# Lesson 10

Longitudes de lado fraccionarias y menores que 1

## Warm-up: Exploración de estimación: ¿Cuál es el área? (10 minutes)

### Narrative

The purpose of this Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to estimate the area of a rectangle when one side is not a unit fraction.

### Launch

• Groups of 2
• Display the image.
• “¿Cuál puede ser el área de la región sombreada?” // “What might be the area of the shaded region?”

### Activity

• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

¿Cuál es el área de la región sombreada?

Escribe una estimación que sea:

muy baja razonable muy alta
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### Activity Synthesis

• “¿Está sombreada más de la mitad o menos de la mitad del rectángulo?” // “Is more than half or less than half of the rectangle shaded?” (More than half)
• “¿Cómo les puede servir esto para estimar?” // “How can you use this to help make your estimate?” (Half of 7 is $$3 \frac{1}{2}$$ so it’s a little more than that.)
• “Después de esta discusión, ¿alguien quiere revisar su estimación?” // “Based on this discussion does anyone want to revise their estimate?”

## Activity 1: Rectángulo con un lado de longitud fraccionaria (20 minutes)

### Narrative

The purpose of this activity is for students to find the area of rectangles with a side length that is a non-unit fraction. Students may use a variety of strategies to find the areas of the shaded region. Monitor for students who are noticing and using the structure of the rectangle and expressions to determine the area (MP7) by:

• grouping the shaded pieces with a fractional area to make whole unit squares
• multiplying the whole number of units by the numerator of the fractional side length and dividing the result by the denominator
Representation: Internalize Comprehension. Begin by asking, “¿Este problema/situación le recuerda a alguien algo que ya hayamos visto/leído/hecho?” // “Does this problem/situation remind anyone of something we have seen/read/done before?”
Supports accessibility for: Memory, Conceptual Processing

### Launch

• Display image from warm-up.
• “Si la altura de la región sombreada fuera $$\frac{5}{6}$$ de una unidad cuadrada, ¿qué expresión pueden escribir para representar el área de la región sombreada?” // “If the height of the shaded region was $$\frac{5}{6}$$ of a square unit, what expression could you write to represent the area of the shaded region?” ($$\frac{5}{6} \times 7$$ or $$7 \times \frac{5}{6}$$)
• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time

### Student Facing

Escribe una expresión de multiplicación que represente el área de cada región sombreada. Después, encuentra el área.

1.
2.
3.

### Activity Synthesis

• Select previously identified students to share their strategies.
• “¿En qué se parecen la segunda y la tercera región sombreada?, ¿en qué son diferentes?” // “How are the second and third shaded regions the same? How are they different?” (They are each 4 units long. There are 12 shaded pieces in each. The 12 pieces in the second example are $$\frac{1}{4}$$ of a unit square. The 12 shaded pieces in the third shaded region are $$\frac{1}{5}$$ of a unit square.)
• Display the expression: $$4 \times 3$$
• “¿Cómo se relaciona esta expresión con la segunda y la tercera región sombreada?” // “How does this expression relate to the second and third shaded regions?” (In both of them, the number of shaded pieces is $$4 \times 3$$.)
• “¿Por qué las áreas de estas dos regiones sombreadas son distintas?” //  “Why are the areas of these two shaded regions different?” (The small pieces are not the same size. They are $$\frac{1}{4}$$ of a unit square in one and $$\frac{1}{5}$$ in the other.)
• Display the expressions: $$12 \times \frac{1}{4}$$ and $$12 \times \frac{1}{5}$$
• “Las expresiones de las áreas de la segunda y la tercera región muestran que en ambas hay 12 partes sombreadas, pero son de tamaños diferentes” // “The expressions for the area show that there are 12 shaded pieces in both but they are different sizes.”

## Activity 2: ¿Cuáles son las longitudes de los lados? (15 minutes)

### Narrative

The purpose of this activity is for students to find the area of rectangles with a fractional side length. The side lengths are not labeled, so students will have to determine them by considering the number and size of the shaded pieces.

In the second problem, students are given expressions and they determine whether or not each expression represents the shaded area in a given diagram. The numbers in the expressions are similar, so students need to consider the structure of the expressions and the shaded regions. While they may match the expressions with the diagrams based on their value, students are encouraged to look for and clearly express how the diagrams represent the different expressions (MP6, MP7).

MLR2 Collect and Display. Circulate, listen for and collect the language students use as they determine the area. On a visible display, record words and phrases such as: length, shaded, fraction, pieces, width, multiply, area, whole, part, expression. Invite students to borrow language from the display as needed, and update it throughout the lesson.

• Groups of 2

### Activity

• 2–3 minutes: quiet think time

• 5–6 minutes: partner work time

• Monitor for students who:

• correctly determine the side lengths of the shaded region.

• use the side lengths to determine a correct multiplication expression.

• refer to the number of shaded pieces and the size of the shaded pieces when determining the area.

### Student Facing

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

2. Estos son dos diagramas. Examina cada expresión y decide si representa el área de la región sombreada de alguno de los diagramas. Prepárate para explicar cómo pensaste.

1. $$\frac{3}{4} \times 5$$
2. $$3 \times \frac{3}{5}$$
3. $$3 \times 4 \times \frac{1}{5}$$
4. $$4 \times \frac{3}{4}$$
5. $$3 \times 3 \times \frac{1}{4}$$
3. ¿Cuál es el área de la región sombreada en cada diagrama?

### Student Response

If students do not write the correct expression, show them the correct expression and ask, “¿De qué manera la expresión representa el área de la región sombreada en el diagrama correspondiente?” // “How does the expression represent the area of the corresponding diagram?”

### Activity Synthesis

• “¿Cómo encontraron las longitudes de los lados de la región sombreada en el diagrama X?” // “How did you determine the side lengths for shaded region in diagram X?” (I looked at the side lengths of the squares and the partitions, then I counted how many are shaded.)
• Display the expression: $$3 \times 4 \times \frac{1}{5}$$
• “¿De qué manera esta expresión representa el área de la región sombreada del diagrama X?” // “How does this expression represent the shaded region in diagram X?” (The vertical rectangle has 3 rows of 4 shaded pieces, so that’s $$3 \times 4$$ shaded pieces. Each one has an area of $$\frac{1}{5}$$ square unit.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy encontramos el área de rectángulos que tenían un lado de longitud entera y un lado de longitud fraccionaria” // “Today we found the area of rectangles with a whole number side length and a fractional side length.”

Display image from first problem in the last activity.

“¿Cuáles son algunas expresiones que representan la región sombreada?” // “What are some different expressions that represent the shaded region?” ($$\frac{2}{3} \times 3$$, $$2 \times \frac{1}{3} \times 3$$, $$\frac{6}{3}$$)

“Escojan una de las expresiones y explíquenle a su compañero de qué manera la expresión representa la región sombreada” // “Pick one of the expressions and explain to your partner how it represents the shaded region.” (There are 3 unit squares and $$\frac{2}{3}$$ of them are shaded. There are two shaded rows and each one is $$\frac{1}{3}$$ of 3 squares. There are 6 shaded parts and each one is $$\frac{1}{3}$$.)