Distributive Property of Multiplication

Task

Lucia uses the picture below to explain the distributive property for the expression (3+1) $\times$ (5+1):

1. Find (3+1) $\times$ (5+1) using the distributive property.
2. Explain how Lucia's picture relates to your calculation in (a).
3. How can Lucia use a picture to find (3-1) $\times$ (5-1) using the distributive property? Explain.

IM Commentary

The goal of this task is to study the distributive property for products of whole numbers, focusing on using geometry to help understand why (-1) $\times$ (-1) = 1. This important property can be viewed as an application of the distributive property to whole numbers. While it is not easy to ''visualize'' taking a product of negative numbers, the pictures studied here add some tangible evidence why (-1) $\times$ (-1) = 1. A more abstract version of this task, dealing with products of more general negative numbers, can be found here: https://www.illustrativemathematics.org/tasks/1667.

Understanding of the distributive property begins in 3.OA.5 and 5.OA.5 as a means to help calculate products of whole numbers: in these settings, only addition is allowed. It continues to be important in high school and pictures like the ones used here provide valuable insight into the quadratic formula: see https://www.illustrativemathematics.org/illustrations/1827.

Solution

1. Using the distributive property several times we find \begin{align}(3+1) \times (5 +1) &=  3 \times (5+1) + 1 \times (5+1) \\ &= (3 \times 5 + 3 \times 1) + (1 \times 5 + 1 \times 1) \\ &= (15+3) + (5+1)\\ &= 24.\end{align}

2. Below, each of the four rectangles making up the larger 4 by 6 rectangle is labeled:

   

   The four labeled rectangles do not overlap so the sum of their areas is the sum of the large rectangle. This tells us that $4 \times 6 = 3 \times 5 + 3 \times 1 + 1 \times 5 + 1 \times 1$ as we also saw in part (a).

3. One way to use the picture to evaluate $2\times 4$ would be to reverse the process of the previous part.  That is, a $2\times 4$ rectangle is a $3\times 5$ rectangle, minus a $1\times 4$ horizontal rectangle, minus a $2\times 1$ vertical rectangle, and finally, minus the $1\times 1$ square in the top right.  But the expression we get from this:  $$(3-1)\times (5-1)=3\times 5-4\times 1-2\times 1-1$$is not the distributive property.  So we look for another way of thinking about this.
   
   First we draw a picture, color coded in a similar way, showing two copies of 3 by 5 rectangle containing a smaller (3-1) by (5-1) rectangle:    

   

   Here the yellow rectangle is (3-1) by (5-1) while the larger surrounding rectangle is 3 by 5. We can get the 2 by 4 rectangle by taking away the orange and red rectangles from the 3 by 5 rectangle. But then we have taken away the top right square twice so we need to add it back in once. Putting in the areas of these rectangles we find $$(3-1) \times (5-1) = 3 \times 5 -3 \times 1 - 1 \times 5 + 1.$$ Expanding the left hand side of this equation using the distributive property as in (a) gives us $$(3-1) \times (5-1) = 3 \times 5 + 3 \times (-1) + (-1 \times 5) + (-1 \times -1).$$ Since $3 \times (-1) = -3 \times 1 = -3$ we deduce that the last terms in these two expressions are equal, that is $-1 \times -1 = 1$.