Lesson 11

This warm-up prompts students to compare four representations of multiplication. It gives students a reason to use language precisely as they talk about characteristics of the items being compared.  During the synthesis, ask students to explain the meaning of any terminology they use, such as strategies, area, and parts.

• Groups of 2
• Display the image.
• “Escojan uno 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

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.
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• “¿Qué número representan los diagramas y la expresión de C?” // “What number do the diagrams and the expression in C represent?” (18) “¿Cómo lo saben?” // “How do you know?” (There are 18 dots in the array. There are 18 squares in the rectangle. If I add up the parts of the expression or the parts of the rectangle, I get 18.) 
• “¿Cuál podría ser la longitud del lado del rectángulo que no está marcado en B? ¿Cómo lo saben?” // “What might be the length of the unlabeled side of the rectangle in B? How do you know?” (3, because the rectangle is the same one as in A, just not showing a grid. 3, because \(3 \times 2 = 6\) and \(3 \times 4 = 12\).
• Consider asking:
  • “Encontremos al menos una razón por la que cada uno es diferente” // “Let’s find at least one reason why each one doesn’t belong.”

The purpose of this activity is for students to find the area of ungridded rectangles using strategies based on the distributive and associative properties. Students represent these strategies on rectangles with no grid. This will be helpful in future lessons as students use area diagrams to represent the multiplication of larger numbers.

• Groups of 2
• “Vamos a encontrar el área de más rectángulos. ¿En qué se diferencian estos rectángulos de los rectángulos con los que trabajamos en la última lección?” // “We are going to find the area of more rectangles. How are these rectangles different from the rectangles we worked with in the last lesson?” (They don’t have a grid in them. We can’t see the squares.)
• 30 seconds: quiet think time
• Share responses.

• “Marquen o coloreen cada rectángulo para encontrar su área. Después, escriban una o más expresiones que representen su trabajo y muestren cómo encontraron el área” // “Mark or shade each rectangle to help you find its area. Then write one or more expressions that represent your work and show how you found the area.”
• 5–7 minutes: independent work time
• “Compartan con su compañero cómo encontraron el área de cada rectángulo. Asegúrense de hacer y responder todas las preguntas que tengan sobre sus estrategias” // “Share how you found the area of each rectangle with your partner. Be sure to ask and answer any questions you have about your partner’s strategy.”
• 3–5 minutes: partner discussion

If students say they aren’t sure where to mark or shade the rectangle because they can’t see the squares, consider asking:

• “¿Qué números estás multiplicando para encontrar el área?” // “What numbers are you multiplying to find the area?”
• “¿Cómo podrías descomponer uno de los factores para encontrar el producto? ¿Cómo mostrarías eso en el diagrama?” // “How could you decompose one of the factors to help you find the product? How would you show that on the diagram?”

• “¿Cuál es la diferencia entre mostrar su estrategia en un rectángulo sin cuadrícula y mostrar su estrategia en un rectángulo con una cuadrícula?” // “How was showing your strategy on a rectangle with no grid different than showing your strategy on a rectangle with a grid?” (I just estimated where I thought I should split the rectangle. I was thinking more about the numbers than counting all the squares.)

In this sorting activity, students identify expressions that could represent the area of the same rectangle and explain their reasoning. To do so, they apply their understanding of properties of multiplication and draw rectangles as needed as they interpret parts of the expressions. Some students may sort expressions based only on the value of the expressions. Encourage them to explain or show how they know, for instance, that \(8 \times 6\) and \(3 \times 6 + 5 \times 6\) can represent the area of the same rectangle (MP2, MP7). Some of the expressions from this activity are used in the synthesis to highlight the commutative, distributive, and associative properties of multiplication.

• Create a set of cards from the blackline master for each group of 2 or 4.

• Groups of 2 or 4
• Give each group a set of pre-cut cards from the blackline master.
• Give students access to grid paper.

• “Este grupo de tarjetas incluye expresiones que representan áreas de rectángulos. Agrupen las expresiones que representan el área del mismo rectángulo” // “This set of cards includes expressions that represent areas of rectangles. Group together expressions that can represent the area of the same rectangle.”
• “Con su compañero, expliquen sus decisiones de clasificación. Si les ayuda, pueden dibujar rectángulos” // “Work with your partner to explain your sorting decisions. You can draw rectangles if you find them helpful.”
• 8 minutes: partner work time

• Invite students to share their sorting results, drawings (if any), and explanations on how they know those expressions go together.

• Record each group of expressions. Discuss the connections between the expressions, illustrating them on a drawing of a rectangle. For instance, for: \(\displaystyle 8 \times 3\) \(\displaystyle (4 \times 3) \times 2\) \(\displaystyle 4 \times (2 \times 3)\)

  ask questions such as:

  • “¿En qué parte de \(4 \times (2 \times 3)\) vemos el 8?” // “Where do we see the 8 in \(4 \times (2 \times 3)\)?”
  • “¿Cuál es el área de este rectángulo? ¿Cuáles podrían ser las longitudes de sus lados?” // “What’s the area of this rectangle? What could its side lengths be?”
  • (Draw and label a rectangle.)

    

    

  • “¿Dónde vemos \((4 \times 3) \times 2\) en el rectángulo?” // “Where do we see the \(2 \times (4 \times 3)\) in the rectangle?”

    

    

  • “¿Dónde vemos \(4  \times (2 \times 3)\) en el rectángulo?” // “Where do we see the \(4  \times (2 \times 3)\) in the rectangle?”