Knowledge in Brief
CP, AP, and DP
The Common Core mathematics glossary lists the formulas of the properties as follows:
Commutative property of multiplication a x b = b x a
Associative property' of multiplication (a x b) x c = a x (b x c)
Distributive property of multiplication over addition a x (b + c) = a x b + a x c
The definition of the CP refers to the multiplication of two numbers while the AP refers to the multiplication of three numbers (even though both can be applied to situations where the multiplication involves more numbers). As is the case with addition, there is a common conflation between the CP and the AP of multiplication. Thus, it is important to stress that the CP deals with changing the order of numbers while the AP deals with changing the order of operations. The use of the AP is indicated by the use of parentheses that changes the grouping from the first two numbers to the latter two numbers.
With regard to the DP, as mentioned in the earlier section, it can be used in two directions: regular or opposite (Ding 8c Li, 2010). The “regular” direction refers to transformation from a x (b + c) to a x b + axe. This is the major use of the DP in the U.S. curriculum. Very often, students are taught to break apart one factor and then use the DP to generate an expanded notation. For instance, 102 x7 = (100 + 2)x7 = 100x7 + 2x7 (elaborated upon in Section 5.5). However, the DP can also be used in an “opposite” direction, in which a common multiple is factored out to transform a x b + a x c into a x (b + c). Being able to flexibly use the DP in the opposite direction is particularly critical for formal algebraic learning (e.g., solving algebraic equations). Koedinger et al. (2008) found that 71% of sampled U.S. undergraduates could not solve an equation like x- 0.15л; = 38.24 even though they could solve an equation with a single variable. If these students were sensitive to the DP, they could apply the property in the opposite direction to transform л: - 0.15x = 38.24 into (1 - 0.15) x = 38.24 and then solve this equation. Unfortunately, elementary mathematics textbooks in the U.S. do not provide sufficient opportunities for students to apply the DP in the opposite direction, which is quite different from the Chinese textbooks (Ding 8c Li, 2010). This may be one of the sources of cross-cultural differences in U.S. and Chinese students’ responses to items like 98 x 7 + 2 x 7 in my project (see Section 5.1).
Expectations from the Common Core
The Common Core expects elementary students to understand the basic properties in two domains: (a) Operations & Algebraic Thinking, and (b) Number & Operations in Base Ten. In the domain “Operations & Algebraic Thinking,” students are expected to understand the properties of multiplication by third grade. Students are also expected to fluently multiply and divide within 100 by applying the basic properties. Finally, they should be able to identify arithmetic patterns and explain them using the properties of operations. Below is a statement from the Common Core (NGAC 8c CCSSO, 2010, p.23),
Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3x5x2 can be found by 3x5 = 15, then 15x2 = 30, or by 5 x 2 = 10, then 3x10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.). (CCSS.MATFI. CONTENT. З.ОА.В.5).
Note that the Common Core does not clearly state whether students in G3 should know the names of the basic properties. In fact, my observations of classroom lessons and textbook presentations indicate that many of these strategies were given specifically non-technical names in U.S. classrooms. For instance, “thinking of 6 x 4 to solve 4 x 6”is called “using known facts to find unknown facts” or “turn-around facts.” Likewise, some teachers describe using related facts for computation (e.g., 6x4 = 24, 3 x8 = 24) as the “doubling and halving strategy,” instead of identifying it as an application of the AP of multiplication: 6 x 4 = (3 x 2) x 4 = 3 x (2 x 4) = 3x 8. Furthermore, application of the DP such as in stating, “when solving 8x7, one may think of 8 x 5 and 8 x 2” is sometimes called “breaking up a number to multiply” but at other times called “using known facts to find unknown facts” (same name as a CP strategy above). Regardless of when (or whether) students should know these property names, I believe that elementary teachers themselves should clearly know what properties undergird each of these strategies.
In addition, the domain “Number 8c Operations in Base Ten” also expects that students in G3 to G5 should use basic properties, along with place value and inverse relations, to solve multiplication or division arithmetic problems. These tasks may vary based on grade level. Below are examples:
G3: 9 x 80;
G4: 3 x 1234, 31 x 34, 1234 + 3;
G5: 384- 32, 12.36-3
Despite differing levels of complexity, solving each of these tasks demands an application of the basic properties.