Algebraic Expressions for Increasing Both Dimensions
The students explore situations in which a new garden is formed by increasing both the length and width of an original garden. First, the students find the area of the new garden when the particular increase is given. Specifically, the length and width of the original garden are increased by 2 m, 4 m, and 7 m. For each given increase, the students employ different methods to find the area of the new garden. Then they generalize their methods for an unknown increase and express their generalizations using algebraic expressions.
Emily and Mauricio create a drawing to find the area of the new garden from Episode 1. Then they write several arithmetic equations to represent the area of the new garden.
The students apply what they learned in Episodes 1-2 (drawing a picture, finding the area of the new garden in different ways, and writing arithmetic equations) when both the length and width of the original garden is increased by 4 meters.
Mauricio and Emilymake sense of a dynamic applet in which they can change the number of meters by which one increases both the length and width of a given garden.
Emily and Mauricioselect an amount by which to increase the length and width of a given garden, use a drawing to find the area of the new garden, and then check with the applet.
The students describe two methods, in words and in writing, for finding the area of the new garden when both the length and width of the original garden are increased by the same unknown number of meters.
Emily and Mauricio reflect on another student’s equation: (7 + x) • (3 + x) = 7 • 3 + 7 • x + 3 • x + x2. They discuss how this equation is similar to and different from the equations they wrote in Episode 7. They also reflect on the meaning of equivalence in the garden context.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Identify two ways to find the area of a new garden, which was formed by increasing both the length and width of a given garden by the same amount. For example, suppose that you increase both the length and the width of a rectangular 5 ft by 4 ft garden by 2 ft. Then you can find the area of the new garden in two ways. One way is to multiply the length of the new garden (which is 5 + 2 = 7 ft) by the width of the new garden (which is 4 + 2 = 6 ft). A second way is to find the sum of the area of the original garden (20 ft2) and the extended areas (5 × 2 = 10 ft2; 4 × 2 = 8 ft2; and 2 × 2 = 4 ft2). This is binomial multiplication and repeated distributivity applied in an arithmetic situation.
Write arithmetic equations to represent different methods for finding the area of a new garden, which is formed by increasing both the length and width of the original garden by the same amount. For example, the situation described in the previous bullet can be represented by (5 + 2) • (4 + 2) = 42 ft2 and by (5 • 4) + (5 • 2) + (4 • 2) + (2 • 2) = 42 ft2.
Understand that the two expressions from the previous bullet can be set equal to each other to create the equation (5 + 2) • (4 + 2) = (5 • 4) + (5 • 2) + (4 • 2) + (2 • 2), because each side of the equation represents the area of the new garden.
Generalize experiences with changing the amount by which the length and width of a garden are increased, to thinking about an increase of an unknown number of feet, x. Then the area of the new garden can be expressed using the equation (5 + x) • (4 + x) = (5 • 4) + (5 • x) + (4 • x) + x2.
Conceive of algebraic expressions from multiple perspectives. For instance, the expression (5 + x) • (4 + x) can be concevied as a process (adding the increase in length to the length of the original garden and then multiplying by the sum of the increase in width and the width of the original garden) and as a single object or entity (the area of the new garden).
Mauricio and Emily multiply binomials, such as (7 + x) • (3 + x), by connecting their meaning to area drawings set in a gardening context. As a result, they are able to make sense of (7 + x) • (3 + x) as equal to (7 • 3) + (7 • x) + (3 • x) + x2.
CCSS.MATH.CONTENT.HSA.SSE.B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Emily and Mauricio explore situations in which a new garden is formed by increasing both the length and width of an original 7 m by 3 m garden. They first explore the specific increases of 2 m, 4 m, and 7 m. For each given increase, Mauricio and Emily create and use drawings like the one they developed in Lesson 2 (in which the area of the new garden is decomposed into four parts). Then they generalize their methods for an unknown increase in length and width and express their generalizations using algebraic expressions. This process allows Emily and Mauricio to produce equivalent expressions: (7 + x) • (3 + x) = (7 • 3) + (7 • x) + (3 • x) + x2. The expression on the left represents the method for finding the area of the new garden by multiplying the length of the new garden by its width. The expression on the right represents the method of finding the area of the new garden by summing the areas of the four parts of the new garden.
CCSS.Math.Content.HSA.SSE.A.1.B. Interpret complicated expressions by viewing one or more of their parts as a single entity.
In this lesson, the students make sense of complicated expressions, such as (7 + x) • (3 + x), from multiple perspectives. In the gardening context 7 + x is the length of the original garden added to the increase in length. As a single entity, it is the length of the new garden. Similarly, Emily and Mauricio think about (7 + x) • (3 + x) as both the length of the new garden times its width, as the area of the new garden.
According to the CCSSM, “Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet…or dynamic geometry software.” This lesson provides an opportunity for students to use an applet tool to be able to understand that the symbol x (as an increase in length and width) can take on any value. The applet can also help students explore how increasing both length and width by x meters affects the area of the rectangle. In Episode 4, Mauricio and Emily utilize the applet, and as they experiment with the slider to adjust these dimensions, they realize this tool provides them with insights they didn’t expect (e.g., noticing the square in the bottom left of the new garden is “isolated” and its size seems to increase quickly [2:25]). By increasing the length and width of the garden simultaneously, they observe significant changes in the garden’s area. This exercise also reveals intriguing insights into how the size of the increments affects the relative proportions of the garden’s four newly formed sections.
Continuing their exploration in Episode 5, Emily and Mauricio decide on specific values for the increases in length and width. They employ the visual aid provided by the applet to calculate the area of the enlarged garden, using this interactive tool to validate their predicted calculations. By returning to the applet after making their predictions, they are able to confirm the accuracy of their findings, thereby solidifying their understanding of the mathematical principles governing the changes in the garden’s dimensions and area. This tool helps the pair make connections between the distributive property, binomial multiplication, and visual representations of a mathematical context.
