple of that unit fraction as well. The fraction 6/3 can represent the whole number two because these lengths line up. The original definition of 6/3 is: 6 x (1/3). Since 3 x (1/3) = 1, by definition of 1/3, we can calculate that 6 x (1/3) = 2 x (3x(1/3)) = 2x1 = 2
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