Lesson 5

The purpose of this Estimation Exploration is for students to estimate the area of a shaded region. In a previous Estimation Exploration, students looked at a shaded region where the length was represented by a unit fraction and the width could be estimated with a unit fraction. For the diagram presented here, the length is a unit fraction but the width can not be since it is greater than \(\frac{1}{2}\). This prepares students for the work of this lesson which is to consider products of a unit fraction and a non-unit fraction.

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿El área de la región coloreada es más o menos que \(\frac{1}{4}\) de unidad cuadrada? ¿Cómo lo saben?” // “Is the area of the shaded region more or less than \(\frac{1}{4}\) square unit? How do you know?” (It’s more because \(\frac{1}{2}\) of \(\frac{1}{2}\) is \(\frac{1}{4}\), and more than that is shaded.)

The purpose of this activity is for students to write expressions for and find the area of shaded regions whose side lengths are a unit fraction and a non-unit fraction. Students look at a series of diagrams with an increasingly large shaded region so they can look for and make use of structure (MP7).

It is important to relate the work here to what students have already learned about the product of a unit fraction and a unit fraction and this is the goal of the synthesis. For example, students have seen that \(\frac{1}{5} \times \frac{1}{3} = \frac{1}{15}\). Since \(\frac{6}{5}\) is 6 \(\frac{1}{5}\)s, the product \(\frac{6}{5} \times \frac{1}{3}\) will be 6 \(\frac{1}{15}\)s or \(\frac{6}{15}\).

• Groups of 2
• Display image A from the student workbook.
• “¿Cuál es una expresión de multiplicación que representa la región coloreada?” // “What is a multiplication expression that represents the shaded region?”(\(\frac{2}{5} \times \frac{1}{3}\), \(2 \times \frac{1}{15}\))
• “¿De qué manera el diagrama representa su expresión?” // “How does the diagram represent your expression?”(There is \(\frac{2}{5}\) of the first row shaded and that row is \(\frac{1}{3}\) of the square. There are 2 shaded pieces and each is \(\frac{1}{15}\) of the whole square.)

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who:
  • notice that the product of the numerators represents how many pieces of the square are shaded.
  • notice that the product of the denominators represents the number of pieces in the whole square.

If students write mathematically correct multiplication expressions that do not represent the product of a unit fraction and a non-unit fraction, write this type of multiplication expression that represents the diagram and ask “¿Cómo esta expresión representa el área de la región coloreada del diagrama?” // “How does this expression represent the area of the shaded region in the diagram?”

• Ask previously selected students to share the patterns they noticed in the table.
• Display the expression from the last problem: \(\frac {6}{5} \times \frac {1}{3} = \frac {6}{15}\).
• “¿Cómo el diagrama muestra \(\frac {1}{3}\)?” // “How does the diagram show \(\frac {1}{3}\)?” (The first row of pieces in a square is \(\frac{1}{3}\) of the square.)
• “¿Cómo el diagrama muestra \(\frac {6}{5}\)?” // “How does the diagram show \(\frac {6}{5}\)?” (There are 6 pieces shaded and each one is \(\frac{1}{5}\) of the row.)
• “¿Cómo el diagrama muestra \(\frac {6}{5} \times \frac {1}{3}\)?” // “How does the diagram show \(\frac {6}{5} \times \frac {1}{3}\)?” (There is \(\frac{6}{5}\) of a row shaded and that row is \(\frac{1}{3}\) of a square unit.)

The purpose of this activity is for students to write multiplication expressions to estimate the area of a shaded region. This builds on the warm-up and the activity launches by asking students to write an expression for the shaded region in the image they considered in the warm-up.

This work in this activity combines the skill of estimation with an understanding that the area of a rectangle relates to its length and width. So far, students have only calculated these areas as products of fractions when at least one side length is a unit fraction. Students may write products of two non-unit fractions. While they have not yet learned how to calculate these products and relate them to areas, these answers are valid when the estimates are reasonable and students will learn in the next lesson how to find the value of these expressions.

• Groups of 2
• Display the image from the warm-up: “¿Cuál expresión de multiplicación podría representar el área de la región coloreada?” // “What multiplication expression might represent the area of the shaded region?” (\(\frac {2}{3} \times \frac {1}{2}\), \(\frac {3}{5} \times \frac{1}{2}\))
• “¿Por qué tienen sentido estas expresiones?” // “Why do those expressions make sense?” (We can see that the shaded region is a fraction of \(\frac {1}{2}\). It is more than \(\frac {1}{2} \times \frac {1}{2}\).)
• “Vamos a observar más diagramas y a escribir expresiones de multiplicación que puedan representar el área de las regiones coloreadas en cada uno” // “We are going to look at more diagrams and write multiplication expressions that might represent the area of the shaded regions in each one.”

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who:
  • use a unit fraction to help them determine reasonable expressions.
  • reason about the size of the shaded region before writing an expression.
  • draw lines to partition the squares.

• Ask previously selected students to share their solutions.
• Display the first diagram.
• “¿Cómo nos ayuda pensar en \(\frac {1}{2}\) a estimar el área de la región coloreada?” // “How does thinking about \(\frac {1}{2}\) help us estimate the area of the shaded region?” (We know the area is less than \(\frac {1}{2}\) because only part of \(\frac{1}{2}\) is shaded.)
• “¿Aproximadamente qué fracción de \(\frac {1}{2}\) está coloreada?” // “About what fraction of \(\frac {1}{2}\) is shaded?” (More than \(\frac {1}{2}\) of \(\frac {1}{2}\). Maybe \(\frac {4}{5}\) of \(\frac {1}{2}\) or \(\frac {3}{4}\) of \(\frac {1}{2}\).)