Lesson 5

Representemos productos como áreas

Warm-up: Cuántos ves: Uno más (10 minutes)

Narrative

The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see. The arrangement of the groups of dots encourages students to see 5 groups of dots in the first image and then 6 groups of dots in the next image. When students use equal groups and a known quantity to find an unknown quantity, they are looking for and making use of structure. (MP7).

Launch

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

Activity

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.

Student Facing

¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?

Activity Synthesis

• “¿Cómo les ayudó la primera imagen a encontrar el número de puntos de la segunda imagen?” // “How did the first image help you find the number of dots in the second image?” (I know that 5 groups of 3 is 15, and one more group of 3 would be 18.)
• “¿Cómo les ayudaron la primera y la segunda imagen a encontrar el número de puntos de la tercera imagen?” // “How did the first and second images help you find the number of dots in the third image?” (I figured out 5 groups of 4 pretty quickly, then added another group of 4.)

Activity 1: Emparejemos expresiones con áreas (15 minutes)

Narrative

The purpose of this activity is for students to directly connect multiplication expressions to equal groups they see within rectangular areas. Students may decompose the rectangular areas in various ways to see equal groups, but they should relate the rows and columns to the factors of a multiplication expression. This will be helpful in future activities when students multiply side lengths to find the area.

Required Materials

Materials to Copy

• Match Expressions and Areas

Launch

• Groups of 3–4
• Sketch a 5-by-3 gridded rectangle, as shown.

• “¿De qué manera podrían describir este rectángulo?” // “What is one way you could describe this rectangle?” (It has 3 rows of 5 squares. There are 5 groups of 3. Its area is 15 square units. There are 15 squares.)
• Share and record responses. Save responses for discussion after the next activity.
• Display rectangles from the blackline master around the room.

Activity

• “Emparejen cada expresión con uno de los rectángulos que están colgados alrededor del salón. Prepárense para explicar su razonamiento” // “Match each expression to one of the rectangles posted around the room. Be ready to explain your reasoning.”
• 5–7 minutes: group work time

Student Facing

Tu profesor ha colgado imágenes de rectángulos alrededor del salón. Empareja cada expresión con un rectángulo que la pueda representar. Prepárate para explicar tu razonamiento.

1. $$9 \times 5$$
2. $$8 \times 2$$
3. $$7 \times 10$$
4. $$3 \times 3$$
5. $$2 \times 6$$
6. $$8 \times 4$$
7. $$5 \times 7$$

Student Response

If students don’t mention the groups in the rows and columns of squares, consider asking:

• “¿Cómo decidiste cuál rectángulo le correspondía a cada expresión?” // “How did you decide which rectangle matched each expression?”
• “¿Dónde vemos grupos iguales en los rectángulos?” // “Where do we see equal groups in the rectangles?”

Activity Synthesis

• “¿Cómo ven cada factor en el rectángulo?” // “How do you see each factor in the rectangle?” (I can see one factor in the number of squares in a row. I can see the other factor as the number of rows. I see one factor as the number of squares in a column. The other factor is the number of columns. It’s like I see the factors in an array, only it’s squares, not dots.)
• “¿Cómo ven el producto en los rectángulos?” // “How do you see the product in the rectangles?” (If we count the squares in each rectangle, it gives us the same number as the product of the factors. The product is the same as the total number of squares in each rectangle.)
• “¿Por qué al multiplicar obtenemos el mismo número que al contar uno por uno?” // “Why does multiplication give the same number as counting one by one?”

Activity 2: Creemos a partir de expresiones (20 minutes)

Narrative

The purpose of this activity is for students to represent multiplication expressions as rectangular areas. Students use a grid to draw the rectangular area that represents a multiplication expression. In the synthesis, students explain how they interpret the multiplication expression, specifically how they see the equal groups in the rows and columns of the rectangular area. Give students access to square tiles if needed. When students draw and relate area diagrams to multiplication expressions they are reasoning abstractly and quantitatively (MP2).

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 5 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills

Required Materials

Materials to Gather

Launch

• Groups of 2
• “Ahora van a dibujar rectángulos que correspondan a algunas expresiones de multiplicación” // “Now you’re going to draw rectangles that match some multiplication expressions.”
• 1 minute: quiet think time

Activity

• 7–10 minutes: partner work time
• Monitor for a rectangle that two students oriented differently.

Student Facing

1. Los números de cada expresión representan el número de filas (o columnas) de un rectángulo y cuántos cuadrados hay en cada fila (o columna).

En la cuadrícula, dibuja cada rectángulo, márcalo con los números y encuentra su área.

1. $$3 \times 4$$
2. $$4 \times 6$$
3. $$6 \times 3$$
4. $$7 \times 4$$
5. $$3 \times 2$$
2. Explica por qué al multiplicar los números de cada expresión obtenemos el área del rectángulo.

Activity Synthesis

• Have 2–3 students share a rectangle for each expression.
• For each student sample ask:
• “¿Cómo corresponde el área de este rectángulo a la expresión?” // “How does the area of this rectangle match the expression?” (I see 4 equal groups of 6 because each row has 6 squares. I see 4 equal groups because each column has 6 squares.)
• “¿Alguien dibujó un rectángulo diferente para esta expresión?” // “Did anyone draw a different rectangle for this expression?”
• Display a rectangle that two students oriented differently.
• “¿Cómo pueden corresponder estos dos rectángulos a la misma expresión?” // “How can both of these rectangles match the same expression?” (They have the same number of squares. They have the same side lengths, just switched. I see the groups in the rows in the first rectangle and in the columns in the second rectangle.)
• Display the 3-by-5 rectangle and descriptions from the launch in the first activity.
• “¿Cuál de las maneras de describir un rectángulo fue la que más les ayudó a dibujar rectángulos en esta actividad?” // “Which way of describing a rectangle was the most helpful to you as you drew rectangles in this activity?” (It was helpful to describe how many rows were in the rectangle and how many squares were in each row. It was helpful to think about one factor as the number of columns and the other factor as the number of squares in each column.)

Lesson Synthesis

Lesson Synthesis

Display or sketch a 2-by-7 gridded rectangle with the side lengths labeled 2 and 7, as shown:

“¿Cómo podrían encontrar el número total de cuadrados que hay en este rectángulo?” // “How could you figure out the total number of squares in this rectangle?” (Count by one. Count by 2. Count by 7. Multiply $$2 \times 7$$. Multiply $$7 \times 2$$.)

“¿En qué se parecen las áreas rectangulares a otras maneras en las que hemos mostrado la multiplicación?” // “How are rectangular areas similar to other ways we’ve shown multiplication?” (We can see rows and columns like arrays. We can see equal groups in the rows. We can see equal groups in the columns.)

“¿En qué son diferentes las áreas rectangulares y las otras maneras en las que hemos mostrado la multiplicación?” // “How are rectangular areas different from other ways we’ve shown multiplication?” (We’re counting spaces instead of objects.)