# Lesson 12

Descompongamos áreas

## Warm-up: Conversación numérica: Productos parciales (10 minutes)

### Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying a mixed number by a whole number. These understandings help students develop fluency and will be helpful later in this lesson when students decompose rectangles with fractional side lengths to find the area.

### Launch

• Display one problem.
• “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

### Activity

• Keep problems and work displayed.
• Repeat with each problem.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$3 \times 20$$
• $$3 \times 24$$
• $$5 \times 2$$
• $$5 \times 2\frac{1}{2}$$

### Activity Synthesis

• “¿Cómo nos ayuda separar los números a encontrar el producto?” // “How does breaking apart the numbers help us find the product?” (It is easier to find the product of big numbers when we decompose the numbers and add the smaller partial products.)

## Activity 1: ¿Cuál jardín es más grande? (20 minutes)

### Narrative

The purpose of this activity is for students to find the area of a rectangle with a whole number side length and a side length that is a mixed number. Students draw diagrams to represent the area of the gardens in the problem. Students should draw diagrams and find the area in a way that makes sense to them. As students work, ask them to explain their strategy for finding the area. In the activity synthesis, students consider multiplication expressions that use the distributive property to represent decomposing the rectangle into two smaller rectangles.

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “Who’s garden do you think is larger, Noah’s or Priya’s?”. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

### Launch

• Groups of 2
• Display: Noah’s garden is 5 yards by $$6\frac{1}{4}$$ yards. Priya’s garden is 6 yards by $$5\frac{1}{4}$$ yards.
• “¿Quién creen que tiene un jardín más grande? ¿Por qué?” // “Whose garden do you think is larger? Why?”
• 1 minute: quiet think time
• 1-2 minutes: partner discussion
• “Vamos a dibujar un diagrama de cada jardín y vamos a decidir cuál jardín tiene un área más grande” // “We are going to draw the diagrams of each garden and determine which garden has a larger area.”

### Activity

• 1–2 minutes: independent work time
• 8–10 minutes: partner work
• Monitor for students who:
• label the side lengths with mixed numbers
• multiply the whole number side lengths, then add the fractional parts
• decompose the larger rectangles into two smaller rectangles and multiply to find the area

### Student Facing

1. El jardín de Noah mide 5 yardas por $$6\frac{1}{4}$$ yardas. En la cuadrícula, dibuja un diagrama del jardín de Noah.
2. El jardín de Priya mide 6 yardas por $$5\frac{1}{4}$$ yardas. En la cuadrícula, dibuja un diagrama del jardín de Priya.
3. ¿Cuál jardín cubre un área más grande? Prepárate para explicar tu razonamiento.

### Activity Synthesis

• Select 2–3 students to share their responses and reasoning about how they determined which garden had a greater area.
•  Display diagrams in sample student responses and the expressions $$(5 \times 6) + (5 \times \frac{1}{4})$$ and $$(6 \times 5) + (6 \times \frac{1}{4})$$.
• “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “How are these expressions the same? How are they different?” (Each expression is a sum of two products. The first part of each expression is 30. The second part is different because one expression has a 5 and the other one has a 6.)
• “¿Cómo podemos decidir cuál jardín tiene un área más grande sin evaluar las expresiones?” // “How can we determine which garden has a larger area without evaluating the expressions?” (The $$5 \times 6$$ is the same as $$6 \times 5$$ but one garden has 5 one-fourths and the other has 6 one-fourths.)

## Activity 2: Diferentes formas de encontrar el área (15 minutes)

### Narrative

The purpose of this activity is for students to interpret different strategies for multiplying whole numbers and fractions greater than 1 to find the area of a rectangle. Students use a diagram to describe different ways to determine the area of a shaded region. Consider having multiple copies of the diagrams available if students want to use a separate diagram for each strategy. Encourage students to draw on the diagram to show how they decomposed the rectangle.
Students use what they have learned about area to construct different reasonable arguments for effective calculations of the area (MP3).

Engagement: Provide Access by Recruiting Interest. Provide choice. Invite students to decide which student to start with when completing the task.
Supports accessibility for: Organization, Attention

### Launch

• Groups of 2
• “Decidan quién va a ser el compañero A y quién el compañero B. Primero, cada uno va a observar cómo unos estudiantes comenzaron a resolver un problema. Después, va a pensar en cómo podrían terminar de resolverlo. Finalmente, van a compartir su trabajo con su compañero” // “Decide with your partner who will be Partner A and who will be Partner B. You’ll each look at how some students started a problem and how they could finish their work. Then, you’ll share your work with your partner.”

### Activity

• 5–8 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who:
• complete Tyler's and Diego's work by calculating the missing or excess area

### Student Facing

Compañero A

Jada: $$4\times5$$

Priya: $$4 \times \frac{1}{2}$$

Tyler: $$6\times4$$

Compañero B

Clare: $$\frac{10}{2}\times4$$

Diego: $$4 \times 6$$

Elena: $$4\times 11$$

1. Cada caso muestra lo primero que hicieron varios estudiantes para encontrar el área de la región sombreada. Explica cómo puede completar su solución cada estudiante para encontrar el área. Muestra en el mismo diagrama cómo pensaste.
2. Comparte tu respuesta con tu compañero. ¿En qué se parecen sus respuestas?, ¿en qué son diferentes?

### Student Response

If students do not complete the steps that were started in the problem, refer to one of the diagrams and corresponding expressions and ask “¿De qué manera el diagrama muestra esta expresión? ¿Qué más nos hace falta averiguar para encontrar el área de la región sombreada?” // "How does the diagram show this expression? What else do we need to find out in order to determine the area of the shaded region?"

### Activity Synthesis

• Select previously identified students to share their solutions.
• “¿En qué se parecen y en qué son diferentes los dos diagramas?” //  “What is the same and what is different between the two diagrams?” (They have the same area and the side lengths are equal, but they are written in different ways. The expressions for the side lengths are equal, but they are written in different ways.)
• “De las estrategias de los estudiantes, ¿cuál prefieren y por qué?” // “Which student’s strategy do you prefer and why?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy encontramos el área de rectángulos que tienen un lado de longitud fraccionaria usando varias estrategias de descomposición. ¿Cómo podemos describir las estrategias que usamos hoy?” // “Today we tried several different strategies for decomposing rectangles with a fractional side length to find the area of a rectangle. How can we describe the strategies we used today?” (We can decompose the rectangle into two smaller rectangles and add the areas. We can find the area of a larger rectangle and then subtract the area of a smaller rectangle.)

Record answers for all to see. Keep the display visible so students can refer to it in future lessons.