Lesson 6

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying three factors, one of which is ten. These understandings help students develop fluency and will be helpful when students apply the standard algorithm to find the product of a three-digit and a two-digit number. 

Students have an opportunity to look for and make use of structure (MP7) because they can use the distributive property to find a product using previous calculations.

• Display one expression.
• “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.

• “¿Cómo influyó en el resultado haber multiplicado todos los productos por 10?” // “How did multiplying all the products by 10 influence the result?” (It made the result ten times as big, so the digits all shift one place to the left and it has a zero at the end.)
• “¿Cómo se relacionan los productos \(2 \times 243\) y \(20 \times 243\)?” // “How are the products \(2 \times 243\) and \(20 \times 243\) related?” (The second one is ten times as big, so the digits shift one place to the left and it has a 0 at the end.)
• “Pueden usar esta idea hoy cuando usemos el algoritmo estándar para encontrar productos de números de 3 dígitos por números de 2 dígitos” // “You can use this idea today when we apply the standard algorithm to find products of a 3-digit number and a 2-digit number.”

The goal of this activity is to use the standard algorithm to find products in which composition of a new unit happens once.  Students first calculate a 3-digit and 2-digit example using a strategy of their choice and then analyze the same example done with composition recorded above the product. Students may use different strategies when they try on their own including

• partial products
• mentally accounting for the hundred that is composed when finding the product \(3 \times 40\)

After students discuss how composing new units is recorded in the algorithm, they find the value of two multiplication expressions using the standard algorithm.

When students interpret a new way of multiplying a 3-digit and 2-digit number, they use their understanding of place value to make sense of the method (MP7).

• Groups of 2
• “Ahora van a aprender cómo componer y registrar nuevas unidades en base diez cuando están encontrando el producto de un número de tres dígitos por un número de dos dígitos” // “You are now going to learn how to compose and record new units for a three-digit and two-digit product.”

• “Trabajen con su compañero en los primeros 2 problemas” // “Work with your partner on the first 2 problems.”
• 2-3 minutes: independent work time
• 5-7 minutes: partner work time

• Invite students to share how they interpret Lin’s work finding \(241 \times 23\).
• Display the image of Lin’s calculation.
• Circle the 2 in the number 723 in Lin’s calculation.
• “¿Qué representa el 2 que está en la posición de las decenas?” // “What does the 2 in the tens place represent?” (It’s 2 of the tens from \(3 \times 40\).)
• “¿Qué hizo Lin con las otras 10 decenas?” // “What does Lin do with the other 10 tens?” (She makes a hundred out of them and puts them together with the other hundreds when she multiplies 200 by 3.)
• Circle the 1 above 241 in Lin's work.
• “¿Qué representa este 1?” // "What does this 1 represent?" (It’s the hundred from \(3 \times 40\).)
• Circle the partial product 4,820.
• “¿Qué representa 4,280 en el cálculo?” // “What does 4,820 represent in the calculation?” (\(20 \times 241\). The 2 from the factor 23 is in the tens place and so it represents 20.)
• “Ahora tómense unos minutos para resolver los últimos dos problemas” // “Now take a few minutes to solve the last two problems.”
• 4-5 minutes independent work time
• Invite students to share the products and ask students what questions they have about the standard algorithm with composition.

The goal of this activity is to multiply numbers with no restrictions on the number of new units composed. Students first multiply a 3-digit number by a 1-digit number and a 3-digit number by a 2-digit number with no ones. They can then put these two results together to find the product of a 3-digit and 2-digit number with many carries. They then solve one more 3-digit and 2-digit example with no scaffold. Because these calculations have new units composed in almost every place value, students will need to locate and use the composed units carefully. It gives students a reason to attend to the features of their calculation and to use language precisely (MP6).

• “Van a encontrar productos en los que se componen muchas unidades en base diez nuevas. Mientras trabajan, piensen con cuidado en dónde poner estos valores” // “You are going to find products with many new composed units. As you work, think carefully about where you place these values.”

• 8–10 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who use the results of the first two calculations to find the third, and for students who correctly compose all the new place values.

• Display the expression: \(647 \times 59\).
• “¿Cómo les ayudaron los primeros dos cálculos a resolver el tercer problema?” // “How did you use the first two calculations to help with the third problem?” (They gave me the two partial products for the product \(647 \times 59\), so I just had to add them up.)
• Invite students to share their responses for the last product, focusing on the newly composed units.