# Lesson 5

Multiplicación de números de varios dígitos

## Warm-up: Exploración de estimación: Un acertijo impreciso (10 minutes)

### Narrative

In this warm-up, students practice estimating a reasonable answer using known information, rounding, and multiplicative reasoning strategies. For example, students may create a diagram and arrive at $$7 \times 7 \times 7 \times 7$$ as an estimate of the number of fish. Then, they may approximate $$(7 \times 7) \times (7 \times 7)$$ with $$50 \times 50$$, or estimate $$(7 \times 7 \times 7) \times 7$$ with $$(50 \times 7) \times 7$$ or $$350 \times 7$$. Some students may notice that the question is vague and asks, “How many are going to the park?” rather than “How many people are going to the park?” and account only for the number of children and teachers in their estimation. If students ask for clarification, ask them to make their own assumptions and explain why they made those assumptions when they share their estimates.

### Launch

• Groups of 2
• Display the description.

### Activity

• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 2 minutes: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

• Siete profesores van al parque.
• Cada profesor lleva a 7 estudiantes.
• Cada estudiante lleva 7 peceras.
• Cada pecera tiene 7 peces.

¿Cuántos van al parque?

Escribe una estimación que sea:

muy baja razonable muy alta
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### Activity Synthesis

• “¿Alguien hizo una estimación menor que 500?, ¿mayor que 5,000?” // “Is anyone’s estimate less than 500? Greater than 5,000?”
• “¿Alguien incluyó el número de profesores y de estudiantes en su estimación?” // “Did anyone include the number of teachers and students in their estimate?”
• Invite students to share their estimation strategies. After each explanation, ask if others reasoned the same way.
• Consider revealing the actual value of 2,457 teachers, students, and fish.

## Activity 1: Retomemos dos métodos (20 minutes)

### Narrative

In this activity, students revisit two algorithms for multiplying numbers. They recall that, in the standard algorithm, the digit in one factor is multiplied by each digit in the other factor, but the partial products are not recorded on separate lines. Rather, the standard algorithm condenses multiple partial products into a single product.

MLR8 Discussion Supports. Synthesis: For each strategy that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Representation: Develop Language and Symbols. Activate or supply background knowledge. Display $$416 \times 2$$ without the solutions. Ask students to tell you what each digit in 416 represents (for example, the 1 represents 10).
Supports accessibility for: Conceptual Processing, Memory, Language

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students access to grid paper, if needed for aligning the digits in a multiplication algorithm.

### Activity

• 2 minutes: independent work time
• Pause to discuss the first set of questions. Display the two algorithms in the first question. Ask students to share responses.
• “¿En qué se parecen los dos algoritmos? ¿En qué son diferentes?” // “How are the two algorithms alike? How are they different?”
• Highlight student responses to emphasize:
• In method A, each partial product is listed separately before being added at the end.
• In method B, only one digit is recorded at a time. The values for any place value unit are added and only one digit is recorded. Any new units are recorded in the next highest place.
• 6–10 minutes: independent work time
• 2–4 minutes: partner discussion

### Student Facing

1. Antes, en este curso, usamos estas dos formas de multiplicar números:

1. En el método A, ¿de dónde vienen el 12, el 20 y el 800?
2. En el método B, ¿de dónde viene el 1 que está encima del 416?
2. Diego usó ambos métodos para encontrar el valor de $$215 \times 3$$, pero terminó con resultados muy distintos.

1. Sin hacer ningún cálculo, ¿puedes saber cuál método muestra el producto correcto? ¿Cómo sabes que el otro producto no es correcto?
2. Explica qué fue correcto y qué fue incorrecto en los pasos de Diego cuando obtuvo el producto incorrecto. Después, muestra el cálculo correcto usando el método B.
3. En cada caso, usa cualquiera de los métodos para encontrar el producto. Muestra cómo razonaste.

1. $$521 \times 3$$
2. $$6,\!121 \times 4$$
3. $$305 \times 9$$

### Activity Synthesis

• Select students to share their responses to the second set of questions.
• If not mentioned in students’ explanations, highlight that:
• The result of $$3 \times 5$$ is 15, a two-digit number, so the 1 ten should be carried over to the tens place and added to the 3 tens that result from $$3 \times 10$$.
• One ten and 3 tens make 4 tens.
• Poll the class on whether their preferred method is A, B, or is dependent on the problem. Select a student from each camp to explain their reason.

## Activity 2: Dos por dos (15 minutes)

### Narrative

Earlier, students compared and made connections between two algorithms for multiplying a multi-digit number and a single-digit number. In this activity, students compare an algorithm that uses partial products with the standard algorithm for multiplying 2 two-digit numbers. As students analyze and critique each method, they practice looking for and making use of base-ten structure of whole numbers (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Access to grid paper, in case needed to align digits when multiplying

### Activity

• 3–4 minutes: independent work time on the first two problems
• 1–2 minutes: partner discussion
• Monitor for students who can explain the numbers in the standard algorithm.
• 5 minutes: independent work time on the remaining questions

### Student Facing

Estas son dos formas de encontrar el valor de $$34 \times 21$$.

1. En el método A, ¿de dónde vienen el 4, el 30, el 80 y el 600?
2. En el método B, escribe cuáles dos números se multiplican para obtener:

1. 34
2. 680
3. En cada caso, usa los dos métodos para mostrar que la ecuación es verdadera.

### Activity Synthesis

• Invite students to share how they used each method to show that $$44 \times 12 = 528$$​ and $$63 \times 21 = 1,\!323$$

## Lesson Synthesis

### Lesson Synthesis

“Hoy examinamos varios métodos para multiplicar un número de varios dígitos por un número de un dígito y también para multiplicar 2 números de dos dígitos” // “Today we looked at several methods for multiplying a multi-digit number by a single-digit number and also multiplying 2 two-digit numbers.“

“Estos son algunos razonamientos o estrategias de cálculo que hemos visto para multiplicar 2 números de dos dígitos” // “Here are some reasoning or calculation strategies we have seen for multiplying 2 two-digit numbers.”

$$33 \times 2 = 66$$

$$33 \times 10 = 330$$

$$66 + 330 = 396$$

“¿Qué conexiones observan entre estas estrategias? Señalen tantas como puedan” // “What connections do you see among these strategies? Point out as many as you can.”

“¿Cuál de estas estrategias tiene más sentido para ustedes o les parece que es la más clara?” // “Which of these strategies makes the most sense or is clearest to you?”