Lesson 10

This Number Talk encourages students to think about the strategies they can use to multiply 2 two-digit numbers. Students can decompose factors by place value to multiply by multiples of ten, or they can use a doubling and halving strategy to create an equivalent expression. The strategies elicited here will be helpful later in developing a flexible sense of numbers and using this sense to make decisions when multiplying mentally.

• 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 les puede ayudar \(30 \times 7\) a encontrar el valor de \(15 \times 14\)?” // “How can \(30 \times 7\) help you solve \(15 \times 14\)?” (\(30 \times 7\) is twice as many groups with half as much in each group, which results in the same product with easier factors to multiply mentally.)

In this activity, students analyze multiplication involving 2 two-digit factors. Students make sense of work where the partial products are recorded, but there is no indication of where each product came from. After making sense of work, students use the same strategy of finding and keeping track of partial products to evaluate \(31 \times 15\).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

• Groups of 2

• “Encuentren el valor de \(64\times87\). Compartan con un compañero cómo razonaron” // “Find the value of \(64\times87\) and share your reasoning with a partner.”

• “Tómense algunos minutos para darle sentido a los cálculos de Tyler. Prepárense para explicar cómo pensaron” // “Take a few minutes to make sense of Tyler’s calculation. Be prepared to explain your thinking.”
• 3–4 minutes: independent work time on the first problem

MLR1 Stronger and Clearer Each Time

• “Compartan con un compañero su análisis de los cálculos de Tyler. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your analysis of Tyler’s calculation with a partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 2 minutes: structured partner discussion 
• Consider displaying these prompts to support students’ conversations:
  • “¿Pueden explicar cómo al multiplicar _____ y _____ se obtiene _____?” // “Can you explain how multiplying _____ and _____ gives _____?”
  • “¿Pueden usar las palabras ‘productos parciales’ en su explicación?” // “Can you use the phrase ‘partial products’ in your explanation?”
• Repeat with 1–2 different partners.
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• “Ahora intenten usar el método de Tyler para completar el último problema y usen un diagrama para comprobar su trabajo” // “Now try using Tyler’s method to complete the last problem and use a diagram to check your work.”
• 5 minutes: independent work time on the last problem

• To clarify the steps in Tyler’s notation, consider itemizing each step and using color coding, as shown here:

  

  

  

  

  

  

  

  

  

  

• Also consider displaying a corresponding diagram for \(64 \times 87\) and inviting students to make connections to an algorithm.

  

  

When using an algorithm that uses partial products, students may be inclined to pay attention only to single digits in each number and pay little attention to the value of the digits. For example, \(32 \times 19\) might sound like “9 times 2 is 18 and 9 times 3 is 27.” In this activity, students analyze this error (MP3) and also look at the commutativity of multiplication when finding partial products.

To help students see that the place value of the digits impacts each partial product, consider displaying a diagram that shows partial products alongside the vertical notation.

• Groups of 2

• “Trabajen con su compañero en el primer problema. Cada uno debe encontrar el valor de un producto” // “Work with your partner on the first problem. Each partner should find the value of one product.”
• 3–4 minutes: partner work time on the first problem
• Pause for a discussion. Select a group whose calculations yield the same product to display their work.
• “¿Los cálculos deberían mostrar el mismo resultado? ¿Por qué sí o por qué no?” // “Should the calculations show the same result? Why or why not?” (Yes, because the same two numbers are being multiplied.)
• “¿En qué se diferencian las dos formas de calcular?” // “How are the two calculations different?” (The partial products are written in different orders.)
• “¿Importa cuál sea el primer número que se escribe y cuál sea el segundo?” // “Does it matter which number is listed first and which is listed second?” (No)
• “Completen el último problema individualmente” // “Complete the last problem independently.”
• 3–4 minutes: independent work time

• See lesson synthesis.