Lesson 10

Exploremos estrategias de multiplicación con rectángulos

Narrative

The purpose of this How Many Do You See is for students to use grouping strategies to describe the quantities they see.

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.
• Keep the images displayed for the launch of the next activity.

Student Facing

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

Activity Synthesis

• “¿Cómo podemos usar las cantidades que vemos rápidamente para encontrar el número total de cuadrados?” // “How can we use amounts that we can see quickly to find the total number of squares?” (We can look for repetition of the number of squares that we can easily see. We can add to or multiply the number of squares we can easily see.)
• “¿Alguien puede expresar con otras palabras la forma en la que _____ vio los cuadrados?” // “Who can restate the way _____ saw the squares in different words?”
• “¿Alguien vio los cuadrados de la misma forma, pero lo explicaría de otra manera?” // “Did anyone see the squares the same way but would explain it differently?”
• “¿Alguien quiere compartir otra observación acerca de la forma en la que _____ vio los cuadrados?” // “Does anyone want to add an observation to the way _____ saw the squares?”

Activity 1: De diagramas a expresiones (20 minutes)

Narrative

The purpose of this activity is for students to analyze different ways of decomposing a gridded rectangle to find the total number of squares in a rectangle. For example, they see that the area of a rectangle that is 3 units by 6 units can be found adding $$3 \times 5$$ and $$3 \times 1$$ and relate that strategy to the expression $$(3 \times 5) + (3 \times 1)$$. The area can also be found by decomposing the rectangle into two halves or finding $$3 \times 3$$ twice, which is represented by $$2 \times (3 \times 3)$$.

The reasoning here allows students to visually make sense of strategies for multiplication that are based on the associative and distributive properties of multiplication. The focus is not on naming the properties, but rather on interpreting the expressions and relating them to the quantities in the diagrams (MP7).

Launch

• Groups of 2
• Display the first problem.
• “Tómense un minuto para entender cómo Andre y Elena encontraron el área de un rectángulo” // “Take a minute to make sense of how Andre and Elena found the area of a rectangle.”
• 1 minute: quiet think time

Activity

• “Discutan con su compañero en qué se parecen y en qué se diferencian las estrategias de Andre y Elena. También discutan cómo se relacionan los números de cada una de sus expresiones con sus diagramas” // “Work with your partner to discuss how their strategies are alike and different, and how the numbers in each of their expressions relate to their diagrams.”
• 5–7 minutes: partner work time
• Share responses.
• “Resuelvan el segundo problema con su compañero” // “Work with your partner to complete the second problem.”
• 3–5 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for and collect ways that students decompose the rectangle and the language students use to describe the strategies they used. Listen for: decomposed, smaller parts, smaller rectangles, $$2 \times 2 \times 9$$, $$2 \times 18$$, $$4 \times 5 + 4 \times 4$$, and $$5 \times 4 + 4 \times 4$$.
• Record students’ diagrams, words, and phrases on a visual display and update it throughout the lesson.

Student Facing

Andre y Elena están hallando el área de este rectángulo.

Andre escribe $$6 \times 3$$.

Él marca el rectángulo así:

Después, Andre escribe:

$$2 \times (3 \times 3)$$
$$2 \times 9 = 18$$

Elena escribe  $$3 \times 6$$

Ella marca el rectángulo así:

Después, Elena escribe:

$$3 \times (5 + 1)$$
$$(3 \times 5) + (3 \times 1)$$
$$15 + 3$$
18

1. Discute con un compañero:

1. ¿En qué se parecen las estrategias de Andre y Elena? ¿En qué son diferentes?
2. ¿Cómo se relacionan los números de las expresiones de Andre con su diagrama?
3. ¿Cómo se relacionan los números de las expresiones de Elena con su diagrama?
2. Este es otro rectángulo.

Podemos encontrar su área hallando $$4 \times 9$$.

1. Marca o colorea el rectángulo de una manera que te ayude a encontrar su área.
2. Escribe una o más expresiones que representen lo que hiciste en el diagrama y muestra cómo encontraste el área.

