Lesson 4

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying unit fractions. These understandings help students develop fluency and will be helpful later in this lesson when students make sense of a unit fraction multiplied by a non-unit fraction.

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

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

• “¿Qué patrones observan?” // “What patterns do you notice?” (The numerators are all 1. The denominators are all even numbers. The fractions are getting smaller. Each time, we find a smaller fraction of \(\frac {1}{2}\).)

The purpose of this activity is for students to draw a diagram representing the product of a unit fraction and a non-unit fraction. Then students use the diagram to represent the product with an expression and find its value. Students may draw many different diagrams that represent the situation. The context of sports fields was chosen to encourage students to divide the square in thirds, vertically or horizontally and, subsequently, to divide the two thirds that represents the sports in half, horizontally or vertically, to represent the part of the sports section that will be used for soccer fields. The activity synthesis focuses on the expressions and equations that represent the area for the soccer fields. Students reason abstractly and quantitatively throughout as they relate their diagram and the expression representing it to the park (MP2). 

• Groups of 2
• “¿Qué tipos de cosas observan y qué cosas hacen en el parque?” // “What kinds of things do you see and do in the park?”
  • play frisbee or other games
  • watch the ducks
  • use the swings
  • have a picnic
  • play with my friends
• 1–2 minutes: independent think time
• 1–2 minutes: partner discussion

• 5 minutes: partner work time
• Monitor for students who label the diagram to show the different parts of the park.

If students do not draw a diagram that represents \(\frac{1}{2} \times \frac{2}{3}\), suggest they draw a diagram to represent the \(\frac{2}{3}\) of the park that will be used for different sports. Ask: “¿Cómo puedes ajustar tu diagrama para mostrar que en \(\frac{1}{2}\) de la sección destinada a deportes se harán campos de fútbol?” // “How can you adapt your diagram to show that \(\frac{1}{2}\) of the section used for sports will be soccer fields?”

• Invite students to share their drawings of the soccer fields.
• Display the image from the student solution or use a student-generated image.
• “¿De qué manera el diagrama representa \(\frac{2}{3}\)?” // “How does the diagram represent \(\frac{2}{3}\)?” (There are two vertical slices of the park on the left that are \(\frac{2}{3}\) of the park.)
• Display expression: \(\frac{1}{2} \times \frac{2}{3}\)
• “¿De qué manera el diagrama representa \(\frac{1}{2} \times \frac{2}{3}\)?” // “How does the diagram represent \(\frac{1}{2} \times \frac{2}{3}\) ?” (The top half of the left \(\frac{2}{3}\) of the square is shaded darkly.)
• “¿De qué manera el diagrama representa cuánto del parque entero será usado para campos de fútbol, que es también el valor de \(\frac{1}{2} \times \frac{2}{3}\)?” // “How does the diagram represent how much of the whole park will be used for soccer fields, which is also the value of \(\frac{1}{2} \times \frac{2}{3}\)?” (It shows 2 pieces that are each \(\frac{1}{6}\) of the whole.)

The purpose of this activity is for students to relate expressions to a diagram in a situation where they represent the product of a unit fraction and a non-unit fraction. Students work with a diagram that represents a different park. Students write expressions, trade with a partner, and interpret their partner’s expressions and match them to a diagram. As students work together, listen for how they explain why the expressions represent the corresponding areas. While the activity focuses on relating expressions and parts of the diagram, in the synthesis students find the value of products and analyze equations in terms of the park (MP2). As students discuss and justify their decisions while looking through each others’ work, they share mathematical claims and the thinking behind them (MP3).

• Groups of 2

• 5–6 minutes: independent think time
• 2–3 minutes: partner work time
• Monitor for students who:
  • can explain why \(\frac{1}{5} \times \frac{1}{2}\) represents the part of the park for the swings.
  • can explain why \(\frac{3}{5} \times \frac{1}{2}\) represents the part of the park for the grass.
  • write \(\frac{4}{5} \times \frac{1}{2}\) to represents the part of the park for the pond.

If students do not identify which section is \(\frac{3}{5}\) of \(\frac{1}{2}\) of the park, suggest they draw a separate diagram to represent each section. Ask: “¿Cómo describirías esta sección con relación al parque entero? ¿Cómo puedes representar lo que describiste con una expresión de multiplicación?” // “How would you describe this section in relation to the whole park? How can you represent what you described with a multiplication expression?”

• Ask previously selected students to share their thinking.
• Display: \(\frac{1}{5} \times \frac{1}{2} = \frac {1}{10}\)
• “¿Qué parte del diagrama está representada por esta ecuación?” // “What part of the diagram does this equation represent?” (It represents the section of the park that is swings. We can see that the swings take up \(\frac{1}{5}\) of \(\frac{1}{2}\) of the park.)
• Display: \(\frac{3}{5} \times \frac{1}{2} = \frac {3}{10}\)
• “¿Qué parte del diagrama está representada por esta ecuación?” // “What part of the diagram does this equation represent?” (It represents the section of the park for grass, which is \(\frac{3}{5}\) of \(\frac{1}{2}\) of the park and it is also \(\frac{3}{10}\) of the whole park.)
• Display: \(\frac{4}{5} \times \frac{1}{2} = \frac {4}{10}\)
• “¿Qué parte del diagrama está representada por esta ecuación?” // “What part of the diagram does this equation represent?” (It represents the section of the park for the pond, which is \(\frac{4}{5}\) of \(\frac{1}{2}\) of the park and it is also \(\frac{4}{10}\) of the whole park.)