Lesson 5

Algoritmo estándar: Números de varios dígitos, sin componer

Warm-up: Conversación numérica: Productos parciales (10 minutes)

Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for mentally multiplying numbers that require composing a new unit. Students apply this understanding in the lesson when they compose a new unit using the standard algorithm.

Launch

  • Display one problem.
  • “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time

Activity

  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Facing

Encuentra mentalmente el valor de cada producto.

  • \(20 \times 3\)
  • \(24 \times 3\)
  • \(120 \times 3\)
  • \(140 \times 3\)

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

Activity Synthesis

  • “Comparen las decenas de \(20 \times 3\) y las decenas de \(24 \times  3\). ¿Qué pueden decir?” // “How do the tens in \(20 \times 3\) compare to the tens in \(24 \times  3\)?” (There is one more ten in \(24 \times 3\) that came from \(3 \times 4\).)
  • “Comparen las centenas de \(120 \times 3\) y las centenas de \(140 \times 3\). ¿Qué pueden decir?” // “How do the hundreds in \(120 \times 3\) compare to the hundreds in \(140 \times 3\)?” (There is one more hundred in \(140 \times 3\) that came from \(40 \times 3\).)

Activity 1: Comparemos dos algoritmos (20 minutes)

Narrative

The purpose of this activity is for students to connect an algorithm that uses partial products to the standard algorithm when multiplying a three-digit and a two-digit number. The standard algorithm shows 2 partial products while the other algorithm shows 6 partial products. While the products are recorded differently, the same 6 partial products are still part of both calculations, and this activity gives students a chance to see this common structure while also appreciating the different way the standard algorithm records the calculations.

Students use the common structure in the two algorithms (MP7) to make sense of the standard algorithm before they use it themselves in the next activity. 

MLR8 Discussion Supports. Display sentence frames to support small-group discussions: “_____ y _____ se parecen porque . . .” // “_____ and _____ are the same/alike because . . .”, “_____ y _____ son diferentes porque . . .” // “_____ and _____ are different because . . . .”
Advances: Listening, Speaking

Launch

  • Groups of 2
  • Display the algorithms.
  • “Hoy vamos a aprender acerca de un algoritmo nuevo” // “We are going to learn about a new algorithm today.”

Activity

  • 1–2 minutes: quiet think time
  • 8–10 minutes: partner work time
  • Monitor for students who notice that:
    • the two algorithms show the same products in the same right to left order.
    • the two algorithms record the results of the products differently.

Student Facing

A continuación se muestran dos algoritmos para encontrar el valor de \(413 \times 21\).

multiply. four hundred thirteen times 21.

paso 1

multiply. four hundred thirteen times 21. 3 rows. First row: four hundred thirteen. Second row: multiplication symbol, 21. Horizontal line. Third row: 3

paso 2

multiply. four hundred thirteen times 21. 3 rows. First row: four hundred thirteen. Second row: multiplication symbol, 21. Horizontal line. Third row: 13

paso 3

multiply. four hundred thirteen times 21. 3 rows. First row: four hundred thirteen. Second row: multiplication symbol, 21. Horizontal line. Third row: four hundred thirteen

paso 4

multiply. four hundred thirteen times 21. 4 rows. First row: four hundred thirteen. Second row: multiplication symbol, 21. Horizontal line. Third row: four hundred thirteen. Fourth row: sixty.

paso 5

multiply. four hundred thirteen times 21. 4 rows. First row: four hundred thirteen. Second row: multiplication symbol, 21. Horizontal line. Third row: four hundred thirteen. Fourth row: two hundred sixty

paso 6

multiply. four hundred thirteen times 21. 4 rows. First row: four hundred thirteen. Second row: multiplication symbol, 21. Horizontal line. Third row: four hundred thirteen. Fourth row: eight thousand two hundred sixty

paso 7

multiply. four hundred thirteen times 21.
  1. ¿En qué se parecen los dos algoritmos? ¿En qué se diferencian?
  2. Explica o muestra en dónde ves cada paso del primer algoritmo en el segundo algoritmo.
  3. ¿En qué se parecen y en qué se diferencian los últimos pasos de los dos algoritmos?

