Recognize area (3.MD.7.d), angle (4.MD.7), and volume (5.MD.5.c) as additive

According to the CCSS, students are expected to recognize that area, angle, and volume are additive. But what does it mean for these attributes to be additive? If a measurable attribute is additive, that means that the measurement of the whole is the sum of the measurements of the (non-overlapping) parts. Thus, the area of an L-shape can be calculated by subdividing the shape into two rectangles or making a large rectangle then subtracting the area of the small rectangle that was added from the area of the large rectangle.

Actually, most measurable attributes studied in elementary schools are additive. In fact, the measurement process - select a unit, 'cover' the object with the unit without a hole or an overlap, and count the number of the unit - only works when an attribute is additive. Thus, other measurement attributes discussed in the K-5 standards - length, capacity (liquid volume), and elapsed time - are also additive. However, attribute such as speed and density studied in upper grades are not additive and their measurements are actually ratios of two other measurement. Therefore, one of the first ideas students need to understand as they study attributes that are ratios of two other measurements is focusing on one attribute is not sufficient.

As elementary school students learn different attributes are additive, we want them to understand about the measurement process explicitly. Whether that happens with area in Grade 3 or later will be a curricular decision.

It is a little curious that the CCSS does not explicitly state that students understand length as additive. However, when 2.MD.4 says, "Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit," implicit in this standard is that we can perform subtraction of two length measurements. Thus, students are indeed dealing with the fact that length is additive.

Speaking of additivity of length, the word "perimeter" does not appear in the CCSS until Grade 3 when a cluster, Geometric Measurement, is introduced. In Grade 3, students are expected to "recognize perimeter as an attribute of plane figures and distinguish between linear and area measures." I often hear US teachers lamenting how students confuse area and perimeter. I think one reason for this confusion is because area and perimeter are often introduced simultaneously. However, the idea of perimeter should be discussed as soon as standard units for length are introduced in Grade 2. Just as students can determine how much longer one object is than another (2.MD.4), they can also find the total length by putting those objects end-to-end. Then, as a special application of this idea, students can think about the total length of sides around different geometric shapes they have seen. Perhaps the term "perimeter" can also be introduced. Then, in Grade 3, when students compare the "spaciousness" of two flat regions - for example, comparing two picnic blankets - we should help them realize explicitly that the perimeter is not an appropriate way to compare spaciousness. We can then introduce "area" as an attribute that measures how much space inside a flat figure. Having this experience at the time area concept is introduced will, I believe, reduce the amount of confusion students have about area and perimeter. Of course, if the focus of area instruction is just on how to calculate the area, then students' understanding of these ideas will continue to be limited.

2.MD.9

2.MD.9
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Recently, I was working on a project in which the above standard (2.MD.9) was one of the foci. Although I had a couple of different questions about this standard, I want to focus on the last part of the standard.

As I grew up in Japan and going through the Japanese elementary and secondary schools, I never came across a line plot as a way to represent data. Granted that the Japanese school mathematics curriculum back then placed a very limited emphasis on data analysis. However, I did learn bar graphs, broken line graphs, histograms, circle graphs, etc. which are pretty standard statistical graphs. In the CCSS are mentioned several times in Grades 3 through 5, and I wondered why. So, I read the progression document on K-5 Measurement & Data. In the document, the authors discuss that there are two paths in the K-5 MD (data analysis portion) standards. One path deals with categorical data and the other with measurement data. The authors state that in the categorical data path, they focus on bar graphs to represent and analyze such data. On the other hand, measurement data are represented on a line plot.

Although I understand the importance of students understanding the nature of data, I found their focus on bar graphs for categorical data and line plot for measurement data to be rather strange. Students should definitely understand that the nature of data influences what analysis is possible. For example, with categorical data, it is not possible to calculate the mean or the median. However, do we always represent measurement data on a line plot? Are there any cases where we might use a bar graph to represent measurement data?

Suppose a second grade class collected the data on the height of the students in the classroom (in nearest cm). In that situation, doesn't it make sense to represent the data (heights) using a bar graph - or anything other than a line plot? In fact, the mean of the data set is obtained when we even out the height of those bars (although second graders won't be calculating the mean). So, saying that a line plot is the primary representation of measurement data seems to be rather strange.

It seems like the distinction being discussed is more about whether or not we are looking at the actual measurements or the frequencies (counts) of measurement data. A line plot is used to represent the frequencies of measurement data - how many students are 134cm tall, how many are 135cm tall, etc.. Eventually, students may create intervals and represent the frequency distribution as a histogram. Or, if we collect data to see how (air) temperature changes during the school day. It seems perfectly reasonable to represent the data using a broken line graph. In a way, the distinction may be whether the measurement data are represented on the horizontal axis or the vertical axis. I am not sure if the distinction is categorical or measurement data. When I asked a statistician, she told me that when we create intervals to make a histogram, we are "categorizing" the data. So, in a certain sense, an interval is like a category, like "strawberry" as the favorite ice cream flavor.

Anyway, I do think it is important that students understand the distinction between categorical data and measurement data. I also think it is important for students to be mindful about whether or not we are interested in the actual values of measurement data or frequencies. However, when we represent frequencies of measurement data, there may be some things we shouldn't do. For example, with categorical data (like favorite ice cream flavors), it is perfectly appropriate to order the data from most frequent to the least frequent ones as we draw a bar graph. However, with measurement data on a horizontal axis (a line plot), such a manipulation is not appropriate.

Double Number Line Diagrams

6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

UPDATE (see below)

I just created a 9-minute video on the basic idea of double number line diagrams. Although double number line diagrams are specifically mentioned in a Grade 6 standard, if we want students to use them to solve ratio and rate problems in Grade 6, they really need to be familiar with the representations by then. That means they should be really introduced in elementary schools.

Anyway, I hope to create additional videos to elaborate how double number lines may be used to represent students' own reasoning, and eventually become their own thinking tools.

UPDATE: February 4, 2012
I just uploaded another video on double number line. I apologize for some background noises. Also, at one point I said something like "to go from 0.8 to 0.1..." when I really should have said "to go from 0.8 to 1..." The app I'm using does not allow me to edit the video, and I didn't want to re-do the whole video. So, please excuse my errors.