Show students the following visual math talk prompt and be prepared to pause the video where indicated:

You’ll notice that a series of four (4) buildings appear and students are asked:

Without counting, determine the number of windows on
the street facing side of each apartment building.

On each of the buildings, students are given the number of rows and columns of windows and they are all (likely) known facts for students.

The goal is for students to calculate the partial products or the number of windows in “parts” of a much larger apartment building that is not a known multiplication fact.

While students may not immediately notice, if they instinctively begin to decompose the factors into smaller, more accessible quantities for multiplying, they may recognize that the number of windows on the first four (4) apartment buildings is actually equivalent to the total number of windows on the larger building.

For example, if students decompose 13 rows into (10 + 3) rows and 16 columns into (10 + 6) columns, the resulting partial products will be:

10 x 10

10 x 6

3 x 10

3 x 6

These smaller rectangular arrays can be recomposed to cover the large 13 x 16 array that represents all of the windows on the street facing side of the purple building.

Although it might not be obvious initially, it is important to note that the standard algorithm for multiplication is essentially leveraging the same partial products strategy that works due to the distributive property of multiplication.

Despite the fact that we are sharing this strategy in a horizontal fashion, the traditional “stacking” method for multiplying 2-digit by 2-digit numbers results in the exact same partial products.

It is also worth noting that this method that emerges quite naturally when students are leveraging an array model to multiply unfriendly numbers is often taught later in middle school and algebra 1 courses as “FOIL” in order to multiply binomials.

Consider reading about the Progression of Multiplication to get a more thorough sense of how the distributive property, partial products and the standard algorithm are connected.