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.
Episode 1: Making Sense
Mauricio and Emily make sense of a new situation in which both the length and width of a given garden are increased by 2 meters.
Episode 2: Exploring
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.
Episode 3: Repeating Your Reasoning
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.
Episode 4: Making Sense
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.
Episode 5: Repeating Your Reasoning
Emily and Mauricio select 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.
Episode 6: Exploring
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.
Episode 7: Exploring
Mauricio and Emily express their written generalizations from Episode 6 using algebra.
Episode 8: Reflecting
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.