Student Response

If students count one-by-one to find the area if the rectangle in the second problem, consider asking:

• “¿Cómo encontraste el área del rectángulo?” // “How did you find the area of the rectangle?”
• “¿Cómo puedes usar un producto que ya conoces para encontrar el área del rectángulo? ¿Cómo puedes mostrar tu estrategia en el rectángulo?” // “How could you use a product you already know to find the area of the rectangle? How could you show your strategy on the rectangle?”

Activity Synthesis

• “¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.

Activity 2: De expresiones a diagramas (15 minutes)

Narrative

In this activity, students are given expressions that represent strategies for finding the area of rectangles. The strategies are based on the distributive property and the associative property of multiplication. Students interpret the expressions by marking or shading area diagrams and connect each expression to the product of two factors (MP2). For instance, they see that to find the value of $$2 \times (2 \times 6)$$ is to find the value of $$4 \times 6$$ or $$6 \times 4$$.

MLR8 Discussion Supports: Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, trace the area showing 5 columns of 3, and write $$5 \times 3$$.
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 2 of the 3 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills

Launch

• Groups of 2
• “Tómense un minuto para leer las instrucciones de la actividad. Después, hablen con su compañero sobre lo que se les pide que hagan” // “Take a minute to read the directions of the activity. Then, talk to your partner about what you are asked to do.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Answer any clarifying questions from students.

Activity

• “Marquen o coloreen cada diagrama para representar cómo encontró el área cada estudiante” // “Mark or shade each diagram to represent how each student found the area.”
• 3–5 minutes: independent work time
• “Compartan con su compañero cómo usaron los rectángulos para mostrar cada expresión” // “Share with your partner how you used the rectangles to show each expression.”
• 3–5 minutes: partner discussion

Student Facing

Estos son algunos rectángulos y algunas expresiones que muestran la manera en la que tres estudiantes vieron el área de los rectángulos.

Noah

$$(5\times 3)+(2 \times 3)$$

Priya

$$2 \times (2 \times 6)$$

Tyler

$$(5 \times 8) + (3 \times 8)$$

1. Escribe los dos factores que se pueden multiplicar para encontrar su área.
2. Marca o colorea cada rectángulo para mostrar la manera en la que cada estudiante vio el área. Prepárate para explicar tu razonamiento.

Activity Synthesis

• “¿Cuáles son los dos factores que pueden multiplicar para encontrar el área del rectángulo de Noah?” // “What are the two factors you can multiply to find the area of Noah’s rectangle?” (7 and 3)
• “¿Cómo se relacionan esos números con la expresión que él escribió: $$(5\times 3)+(2 \times 3)$$?” // “How are those numbers related to the expression that he wrote: $$(5\times 3)+(2 \times 3)$$?”  ($$7 \times 3$$ is 21. Finding $$5\times3$$, which is 15, then adding $$2\times3$$, which is 6, also gives 21.)
• “¿En qué parte de su expresión ven los dos factores?” // “Where do you see the two factors in his expression?” (The 7 is 5 and 2 combined. The 3 is in the $$5 \times 3$$ and $$2 \times 3$$.)
• Repeat the line of questioning with Priya’s rectangle and expression.

Lesson Synthesis

Lesson Synthesis

“Hoy usamos diagramas para encontrar el área de varios rectángulos. Descompusimos los rectángulos de distintas maneras y escribimos expresiones diferentes” // “Today, we used diagrams to find the area of rectangles with certain side lengths. We decomposed the rectangles in different ways and wrote different expressions.”

“¿Cuáles fueron algunas estrategias para descomponer los rectángulos y así encontrar sus áreas?” // “What were some strategies for decomposing the rectangles to find their areas?” (Partition one side into smaller parts and find the area of smaller rectangles within the original one. Partitioning the rectangle into two halves and finding the area of each half and then doubling it.)

“¿Cómo nos podrían ayudar estas estrategias a multiplicar dos números?” // “How might these strategies help us multiply two numbers?” (They show that we can break apart or decompose one of the numbers and multiply smaller numbers and then combine the results. Using diagrams and writing expressions can help us see and record the parts.)