In particular, different tasks were chosen in order to present particular concepts in different ways, such as those tasks about the distributive property (a critically important understanding that numbers can be partitioned to make operating with them easier, for example 34 x 7 can be considered as [30 x 7] + [4 x 7]) and inverse relationship (another important understanding that sometimes it is easier and more efficient to use division even though the problem appears to be a multiplication one, for example, 24 x ?

Another aspect of multiplicative thinking that has been explicitly taught is the distributive property and this has been done by exposing students to a variety of strategies and representations.

Knowing the technical vocabulary, such as

distributive property, is not as important as realizing how such a property is used.

Moreover, the structure of the Smarties-Box Challenge rewards possessing a comprehensive working knowledge of the

distributive property (see Will's group), which has been described as being at the core of multiplication (Kinzer & Stafford, 2014).

The

distributive property is much more visible, as is the place value of all the numbers involved in the problem:

The final two strategies are considered to be multiplicative strategies as they reflect both an understanding of the composite unit and

distributive property.

There was no depiction of the desired representation of six groups of 17 sticks (Figure 2) which might reflect an understanding of the standard place value partition and/or the

distributive property.

The

distributive property can be equally well shown with an array (see Figure 4).

For example, with the following strings, I aim for the students' discovery of doubles and halves in the first set and of the

distributive property in the second set.

It also encourages the development of the

distributive property (that is that 14 X 3 is equivalent to 10 X 3 + 4 X 3, another important mental computation strategy) and the link between multiplication and area.

Their solutions for solving these different recipe problems illustrate an implicit understanding of the

distributive property.

To overcome these procedural and conceptual errors, Crystal needed to understand the

distributive property of multiplication.