Mathematics

4.MD.A.2 Transcript

This is Common Core State Standards support video in mathematics. The standard is 4.MD.A.2. This is a pretty long standard. It states: use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

In addressing this standard, one of the things that we need to do is well, what's our limitations? In looking at the fourth grade standards, there's 4.OA.A.3, and it talks about solving multistep word problems posed with whole numbers and having whole number answers.

So, we know that multistep problems are fair game. And then, what kind of numbers are we limited to? In this same grade level standard, 4.NBT.B.4 talks about fluently adding and subtracting multidigit whole numbers. There's also standard 4.NBT.B.5 that talks about multiplication, but it looks like we're limited to whole numbers of up to four digits being multiplied by a one-digit whole number or multiplying two two-digit numbers.

Standard 4.NBT.B.6 talks about division. And here it states that we need to find whole number quotients and remainders with up to four-digit dividends and one-digit divisors. So, it looks like we're limited to one-digit divisors in particular when it comes to dividing. Okay, this standard also involves fractions. Well, there's standard 4.NF.B.3c that talks about adding and subtracting mixed numbers with like denominators, and there's standard 4.NF.B.4c that talks about solving problems with multiplication of a fraction by a whole number. So it looks like we have to limit it to that; I don't think we can multiply a fraction by a fraction.

And then, standard 4.NF.B.3d talks about adding and subtracting fractions, referring to the same whole, and having like denominators. So we have to use like denominators. At this grade level, they're not expected to have unlike denominators and have to get a common denominator and all that. The standard also talks about decimals. In looking at the fourth-grade standards, there's 4.NF.C.6, which talks about using decimal notation for fractions. But further investigation reveals that there are no standards in grade four that specifically address expectations regarding operations with decimals. It's not until grade five with standard 5.NBT.B.7 that we see decimals. It talks about adding, subtracting, multiplying, and dividing decimals to hundredths. So we're left with a little bit of a dilemma here, but it would be pretty safe to assume that we need to keep it pretty simple, maybe even deal primarily with just tenths.

The standard talks about measurements. In the same grade level, standard 4.MD.A.1 talks about knowing relative sizes of measurement units within one system of units, such as kilograms and so forth. And again, within a single system of measurement is the focus. So students will not be expected to convert, say, from kilograms to pounds or anything. We have to keep it either all metric or all within the English system.

Now there's a lot to this standard, and we can't do your four operations with each one of these different contexts, but we'll just have to do enough to give you a reasonable idea of what the expectations might be. So let's take this problem involving distance. Charles is building a wall with cinder blocks and using mortar to hold the blocks together. Each block has a height of three fourths of a foot. Also, one half of an inch of space is used between the blocks for the mortar. How tall will the designed wall be? Well, if we look at the diagram and we count, we can see that there's five of these spaces where the mortar's going to go to hold the blocks together. So there's five increments of one half inch each.

And of course, students can use a physical model; over here we can take those halves, we can combine those two halves to be one, another two halves to be one, and we have one half left over here. So, using a physical model, students can easily determine that we would have two and a half inches total for the mortar. So, using the physical model, students can pretty easily tell and determine that we would have two and a half inches total that we would need to allow for the mortar spaces.

Students can also look at this as five over two when you use the standard algorithm and multiply. So, over here we have five. But each of these represents one half. So again, we can do the same thing as before and determine that we have two and a half inches. We've taken care of that part of the problem. Now we need to concentrate on the height of the blocks, which is three fourths of a foot. Here we have a physical model of three fourths, and we have six of them. But that's going to be pretty difficult to do.

Notice that we have a little bit of a problem here, because we have two different units of measure. We have inches and we have feet. Now, notice that in the standard, it does talk about problems that require expressing measurements given in a larger unit in terms of a smaller unit. So what we can do here is, let's change the feet to inches so that we're dealing with nothing but inches.

So let's refer to our regular ruler, 12 inches. Now when we do this, we're also addressing this last part of the standard that talks about representing measurement quantities using diagrams, such as number line diagrams that feature a measurement scale. Okay, so we have 12 inches total. Now we're talking about three fourths of a foot. So it makes sense that we need to split this up, the 12 inches, into four equal parts. Twelve divided by four is three. So we know that every three inches will be one fourth of a foot, and if we just count off, we have one, two, three fourths, so we would be at 9 inches.

