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Seidel's distributive property

Task Statement

Ms. Seidel is introducing the distributive property. To motivate her students, she wants to give them an example that will focus their attention on how using the distributive property can simplify computation. In which of the following examples will the use of the distributive property most simplify the computation?

a) 12 x 29 + 12 x 38 = ___  

b) 17 x 37 + 17 x 63 = ___

c) 13 x 13 + 15 x 15 = ___

d) 16 x 24 + 16 x 24 = ___

Task of Teaching

Choosing examples: selecting a problem for an exercise

IM Commentary

Examples shape instructional opportunities, however crafting and choosing good examples requires mathematical dexterity and skill in doing mathematical problems while tracking on instructional goals. This task asks for an example in which the distributive property can be used to simplify computation significantly. The purpose of this task is to see whether teachers flexibly consider different ways of evaluating the expressions using the distributive property and, simultaneously, what these imply for efficiency of the computation. It requires recognizing that the distributive property does not avoid multiplication, but does allow for regrouping quantities into powers of 10, which greatly simplifies multiplication in a base ten system. The task is currently written as a multiple-choice item for assessment. But it also can be used for launching a discussion about the nature of examples for which the distributive property is useful.

The mathematical task of teaching is choosing examples, but the teaching scenario needs to create a realistic need for choosing an example that requires the distributive property. In this scenario, the pedagogical purpose is to motivate learning of the distributive property. In particular, the scenario proposes motivating the distributive property by giving an example that will focus students’ attention on how using the distributive property can simplify computation. This means that the example needs to provide a sharp contrast in the extent to which the computation is simplified by using the property relative to not using it. The examples given in the options in this task are selected to create such a contrast, where only option (b) significantly reduces the complexity of the multiplication.  The instructional setting of introducing the distributive property contributes to a sense that the scenario is realistic.