Mathematics

3.MD.A.2 Transcript

This is Common Core State Standards support video in Mathematics. The standard is 3.MD.A.2

The standard reads: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. Let’s first focus on the initial statement that deals with measuring and estimating liquid volumes and masses. Now there might be some challenges in finding measurement instruments that meet the limitations that exist at this grade level.

Let's see what those limitations are; first let's look at measuring and estimating masses. Now a lot of the scales that are out there are probably digital. The digital scales are going to use decimals, and that poses a real challenge, because decimals do not appear in the Common Core State Standards until grade 4. At grade 3, the expectation is still that they're going to be working with whole numbers. You also have these types of scales, the spring type scales, but a lot of them are going to use English units of measure—pounds and ounces. But we're supposed to use grams and kilograms; we’re in the metric system. Even if you're able to find this type of scale that is marked off in grams and kilograms, there's going to be a problem with the increments, because the likelihood is that they're going to be marked off in tenths to match the metric system, being a base-10 or decimal system. Let's say you want to go with a balance type of a scale, and so you're going to have weights in 1-kilogram increments that you can use. And let’s say you want to weigh these three books. So we put them on the scale, and then we're going to put some of those 1-kilogram weights on there. But since that's not very precise, you know 1-kilogram is 2.2 pounds. So more than likely, it’s going to be unbalanced, and the temptation is going to be, well we have some smaller weights, like maybe you have some that are 100 grams each. So let’s use some of these smaller ones, and we'll put them on there until we get this to balance out. Now we have a problem, because when students solve problems for this standard, we have to be using the same units. Plus students are not expected to be able to convert between grams and kilograms. Now we do have these types of scales, and you will be able to find them in the metric system, using grams and kilograms. And in fact, let’s say you have one like this, and it’s marked off in increments of 50 grams each. You can probably find some scales that are also calibrated to be more precise, say something like this second one that is marked off in increments of 5 grams. And if you want to weigh some heavier objects, you might be able to find some in kilograms that are marked off to certain degrees of precision. Again keep in mind though that whatever problem students have to do, you have to be dealing with the same units.

Now let’s look at measuring and estimating liquid volumes. When dealing with liquid volumes, we’re actually addressing standard 3.MD.B.4 that states, “generate measurement data by measuring lengths using rulers marked with halves and fourth of an inch.” Now how did these connect? Well if you look at your ruler, typically you measure length horizontally, but if you have students measure length vertically, you know something like this, so we again rotate our ruler, then it's easy to see that the increments that are marked off on your ruler are going to correspond directly to the increments that are marked off on measuring cups. So let's look into what might happen with measuring cups using the metric unit of liters. Well 4 liters is about 4 quarts, which is a gallon. And so you we could have a measuring cup of 4 liters, however the limitation to whole numbers would necessitate increments of 1 liter, which again is not very precise. But the fact that this is just part of it in liters, that's okay; it’s not real precise. But notice that the standard says students are expected to estimate. So for example, but let's say they poured some liquid into this measuring cup, and let’s say it came up to about here. So the student can estimate, well it's close to 3 liters. And that’s what they use for their measurement, and they can then solve whatever the problem was using whole numbers. Let's say you have a measuring cup and it's 1 liter; well 1-liter capacity that makes sense. But the increments might be problematic. There's a standard, 3.NBT.A.2, that states that, “students are supposed to fluently add and subtract within 1000.” Now let's say a student had a problem where they measure of a liquid volume, and let’s say it was 600 milliliters. And then they had another cup, and they measured that same liquid; and let’s say it was 800 milliliters, and they’re supposed to combine them together. Well 600 and 800, that’s 1400. And so you violated the restriction that's posed by this standard, 3.NBT.A.2, because they are supposed to stay within 1000. Now there's a question when you get to 1000 milliliters, because that is equivalent to 1 liter. And again students need to be working within the same units, so we can't be mixing up liters and milliliters. But here's a catch: liters are mentioned in the standard, but you have a measuring cup that’s measured also with increments of milliliters. But this standard seems to restrict measurement to liquid volume to liters, so that eliminates the use milliliters as units. It’s interesting that for mass, we have grams and kilograms, but for liquid volume we only have a liter, so we’re restricted to that. So in looking at the first part of this standard, we're going to have two problems here: first the types of measurement instruments available, that might be a real problem for you, but the big headache here is that there's no decimals at third grade. So there seems to be a little bit of a mismatch, because the metric system is a base-ten system, but students at this level are only using whole numbers.

Now let’s look at the second part of the standard that deals with the solving of one-step word problems using the four operations of addition, subtraction, multiplication, and division. Again we have to keep this standard in mind, 3.NBT.A.2, that the addition and subtraction has to stay within 1000. But we will be addressing a second standard here, 3.OA.A.3. That standard states you use multiplication and division within 100 to solve word problems and situations involving measurement quantities, and that's what we are doing when we address 3.MD.A.2. So let's look at a few examples of word problems here. Let's take this one, you’ve poured 60 liters of water into a barrel, you return to it days later and realize that the barrel has a leak. It now has 40 liters of water in it. So the first question here would be well how much water leaked out? So it would be a basic subtraction problem, where they would subtract 40 from 60. But you could add a second question to this, if this leaking occurred at a constant rate for 5 days, about how many liters did it leak each day? So again, they would have to take their solution of 20 liters, and it's a basic division problem where they have to figure out. Well they split that out among 5 days, which would be 5 ÷ 20, which would be 4. So it would leak about 4 liters each day. Let’s do a second example. You have a farmer, and he takes a cow to the auction. The weight of the cows is determined by the weight of the trailer and the cow on a freight scale. Then that cow is unloaded, and the empty trailer is weighted. So the difference between the two is going to be how much the cow weighs. So the weight of the trailer and the cow together was 863 kilograms, and the weight of the trailer empty is 585 kilograms, how much does the cow weigh? So it would be a subtraction problem within 1000, where you simply subtract 585 from 863 to get the weight of the cow. Let's take another example, David was curious as to how much the books weight that he has to carry in his backpack, so he weighed them using this type of scale. And if you look, it looks like it's about 4 kilograms. And so the question is, if 9 students in the class have to carry that same sets of books, what's the total weight of all those books? Well let’s see, each one is caring about 4 kilograms there’s 9 of them, so it's a basic multiplication problem. In looking at the activities and the problems that you would use for this standard, if we look at the standards for mathematical practice, from the first four, we would be utilizing and addressing a number one, which says make sense of problems and persevere in solving them. And when we look at the rest of the standards for mathematical practice, we would be using tools strategically. We’d be measuring mass with the scales, and we'd be measuring liquid volume with measuring cups. And then we would be looking for making use of structure, looking for patterns, and so on.