Mathematics

7.NS.A.1b Transcript

This is Common Core State Standards support video for mathematics, standard 7.NS.1b. Okay, this is pretty lengthy. The introductory statement for this standard says: apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. And then, specifically, part b states: understand p plus q as a number located a distance absolute value of q from p, in the positive or negative direction depending on whether q is positive or negative; show that a number and its opposite have a sum of zero (are additive inverses); interpret sums of rational numbers by describing real-world contexts.

Now, that’s quite a bit, so let’s take it a little bit at a time. In the preliminary statement for standard 7.NS.1, it talks about previous understandings of addition and subtraction. Now, in Grade 6, students extended their previous understanding of numbers to the rational numbers, and in Grade 6, negative numbers were introduced in standards 6.NS.5 and 6.NS.6. Now, a little bit of clarification here; notice what’s in red in standard 7.NS.1; apply and extend previous understandings of addition to add rational numbers; represent addition on a horizontal or vertical number line. Now, the focus here is addition because the standard after this one, subtraction of rational numbers is covered in standard 7.NS.1c. So, again, the focus is going to be addition of rational numbers.

Now let’s look specifically at 7.NS.1b, and let’s look at the first part; understand p plus q as the number located at a distance absolute value of q from p in a positive or negative direction depending on whether q is positive or negative. So, let’s take the most simple example where p would be zero. So, when we’re talking about p plus q, well, q could be, if we’re talking just the distance, it could be a distance of q in the positive direction or it could be a distance of q in the negative direction from that starting point. So, if we take that scenario, and let’s say that p is zero, and that’s our starting point, and the absolute value from that point is four. Well, again, we could be going four in this direction, which would put us at a positive 4, or we could be going a distance of four in this direction, which would put us at a negative 4.

Now let’s try a different starting point other than zero. Let’s say p was 7, so that’s our starting point, and the absolute value of q was 4. So, I’m going to travel a distance of four from 7. Now we need to consider both contexts. If we travel 4 in the positive direction, then obviously, we start at 7, and we go four this way, which would put us at 11. Now, if we travel four from 7, but in the opposite direction, then when we’re talking about location, then I’ve got to go four to the left, which would put me at 3.

Let’s try another example. Our starting point for p would be negative 3, and the distance that we want to travel from that point is five. So, again, let’s say we start at, we’re starting at negative 3 we’re going to go five in this direction. Well, let’s see, where would that put us? One, 2, 3, 4, 5, so that’s a negative 2, negative 1, 0, 1, and 2. So, that puts us at a positive 2. But, let’s say our distance is going to be in the opposite direction. So, now, we would have negative 3 as our starting point, but the distance that we want to travel is five. We want to go in the opposite direction. So, where is that going to put us? Let’s see, that would be negative 4, that’s going to put us at negative 8 because we went in the opposite direction. So, of course, we want to have students write these out so they start connecting the symbolism to what they’re really doing on the number line.

Now let’s look at the second statement in this standard 7.NS.1b; show that a number and its opposite have a sum of zero, or they’re additive inverses. Now, for simplicity, let’s assume that p is a positive number so that there’s no confusion with the signs and so forth. So, if we’re starting at zero, and then we go some distance p, and we’re assuming that’s a positive number initially, then I would go to the right a distance of p from zero. But then, when I go in the opposite direction, where again, I’m adding the opposite of p, well, guess what? I’m going to end up right back where I started. It would be sort of like, if I went 10 miles away from home, and then I traveled 10 miles back to the house. I’m back where I started, so I’m at zero.

Now let’s apply the commutative property. Now, let’s say that we’re going to go in the opposite direction this time. We’re going to go a distance of p to the left to begin with, and then we go a distance of p, but back to the right. So, again, I’m going to end up back where I started, at zero. So, again, this is the idea of the additive inverse, that any number plus its opposite will always be zero.

Now let’s look at the last statement in this standard; interpret sums of rational numbers by describing real-world contexts. Now, teachers can design a whole bunch of real-life contexts that involve sums of rational numbers. What you have to be careful about, the key is that the result indicates location, not the total distance traveled, which really would be focusing on absolute value, or that it involves the distance between two points because that’s really subtraction, which is the next standard 7.NS.1c.

So, with that in mind, let’s look at some examples. Let’s say Josie is 35 miles from home. So, this is her starting point right here, 35 miles from home, assuming that home was zero. She travels 20 miles farther from home. So, how far from home is she now? Okay, so she’s starting at 35. We’re going to go a distance of 20 further away from home. So, I’m actually going the positive direction. Let’s see, these are in fives, so they’d be at 5, 10, 15, 20. So, we would be here, and that would be 40, 45, 50, 55. So, we would be a distance of 55 miles away from home.

