Students reason about the area of a rectangle with a variable side length. They express the area for different values of the variable and when the value is unknown. Students review symbolic notation for showing multiplication as they express the product of a number and a variable. The reasoning here will be helpful later in the lesson when students apply the distributive property in the context of finding the areas of rectangles whose side lengths are expressions with variables.
Student Lesson in Spanish
Launch
Allow students 2–3 minutes of quiet work time, followed by a whole-class discussion.
Activity
None
Student Task Statement
A rectangle has a length of 4 units and a width of units. Write an expression for the area of this rectangle.
What is the area of the rectangle if is:
3 units?
2.2 units?
unit?
Could the area of this rectangle be 11 square units? Explain your reasoning.
Student Response
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Building on Student Thinking
If students indicate they are not sure how to start and haven't drawn a diagram of a rectangle, suggest that they do so.
Activity Synthesis
Select students to share their response to each question. Consider displaying a diagram of a rectangle and annotating it to illustrate students’ responses or explanations. Highlight the following points:
Rectangle areas can be found by multiplying length by width.
Both and are expressions for the area of this rectangle. These are equivalent expressions.
Lengths don’t have to be whole numbers. Neither do areas.
10.2
Activity
10 mins
Partitioned Rectangles When Lengths Are Unknown
Standards Alignment
Building On
Addressing
6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression to produce the equivalent expression ; apply the distributive property to the expression to produce the equivalent expression ; apply properties of operations to to produce the equivalent expression .
In this activity, students use expressions with variables to represent lengths of sides and areas of rectangles. These expressions are used to help students understand the distributive property and its use in creating equivalent expressions.
Launch
Arrange students in groups of 2–3. Give students 3–4 minutes of group work time, followed by a quick whole-class discussion.
MLR6 Three Reads. Keep books or devices closed. Display only the description of the two rectangles, without revealing the instructions to write expressions. “We are going to read this question 3 times.”
After the 1st read: “Tell your partner what this situation is about.” (The problem is about two rectangles with some side lengths given.)
After the 2nd read: “List the quantities. What can be counted or measured?” (The width of both rectangles is 5. The length of one rectangle is 8, and the other rectangle’s length is .)
For the 3rd read: Reveal and read the instructions to write expressions. Ask, “What are some ways we might get started on this?”
Advances: Reading, Representing
Activity
None
Student Task Statement
Here are two rectangles. The length and width of one rectangle are 8 and 5 units. The width of the other rectangle is 5 units, but its length is unknown so we labeled it .
Write an expression for the sum of the areas of the two rectangles.
The two rectangles can be composed into one larger rectangle, as shown.
What are the length and width of the new, larger rectangle?
Write an expression for the total area of the new, larger rectangle as the product of its width and its length.
Student Response
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Building on Student Thinking
Activity Synthesis
Solicit students’ responses to the first and third questions. Display two of the expressions, as shown. (Expressions that are equivalent to these are fine.) Ensure that everyone agrees that one expression is an acceptable response to the first question and the other is an acceptable response to the third question.
Ask students to look at the two expressions and invite them to share something they notice and something they wonder. Here are some things that students might notice.
The 5 appears twice in one expression and only once in the other.
These look like an example of the distributive property, but with a variable.
These expressions must be equivalent to each other, because they each represent the area of the same rectangle.
If no students mention the last point—that the expressions are equivalent—ask them to discuss this idea.
10.3
Activity
20 mins
Areas of Partitioned Rectangles
Standards Alignment
Building On
Addressing
6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression to produce the equivalent expression ; apply the distributive property to the expression to produce the equivalent expression ; apply properties of operations to to produce the equivalent expression .
In this activity, students are presented with several partitioned rectangles. They identify the length and width for each rectangle, and then write expressions for the area in two different ways, as:
The product of the length and the width, one of which is expressed as a sum of partial lengths.
The sum of the areas of the smaller rectangles that make up the large rectangle.
Students reason that these two expressions must be equal since they both represent the total area of the partitioned rectangle (MP2). In this way, students see several examples of the distributive property. Students may choose to assign values to the variable in each rectangle to check that their expressions for area are equal.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Keep students in groups of 2–3. Give students 10 minutes of group work time, followed by a whole-class discussion.
