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.