Algebraic Expressions for Decreasing One Dimension
Emily and Mauricio explore situations in which a new garden is formed by decreasing 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 decreased by 4 ft and then by 7 ft. For each given decrease, the students employ different methods to find the area of the new garden. Then they generalize their methods for an unknown decrease in length and express their generalizations using algebra.
Episode 1: Making Sense
Mauricio and Emily make sense of a new situation in which the length of a given garden is decreased by 4 feet. They make a prediction about how this decrease 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. This is challenging, because they have to figure out how to express subtraction in a drawing.
Episode 3: Reflecting
Emily and Mauricio reflect on their drawing from Episode 2 and write equations to express different ways to find the area of the new garden. They explore whether or not they can distribute multiplication over subtraction.
Episode 4: Repeating Your Reasoning
Mauricio and Emily 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 decreased by 7 feet.
Episode 5: Exploring
The students draw a picture of a new garden when the length of the original garden is decreased by an unknown amount. They explore how to use algebra to express their two general methods for finding the area of the new garden.
Episode 6: Exploring
Emily and Mauricio explore whether or not they can set two algebraic expressions from Episode 5 equal to each other by substituting 2 ft for the variable representing how much the length of the original garden is decreased by.
Episode 7: Reflecting
Mauricio and Emily reflect on the meaning of equivalence in another student’s equation: (12 − y) • 4 = 12 • 9 − y • 9. They identify what each symbol and term in the equation means in the garden context.