Algebraic Expressions for Increasing One Dimension
Mauricio and Emily explore situations in which a new garden is formed by increasing the length of an original garden. First, the students find the area of the new garden when the particular increase in length is given. For example, the length of a garden is increased by 3 ft, 2.5 ft, 3.5 ft, and 5 ft. 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 in length and express their generalizations using algebraic expressions.
Episode 1: Making Sense
Emily and Mauricio make sense of a new situation in which the length of a given garden is increased by 3 feet. They make a prediction about how this increase will affect the area of the new garden.
Episode 2: Exploring
The students make a drawing to find the area of the new garden from Episode 1. Then they talk about the meaning of each number in the drawing.
Episode 3: Reflecting
Mauricio and Emily reflect on their drawing from Episode 2 and write equations to express different ways to find the area of the new garden. The idea of distributing multiplication over addition comes up.
Episode 4: Repeating Your Reasoning
Emily and Mauricio apply what they learned in Episodes 1-3 (drawing a picture, finding the area of the new garden in different ways, and writing arithmetic equations) when the length of the original garden is increased by 2.5 feet.
Episode 5: Exploring
The students explore an applet to think about the area of a new garden when the length of the original garden is increased by 5 ft, 2 ft, and 3.5 ft. They provide a verbal explanation of a general method for finding the area.
Episode 6: Exploring
Mauricio and Emily draw a picture of a new garden when the length of the original garden is increased by an unknown amount. They explore how to use symbols and letters to represent an unknown length and an unknown area.
Episode 7: Exploring
Emily and Mauricio choose variables and write algebraic equations expressing the area of the new garden from Episode 6 in several ways.
Episode 8: Reflecting
Mauricio and Emily reflect on another student’s equation: (5 + x) • 4 = 5 • 4 + x • 4. 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.