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.
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.
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.
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.
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.
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.
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.
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.
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 decreasing the length of a given garden. For example, if you decrease the length of a rectangular 5 ft by 4 ft garden by 2 ft, then you can find the area of the new garden by either finding the area of the original garden (5 × 4 = 20 ft2) and taking away 8 ft2 (2 × 4), or you can subtract the amount of decrease from the length of the original garden (5 – 2 = 3 ft) and multiply that new length by the width (3 × 4 = 12 ft2). This is distributive thinking applied in the situation.
Write arithmetic equations to represent different methods for finding the area of a new garden, which is formed by decreasing the length of the original garden. For example, the situation described in the previous bullet can be represented by (5 – 2) • 4 = 12 ft2 and by (5 • 4) – (2 • 4) = 12 ft2.
Understand that the two expressions from the previous bullet can be set equal to each other to create the equation (5 –2) • 4 = (5 • 4) – (2 • 4), because each side of the equation represents the area of the new garden.
Generalize experiences with changing the amount by which the length of a garden is decreased to thinking about a decrease in length by some unknown number of feet, x. Then the area of the new garden can be expressed with the equation (5 – x) • 4 = (5 • 4) – (x • 4)
Conceive of algebraic expressions from multiple perspectives. For instance, 5 – x can be conceived of as a process (subtracting some unknown number of feet from 5 ft) and as a single object or entity (the length of the new garden).
Common Core Math Standards
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 decreasing the length of an original garden, which is 12 ft long and 9 ft wide. Although the structure of this lesson is similar to that of Lesson 3, it poses a unique challenge, namely for students to figure out how to express subtraction in an area drawing. They resolve this challenge while working with the specific decreases in length of 4 ft and 7 ft. For each given decrease, Mauricio and Emily adapt their two different methods from Lesson 3 to find the area of the new garden. Then they generalize their methods for an unknown decrease in length and express their generalizations using algebraic expressions. This process allows Emily and Mauricio to produce equivalent expressions: (12 – x) • 9 = (12 • 9) – (x • 9). 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 starting with the area of the original garden and subtracting the area that is being removed.
CCSS.MATH.CONTENT.6.EE.A.3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x.
In this lesson, the students apply the distributive property to expressions like (12 – x) • 9 to produce the equivalent expression (12 • 9) – (x • 9). Unlike Lesson 3, where they found distributing more straigthforward, in this lesson, the students have to explore whether or not it makes sense to distribute multiplication over subtraction. In this exploration, it helps Emily and Mauricio to try to connect the meaning of each number, symbol, and term to an area model set in a gardening context.
According to the CCSSM, “Mathematically proficient students look closely to discern a pattern or structure…. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.” In the first two episodes, Emily and Mauricio notice similarities in the structures of two garden contexts. They extend their prior understanding of a garden being increased in size and apply it in a context in which the garden’s size is decreasing. For example, in Episode 2, they draw a rectangle and label its dimensions before partitioning it to show a new garden with new labels [2:26]. This is similar to the structure they used in previous lessons where the garden was increasing in size. In Episodes 3 and 4, the pair explore several specific instances in which the garden’s length decreases by known amounts [e.g., Episode 3, 2:21; Episode 4, 1:19]. This results in a model of the garden context, which they use in Episode 5 to generalize their method to a decrease by an unknown length [1:12].In the final episodes, they leverage the structure of their pictorial model to make connections between algebraic symbols and verbal representations of their method [e.g., Episode 7, 0:27].
