This is Common Core State Standards Support Video for Mathematics; the standard is 2.MD.A.2. This standard states: Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
Students should already have some experience with the idea of the length of an object. Back in first grade, there was standard 1.MD.A.2, where students were expected to express the length of an object as a whole number of length units by laying multiple copies of a shorter object end to end with no gaps or overlaps. What is probably new in this standard is the second statement where students need to describe how the two measurements relate to the size of the unit chosen. Note that we need to measure the length of an object twice using two different units of length. So let’s use a new crayon as our length measure. So we take our crayon. We get some additional ones. We lay them end-to-end with no overlaps, and we see that the length of this segment is 3 crayons. But we have to measure this twice. So let’s take a paper clip as our unit of measurement. Get several paper clips. Lay them end to end, no overlaps. Count them; let’s see, 1, 2, 3, 4, 5, 6, 7. So the length is 7 paper clips.
Now we need to pay attention to the second part of the standard where students have to talk about how the two measurements relate to the size of the unit chosen. Well notice that the crayons, those are bigger units, and the paper clips were smaller. And note that we only use 3 crayons, but we had to use 7 paper clips. So after several iterations of such exercises, students should be able to determine that the larger the unit of measure, the fewer of those objects or units that are needed to measure a given length. So again, it’s an inverse relationship; the smaller the unit of measure, the more of them that you’re going to need. All of this lays an informal foundation for the idea of proportionality. However, the concept of proportional relationships isn’t introduced in the standards until Grade 6. Note that the conceptual focus here is that of inverse proportionality, which isn’t even cited in the standards.
Students will need quite a bit of experience with this, again this whole idea of relating the size of the unit chosen to the number of those units that are going to be needed. So let’s say for example, we have this length here—turned out to be 2 large paper clips. And if we use a smaller unit, these smaller paper clips, we need more of them. We need four to measure that same distance. And in turn, we take an even smaller unit, let’s say these spaghetti sticks, and we needed six of those. So we need more of them because the size of the unit was smaller. And then take it one more step; let’s say we’re using these coins or these checkers. We needed eight of those, because again, we had a smaller unit, and we need more of them to measure the same length.
In second grade, there’s a standard that precedes this one, 2.MD.A.1, and it deals with measuring the length of an object by selecting tools such as rulers, yardsticks, meter sticks, and measuring tapes. And then the standard that follows this one, 2.MD.A.3, talks about estimating lengths using units of inches, feet, centimeters, and meters. So note that this cluster is focused on lengths in standard units. A while ago, we did measurements using non-standard units like paper clips or crayons, and so forth. But now we are going to use standard units. Note that we have the English standards of unit with inches and feet, but we’re also dealing with the metric system with centimeters and meters. Now centimeters and meters, we might get into some larger numbers. But if we look at the second grade standards, there’s standard 2.NBT.A.2, where students have to count within 1,000, and they can skip count by fives, tens, and 100s. So we might get into some large numbers with the centimeters, but students are expected to handle numbers within 1,000. So we should be okay as long as we keep them to three digits.
In the next grade level, Grade 3, we have this standard, 3MD.B.4, and it talks about generating measurement data by measuring lengths using rulers marked with halves and fourths of an inch. But at this grade level, second grade, number knowledge is limited to whole numbers. So students will probably need to estimate to the nearest whole unit, because again, we’re not dealing with fractions or mixed numbers. Here’s your typical ruler. But we have a problem though, because it’s going to be marked of in fractions of an inch, and that might be a little bit confusing to students. So it might be necessary to construct your own rulers using a regular ruler as a template. But you need to mark it off where we just have whole numbers. We just have whole numbers for the number of inches. Be careful with the beginning and the end. You may not have enough room to actually write the numbers, but students need to make sure that they understand that at the beginning, that’s understood to be 0, and at the end that’s understood to be a 12, although typically you will see the 12 there marked at the end.
So we have our homemade rulers, and again, we’re going to go back and apply the previous standard where we have to measure the length of an object. But we have to make sure though, in order to handle standard 2.MD.A.2, that students do in fact know how to measure the length of an object. We must ensure that students understand the measurement basics. For example, they have to know what each mark represents. So in our homemade ruler here, each mark represents 1 inch. They have to know that you always start at 0 to do a measurement, and things won’t always measure exactly. We might have to approximate, round it to the nearest whole number in this case. So let’s say we have this pencil. We measure it, and we’re pretty lucky because it looks like it’s exactly 3 inches. But we’re not so lucky with this pencil. It’s not exactly 4 inches. It’s more than that, but it’s closer to 4 inches than it is 5. So we would approximate this to 4 inches. So let’s say we’re going to measure this real length object that’s a flute. And we take our homemade ruler, and so we start at 0, and we put it at the beginning of the flute. We can either slide this ruler over or get a second ruler and lay it end to end, and we notice that this is exactly 24 inches.
Now let’s use another measurement tool, but this one is strictly marked off as just 1 foot. So we have 1 foot here, and then we can either mark that spot and slide it over or just take a second ruler and lay that end to end, and we see that this is 2 feet. We need to emphasize the second part of the standard, again, this idea or relating the size of the unit chosen to how many of them we needed. And in this case well, let’s see, feet. We only needed two of them. We needed 24 inches. A foot is a much larger unit of measure than an inch, so it stands to reason that we needed fewer feet than we did inches to measure the same object, this flute.
Now let’s say we’ve constructed a yardstick, and it’s marked off in feet. And then we also have a yardstick that’s marked off in inches, 36 of them, of course. Let’s say we’re going to measure the width of a door. So we use our yardstick that’s marked off in feet, and we see that well, it’s not quite 3 feet, but that’s close enough. We approximate it to a whole number. So this is 3 feet for our measurement. Then we use a yardstick that’s marked off in inches, and it looks like this is about 35 inches. So that’s our measurement. Now we pay attention to the second part of the standard, and we need to look at the relationship of the units here. A foot is a much bigger unit than an inch, so it stands to reason that yes, we needed much fewer feet here to measure the same distance than we did inches. Now let’s work with the metric system. We have a meter stick and then we have a second meter stick that’s measured off in centimeters. Let’s say the task is for the students to measure the teacher’s desk. So let’s say they took the meter stick that’s measured off just as meters, and the measurement turned out to be approximately 2 meters. Then we use the meter stick that was marked off in centimeters, and we estimate that it was 187 centimeters. Paying attention to the second part of the standard, meter as a unit is much larger than a centimeter. So again, it stands to reason that we needed much fewer meters to measure the desk than we did centimeters.
With respect to our standards for mathematical practice, if we inspect the first four, if we do the activities in this standard, students would be reasoning abstractly and quantitatively. If we have them work in groups, they would construct viable arguments and critique the reasoning of others. Looking at the last four of those standards, students would be using tools strategically, yardsticks and rulers and so forth. They would have to attend to precision, and they would have to look for and make use of structure.