Multiplying Binomials
In Lesson 5, the students started with a rectangle of given dimensions (e.g., 7 m by 3 m) and increased the length and width by the same unknown amount. In this lesson, the students start by thinking about a square of unknown side length. Then they increase its length by 4 inches and its width by 3 inches. They make sense of binomial multiplication, such as (y + 4) • (y + 3) = y2 + (4 • 7) + (3 • y) + 12, as an expression of two different ways to find the area of the rectangle that is created. Mauricio and Emily also reflect on the repeated use of the distributive property in binomial multiplication.
Episode 1: Making Sense
Emily and Mauricio create a drawing of a square of fabric that is some unknown number of inches long on each side. They select variables to represent the length, width and area of the fabric.
Episode 2: Exploring
Mauricio and Emily rewrite their equation from Episode 1, this time using only two variables. They test their new equation using a length of 4 inches for each side of the square piece of fabric.
Episode 3: Exploring
The students create a drawing of a new rectangular piece of fabric by starting with the square fabric of unknown side length from Episode 2 and increasing its length by 4 inches and its width by 3 inches. They find the area of the new piece of fabric using two different methods.
Episode 4: Reflecting
Emily and Mauricio reflect on the two algebraic expressions that they wrote in Episode 3 and connect each expression to its meaning in the fabric context.
Episode 5: Exploring
Mauricio and Emily explore whether or not their two algebraic expressions from Episode 4 are equivalent. Specifically, they provide justifications for forming the following equation: (y + 4) • (y + 3) = y2 + (4 • 7) + (3 • y) + 12.
Episode 6: Reflecting
The students reflect on the meaning of the equation from Episode 5: (y + 4) • (y + 3) = y2 + (4 • 7) + (3 • y) + 12. They let y = 2 inches and create a drawing that shows both the original and new piece of fabric. Then they reflect on the meaning of the equation in terms of lengths, widths, and areas.
Episode 7: Reflecting
Emily and Mauricio discuss the meaning of distributivity in the equation, drawing from Episodes 5 and 6.