So with the help of using whole numbers and using this representation, this ruler, students can determine that three fourths of a foot would be 9 inches. And now we can convert this over. Instead of having six times three fourths of a foot, we would have six times 9 inches, which would be 54 inches. And now, we can take on the rest of the problem. Don't forget we've already figured out the mortar space, five of those one-half inches, which was two and a half. Now we combine those together, and we have 56 and a half inches for the total height to the wall.

Let's take a problem involving time. Michelle fell asleep at 10:30 p.m., 10:30 at night. A storm came in, and she was awakened by thunder at 5:15 in the morning. She was not able to go back to sleep. How much sleep did she get that night? Here again, we can address the last part of the standard, again, that talks about representing measurement quantities using diagrams, such as a number line diagram that features a measurement scale. So, we're taking this timeline, and we're starting off with a 10 to represent 10 p.m., 10 at night, and we did it enough to where we go all the way to 6:00 in the morning. So if we count off, let's see, we're going to start at 10:30, and if we go in one-hour increments, let's see, 10:30, 11:30, 12:30, 1:30, and so forth. We go all the way up to 4:30, but if we go another hour, we'll be at 5:30, which is too far, because we need to stop at 5:15.

So, so far we got 6 whole hours. Now each hour is 60 minutes, and if we take 60 minutes and split it up four equal ways, each one of these increments will be 15 minutes. So now we start at 4:30, and we need to go one, two, three increments of 15, which should be 45 minutes. So now we've solved the problem that she's going to get 6 hours and 45 minutes of sleep. And again we did it pretty much by using a timeline, and we addressed the last part the standard.

Let's do another time problem. It takes Julia 1 hour and 50 minutes to drive to her parents' house. It usually takes her 3 hours and 10 minutes to drive to her grandparents' house. How much longer does it take her to drive to her grandparents than to her parents? Okay, well, we have to find the difference between the two, so you know it's a subtraction problem, but more than likely here's what students are going to do. They're so used to working with place value that, okay, well I can't take 50 from 10 so I have to decompose the 3 hours to 2 hours, but then they're liable to do this–just put the one there and think that they have a 110 minutes minus 50 or 60 minutes and so forth, and of course, they've just solved this incorrectly.

Let's go back and start over. All right, we're going to do our subtraction. But wait a minute. If we decompose this hour, well that's not 100 minutes, that's 60 minutes. Okay, that 1 hour is going to become 60 minutes that then I combined with the 10 minutes that I had already. So, I know that I have 70 minutes. So now I can solve the problem–70 minus 50 is 20, and then 2 minus 1 is 1. So, the difference in the time as far as driving is going to be 1 hour and 20 minutes. This is the type of problem that students need to give them a little bit more insight as to what happens when you compose and decompose numbers. And again, this is not place value. They have to pay a lot of attention as to what these numbers really represent. Like in this case again, the 1 hour isn't 100 minutes, it's 60 minutes.

Let's look at liquid volume. Let's say you have a container, and it's 1 liter and it’s marked off in100-milliliter increments. And here's our problem. Michael places a beaker in the sink and notices that it contains approximately 250 milliliters of water. He returns 5 hours later and notices that the beaker has more water in it now. He realizes that the faucet has a slow leak. About how many milliliters is the faucet leaking each hour?

Now this is a multistep problem, because first of all, well how much did it leak total? Well, let's see, the reading now is 600. It started off at 250. Take the difference between the two. That's 350 milliliters. But we're not there yet, because the question is asking about how much did it leak per hour? Well, this took place over 5 hours, so now we have to take the 350 milliliters and split it up into five equal parts, and of course we do that by dividing: 350 divided by five, which gives us 70. So, the answer to the problem is that the faucet is leaking about 70 millimeters each hour.

Now let's look at an example using masses of objects. Since modern scales are digital, problems involving mass would be a good starting point for operations with decimals. Let's say Lena placed her full backpack on a scale, and it registered 6.3 kilograms. Okay, then Lena emptied it out and placed it back on the scale with nothing inside of it. How much do the contents of the backpack weigh?

