Lesson 13

Encontremos el área de algunas figuras

Warm-up: Conversación numérica: Extendamos formar una decena (10 minutes)

Narrative

The purpose of this Number Talk is to elicit strategies students have for adding two numbers when one number is close to a whole number of tens. These understandings help students develop fluency in addition. Students may look for and make use of structure (MP7) in a number of ways. For example, they may add 1 to the first addend to make a full ten and subtract 1 from the second addend to find each sum. They may also notice how the addends compare to those in the previous expression and use the change to find the new sum.

In this string, students may also add the tens and ones separately to find the sum. Adding by place value is the focus of upcoming work. This Number Talk also enables the teacher to learn the strategies students currently have for addition.

Launch

  • 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

Activity

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

Student Facing

Encuentra mentalmente el valor de cada expresión.

  • \(109 + 4\)
  • \(109 + 14\)
  • \(209 + 34\)
  • \(219 + 34\)

Student Response

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Activity Synthesis

  • “¿Qué observan sobre cómo se relacionan las dos primeras expresiones? ¿Qué observan sobre cómo se relacionan las dos últimas expresiones?” // “What do you notice about how the first two expressions are related? What about how the last two expressions are related?” (The second sum is 10 more than the first. The fourth sum is 10 more than the third.)

Activity 1: Adiós a los cuadrados (20 minutes)

Narrative

The purpose of this activity is for students to find the area of figures that are composed of rectangles but are not fully gridded with squares. Partially gridded figures help to prepare students to find the area of figures with only side length measurements. Students should be encouraged to find side lengths and multiply, rather than rely on counting, as the grids disappear. If students continue to draw in the squares, ask them if there is another way to find the area.

Engagement: Develop Effort and Persistence: Differentiate the degree of difficulty or complexity. Some students may benefit from starting with a smaller figure, one with more accessible values.
Supports accessibility for: Social-Emotional Functioning, Visual-Spatial Processing

Launch

  • Groups of 2
  • Sketch or display a rotated L-shape figure as shown.
  • “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: The figure is not a rectangle. It could be split into smaller rectangles. Students may wonder: Why are there no squares inside? How can I find out how many squares will cover that shape?)
  • 1 minute: quiet think time
  • Share and record responses.
  • “¿Qué información les ayudaría a encontrar el área de esta figura?” // “What information would help you find the area of this figure?” (The side lengths. Being able to see the squares inside the figure.)
  • 1 minute: quiet think time
  • Share responses.
  • Display image from the first problem.
  • “¿Qué información, de la que se da en la figura, podría ayudarles a encontrar el área?” // “What information is given in this figure that could help you find the area?” (Grid lines. The side lengths. Some of the squares.)
  • Share responses.

Activity

  • “Ahora, encuentren el área de esta figura con su pareja” // “Now work with your partner to find the area of this figure.”
  • 5 minutes: partner work time
  • Monitor for strategies for finding the side lengths and decomposing into rectangles.
  • “Examinemos la primera figura” // “Let's look at the first figure.”
  • Have students share strategies for finding the side lengths and area of figures with a partial grid.
  • “Observen la siguiente figura. Piensen en cómo podrían encontrar el área de esta figura” // “Take a look at the next figure. Think about how you could find the area of this figure.”
  • 1 minute: quiet think time
  • “Con su pareja, encuentren el área de esta figura” // “Work with your partner to find the area of this figure.”
  • 5 minutes: partner work time
  • Monitor for strategies for finding the side lengths.

Student Facing

¿Qué observas? ¿Qué te preguntas?

A figure for finding area.

Encuentra el área de cada figura. Explica o muestra tu razonamiento.

  1.  
    A figure for finding area.

  2.  
    T-shaped figure. If split horizontally, 2 rectangles. Top, 8 tick marks by 3 tick marks. Bottom, 4 tick marks by 2 tick marks. 

Student Response

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Activity Synthesis

  • “¿Cuál fue su estrategia para encontrar el área de la segunda figura?” // “What was your strategy for finding the area of the second figure?”
  • “¿Qué les ayuda a encontrar el área de figuras como estas, que no están completamente cubiertas con cuadrados?” // “What helps you find the area of figures like these where the shape is not fully covered with squares?” (Imagining where the squares would be so I can count them or find the side lengths. Finding the side lengths and multiplying to find the area of rectangles in the figure.)

