Concept #5: Placing Fractions on the Number Line
To get my students to understand how to place fractions, they will go through a series of placement prompts. First, I want them to work together to place multiples of unit fractions between 0 and 1. The general fraction was already defined above and will be practiced by the students using the Table Mats activity outlined in the appendix. Below in Figure 16, students will be asked to decide what fractions the different colored arrows represent. This will be repeated throughout the activity using a variety of different denominators and arrow locations.
Figure 16
A more difficult problem set will be similar to Figure 17 and Figure 18. This will be useful if a group is progressing quickly through the problem set or are in need of more challenging material.
Figure 17
Figure 18
Similar problems to Figure 18 will have my students use reasoning to justify what each letter represents. These types of problems can be done in a math journal entry or even a Math Talk. I want my students to do this same procedure as before; however there will be some missing information just as there was in the integer placement section of this unit. Missing information will promote justification of what fraction each arrow represents and discussions surrounding these decisions. The students will then create their own number lines with missing fractions on them and trade them with a partner to complete. This will allow for student generated problems that can be later used on a future Math Talk or assessment.
The students will now have examples locating fractions on the number line using approximation. I will use that to approximate placement of 13/5 to illustrate how I can teach a general placement of this improper fraction. First, we know that the nearest whole number can be determined by dividing 5 into 13, getting 2 with a remainder of 3. The leftover will be described as follows: 2 x 5 = 10 and 3 x 5 = 15, so 13/5 = 2 + 3/5 and must be between 2 and 3. As outlined before, the partitions between 2 and 3 will match the denominator and we can split the interval evenly into five parts. The leftover will have to be placed between these whole numbers.
Figure 19
Another strategy for students to use is simply lay off 13 copies of 1/5. As mentioned above in Figure 15, this would be an easier method. I would want my students to use both methods during their explanations to the problem sets. Students will be asked to place a series of improper fractions and mixed numbers located in the Appendix labeled A2.
Orientation and Negative Sign
The above-mentioned activities with fractions have all been on the right side of the number line, or only in the positive direction. Now we need to make use of the entire number line, just as we did for whole number placement. We will simply reflect the positive rational number across zero to obtain the negative number. Reflections show numbers and their opposites. I would want my students to realize that the distance from zero is the same for both some number d and –d. This can be done as an activity as well with my students. They will be asked to reflect a series of numbers across the zero and can arrive at this conclusion on their own.
My students have a very difficult time placing the negative sign in front of rational numbers. I need to make a series of prompts where negative rational numbers will be seen three different ways. For instance, the fraction -1/3 might be written as (-1)/3 or on rare occasions 1/(-3). Students will see a mix of these signs when they discuss the prompts in class.
Lastly, students will be given a series of fractions with different denominators and be asked to order them. These problem sets can be found in the appendix.
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