Area Models with Whole Numbers
I will introduce polynomial multiplication with my students by first getting them acclimated with multiplying whole numbers using the box method. We will study multiplication of two-digit numbers using the area model, then transition to the box method for multiplication when one (or both) factor has three or more digits. The area model is a great way of illustrating to students how quickly the value represented by each digit increases when moving to the left, place by place, and to help them visualize how each part of the number contributes to the overall product. I will emphasize to my students to make notes and observations as the powers of ten increase. I would explicitly like my students to observe the relationships within the antidiagonals & increasing powers of ten using the box method. As they transition to using this method with variables, I want them to notice how those general ideas around the powers of ten are justified the same way, based on the rules of arithmetic.
These models will be used in rich discussions and activities as the unit progresses, there are a few explicit examples below.
One by Two (1 × 2) Digit Multiplication
In the examples below, the area model is set up by having one row and two columns. This model is consistent with the number of digits/powers of ten that are available in the two factors being multiplied. Each factor is broken into its respective place value parts, which are and then multiplied by the place value parts of the other factor, one place value part at a time. This process is repeated until all parts have been multiplied. The final product is then determined by the sum of all parts. In the first three examples, the left-hand factor is a single digit, so has only one place value part.
In the above examples you can see how the product of the factors in the ones place, play a significant role in the final result. Students will be able to visualize how quickly and easily the powers of tens can increase using this box method.
Two by Two (2 × 2) Digit Multiplication
In order to introduce the box method of multi-digit numbers, I will use the basic area model in detail for my students. Using the area model in parallel with the box method will allow students to see how the box method is the shorthand way of multiplying multi-digit numbers. The following examples show a similar process with factors that both have more than one digit. This is visually represented by adding an additional row to allow for the additional digit in the first (red) factor.
I will ask students what observations did they make as they moved to multiplying two-digit factors? How did the area model/box change? What do you notice about the products inside each box? I’m looking for students to make connections with the powers of ten increasing as you move up the columns, or move to the left in the rows. I hope that they will also observe that the same powers of ten appear along the antidiagonals from each other (as illustrated in the example below).
Three by Three (3 × 3) Digit Multiplication
As with the previous examples of box models, as the number of digits/place value parts in one of the factors increases, so does the number of rows or columns of the box. This model also offers a very helpful view of numbers with the same order of magnitude being directly diagonal to one another. I would use this example as an opportunity to really have a deep discussion as to how this way of multiplication allows for such organization.
Can you recall a time where you identified something similar? I’d point out to my students that in fact looking at the basic multiplication chart offers such a pattern also. Due to the associative and commutative rules, the products of the place value parts are just the products of the base ten parts times the products of the digits.
I would like to take a moment to point out that with this example in particular you can really begin to introduce the concepts of polynomial multiplication by way the of the fourth stage of the 5 Stages of Place Value. Below I will take a moment to demonstrate with my students the possibility to connect the process of multiplying whole numbers with multiplying in polynomial form.
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