Lesson 7

Multipliquemos números de tres y de cuatro dígitos por números de un dígito

Warm-up: Exploración de estimación: Área desconocida (10 minutes)

Narrative

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. Listen for the different ways students use their understanding of place value to estimate the area and explain why an estimate is too low or too high.

Launch

  • Groups of 2
  • Display the image.
  • “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
  • 1 minute: quiet think time

Activity

  • “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Record responses.

Student Facing

¿Cuál es el área del rectángulo?

Rectangle. Horizontal side, 3 hundred ninety 5. Vertical side, 6.

Escribe una estimación que sea:

muy baja razonable muy alta
\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)

Student Response

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Activity Synthesis

  • “¿Su estimación es mayor o es menor que el producto exacto?” // “Is your estimate greater or less than the actual product?” (greater)
  • “¿Cómo podríamos encontrar el producto exacto?” // “How could we find the actual product?” (\(6 \times 400 - 6 \times 5\), or subtract \(6 \times 5\) from 2,400)

Activity 1: Multipliquemos números más grandes (15 minutes)

Narrative

In this activity, students use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Though students may decompose the multi-digit factor in different ways, the activity is designed to encourage them to decompose it by place value—into hundreds, tens, and ones.

MLR2 Collect and Display. Synthesis: Direct attention to words collected and displayed from the previous activity. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading

Launch

  • Groups of 2

Activity

  • “Trabajen con su compañero en el primer problema. Luego, intenten hacer el resto de la actividad individualmente” // “Work with your partner on the first problem. Then, try the rest of the activity independently.”
  • 2 minutes: group work time on the first problem
  • 5–7 minutes: independent work time on the rest of the activity
  • 3 minutes: partner discussion
  • Monitor for students who:
    • write expressions that show the multi-digit factor decomposed by place value
    • draw diagrams that partition the multi-digit factor by place value

Student Facing

  1. Clare dibujó este diagrama.
    Diagram, rectangle partitioned vertically into 3 rectangles.
    1. ¿Qué expresión de multiplicación puede estar representada por el diagrama?

    2. Encuentra el valor de la expresión. Muestra tu razonamiento.
  2. Piensa en la expresión \(6 \times 252\).

    1. Dibuja un diagrama para representar la expresión.
    2. Encuentra el valor de la expresión. Muestra tu razonamiento.

  3. Lin dibujó un diagrama para representar \(3 \times 2,\!135\).

    Diagram, rectangle partitioned vertically into 4 rectangles.

    1. Completa el diagrama de Lin.
    2. En cada parte del diagrama, escribe una expresión que represente el valor de esa parte.

    3. Encuentra el valor de \(3 \times 2,\!135\). Muestra cómo razonaste.

Student Response

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Activity Synthesis

  • Invite several students to share diagrams they drew to represent \(6\times252\).
  • As each student shares, consider asking the class:  
    • “¿Dónde ven las partes del factor que se descompuso?” // “Where do you see the parts of the factor that was decomposed?”
    • “¿Qué expresiones están representadas en el diagrama?” // “What expressions are represented in the diagram?”
  • Record expressions as students share.

Activity 2: Los errores de Jada (20 minutes)

Narrative

This activity extends students' work with multiplication to include a factor with up to four digits. Students begin to generalize that they could decompose any number into parts and multiply the parts. In this activity, students analyze a common error when multiplying. The work they look at does not apply place value understanding and therefore represents a product that is unreasonable for the given expression. When students analyze Jada's work, find her errors and explain their reasoning, they critique the reasoning of others (MP3).

Students see partial products as a way to describe the sub-products when a factor is decomposed and multiplied using the distributive property. Continue to refer to partial products in students’ diagrams and calculations by their mathematical name to build students’ intuition for their meaning (though students are not expected to use them in their reasoning).

