USING CONTEXT TO UNDERSTAND THE RATIONAL NUMBER SYSTEM
By: Beau Bailey and Erika Silva
The introduction of the rational numbers in Grade 6 requires that we broaden our outlook on the number line when making comparisons of numbers and quantities. Simply explaining to students that the numbers to the left of zero are negative values is not enough to develop true understanding of why these numbers are mathematically necessary. In our world we experience applications of the number line on both sides of zero, and these applications will aid our students in understanding what those negative values represent and how we can use them mathematically.
Ordering magnitudes, for a sixth grade student, is really nothing new once the magnitudes have been meaningfully determined. They have previously compared numbers and quantities by their magnitudes as those numbers and quantities represented very specific contexts, such as the number of different animals in the zoo, or the heights of their classmates. However, in many real-world and mathematical situations, comparison of magnitudes is insufficient. For example, let’s compare deposits into an account; if we compare the magnitudes of these deposits we will find the greatest deposit made. Similarly, we can compare withdrawals made from an account; if we compare the magnitudes of these withdrawals we will find the greatest withdrawal from the account. If we look at the overall history of account transactions and compare their magnitudes, we can find the greatest transaction; however, it will not provide us with the most favorable transaction. This is because the actions of depositing and withdrawing money have opposite effects on the account balance. Deposits live on the positive side of the number line and withdrawals live on the negative side. It is for this reason that in such situations, we must instead compare numbers via their positions along the entire number line, not just to one side of zero in the case of comparing magnitudes.
There are countless applications involving numbers on both halves of the number line. There are also countless applications of comparing magnitudes where the numbers coexist on one-half of a number line. It is the context of the situation that helps us to determine which of these two outlooks is more appropriate. For instance, we can compare depths of objects below sea level using magnitude since each object is described as being a given distance, which must be positive, below sea level. How would we describe the depth of a position on a dock that lies above sea level? If I describe such a position as being x feet above sea level, I’ve created a new context by changing the description to above sea level which requires a distinct new number line. It is often more useful in this situation to consider the elevations of the given objects with a set reference point at sea level. In this case, the objects are considered as positions along the number line with a fixed reference point of zero. The sign of the number representing the elevation of an object determines whether the object is positioned above or below that reference point. Absolute value tells us exactly how far the number is from the reference point; however it is the sign of the number that sets our description of the number’s position in reference to zero. What is important for us to realize is that the context of a situation assigns direction to the quantities involved, and that direction may not always be the same.
Consider a football coach who is comparing his plays from a game and which of them are more effective than others. If the coach is focused on his offense, he considers the plays which yield the greatest gains in yardage as favorable, but not every play will result in a gain. Sometimes the offensive team is pushed in the opposite direction. A play that yields a 7-yard gain is far more favorable than a play resulting in a 7-yard loss. However, there is great meaning in both of these plays to the coach however. The number representing it on the number line is what provides that meaning. Being able to compare forward motion of the plays along the number line not only helps him to see which plays are effective, but also shows him which plays are not unfolding as they were designed. If the coach is focused on his defense, then he should take the opposite outlook on those numbers since a 7-yard gain from the opposing team’s offense is instead a 7-yard loss by his defense.
Going one step further, sports commentators often provide viewers with an average yards per play statistic for a team’s offense in a given time period. Is this statistic calculated by magnitudes only? Do the statisticians insert the descriptions of gain and loss into the calculation? Of course not. Using both sides of the number line is what provides the most accurate statistic about a team’s offensive capabilities
Before any discussion about using operations with signed numbers, it is important that students are fluent in understanding how to represent quantities in given situations using signed numbers. They should be able to accurately locate those numbers along the number line, use the locations of those numbers on the number line for comparison, and be able to contextualize such a comparison to make meaning from the mathematical model.
This is a post by Eureka Math team members Beau Bailey, writer for Grades 6, 7, & 10, and Erika Silva, writer for Grades 6–8.