Lesson 13

Area and Properties of Operations

Warm-up: Number Talk: Parentheses (10 minutes)

Narrative

This Number Talk encourages students to think about equivalent expressions and to rely on the properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students match diagrams to expressions.

Launch

  • Display one expression.
  • “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time

Activity

  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Facing

Find the value of each expression mentally.

  • \(5 \times (7 + 4)\)
  • \((5 \times 7) + (5 \times 4)\)
  • \((5 \times 7) + (5 \times \frac{1}{4})\)
  • \((5 \times 7) - (5 \times \frac{1}{4})\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

“What is the same about the last two expressions? What is different?” (They both have the same two products but one of them is the sum and the other is the difference)

Activity 1: Card Sort: Diagrams and Expressions (20 minutes)

Narrative

The purpose of this activity is for students to analyze area diagrams and use the properties of operations to interpret expressions. The diagrams are decomposed in different ways and the expressions all have a fractional part but sometimes it is written as a mixed number and sometimes the whole number and fraction are separated using the distributive property. The numbers in the diagrams, both the whole number part and the fractional part, are deliberately chosen to resemble one another so students need to analyze the expressions carefully to make matches. 

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed ___ , so I matched . . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing, Representing

Required Materials

Materials to Copy

  • Card Sort: Diagrams and Expressions

Required Preparation

  • Create a set of cards from the blackline master for each group of 2.

Launch

  • Groups of 2
  • Give each group a set of cards from the blackline master.

Activity

  • 1–2 minutes: independent work time
  • 8–10 minutes: partner work

Student Facing

Your teacher will give you and your partner a set of cards.

  1. Sort the cards in a way that makes sense to you.
  2. Match each expression to an appropriate diagram. Some diagrams match more than one expression.
  3. Work with your partner to find the area of each shaded region. Explain or show your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Display Diagram C and Expressions J and O.

\(3 \times 5 \frac{2}{5}\)

\((3 \times 5) + (3 \times \frac {2}{5})\)

  • “How does each expression represent the area of the shaded region?” (There are 3 rows and each row has an area of \(5\frac{2}{5}\) square units.)
  • Display Diagram B and Expression L.

\((5 \times 3) -\left(5 \times \frac{2}{5}\right)\)

  • “How does the expression represent the area of the shaded region?” (\(5 \times 3\) is the area of the full rectangle and then I take away the unshaded part, \(5 \times \frac{2}{5}\).)

Activity 2: Write Expressions (15 minutes)

Narrative

The purpose of this activity is for students to write expressions that represent the area of shaded regions. Monitor for students who are writing a variety of expressions that represent the distributive property.

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for writing an expression to match the shaded area in each diagram before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

Launch

  • Groups of 2

Activity

  • 1–2 minutes: quiet think time
  • 5–7 minutes: partner work time

Student Facing

Write as many expressions as you can to match the area of the shaded region in each diagram.

  1.  
    Area diagram. Length, 5 and 1 third. Width, 4.

  2.  
    Area diagram, Length, 7. Width, 3 and 3 fourths.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students only write one expression for a diagram, partner them with a student who found a different expression and ask:
  • “How does this expression represent the shaded region?”
  • “How can two different expressions represent the same shaded region?”
  • “What other expressions can we write to represent the shaded region?”

Activity Synthesis

  • Select 2–3 students to share their expressions for the first problem.
  • If not mentioned by students, display:

    \(4 \times 5\frac{1}{3}\)

    \(4 \times \frac{16}{3}\)

  • “How does each expression represent the first problem?” (There are 4 rows and each row has \(5 \frac{1}{3}\) square units or \(\frac{16}{3}\) square units.)
  • “Did your approach change when writing expressions for the last diagram? If yes, how?” (It was easier to see the square units and apply the distributive property.)
  • If not mentioned by students, display these expressions for the area of the last diagram in square units:

    \((7 \times 3) +\left(7 \times \frac{3}{4}\right)\)

    \((7 \times 4) - \left(7 \times \frac{1}{4}\right)\)

Lesson Synthesis

Lesson Synthesis

“What’s your favorite way to find the area of a shaded region? What is a new way that you are excited to try or learn more about?”

Consider having students write their responses in their journals.

Cool-down: Equivalent Expressions (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.