Representation: Access for Perception. Use base-ten blocks to demonstrate Jada’s error and how her work should be corrected. Invite students to discuss how they might avoid such an error (for example, by estimating the product first, or by visualizing the value in base-ten blocks).
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Launch

  • Groups of 2

Activity

  • “Tómense unos minutos en silencio para analizar los errores de Jada. Luego, compartan con su compañero cómo pensaron” // “Take a few quiet minutes to analyze Jada’s errors. Then, share your thinking with your partner.”
  • 2 minutes: independent work time on the first problem about Jada’s diagram
  • 2 minutes: partner discussion
  • Pause for a discussion.
  • “¿Qué hizo Jada correctamente?” // “What did Jada do correctly?” (She multiplied the non-zero digits correctly. She decomposed the 6,489 by place value, which is helpful.)
  • “¿Qué le faltó hacer a Jada?” // “What did Jada miss?” (She multiplied only the non-zero digits. She didn’t account for the place value of the digits being multiplied. For example: The partial product of \(3 \times 6,\!000\) is 18,000, not 18.)
  • Display and correct Jada’s diagram as a class.
  • “Ahora completen los demás problemas individualmente” // “Now complete the remaining problems independently.”
  • 5 minutes: independent work time on the last two problems
  • Monitor for students who:
    • draw diagrams that partition the multi-digit factor by place value
    • write expressions that show the multi-digit factor decomposed by place value

Student Facing

  1. Jada usó un diagrama para multiplicar \(3 \times 6,\!489\) y cometió algunos errores.
    Diagram, rectangle partitioned vertically into 4 rectangles.
    1. Explica qué errores cometió Jada.
    2. Encuentra el valor de \(3 \times 6,\!489\). Muestra cómo razonaste.

  2. Encuentra el valor de \(5 \times 699\). Muestra cómo razonaste.
  3. Encuentra el valor de \(8 \times 4,\!973\). Muestra cómo razonaste.

Student Response

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Activity Synthesis

  • Select 1–2 students to share their solutions and reasoning for finding the value of \(8\times4,\!973\).
  • Consider asking students to write expressions that represent the way they found each product. For example:
    \(\displaystyle 8 \times 4,\!973\\ 8 \times (4,\!000 + 900 + 70 + 3)\\ (8 \times 4,\!000) + 8 \times 900 + 8 \times 70 + 8 \times 3\\ 32,\!000 + 7,\!200 + 560 + 24\\ 39,\!784\)

Lesson Synthesis

Lesson Synthesis

“Hoy usamos diagramas para multiplicar números de tres y de cuatro dígitos por números de un dígito. Comparemos estos diagramas que representan \(7 \times 2,\!129\) y \(7 \times 129\)” // “Today we used diagrams to multiply three- and four-digit numbers by one-digit numbers. Let’s compare diagrams that represent \(7 \times 2,\!129\) and \(7 \times 129\).”

Display: \(7 \times 2,\!129\) and \(7 \times 129\)

area diagram

\(14,\!000 + 700 + 140 + 63 = 14,\!903\)

area diagram

\(700 + 140 + 63 = 903\)

“¿En qué se parecen las representaciones y en qué son diferentes?” // “How are the representations alike and how are they different?” (Sample responses:

  • Alike: They both show the expanded form of the multi-digit factor, have 7 as one of the factors, and show partial products. They both show one factor decomposed by place value or written in expanded form.
  • Different: One diagram is decomposed into four parts because the factor 2,129 has four digits. The diagram shows 4 partial products. The other diagram is decomposed into 3 parts, because 129 has three digits. The diagram shows 3 partial products.)

“¿Cómo encontrarían el valor de \(7 \times 2,\!039\)?” // “How would you find the value of \(7 \times 2,\!039\)?” (Think of 2,039 as \(2,\!000 + 30 + 9\) and find \((7 \times 2,\!000) + (7 \times 30) + (7 \times 9)\).)

Cool-down: El valor del producto (5 minutes)

Cool-Down

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