This is Common Core State Standards support video for mathematics. The standard is 3.MD.B.4. This standard states: generate measurement data by measuring links using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
First, let's focus on this idea of using rulers marked with halves and fourths of an inch. The reality is that this is the typical type of ruler that you would run into, where it's not marked off in fourths or halves. Typically, it's going to be a little more precise, probably eighths, or sixteenths of an inch. So, you might have to manufacture your own, where it is marked off in fourths. Your students may not be too familiar with what the marks mean and what it means to have it marked off in fourths. So, it might be a good idea to also label what the marks represent, to get them used to this idea of, again, that these marks mean one-fourth of an inch each. Of course, at the ends, they have to understand that at the beginning, we have our number zero, and at the end in this case, it's 12.
It would be a good idea to also address another standard here, standard 3.NF.A.3.B, dealing with recognizing and generating simple equivalent fractions. In this case, one half is equal to two fourths. Now, instead of marking this off in halves, some of you might have decided to mark it off as two-fourths and then talk about this being equivalent to one-half.
Now, let's focus on this idea of measuring lengths. Now, students must understand measurement basics. It just depends on the experience of your students at this point. Of course, they have to know what each mark represents. They have to understand that you always start at zero, at least at this grade level. And then they need to approximate when necessary, because things won't always fall exactly on one of the marks. Let's say, for example, we're measuring pencils. In this case, this is nice, because this one is exactly 3 inches long. But, what if we have something like this? That looks like it's about four and one fourth. It's about four and a quarter inches long.
What we have to address is the fact that mixed numbers are not mentioned in third grade Common Core State Standards. In fact, mixed numbers don't appear in the standards until fourth grade in this standard, 4.NF.B.3c, that talks about adding and subtracting mixed numbers with like denominators. Again, depending on where your students are, you might have to go back and explain and teach them what a mixed number is.
So, let's do that very quickly. So, in this case, we had a pencil that was four and a quarter inches long. So, we have this number four and one fourth. That consists of, in this case, 4 inches plus one fourth of another one. If we use another example, we have four shaded circles, and that's our whole number. And, let's say that we have another circle where one fourth of it is shaded.
So, here is plus one fourth, and that's our proper fraction. If we put these thoughts together, when you have a whole number plus a proper fraction, that is a mixed number. Now, it's also important for kids to interpret four and one fourth as four plus one fourth, because that'll come in handy in the future. What's also necessary here, it might be confusing. You need to explain a little bit more here because four and one fourth, a mixed number, it goes against what they've seen with place value, because a mixed number, it doesn't fit exactly with the idea of place value.
Here's another example using rectangles. In this case, we have three rectangles that are shaded plus three fourths of another one. So, we have our whole number is three, our proper fraction is three fourths, and we would combine them together, we get our mixed number of three and three fourths.
Now, students can get the idea that we're only talking about shaded things, so, let's use this example. We have three full rectangles, but we only want three fourths of another one, so we need to knock out one fourth so that this is what we have. We have three and three fourths rectangles. Same idea with the circles, instead of dealing with shaded circles, let's just say, well, we just want four and one quarter circles. So, we take our circle here, and knock out three fourths so that we only have one fourth, just one more example of how we can model four and a quarter.
Now, we can focus on measuring lengths. The reality is that students aren't going to have rulers that are marked off where each of the marks is labeled. This is what they're going to be dealing with. So, again, the sooner the students learn what each of the marks represents, the better, because again, that's what they're going to run into in real life. They're going to have to determine that.
Now, let's look at the second part of the standard that deals with showing the data by making a line plot where the horizontal scale is marked off in appropriate units, whole numbers, halves, or quarters. So, we make our line plot, and in this case, the pencil is four and a quarter inches long. So we mark off at that position, four and a quarter. Note the different use of the terms here in the standard. They talk about fourths of an inch, but they also talk about quarters. So, it's important that students realize the meanings here, that they are synonymous. We can refer to a fourth of an inch or a quarter of an inch.
So, we have another pencil, let's see, this one looks like it's about five and a half incheslong. So, we mark that off on our horizontal plot. And, let's say the class, the students have gone through and measured all of their pencils, and here's our final data. Now, of course, to really master this standard, students have got to understand what the numbers and the marks represent on this plot. They have to understand that on the horizontal axis, what we have there is a measured length of pencils in inches.
They have to realize, of course, that not all the numbers are labeled. So, in this case, they’d have to know that that mark right there would be nine and a half. They have to understand that though this horizontal axis is marked off in increments of a quarter-inch, so each one of those marks means one fourth of an inch. And then, of course, to interpret and understand what the graph is saying; in this case, these two marks here say that there's two pencils that are each five and a half inches long.
This standard is an opportunity to lay the foundation for other things. For example, there's a multitude of items that can be measured by students, and they don't have to be limited to one dimension, for example. We can have them measure, say, a rectangle. So, let's do that. So, we take our ruler; let's measure off this length. That looks like it's three and a half. So, we mark it, then we measure the other one; that's three and a half also. Then we need to measure the vertical sides. Let's see, that looks like it's about one and a quarter. We measure the other side, and that looks to be one and a quarter.
So, what we're doing here is laying the foundations for the idea that the opposite sides of a rectangle are equivalent. We're not telling them that, but they'll figure it out. So, when they do their plot, this is what the data will show. And, again, it'll show that I've got two here that are one and a quarter and two that were three and a half.
In addition, items to be measured don't necessarily have to be linear. For example, they can measure cylinders. All we need is some string, and then, of course, we just measure around the cylinder. Then we can take our string and lay it out flat on our ruler, and that'll determine what the distance around the cylinder was. And then, of course, they would take the measurements from the different cylinders, place the data on their horizontal plot, and here's the data. So, we have a horizontal scale that we marked off the distance around different cylinders.
Some of the standards for mathematical practice can be addressed by employing activities, such as those employed in this discussion. What will also help too, is that if we use these instructional strategies with students working in groups, we can do even more. So, here’s four of our standards for mathematical practice. Well, we did have the students reason abstractly and quantitatively. They can construct viable arguments and critique the reasoning of others. For example, they might argue a measurement was six and a half versus six and three quarters. We are definitely modeling the mathematics. We are using appropriate tools strategically, which in this case would be our rulers. And, we are definitely attending to precision. So, by using thoughtful strategies, in addition to addressing this standard, you can also address the standards for practice.