Lesson 3

The purpose of this Estimation Exploration is for students to estimate the area of a shaded region. In the synthesis, students discuss whether the product is greater or less than the expression \(\frac{1}{2} \times \frac{1}{6}\). This allows them to connect the shaded area to their previous work with multiplication expressions (MP7).

• 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 aŕea de la región coloreada es más o menos que \(\frac{1}{2}  \times  \frac{1}{6}\)? ¿Cómo lo saben?” // “Is the area of the shaded region more or less than \(\frac{1}{2}  \times  \frac{1}{6}\)? How do you know?” (More. It looks like it is \(\frac{1}{2}\) the length and more than \(\frac{1}{6}\) the width.)
• “¿Cuál es el valor de \(\frac{1}{2} \times \frac{1}{6}\)?” // “What is the value of \(\frac{1}{2} \times \frac{1}{6}\)?” (\(\frac{1}{12}\))

The purpose of this activity is for students to notice structure in a series of diagrams and the expressions that represent them. They investigate how these expressions vary as the number of rows and columns in the diagram change. Students see how the diagram represents the multiplication expression and also how the diagram helps to find the value of the expression (MP7). Through repeated reasoning they also begin to see how to find the value of a product of any two unit fractions (MP8).

• Groups of 2
• Display the images from the task.
• “¿En qué se diferencian estos diagramas?” // “What is different about these diagrams?” (The number of rows increases by 1. The blue shaded piece gets smaller.)
• “¿En qué se parecen?” // “What is the same?” (The size of the big square. There are always 4 columns. Only one piece is shaded.)
• 1 minute: quiet think time
• Share and record responses.

• 1–2 minutes: independent work time to complete the first problem
• 1–2 minutes: partner discussion
• “Ahora, con su compañero, completen el resto de los problemas” // “Now, complete the rest of the problems with your partner.”
• 5 minutes: partner work time
• Monitor for students who:
  • choose different diagrams to represent with multiplication expressions
  • represent the same diagram with different multiplication expression, for example, \(\frac{1}{2}\times\frac{1}{4}\) and \(\frac{1}{4}\times\frac{1}{2}\)

If students do not write correct expressions to represent the diagrams, write the correct expressions and ask, “¿De qué manera las expresiones representan el área de la parte coloreada del diagrama?” // “How do the expressions represent the area of the shaded piece of the diagram?”

• Display the diagrams from the student workbook.
• Select previously identified students to share.
• As students explain where they see the multiplication expression in each diagram, record the expressions under the diagram for all to see.
• Refer to the diagram that shows \(\frac{1}{2}\times\frac{1}{4}\) or \(\frac{1}{4}\times\frac {1}{2}\).
• Display both expressions.
• “¿De qué manera este diagrama representa estas dos expresiones?” // “How does this diagram represent both of these expressions?” (It shows one half of one fourth shaded in and it also shows one fourth of one half shaded in.)
• Represent student explanation on the diagrams.
• “¿Por qué el área de la región coloreada se vuelve más pequeña en cada diagrama?” // “Why is the area of the shaded region getting smaller in each diagram?” (Because we are shading a smaller piece of \(\frac{1}{4}\) each time.)
• Display:
  \(\frac{1}{2} \times \frac{1}{4}= \frac{1}{2\times 4}=\frac{1}{8}\)
  \(\frac{1}{3} \times \frac{1}{4}= \frac{1}{3\times 4}=\frac{1}{12}\)
  \(\frac{1}{4} \times \frac{1}{4}= \frac{1}{4\times 4}=\frac{1}{16}\)
  \(\frac{1}{5} \times \frac{1}{4}= \frac{1}{5\times 4}=\frac{1}{20}\)
• “Estas ecuaciones representan los diagramas. ¿Qué patrones observan?” // “These equations represent the diagrams. What patterns do you notice?” (They all have \(\frac {1}{4}\) in them. The denominators in the first fractions go 2, 3, 4, 5. The denominators in the middle fractions are multiplication expressions, the denominators in the middle are all multiplied by 4, the denominator in the fraction that shows the value of the shaded piece goes up by 4 each time.)
• “¿De qué manera los diagramas representan \(\frac{1}{2\times 4}\), \(\frac{1}{3\times4}\), \(\frac{1}{4\times4}\), \(\frac{1}{5\times4}\)?” // “How do the diagrams represent \(\frac{1}{2\times 4}\), \(\frac{1}{3\times4}\), \(\frac{1}{4\times4}\), \(\frac{1}{5\times4}\)?” (The numerator tells us that there is 1 piece shaded and the denominator tells us the size of the piece. The denominator also tells us the rows and columns that the whole is divided into.)

The purpose of this activity is for students to use the structure of diagrams to calculate products of unit fractions. They also represent their work using an equation. As students become more familiar with this structure they may not need diagrams as a scaffold to find these products. Drawing their own diagrams, however, will also reinforce student understanding of how to calculate products of unit fractions.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

• Groups of 2

• 3–5 minutes: independent work time
• 1–2 minutes: partner discussion

If students do not refer to the rows and columns when they explain how the diagram represents the equation \(\frac{1}{5} \times \frac{1}{3} = \frac{1}{15}\), ask “¿De qué manera están las filas y las columnas del diagrama representadas en la ecuación?” // “How are the rows and columns in the diagram represented in the equation?”

• Display: \(\frac{1}{2}\times\frac{1}{4}=\frac {1}{(2\times4)} = \frac {1}{8}\) and the corresponding diagram.
• “¿De qué manera esta ecuación representa el diagrama?” // “How does this equation represent the diagram?” (One fourth of one half is shaded which is the same as 1 piece of the whole square that is divided into 2 columns and 4 rows so one eighth of the whole square is shaded.)

MLR1 Stronger and Clearer Each Time

• “Compartan con su compañero su explicación sobre cómo el último diagrama representa \(\frac{1}{5} \times \frac{1}{3}= \frac{1}{15}\). Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your explanation about how the last diagram represents \(\frac{1}{5} \times \frac{1}{3}= \frac{1}{15}\) with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion 
• Repeat with 2–3 different partners.
• If needed, display question starters and prompts for feedback.
  • “¿Pueden dar un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
  • “¿Pueden usar la palabra _____ en su explicación?” // “Can you use the word _____ in your explanation?”
  • “La parte que mejor entendí fue . . .” // “The part that I understood best was . . . .”
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time.