This is Common Core State Standards support video for mathematics; this is standard 6.NS.7d. This standard states: understand ordering and absolute value of rational numbers; distinguish comparisons of absolute value from statements about order.
Let’s look at the introductory statement for this standard; understand ordering and absolute value of rational numbers. Now, the idea of ordering of numbers is covered in earlier standards. For example, fractions, ordering them, is addressed back in the fourth grade in standard 4.NF.2. So, this is where students learn how to distinguish between, say, which is larger, three fourths or two thirds. Going on to some other types of rational numbers, ordering decimals is covered in Grades 4 and 5. In Grade 4, it’s in standard 4.NF.7, and in Grade 5, it’s in standard 5.NBT.3b, where, again, students learn to order decimals such as 7.6 compared to 7.34. And then here, in Grade 6, the notion of integers is introduced. So, students learn in previous standards, in this grade, to order integers in standard 6.NS.6c and standard 6.NS.7a, where again, they learn to distinguish, for example, that negative 12 will be less than negative 4.
Let’s look at the idea of absolute value. This actually occurs right before this standard in standard 6.NS.7c, and it states: understand the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. So, basically, what this is saying is that absolute value simply refers to the distance from zero regardless of the direction, and there are real-world contexts for this. So, here we have two numbers, negative 4 and 4. Both are the same distance from zero. But we have a little bit of a problem because, obviously, this is not a true statement. Negative 4 is not equal to positive 4.
So, mathematically we need some type of symbolism, something to adjust this, and the symbol for absolute value would be two bars, something like this that again, indicates that we’re talking about the distance of 4. So, then, that would actually make this a true statement because we are talking only about the distance from zero, and both of these numbers are four away from zero, just in different directions.
Let’s look at a real-life context. Let’s say I’m at home, and I drive 8 miles to work. So, let’s let home be zero. So, if we are talking about location, I’m starting at zero. I’m starting at home, and then I’m going to drive 8 miles to work. But then, I’m going to drive 8 miles in the opposite direction and go back home. So, I’m back where I started. I’m back at zero. I’m back home. Now, that’s a little bit different situation as opposed to distance traveled. This would be like, okay, the odometer on my car. Well, if I drive 8 miles to work and then 8 miles back, my odometer isn’t going to register that I didn’t travel. There’s no more distance, no more mileage on my car. It would be nice if that was the case, but it isn’t.
So, what happens here is that, okay, I started home. I go 8 miles to work, but then, when I go back home, I’m still traveling 8 miles even though it’s in a different direction. So, here, I have to indicate that I want just the actual distance traveled. And so, my odometer is going to say that I traveled 16 miles, not zero miles.
Now let’s look at the specific standard here, d: distinguish comparisons of absolute value from statements about order. Now, when we’re dealing with positive rational numbers, there really isn’t any confusion between the two ideas, absolute value versus order, because they will be one and the same. So, for example, let’s say we’re talking about a distance of 5. . . 1, 2, 3, 4, 5. So, a distance of 5 and 5 being compared to say, zero, there’s no confusion. Where it does get a little bit confusing for students would be when we’re talking about absolute value and order for negative rational numbers. And the best way to really get a handle on what this is saying is to just look at a few real-life contexts.
Okay, in a real-life context, debt implies negative numbers. So, let’s say, on one credit card, I owe $400, and on another credit card I owe $1,000. Now, mathematically, negative 1,000 is less than negative 400. But, here’s where it gets a little bit sticky. A real- life statement might be something like a debt of $1,000 is greater than a debt of $400. Now, here’s where again, there might be a little bit of confusion because we’re saying greater than, even though back here, the comparison between those two numbers is less than, that again, negative 1,000 is less than negative 400.
But what you’re really dealing with here in this statement is absolute value because we’re talking about the distance from zero. A debt of $1,000 is going to put me further away from zero than a debt of $400 is. So, again, students need to be aware of the distinction here, that, again, negative 1,000 isn’t greater than negative 400, but the distance from zero is.
Let’s take a horizontal number line, and let’s make it vertical and apply a temperature context. Now, colder implies a lower temperature. So, for example, here, negative 40 degrees is less than negative 20 degrees. But a real-life statement might be that 40 below zero is more cold than 20 below zero; again, the confusion with this idea of more cold, even though negative 40 is less than negative 20. But again, what we’re dealing with here is absolute value. We’re talking about the distance from zero where, if the temperature is minus 40 degrees, I am, in fact, further away from zero than I would be at negative 20 degrees.
Let’s take one more context, again, a vertical number line application where we’re talking about depth. Now, depth implies the distance from the surface of the water in some contexts. So, in this scenario, a depth of negative 4,000 is less than negative 2,000. But then, when we talk about this in a real-life kind of way, we say that 4,000 feet below sea level is a greater depth than 2,000 feet below sea level; again, this idea of greater. But again, we are dealing with absolute value even though absolute value is never actually stated in the sentence. But, again, what this is really saying is that negative, a depth of 4,000 is further away from zero, from the surface, than a depth of 2,000.
So, again, these are just a few examples of the care that needs to be taken when you’re dealing with comparisons from an absolute value perspective versus talking about comparing the two numbers numerically. Again, just be more careful with negative rational numbers because that is where it might get a little bit confusing. Students understand that in one case, you’re dealing with absolute value; in the other case, you’re making a comparison between the two numbers.