## IM Commentary

The purpose of this task is to help students picture the multiplicative structure of the number 30 in different ways. Some of this structure is easier to communicate using grouping symbols, as it is easier to interpret $2\times(3\times5)$ than $2\times3\times5$ in terms of equal-sized groups even though they both mean "2 times 3 times 5." Because there are many different ways to draw the pictures called for in this task and it is the interpretation of the pictures that matters most, the task is very well suited for a whole-group discussion where students show and explain the pictures they have drawn to their classmates.

The pictures shown in the solution are increasingly more abstract as they go on. The first pictures in each part try to make the groups more explicit by using colors or divisions within the array; as the solution continues, the partitions of the array are described verbally. In this way, students can see that if they start with the same array, they can partition it in different ways to bring out different factors. If the teacher has 1/4-inch graph paper and colored pencils on hand, it will be much easier for students to draw the pictures called for in this task than if they are required to draw them free-hand.

The term "factor" has two different meanings: in the context of whole-number multiplication, a factor of a whole number is another whole number that divides it evenly (that is, with remainder 0). On the other hand, in the context of any explicitly defined multiplication problem

$$A\times B = C$$

we call $A$ and $B$ the factors and $C$ the product, even when $A$, $B$, and $C$ are not whole numbers. In fifth grade, students extend their understanding of multiplication to include factors that are not whole numbers, but this task invokes the first meaning, namely, whole-number factors. This task is a good precursor to 6.NS.B.4, where students are expected to find common factors and multiples, and thus need to be very comfortable with equations like

$$70 = 2\times5\times7$$

One of the best geometric representations of a number with three factors is as the volume of a rectangular prism, which is also important in 5th grade. See 5.MD,OA You Can Multiply Three Numbers in Any Order for a task that connects the different ways of multiplying the prime factors of 30 to the rectangular prism with dimensions 2, 3, and 5 units.