Use Collect and Display to direct attention to words collected and displayed from an earlier lesson. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Action and Expression: Internalize Executive Functions. To remind students how to write expressions for the length, width, and total area of a rectangle, begin with a small-group or whole-class demonstration and think-aloud that shows how to complete the first row of the table. Keep the worked-out calculations on display for students to reference as they work. Supports accessibility for: Memory, Conceptual Processing
Activity
None
Student Task Statement
For each rectangle, write an expression for the width, an expression for the length, and two expressions for the total area. Record them in the table. Check your expressions in each row with your group and discuss any disagreements.
A
Rectangle A is partioned by a vertical line segment into two smaller rectangles. The vertical side is labeled 3 and the top horizontal side lengths are labeled "a" and 5.
B
Rectangle B is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled one third and the top horizontal side lengths are labeled 6 and x.
C
Rectangle C is partitioned by 2 vertical line segments into three equally sized rectangles. The vertical side is labeled r and the top horizontal side lengths are each labeled 1.
D
Rectangle D is partitioned by 3 vertical line segments into 4 equally sized rectangles. The vertical side is labeled 6, and the top horizontal side lengths are each labeled 4.
E
Rectangle E is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled m and the top horizontal side lengths are labeled 6 and 8.
F
Rectangle F is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled 5 and the top horizontal side lengths are labeled 3 x and 8.
rectangle
width
length
area as a product of
width times length
area as a sum of the areas
of the smaller rectangles
A
B
C
D
E
F
Student Response
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Building on Student Thinking
Activity Synthesis
Select students to share their expressions for the areas of Rectangles D and F. Ask them to explain how each expression relates to the diagram. Display and annotate the diagrams, if possible. Ask students:
“How do you know if the pair of expressions for the area of each rectangle are equivalent?” (They represent the area of the same figure. Applying the distributive property in one expression gives the other expression, so we know they are equivalent.)
“Suppose we know the value of the variable . What can you predict about the values of the two expressions?” (They would be equal.) “How would you test your prediction?” (Substitute the value for the variable and do the computation.)
Display the expressions from the rows for Rectangles D and F:
D
6
F
5
Tell students that as they work with a greater variety of expressions, it is helpful to be able to refer to the parts in the expression (just as students learned to use “factors” and “product” to refer to the parts in a multiplication, and “dividend,” “divisor,” and “quotient” for division.)
Introduce the word “term” to students. Explain that a term is a part of an expression. A term can be a single number, a single variable, or a product of numbers and variables. In the displayed expressions, 5, , , and are terms. Invite students to identify a few other terms in their completed table.
Add “term” to the display of other vocabulary words from the unit (or revise similar words or phrases students used previously to name the same concept).
Lesson Synthesis
Display a pair of equivalent algebraic expressions from this lesson (such as and ) and a pair of equivalent numerical expressions from a previous lesson (such as 33(10 + 2) and ).
Ask students to compare the expressions. Discuss questions, such as:
“How are they alike? How are they different?”
"How can you use a rectangular diagram to show that the distributive property applies to both expressions with numbers and those with variables?"
Students should see that their work with expressions containing variables is an extension of the work they did with expressions with numbers.
Student Lesson Summary
The distributive property can also help us write equivalent expressions with variables. We can use a diagram to help us understand this idea.
Here is a rectangle composed of two smaller Rectangles A and B.
A rectangle is partitioned by a vertical line segment creating two smaller rectangles, A and B. Rectangle A has a vertical side length of 3 and horizontal side length of 2. Rectangle B has a horizontal side length of x.
Based on the drawing, we can make several observations about the area of the large rectangle:
One side length of the large rectangle is 3 and the other is , so its area is .
Since the large rectangle can be decomposed into two smaller rectangles, A and B, with no overlap, the area of the large rectangle is also the sum of the areas of rectangles A and B: or .
Since both expressions represent the area of the large rectangle, they are equivalent to each other. is equivalent to .
We can see that multiplying 3 by the sum is equivalent to multiplying 3 by 2 and then 3 by and adding the two products. This relationship is an example of the distributive property.
When working with expressions of all kinds, it helps to be able to talk about the parts. In an expression like , we call the 6 and “terms.”
A term is a part of an expression. A term can be a single number, a single variable, or a product of numbers and variables. Some examples of terms are 10, , , and .
Standards Alignment
Building On
Addressing
6.EE.A.2
Write, read, and evaluate expressions in which letters stand for numbers.
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions and are equivalent because they name the same number regardless of which number stands for.
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions and are equivalent because they name the same number regardless of which number stands for.
