Nancy asked few if any questions of her students because her focus was on directing students on how to use the tiles. However, she could have used questioning to engage students in the SMP. Suppose she has base-10 blocks and algebra tiles available for the students. The initial problems should be restricted to positive numbers so that students detect patterns and will get correct answers to help in visualizing the distributive property. Consider the following questions with possible student answers after each question. Not all of the following questions need be posed if students can see how to proceed on their own based on prior experiences with base-10 blocks.
How can we represent the product of (5)(3) geometrically?
Use blocks to make a rectangle with length 5 units and width 3 units. Its area has 15 unit squares = 15 square units.
Represent the product of (12)(13) geometrically).
Some students may use the base-10 blocks to show repeated addition of 13, twelve times. Applaud the process, and then ask them to try it with the fewest number of blocks. This results in a 12 x 13 rectangle or array that looks like Figures 3.2 and 3.3 when x = 10. In that case its area is 100 + 30 + 20 + 6 or 156 square units.
Can we always represent the product of two numbers as the area of a rectangle? Why?
Yes. Because multiplication is repeated addition, it can be represented as an array or rectangle.
How can we represent the product of (10 + 2)(10 + 3) geometrically?
This is the same as using the fewest blocks for 12 x 13.
What properties for multiplication are illustrated in (10 + 2)(10 + 3)?
Distributive property; combining like terms with the distributive property.
How can we show the product of (x + 2)(x + 3) geometrically? What is the product? How does that connect to previous examples? How does that illustrate the distributive property?
Students should connect this product directly to (10 + 2)(10 + 3).
In Figure 3.2, the dimensions of the rectangle are shown, and its area, (x + 2)(x + 3) square units, is what students must understand to be that which they are seeking. Once students place tiles in the space to fill the entire area of the rectangle (see Figure 3.3), they will have determined its area to be x2 + 3x + 2x + 6, or x2 + 5x + 6. Connection to the distributive
FIGURE 3.2 Preparing to Multiply Binomials with Algebra Tiles Using the Area Model for Multiplication
FIGURE 3.3 Area Model for Multiplication Using Algebra Tiles
property is fundamental to this process and should be reinforced at this point so that students see how it connects to the area model. It is also important to see the abstraction to the distributive to justify why we can add like terms: 2x + 3x = (2 + 3)x = 5x (SMP7).
Students are now prepared to use distributive property to multiply binomials with the tiles (SMP2, SMP4) and to look for patterns to explain a process for multiplying binomials (SMP3, SMP8).