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

Advancing Student Thinking

If students do not explain where each of the partial products are in the standard algorithm, ask, “¿Cómo podemos usar expresiones de multiplicación para mostrar dónde está cada producto parcial en el algoritmo estándar?” // “How can we use multiplication expressions to show where each partial product is in the standard algorithm?”

Activity Synthesis

  • Invite students to share what was alike in the two algorithms, highlighting:
    • all six partial products are calculated in both.
    • they are calculated in the same order.
    • they both need to add up their partial products at the end.
  • Invite students to share what was different in the two algorithms highlighting:
    • one algorithm lists each partial product on a separate line while the standard algorithm lists some of them on the same line
  • Circle the first partial product (413) in the standard algorithm.
  • “¿Qué representa el primer producto parcial (413)?” // “What does the first partial product 413 represent?” (It's \(1 \times 413\).)
  • Circle the second partial product (8,260) in the standard algorithm.
  • “¿Qué representa el segundo producto parcial (8,260)?” // “What does the second partial product 8,260 represent?” (It's \(20 \times 413\).)

Activity 2: Usemos el algoritmo estándar (15 minutes)

Narrative

The purpose of this activity is for students to practice multiplying a two-digit and a three-digit number using the standard algorithm. The problems do not involve composing new units so that students can practice the procedure of multiplying each place in one factor by each place in the other factor. In the last problem, students look at incorrect work where the value of the digit in the tens place is not accounted for. This problem encourages them to use estimation to assess the reasonableness of their answers and is the focus of the lesson synthesis.

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were needed to solve the problems using the standard algorithm. Display the sentence frame, “La próxima vez que use el algoritmo estándar, prestaré atención a . . .” // “The next time I use the standard algorithm, I will pay attention to . . . .”
Supports accessibility for: Attention, Conceptual Processing

Launch

  • Groups of 2

Activity

  • 8-10 minutes: independent work time
  • 2-3 minutes: partner discussion

Student Facing

Usa el algoritmo estándar para encontrar el valor de cada expresión.

  1. \(202 \times 12\)
  2. \(122 \times 33\)
  3. \(321 \times 24\)
  4. Diego encontró el valor de \(301 \times 24\). Este es su trabajo. ¿Por qué no tiene sentido la respuesta de Diego?

    multiply. three hundred one times 24.

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

Activity Synthesis

  • Invite students to share their solution for \(122 \times 33\).
  • “¿En qué se parece multiplicar 122 por el 3 que está en la posición de las unidades de 33 a multiplicar 122 por el 3 que está en la posición de las decenas?” // “How is multiplying 122 by the 3 in the ones place of 33 the same as multiplying 122 by the 3 in the tens place?” (In both cases I get 366.)
  • “¿En qué se diferencia?” // “How is it different?”(The 3 in the tens place represents 30 and so the 366 needs to shift one place to the left because it is really 366 tens or 3660.)

Lesson Synthesis

Lesson Synthesis

“Hoy usamos el algoritmo estándar para multiplicar un número de dos dígitos y un número de tres dígitos” // “Today, we used the standard algorithm to multiply a two-digit number and a three-digit number.”

Display Diego’s work from the last problem.

“¿Por qué no tiene sentido la respuesta de Diego?” // “Why doesn’t Diego’s answer make sense?” (The product is too small. \(300\times20\) is 6,000, so the product is greater than that.)

“¿Qué consejo le darían a Diego para que reconsidere lo que pensó?” // “What advice would you give Diego to revise his thinking?” (Remember that the 2 in 24 is 2 tens. So 2 tens times 1 should be 20, so you need to write the 2 in the tens place.)

Cool-down: Algoritmo estándar sin componer una nueva unidad en base diez (5 minutes)

Cool-Down

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs.