Lesson 9

Relacionemos el área con la multiplicación

Warm-up: Cuál es diferente: Área (10 minutes)

Narrative

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology to describe the characteristics of area.

Launch

• Groups of 2
• Display the image.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

Activity

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

Student Facing

¿Cuál es diferente?

Activity Synthesis

• “Encontremos al menos una razón por la que cada una es diferente” // “Let’s find at least one reason why each one doesn’t belong.”

Activity 1: Encontremos el área (15 minutes)

Narrative

The purpose of this activity is for students to find the area of rectangles with one fractional side length and one whole number side length. Students begin by considering a rectangle with whole number side lengths and then look at a series of rectangles with unit fraction side length. All of the rectangles have the same whole number width to help students see how the area changes when the fractional width changes. Students should use a strategy that makes sense to them. These strategies might include counting the individual shaded parts in the diagram or thinking about moving them to fill in unit squares. Some students may use multiplication or division. These ideas will be brought out in future lessons. During discussion, connect the different strategies students use to calculate the areas. As they choose a strategy, they have an opportunity to use appropriate tools, whether it be expressions that represent the shaded area or physical manipulations of the diagrams, strategically (MP5).

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for finding the area of a rectangle that has one side length that is a fraction. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Memory, Language

Launch

• Groups of 2
• Display the images of the shaded rectangles.
• “¿En qué se parecen todos los rectángulos?, ¿en qué son diferentes?” // “What is the same about all of the rectangles? What is different?” (They are all shaded. They have different amounts shaded. They have different widths.)
• “Vamos a averiguar cuánto de cada rectángulo está sombreado. Esto se llama encontrar el área de la región sombreada. ¿Qué estrategias podemos usar para encontrar el área de cada una de las regiones sombreadas?” // “We are going to figure out how much of each rectangle is shaded. We call this finding the area of the shaded region. What are some strategies we could use to find the area of each of the shaded regions?” (Move the pieces around to make full squares, count the number of blue pieces and multiply the number of pieces by their size.)

Activity

• 5–7 minutes: partner work time
• Encourage students to find the area in a way that makes sense to them.
• Monitor for students who:
• count the number of shaded parts and multiply the total number of parts by their fractional area.
• visualize moving the shaded parts to fill whole unit squares.

Student Facing

Encuentra el área de la región sombreada. Explica o muestra tu razonamiento.

1.

2.

3.

4.

Student Response

If students do not find the area of the shaded region, ask “¿Cómo podemos usar el rectángulo que tiene 6 cuadrados unitarios sombreados para encontrar el área de las otras regiones sombreadas?” // “How can you use the rectangle that has 6 unit squares shaded in to help you find the area of the other shaded regions?”

Activity Synthesis

• Ask previously selected students to share their reasoning.
• “¿En qué se parecen las estrategias?, ¿en qué son diferentes?” // “What is the same about the strategies? What is different?” (They all counted the number of shaded parts, but they counted them in different ways. Some people multiplied and some people moved the parts to make whole unit squares.)
• Display image from final problem.
• “¿De qué manera la expresión $$6 \times \frac{1}{4}$$ representa el área sombreada, en unidades cuadradas?” // “How does the expression $$6 \times \frac{1}{4}$$ represent the shaded area in square units?” (There are 6 shaded parts and each one has an area of $$\frac{1}{4}$$ square unit.)
• “¿De qué manera la expresión $$\frac{1}{4} \times 6$$ representa el área sombreada, en unidades cuadradas?” // “How does the expression $$\frac{1}{4} \times 6$$ represent the shaded area in square units?” (There is a rectangle whose area is 6 square units and $$\frac{1}{4}$$ of the rectangle is shaded.)

Activity 2: Dibujemos rectángulos (20 minutes)

Narrative

The purpose of this activity is for students draw and shade rectangles with a unit fraction side length and a whole number side length. Then they find the areas of the shaded regions. The tactile experience of drawing and shading encourages students to count the number of shaded parts and then either reason about their size or think about moving them to make full unit squares. They also consider a diagram where not all of the unit squares are shown. Students estimate how many of the unit squares are hidden. This helps to highlight that finding the total area can be done with multiplication where one factor is the area of each shaded part and the other factor is the total number of shaded parts (MP7).

Required Materials

Materials to Copy

• Grid Paper 5

Launch

• Groups of 2
• Give students grid paper.

Activity

• 5–7 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• use the grid structure on the paper to draw their rectangles
• count the number of unit squares that might be hidden under the yellow rectangle

Student Facing

• $$\frac{1}{2}$$ unidad por 1 unidad
• $$\frac{1}{2}$$ unidad por 2 unidades
• $$\frac{1}{2}$$ unidad por 3 unidades
• $$\frac{1}{2}$$ unidad por 4 unidades
2. Encuentra el área de cada rectángulo que dibujaste.
3. ¿Qué información necesitas para encontrar el área de la región sombreada?
4. ¿Cuál podría ser el área de la región sombreada? Explica o muestra tu razonamiento.

Student Response

If students do not draw the rectangles correctly, show them the shaded region from the previous activity with side lengths 6 units and $$\frac{1}{2}$$ unit and ask “¿Cómo podrías adaptar este diagrama para mostrar un rectángulo de 4 unidades por $$\frac{1}{2}$$ unidad?” // “How could you adapt this diagram to show a rectangle that is 4 units by $$\frac{1}{2}$$ unit?”

Activity Synthesis

• Display a student generated image of the $$\frac{1}{2}$$ unit by 4 unit rectangle.
• Ask previously selected students to describe how they drew they 4 by $$\frac{1}{2}$$ rectangle.
• Display image from student workbook of the rectangle that is partly covered.
• “¿Qué necesitan saber para encontrar el área de la región sombreada?” // “What do you need to know to determine the area of the shaded region?” (We need to know how many unit squares are under the yellow rectangle.)
• “¿Cuántos cuadrados unitarios creen que forman el rectángulo?” // “How many unit squares do you think make up the rectangle?” (6 or 7)
• “¿Cómo usaron el número de estos cuadrados unitarios para estimar el área de la región sombreada?” // “How did you use the number of these unit squares to make an estimate for the shaded region?” (I know that each shaded blue rectangle is $$\frac{1}{2}$$ of a square so if there are 6 or 7 of those, that would be $$\frac{6}{2}$$ or $$\frac{7}{2}$$ square units.)

Lesson Synthesis

Lesson Synthesis

Display the shaded rectangle that has an area of 6 whole units:

“¿Qué estrategias usamos para encontrar el área de rectángulos en los que ambos lados tienen longitudes enteras?” // “What strategies do we use to find the area of rectangles with 2 whole number side lengths?” (We can count the number of squares. We can multiply the side lengths.)

“¿Qué estrategias usamos hoy para encontrar el área de rectángulos en los que una longitud de lado es entera y la otra es una fracción unitaria?” // “What strategies did we use today to find the area of rectangles with a whole number side length and a unit fraction side length?” (We counted the number of unit squares and multiplied by the size of the shaded region in each unit square.)

“¿En qué se parecen y en qué se diferencian la estrategia que usamos para encontrar el área de rectángulos con ambas longitudes de lado enteras y la estrategia que usamos para rectángulos con una longitud de lado entera y la otra fraccionaria?” // “How are the strategies we used to find the area of rectangles with whole number side lengths the same as and different from the strategies we use to find the area of rectangles with a whole number side length and a fractional side length?” (We can use the same strategies, but we count area that is less than one unit square. We are still multiplying, but one of the numbers is a fraction.)