The purpose of this Warm-up is to reinforce the meaning of an area diagram, writing the area in two distinct ways to illustrate the distributive property. In this lesson, students will use the distributive property for increasingly complex expressions, and the area diagram is a useful support.
Launch
Give students 2 minutes to write as many expressions as they can think of.
Activity
None
Write as many expressions as you can that represent the area of this rectangle.
Large outer rectangle partitioned into 2 smaller rectangles, vertical side 10. First rectangle top side length 6. Second rectangle top side length 3 x.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to recall different ways to think about the area of the entire rectangle in order to write equivalent expressions. Ask a student to share an expression and record it for all to see. Ask if students agree that the expression represents the area of the rectangle, and if not, how they would amend it. Then ask for additional expressions and record them. Ensure at least one expression involves the height by the total width, such as , and another expression involves the sum of the areas of the two smaller rectangles, such as .
2.2
Activity
Standards Alignment
Building On
Addressing
7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
In this activity, students practice rewriting expressions using the distributive property. Each expression involves subtraction or negative numbers.
Before completing the task, students are given an example student statement that is intentionally unclear, incorrect, or incomplete. Students critique the statement and improve it by clarifying meaning, correcting errors, and adding details (MP3).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Arrange students in groups of 2. Instruct them to take turns writing an equivalent expression for each row. One partner writes the equivalent expression and explains their reasoning, while the other listens. If the partner disagrees, they work to resolve the discrepancy before moving to the next row.
Draw students’ attention to the organizers that appear above the table, and tell them that these correspond to the first three rows in the table. Let students know that they are encouraged to draw more organizers like this for other rows, as needed.
Use Critique, Correct, Clarify to give students an opportunity to improve sample reasoning for why an expression is equivalent by correcting errors, clarifying meaning, and adding details.
Display this first draft:
“The expression is equivalent to because you times by .”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Display and review these criteria:
Using “distributed” or “distributive property”
Using “factor,” “expand,” or “terms.”
Rewriting subtraction as adding the opposite
Rearranging terms that are being added
Labeled an area diagram
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code each term of the factored and expanded expressions, in both the area diagram and the first row of the table. Supports accessibility for: Visual-Spatial Processing
Activity
None
In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.
factored
expanded
Student Response
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Building on Student Thinking
If students are unsure how to proceed, consider asking them to draw an organizer to represent the terms in the expression or to think about how subtraction can be rewritten as adding the opposite.
Activity Synthesis
Much of the discussion will take place in small groups. The purpose of the discussion is to use diagrams and carefully rewrite expressions to understand why expressions are equivalent.
Display the correct equivalent expressions and work to resolve any discrepancies. Expanding the term may require particular care. One way to interpret it is to rewrite as . If any confusion about handling subtraction arises, encourage students to employ the strategy of rewriting subtraction as adding the opposite.
To wrap up the activity, ask:
“Which rows did you and your partner disagree about? How did you resolve the disagreement?”
“Which rows are you the most unsure about?”
“Why are all the expressions in this column called ‘factored expressions’? Why are these called ‘expanded expressions’?”
“Describe a process or procedure for taking a factored expression and writing its corresponding expanded expression.”
“Describe a process or procedure for taking an expanded expression and writing its corresponding factored expression.”
2.3
Activity
Standards Alignment
Building On
Addressing
7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
In this activity, students factor more complex expressions. This activity is an opportunity to notice and make use of structure (MP7) in order to apply the distributive property in more sophisticated ways.
Launch
Display the expression , and ask students to calculate as quickly as they can. Invite students to explain their strategies. If no student brings it up, ask if the three numbers have anything in common. (They are all multiples of 9.) One way to quickly compute would be to notice that can be written as or which can be quickly calculated as or 0. Tell students that noticing common factors in expressions can help us write them with fewer terms, which can make the expressions easier to use.
If needed, remind students that the instruction “Factor each expression” means to apply the distributive property to rewrite each sum as a product: . The result is an equivalent expression with fewer terms.
Keep students in the same groups. Give them 5 minutes of quiet work time and time to share their expressions with their partner, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy, such as “I see a common factor is _____, so . . . .” “I noticed _____, so I . . . .” “What do you notice about . . . ?” or “I agree/disagree because . . . .” Supports accessibility for: Language, Social-Emotional Functioning
Activity
None
Factor each expression. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to highlight the use of the distributive property. For each expression, invite a student to share their process for rewriting it with fewer terms.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
Lesson Synthesis
Share with students, “Today we learned to factor and expand expressions by using the distributive property.”
To review this new vocabulary, consider asking students:
“What does it mean to expand an expression?” (Multiply the factor outside the parentheses by each term inside the parentheses.)
“Give an example of expanding an expression.” ()
“What does it mean to factor an expression?” (Divide each term in the expression by a common factor and write that factor outside a set of parentheses.)
“Give an example of factoring an expression.” ()
Student Lesson Summary
Properties of operations can be used in different ways to rewrite expressions and create equivalent expressions. For example, the distributive property can be used to expand an expression such as to get .
The distributive property can also be used in the other direction to factor an expression such as . In this case, we know the product and need to find the factors.
The terms of the product go inside:
Think of a factor each term has in common: and each have a factor of 4. The common factor can be placed on one side of the large rectangle:
Now think: "4 times what is 12x?" and "4 times what is -8?" Write the other factors on the other side of the rectangle:
So, is equivalent to .
Standards Alignment
Building On
7.NS.A
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
