Lesson 6

The purpose of this warm-up is for students to compare different shaded regions in order to introduce the new type of region that will be considered in this lesson, namely regions where neither side length is a unit fraction. The focus of the discussion is on diagram A where neither side length is a unit fraction. 

• Groups of  2
• Display the image.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• 2–3 minutes: partner discussion
• Share and record responses. 

• “¿Por qué la imagen A es diferente?” // “Why doesn’t image A belong?" (It’s the only one where neither side length is a unit fraction.) 
• “¿Cuál es el área de la región coloreada en la imagen A? ¿Cómo lo saben?” // “What is the area of the shaded region in image A? How do you know?” (\(\frac{6}{12}\) because there are 6 shaded pieces and there are 12 pieces in the whole square.)

The purpose of this activity is for students to relate the structure in an expression to an area diagram (MP7). As students work with their partners, make sure both partners have an opportunity to verbally explain how the diagram represents each expression.

• Groups of 2

• 2–3 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who can explain how each expression is represented in the diagram.

If students do not explain how each expression represents the area of the shaded region, ask: “¿Cómo describirías el área de la región coloreada?” // “How would you describe the area of the shaded region?” Connect students' explanations to the given expressions.

• Ask previously selected students to share their thinking.
• “¿De qué manera \(\frac {8}{30}\) representa el diagrama?” // “How does \(\frac {8}{30}\) represent the diagram?” (There are 8 pieces shaded and each piece is \(\frac {1}{30}\) of the square.)
• “¿De qué manera la expresión \(2 \times 4 \times (\frac {1}{5} \times \frac {1}{6})\) representa el diagrama?” // “How does the expression \(2 \times 4 \times (\frac {1}{5} \times \frac {1}{6})\)\(\) represent the diagram?” (The shaded region is a 2 by 4 array and each of the pieces in the array is \(\frac {1}{5}\) of \(\frac {1}{6}\) of the whole square.)

The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators. 

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

• Groups of 2

• “Empiecen a completar la tabla de manera independiente. Después de un par de minutos, terminarán de completar la tabla y responderán el resto de las preguntas con su compañero” // “Start working on completing the table independently. After a couple minutes, you'll work with your partner to complete the table and answer the rest of the questions.”
• 1–2 minutes: independent work time
• 5-8 minutes: partner work time
• Monitor for students who:
  • notice the area of the shaded regions is always twentieths
  • write the expression \(\frac{6}{5}\times\frac{4}{5}\) to represent the shaded region of the last diagram in the table
  • explain that the expression \(\frac{6\times4}{5\times4}\) represents the shaded part of the last diagram in the table because \(6\times4\) represents the number of pieces that are shaded and \(4 \times 5\) represents the number of those pieces in the unit square

If students do not explain that the product of the numerators represents the number of pieces in the shaded region and the product of the denominators represents the number of pieces in the whole, consider asking: “¿En qué se parecen y en qué son diferentes los diagramas A y B?” // “What is the same and what is different about diagrams A and B?”

• Ask previously selected students to share their reasoning.
• “¿De qué manera las expresiones de la tabla representan el número de partes coloreadas y el tamaño de las partes coloreadas?” // “How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
• Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (that is, the number of those pieces in the whole).