Activity 2: ¿Cuántos adoquines necesitamos? (15 minutes)

Narrative

The purpose of this activity is for students to find the area of a figure composed of rectangles given only their side lengths. The context of paving a patio provides students a link to their experience with squares of various sizes and should help them imagine how the diagram of the patio could be covered with squares. Students decompose the patio into rectangles and can multiply to find the area of the patio, but they should make the connection that the number of pavers needed to cover the patio is the same as the area of the patio. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

MLR7 Compare and Connect. Synthesis: Invite groups to prepare a visual display that shows the strategy they used to figure out the number of tiles and the area of the floor. Encourage students to include details that will help others interpret their thinking. Give students time to investigate each others’ work. During the whole-class discussion, ask students, “¿Por qué resultó la misma área con todos los métodos?” // “How did the same area show up in each method?” “¿Por qué las diferentes estrategias llevaron al mismo resultado?” // “Why did the different approaches lead to the same outcome?” “¿Alguien resolvió el problema de la misma forma, pero lo explicaría de otra manera?” // “Did anyone solve the problem the same way, but would explain it differently?”
Advances: Representing, Conversing

Launch

  • Groups of 2
  • Display the image.
  • “Este problema se trata de la construcción de un patio con adoquines. Los adoquines son rocas, ladrillos o bloques que se ponen sobre el suelo para hacer un camino o un área adoquinada. Este es un patio hecho de adoquines” // “This problem is about making a patio using pavers. Pavers are stones, bricks, or blocks that are put on the ground to make a path or paved area. This is a patio that’s made of pavers.”
  • “Antes hablamos sobre distintos tipos de unidades que podemos usar para medir el área. En este caso, Noah usa adoquines de 1 pie cuadrado. Eso significa que el adoquín es de 1 pie por 1 pie. Ayúdenlo a descubrir cuántos adoquines necesita para cubrir el patio. Piensen en la situación durante un minuto” // “Earlier, we’ve talked about several different types of units that we can use to measure area. Here, Noah is using pavers that are 1 square foot. That means the paver is 1 foot by 1 foot. Help him figure out how many pavers he needs to cover the patio. Take a minute to think about the situation.”
  • 1 minute: quiet think time

Activity

  • “Ahora, respondan las preguntas con su pareja” // “Now work with your partner to answer the questions.”
  • 5 minutes: partner work time
  • Monitor for multiplication expressions students use to represent the area of the rectangles within the figure, such as:
    • \(9 \times 4 = 36\), \(3 \times 2 = 6\), \(36 + 6 = 42\)
    • \(6 \times 3 = 18\), \(6 \times 4 = 24\), \(18 + 24 = 42\)
    • \(9 \times 4 + 3 \times 2\) or \((9 \times 4) + (3 \times 2)\)
    • \(6 \times 3 + 6 \times 4\) or \((6 \times 3) + (6 \times 4)\)

Student Facing

Noah quiere usar adoquines de 1 pie cuadrado para hacer un patio pequeño en la huerta comunitaria. Este es un diagrama del patio.

6-sided shape. Straight sides. All side lengths meet at right angles. Bottom, 3 ft. Right side rises 2 ft, then goes right 6 ft, up 4 ft. Top side length, 9 ft. Left side length, 6 ft.  

  1. ¿Cuántos adoquines de 1 pie cuadrado necesitará Noah para cubrir todo el patio?
  2. ¿Cuál es el área del patio? Explica o muestra tu razonamiento.

    Photograph of patio. Floor paved with square tiles. 

Student Response

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Advancing Student Thinking

If students add numbers that indicate they tried to find the area by adding the areas of rectangles that overlap, consider asking:

  • “Dime cómo encontraste el área de la figura” // “Tell me about how you found the area of the figure?”
  • “¿Cómo influiría la superposición de los rectángulos en el número de cuadrados necesarios para cubrir la figura?” // “How would overlapping the rectangles affects the number of squares it would take to cover the figure?”

Activity Synthesis

  • “¿Cómo descubrieron cuántos adoquines necesitaría Noah?” // “How did you figure out how many pavers Noah would need?”
  • Select students to share a variety of strategies. Display any expressions students used in their explanations.
  • During each explanation, ask the class which measurements were and were not used and why.
  • “¿Cómo representa cada expresión una forma de encontrar el área de la figura?” // “How does each expression represent a way of finding the area of the figure?” (The expression \((9 \times 4) + (3 \times 2)\) shows the figure decomposed into a big rectangle across the top and a small rectangle below. Each part of the expression in parentheses represents one of the smaller rectangles in the figure. The expressions \(6 \times 3 = 18\), \(6 \times 4 = 24\), and \(18 + 24 = 42\) show how the figure was decomposed into a 6-by-3 rectangle on the left and a 4-by-6 rectangle on the right, then the areas were added.)

Lesson Synthesis

Lesson Synthesis

“En esta lección, encontramos el área de algunas figuras aunque no tuvieran dibujada la cuadrícula completa. ¿En qué tuvieron que pensar al encontrar el área de una figura cuando solo tenían las medidas de las longitudes de los lados?” // “In this lesson, we found the area of figures even if they were not fully gridded with squares. What did you need to think about when finding the area of a figure with just side length measurements?” (I can imagine it being filled with squares and count them. I can break the shape into rectangles and multiply the side lengths and then add those areas together.)

Cool-down: Encuentra el área (5 minutes)

Cool-Down

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