Concept #2: Placing Positive/Negative Integers on the Number Line and Introduction of Numbers as Vectors.
Placing Positive/Negative Integers on the Number Line
Every integer has an exact location on the number line. Whether the integer to be located is positive or negative, the idea is the same. To locate a number on the number line, my students will simply count the unit intervals from zero in the appropriate direction, as indicated by the sign of the number. This concept will not require a whole lot of time nor attention.
Figure 4 shows locating 4 on the number line. Almost all of my students will be able to do this before entering my class. The next series of steps will be taken to help increase fluency of placement of whole numbers.
Figure 4
In order for this concept to be relevant and grade appropriate to my students, missing or incomplete information will be provided to them. This will be delivered in two types. First a number line with unlabeled partitions and labeled endpoints will be outlined below. Students will be asked to identify the point with the arrow. Answer to below: -16.
Figure 5
Second, students will be asked to fill in the incomplete sections of the number line. This will increase familiarity of the succession of integers and build off of counting principles that were learned at an earlier age. Through practice of this kind, my students will better understand placement and order of integers.
Figure 6
I will adjust the remainder of the activities when appropriate. For instance, I could give my students the same number line and ask them to locate the final integer 3 units in the positive direction from 10. Or I could use the terminology 3 greater than 10. I would like to change the terminology and give prompts that change sign as well. Also, changes in scale might be useful when these are revisited later in the year. These multiple forms of number placement will be a continuing idea that could work its way into a Math Talk. Further extensions and examples of number lines with missing information can be found in the Appendix section labeled A2.
Introduction of Numbers as Vectors
Before moving forward with comparing and ordering whole numbers, I need to lay the foundation of representing whole numbers as vector arrows on the number line. This needs to be established now for operations in the next unit. Numbers have direction and magnitude. Figure 7 is a representation of these vector arrows pointing to both positive and negative 4. The base of the arrow will start at the origin and end at the number’s location. This will help students further understand that numbers are a distance from zero, but not simply a length: an oriented or directed distance, or vector.
Figure 7
Orientation is something that confuses my students a great deal. Using vector model prompts similar to Figure 7 and asking my students to express signed numbers as vectors, they will understand that the measurement equivalent of sign is orientation. More vector placements are found in the appendix.
Comments: