Lesson 5

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.

• 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

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

• “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 \(40 \times 3\).)

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. 

• 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.”

• 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.

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?”

• 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 \(20 \times 413\).)

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.

• Groups of 2

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

• 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.)