Binomials Lesson 3 (Teachers)

This lesson can help students:

• Identify two ways to find the area of a new garden, which was formed by increasing the length of a given garden. For example, if you increase the length of a rectangular 5 ft by 4 ft garden by 3 ft, then you can find the area of the new garden by either finding the sum of the area of the original garden and the increased area, or you can add the length of the original garden to the increase (for a total of 8 ft, in this case) and multiply that by the width. 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 increasing the length of a given garden. For example, the situation described in the previous bullet can be represented by (5 + 3) • 4 = 32 ft² and by (5 • 4) + (3 • 4) = 32 ft². 
• Understand that the two expressions from the previous bullet can be set equal to each other to create the equation (5 + 3) • 4 = (5 • 4) + (3 • 4), because each side of the equation represents the area of the new garden.
• Generalize experiences with changing the amount of increase in the length of a garden to increasing the length of the original garden 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 (adding some unknown number of feet to 5 ft) and as an entity (the length of the new garden).

• 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. 
  
  Mauricio and Emily explore situations in which a new garden is formed by increasing the length of an original garden, which is 5 ft long and 4 ft wide. 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 two 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. This process allows Emily and Mauricio to produce equivalent expressions: (5 + x) • 4 = (5 • 4) + (x • 4). 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 adding the area of the original garden to the area by which the original garden is increased. 
  
• 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 (5 + x) • 4 to produce the equivalent expression (5 • 4) + (x • 4). They connect the meaning of each number, symbol, and term to an area model set in a gardening context, which helps deepen their understanding of the distributive property. 

CCSS.MATH.PRACTICE.MP3. Construct viable arguments and critique the reasoning of others.

According to the CCSSM, “Mathematically proficient students … justify their conclusions, communicate them to others, and respond to the arguments of others. They … distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.” This lesson provides opportunities for students to justify the equivalence of algebraic expressions representing area, first by appealing to the quantities in a drawing and then by connecting that quantitative meaning with more abstract properties and operations (such as the distributive property). In Episode 1, Emily and Mauricio make different predictions about the area of a new garden when the length of a 5 ft by 4 ft garden is increased by 3 ft. They take turns explaining and listening to each other’s arguments and eventually find an error in Emily’s logic [3:00]. In Episodes 1 and 2, the students develop two methods for finding the area of the new garden. In Episode 1, they calculate the area by multiplying the new length (5 ft + 3 ft) by the width (4 ft). In Episode 2, they collaborate to create a new drawing and develop a second method, adding the area of the original garden (20 ft²) to the area of the “extended part” (12 ft²). This provides opportunities for the pair to compare and contrast both methods.

Later, Emily and Mauricio create two arguments to demonstrate the equivalence of their methods, showing that 4 • (5 + 3) = 4 • 5 + 4 • 3. One justification the pair provides is based on their drawing and its context in the garden, while another justification relies on the distributive property [Episode 3]. In the remaining episodes, they work to generalize their methods to account for an unknown increase in length and extend both justifications for the equivalence of expressions to the equation (5 + x) • 4 = 5 • 4 + x • 4, using both contextual quantities and the distributive property. Finally, the pair disagree about the meaning of the symbol “?”, with Emily thinking it should represent the same quantity and Mauricio thinking it can represent any unknown quantity. They discuss their ideas and critique each others’ reasoning [Episode 6, 5:45]. Throughout, Emily and Mauricio construct arguments for why their methods work and how each method produces the same area.