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.
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.
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.
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.
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.
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.
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.
Emily and Mauricio discuss the meaning of distributivity in the equation, drawing from Episodes 5 and 6.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Make meaning for x2 as the area of a square with unknown side length, x.
Make sense of situations in which you start with an unknown, rather than the unknown being the change in some dimension. For example, in Lesson 5, binomial multiplication like (4 + x)(3 + x) represented a situation in which you start with a 4 ft by 3 ft garden and increase both the length and width of the original garden by some unknown amount, x, to form a new garden and find its area. In contrast, the binominal mutliplication encountered in this lesson, (x + 4)(x + 3), represents a situation in which you start with a square piece of fabric that is some unknown number of inches long (x) on each side. Then a new piece of fabric is created by increasing the length by 4 inches and the width by 3 inches, and (x + 4)(x + 3) represents the area of the new piece of fabric.
Identify two ways to find the area of a new piece of fabric, which was formed by increasingthe length of a square piece of fabric (with a side length of x inches) by 4 inches and increasing the width by 3 inches. Then you can find the area of the new piece of fabric in two ways. One way is to multiply the length of the new piece of fabric (which is x + 4) by the width of the new fabric (which is x + 3). A second way is to find the sum of the area of the original garden (x2 in2) and the extended areas (x • 4 in2; x • 3 in2; and 4 • 3 = 12 in2). This is binomial multiplication and repeated distributivity applied in an arithmetic situation.
Write an algebraic equation that represents the two different methods for finding the area of a new piece of fabric: (x + 4)(x + 3) = x2 + (4 • x) + (3 • x) + 12.
Apply the distributive property twice in binomial multiplication to see that (x + 4)(x + 3) = x2 + (4 • x) + (3 • x) + 12.
Mauricio and Emily multiply binomials, such as (y + 4) • (y + 3), by connecting their meaning to area drawings set in a gardening context and by applying the distributive property twice. As a result, they are able to make sense of (y + 4) • (y + 3) as equal to y2 + (4 • y) + (3 • y) + 12.
In this lesson, the students interpret each number, symbol, and expression in the equation (y + 4) • (y + 3) = y2 + (4 • y) + (3 • y) + 12, in terms of quantities in a fabric context. For example, y is the unknown length of each side of a square piece of fabric; 4 is the number of inches that the length of the square piece of fabric is increased by; y + 4 is the length of the new piece of fabric; y + 3 is the width of the new piece of fabric; and (y + 4) • (y + 3) is the area of the new piece of fabric.
According to the CCSSM, “Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” This lesson offers students the opportunity to develop rules for operating abstractly with algebraic symbols to re-express or simplify algebraic expressions by using a real-life context of cutting fabric and finding its area. Mauricio and Emily decontextualize a situation involving fabric by writing expressions and equations using variables to account for unknown quantities. For example, in Episode 1, the pair draw a square to represent the fabric being cut and assign variables for each side and the area of the fabric [2:43]. In Episode 2, they revise their model by taking into account that square pieces of fabric have the same length and width, so they re-represent the context using only one variable.
Emily and Mauricio also contextualize complex algebraic expressions and equations by linking the meaning of each term and symbol to quantities in the fabric context. For example, in later episodes, the students reason quantitatively with the equation (y + 4) • (y + 3) = y2 + (4 • y) + (3 • y) + 12, first by letting y = 2 and by creating a drawing that shows both the original and new piece of fabric [Episode 5, 4:42]. They reflect on the meaning of the equation in terms of length, width, and area and ground these quantities in the context of cutting fabric [Episode 6, 8:08]. In Episode 7, they connect the abstract symbolic form of the distributive property used in binomial multiplication to the concrete example of increasing the size of a piece of fabric and expressing the area of the new piece of fabric.
