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 greater numbers.

• “¿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, students apply their understanding of properties of multiplication and draw rectangles as needed in order to interpret parts of the expressions. Some students may sort expressions based only on their values. Encourage those students to explain or show how they know, for instance, that and can represent the area of the same rectangle (MP7). Some of the expressions from this activity are used in the Synthesis to highlight the commutative, distributive, and associative properties of multiplication.

• 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. Example:

  Ask questions, such as:

  • “¿En qué parte de  vemos el 8?” // “Where do we see the 8 in ?”
  • “¿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  en el rectángulo?” // “Where do we see the in the rectangle?”

    

  • “¿Dónde vemos  en el rectángulo?” // “Where do we see the in the rectangle?”