# Lesson 11

Lados de longitudes fraccionarias y mayores que 1

## Warm-up: Verdadero o falso: Tercios (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate strategies they have for relating division of two whole numbers to multiplication of a fraction by a whole number. The reasoning students use here helps to deepen their understanding of the relationship between multiplication and division. It will also be helpful later when students find the area of rectangles with mixed number side lengths.

### Launch

• Display one equation.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide si cada afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.

• $$10 \div 3 = 10 \times \frac{1}{3}$$
• $$10 \div 3 = 10 \frac{1}{3}$$
• $$\frac{10}{3} = 5 \times \frac{2}{3}$$

### Activity Synthesis

• “¿Cómo pueden explicar su última respuesta sin encontrar el valor de ambos lados?” // “How can you explain your answer to the last statement without finding the value of both sides?”

## Activity 1: Mayor que uno (20 minutes)

### Narrative

The purpose of this activity is for students to multiply a whole number by a fraction greater than 1 in a way that makes sense to them. Monitor for students who:

• can explain why the area of the shaded region is $$18 \frac{2}{3}$$ or $$16 \frac{8}{3}$$ by counting the number of shaded whole square units and then counting the number of shaded third of a square units
• can explain how to use the expression $$4 \frac{2}{3} \times 4$$ to find the area of the shaded region
• can explain how to use the expression $$\frac{14}{3} \times 4$$ to find the area of the shaded region

As students work with fractions greater than 1, they may choose to write or rewrite them as mixed numbers. Students may also relate the expressions to the diagrams in different ways. Encourage them to interpret the diagram and find the area of the shaded region in whatever way makes sense to them. During the synthesis, show the relationships between the different ways of finding and representing the area.

Identifying which expressions represent the area of the rectangle requires careful analysis of the expressions and the figure and the correspondences between them (MP7).

MLR8 Discussion Supports. Display sentence frames to support small-group discussion: “Me pregunto si . . . ” // “I wonder if . . . ”, “_____ y _____ se parecen porque . . .” // “_____ and _____ are the same because. . . .”, and “_____ y _____ son diferentes porque . . . ” // “_____ and _____ are different because . . . .”

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 5 minutes: partner work time
• As students work, consider asking:
• “¿Cómo calcularon el área de la región sombreada?” // “How did you calculate the area of the shaded region?”
• “¿Cómo saben que su respuesta tiene sentido?” // “How do you know your answer makes sense?”

### Student Facing

1. $$4 \frac{2}{3} \times 4$$
2. $$16 \times \frac{8}{3}$$
3. $$\frac{14}{3} \times 4$$
4. $$\frac{56}{3}$$
5. $$4 \times \frac{5}{3}$$

### Student Response

If students do not interpret the factors as side lengths of rectangles, encourage them to listen to a partner explain where they see the multiplication expression and then describe what their partner says in their own words.

### Activity Synthesis

• Ask selected students to share in the given order.
• “¿De qué manera $$\frac{56}{3}$$ representa el área de la región sombreada?” // “How does $$\frac{56}{3}$$ represent the area of the shaded region?” (There are 56 pieces shaded in and each piece has an area of $$\frac{1}{3}$$ of a unit square.)
• Display: $$4 \frac{2}{3} \times 4= \frac{14}{3} \times 4$$
• “¿Cómo sabemos que estas expresiones son iguales?” // “How do we know these expressions are equal?” (They both represent the shaded area in the diagram or I know that 3, 6, 9, 12 thirds is 4 wholes and then there are two thirds left.)

## Activity 2: Diagramas y expresiones para el área (15 minutes)

### Narrative

The purpose of this activity is for students to find areas of rectangles where one side is a whole number and the other side is a fraction that is greater than 1.  Students should solve the problems in a way that makes sense to them. Ask students to explain how the diagrams show the multiplication expressions.

Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share how they use the diagram to calculate the area of the desk or the garden and the method they used to do so with a classmate.
Supports accessibility for: Conceptual Processing, Attention

• Groups of 2

### Activity

• 5 minutes: individual work time
• 5 minutes: partner work time

### Student Facing

1. Escribe una expresión de multiplicación que represente el área de la región sombreada.
2. ¿Cuál es el área de la región sombreada?

1. Escribe una expresión de multiplicación que represente el área de la región sombreada.

2. ¿Cuál es el área de la región sombreada?

### Student Response

If students do not write multiplication expressions to represent the area of the shaded region, prompt them to explain how they found the area of the shaded region. Then, write expressions to represent the student’s strategy and ask, “¿De qué manera estas expresiones representan tu estrategia?” // “How do these expressions represent your strategy?”

### Activity Synthesis

• Ask several students to share their responses to the second problem.
• Display:
• $$3 \times 3\frac{3}{4}$$
• $$3 \times \frac{15}{4}$$
• “¿De qué manera estas expresiones representan el área de la región sombreada?” // “How do these expressions represent the area of the shaded region?” (They are the width and length of the shaded region. In one, the length is a whole number and some fourths, and in the other, it is all fourths.)
• “¿En qué se diferencian las maneras de encontrar el valor de estas dos expresiones?” // “How is finding the value of these two expressions different?” (For the first one, I can multiply 3 by 3 and then 3 by $$\frac{3}{4}$$ and add them together. For the second one, I can see how many fourths I have using multiplication.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy aprendimos que podemos aplicar nuestra compresión de la multiplicación para encontrar el área de un rectángulo en el que una longitud de lado es una fracción mayor que 1” // “Today we learned that we can apply our understanding of multiplication to find the area of a rectangle with a side length that is a fraction greater than 1.”

Display the image.

Display the expression: $$2 \times 9 \times \frac{1}{4}$$
“¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (There are $$2 \times 9$$ small pieces and each one has an area of $$\frac{1}{4}$$ square unit.)
Display the expression: $$2 \times \frac{9}{4}$$
“¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (There are two rows of shaded pieces and each row has area $$\frac{9}{4}$$ square units.)
Display the expression: $$(2 \times 2) + (2 \times \frac{1}{4})$$
“¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (There's a 2 by 2 array of whole square units and then there are 2 shaded pieces each having area $$\frac{1}{4}$$ square unit.)