Let’s take that same problem, almost, except now she’s 35 miles from home, but she travels 20 miles back toward home. So, notice that, again, I’m going an absolute value 20 miles, a distance of 20 miles, but this time it’s in the opposite direction. So, now I’m going to start at 35, but I’m going to go back toward the house a distance of 20 miles, and this is in fives, so that’s 5, 10, 15, 20, so I would be here. And that would put me at, let’s see, 30, 25, 20, 15. So, that would put me 15 miles from home. But, again, note the difference, and it was very easy to manipulate this question, this scenario where I could, just by changing the numbers a little bit, make it both contexts, where I’m traveling a distance in one direction, and in the other problem, I’m traveling the same distance, but in the opposite direction.

Let’s try another problem. Manny has $250 in his savings account. He deposited $75 , so what’s his new balance? Now, deposit means that you’re going to add $75 to this, so addition of a positive number is going to take me in this direction. Let’s see, these are in 25s, so that would be, that would put me here, which should be 275, 300, 325. So, the new balance would be $325. Now, simply by changing the wording a little bit, this time he has $250 in savings, but he withdrew $75. So, what’s his new balance? So, again, we’re traveling the same distance, 75, but this time we’re going in the opposite direction because he took money away from his account. So, let’s see, we’re going to go 25, so we will end up here. So, that would be 225, 200, 175 is where his new balance would be. So, again, basically the same problem, just change it a little bit to change the context, where you’re going $75 in the opposite direction.

Let’s try another example, temperature. Let’s see, the temperature in Anchorage is minus 20 degrees Fahrenheit. It then rose 35 degrees. What’s the new temperature? Well, let’s see, starting point is here at negative 20. It’s going to rise 35 degrees. So, we’re going to go a distance of 35 this way. Let’s see, these are in fives, so that’s 5, 10, 15, 20, 25, 30, 35, so we’re here, which would put us at a positive 15 degrees Fahrenheit.

We can take this same exact scenario and just change the wording a little bit where it started at 20 degrees, well, negative 20 degrees Fahrenheit, and it dropped 35 degrees. So, this is the current temperature, but it’s going to drop 35 degrees. I’m going to have to make up a little bit more. So, let’s see, we go 5, 10, 15, 20, 25, 30, 35, so we would be here. Well, let’s just count. Let’s see, that would be 25, 30, 35, 40, 45, 50, 55. So, we would be at negative 55. So, it’s definitely pretty cold in Anchorage because we started at 20 degrees below zero and it dropped 35 degrees, so we’re now at minus 55 degrees.

Another similar scenario, let’s talk about depth. A submarine is at a depth of 1600 feet. It rises 400 feet. What’s the new depth? So, if we start at negative 1600 and we go upwards 400 feet, these are in increments of 200, so we would be, negative 14, negative 1200. So, we would be at negative 1200 for our new depth. Now, we can take that same exact problem, tweak it a little bit, but instead of rising 400 feet, the submarine is going to dive 400 feet. So, we’re going to travel an absolute value of 400 but in the opposite direction. So, in this case, okay, we’re going to drop. So, that’s 18, 2000, so then, our new depth for the submarine would be negative 2000 or 2000 feet below the surface.

Now, it should pretty much cover the standard, but, again, the focus is from a number line perspective. So, again, if I’ve got some number, and I can go a positive distance, I can go a distance of q this way, where this would be p plus q. But, we also have scenarios where I’m going to go a distance of q this way. So, that would actually be p plus the opposite of q. But, you might want to throw in other ways of looking at adding a distance of q either in a positive or a negative direction, and that’s by using some kind of manipulatives, sort of like this, where, let’s say, the black checker would be a positive 1 and the red is a negative 1.

And this enables you to set up quite a few different types of scenarios, let’s say something like this. Now, the idea is to pair these up in such a way where I have a 1 and a negative 1. And we just learned that those would be additive inverses. Those would be zero. So, we have another 1 plus negative 1, which would give me a zero, and I have another, well, I guess what we would call a zero pair. So, what do we have left here? We would have this. Now, it’s a good idea to always take what you’re doing with the concrete manipulatives and connect it back to what it looks like symbolically. So, in this case, the 5 plus negative 3 ended up with the manipulatives to be 2 plus 0 plus 0 plus 0, which gives us our final solution of 2. Now, this is important, this whole idea of an additive inverse, because it’s one of the founding ideas and building blocks for algebra. So, adding zero is very important. So, again, doing something like this would help to start ingrain that idea of the additive inverse.

And so, there are other scenarios that you can apply to this. Let’s say you start off with a negative 5 and you’re going to add a positive 3. Well, again, just like before I want to pair up a plus 1 and a minus 1 because we know that that’s 0, and this is what you would end up having. But, again, the idea that nothing really gets cancelled or thrown away, it’s just the idea, again, that I have these zeros, these zero pairs. But, again, it’s important that the negative 2 doesn’t just magically appear. Again, you actually have a negative 2 plus 3 zeros, which still gives you a negative 2; again, very important for laying the foundation for solving equations, the idea of the additive inverse.

And just one more example; let’s say they are all the same. Let’s say I have a negative 5 plus a negative 3. Well, this just involves throwing them all together. So, I end up with 8 of them, but they’re all negatives. Nothing changed as far the signs. So, we have negative 5 plus negative 3 would be a negative 8.