Well, let's see, logic dictates, let's see, full it weighed 6.3, and now looking at our scale, empty it weighs 1.5 kilograms. So if we take the difference of the two; wait a minute, you can't just arbitrarily just write down these measurements any way that you want. Remember that basic idea that you can only combine like items? So what we have to do here is, we have to line up the decimals. That's the reason behind this. We have to make sure that in this case, we're subtracting tenths from tenths and ones from ones.

So now that we've ensured that, well let's see, I couldn't take five from three so I have to borrow a one, and the one is 10 tenths, so now instead of 3 tenths, we have 13 tenths. Now we can finish it out; 13 minus 5 is 8; 5 minus 4 is 1. At this point, students should have already had some practice with what to do with the decimal point. And so, at this stage, students should know that it's going to be 4.8. We need to put our decimal straight down and put it between the four and the eight. So we have 4.8 kilograms. That's the solution to the problem. That's how much the contents of the backpack weighed.

Let's take a second example illustrating masses of objects. Manny has a penny, and places it on the digital scale. And this is a different scale—it's a little bit more precise. He sees that the reading is 2.5 grams. How much would 5 pennies weigh? Well let's see, 1 penny weighs 2.5. One approach would be, since we only have five, is to just use addition. So, if we take the 2.5 and write it down five times; make sure that we lined up to the decimals.

Now it's simply a matter of adding. So we add up all our fives, that's 25. Then we add up all our twos, that's going to be 12. And again, students should have already had some experience with what to do with the decimal point. So 5 pennies would weigh 12.5 grams. By doing the addition first, it gives us a little bit of a head start on what to do if we did this with multiplication. If we approached this as a multiplication problem, 2.5 times five, well, we do our multiplying five times five is 25. And five times two is 10 plus another two, that's 12. And again, based on already having done this with addition, we know what to do with the decimal point, that it should be 12.5 grams.

Okay, let's solve something with money. Now, standard 5.NBT.B.7, in the next grade, deals with decimals—adding, subtracting, multiplying, dividing decimals to hundredths. Now, since money goes out to the hundredths place, using those kind of contexts makes sense as far as maybe laying the groundwork for this standard in fifth grade where they'll already have a little bit of some basic experience dealing with hundredths.

So, let's say that Jerry found some loose change in his pocket. He's got 2 pennies, 3 nickels, 1 dime, and 2 quarters. How much money does he have in loose change? Well, let's rearrange the pennies. Let's group them up to where we have the same type together. We've got 2 pennies, which of course is 2 cents. We've got 3 nickels, which is 15 cents. We've got 1 dime, which is 10 cents, and then we have 2 quarters, which should be 50 cents. And again, we can't just do this willy-nilly. Remember we can only add like items. So we're going to have to line up the decimals to again ensure that we're adding the same stuff, where in this case, we're adding tenths with tenths and hundredths with hundredths.

So, now we can do our addition problem. We add up our hundredths, which should be seven. Then we add up our tenths, one, one, and five, which would also be seven. Using a little bit of logic here, well that's 77 cents. And we know that the representation for that will be .77. And it also follows the same pattern as before that when we're adding numbers that are decimals, we end up just bringing the decimal point down.

If we look at the standards for mathematical practice, there was a whole lot to this standard. There's so many things that we can do here, and there's so many other standards that it connects to that you could argue that in doing these types of activities, that we've addressed all four of these initial ones: making sense of problems and persevering in solving them, reasoning abstractly and quantitatively, constructing viable arguments, andcritiquing the reasoning of others, and modeling with mathematics. And the same would apply to the last of these standards. I think we could argue that each one of these should be addressed with these different activities for standard 4.MD.A.2. They would be using appropriate tools, they would be attending to precision, they'd be looking for and making use of structure, and they'd be looking for and expressing regularity in repeated reasoning.

Again a fairly extensive standard; it takes a whole lot to address this, and there's also quite a bit of prerequisite knowledge that is needed. There's some uncertainty as to what to do with the decimals, but again we should keep it simple primarily with tenths, but I think if you work with money, you can also get away with working a little bit into the hundredths place.