Lesson 8

This Number Talk encourages students to use multiples of 10 to mentally multiply two-digit numbers that are close to multiples of 10. Students can use place value reasoning and the distributive property of multiplication over addition or subtraction to find the value of the products. The work here prompts students to think flexibly about how numbers can be decomposed strategically when multiplying.

• 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 nos puede ayudar \(20 \times 60\) a encontrar el valor de \(19 \times 60\)?” // “How can \(20 \times 60\) help us find the value of \(19 \times 60\)?” (\(19 \times 60\) is one group of 60 less than \(20 \times 60\), so we can subtract 60 from \(20 \times 60\).)

In this activity, students use rectangular diagrams and similar reasoning as in earlier activities to represent the multiplication of 2 two-digit numbers. They analyze a progression of diagrams, starting with those that represent multiplication of two-digit and one-digit numbers (18 and 6), a two-digit number and a ten (18 and 10), and then 2 two-digit numbers (18 and 16).

Students may decompose factors in different ways. For example, those who are familiar with multiples of 25 may find it intuitive to decompose \(25 \times 46\) as \(25 \times 40\) and \(25 \times 6\), rather than decomposing 25 into \(20 + 5\).

• Display the three diagrams in the activity.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion

• “En silencio, trabajen unos minutos en el primer problema. Luego, compartan con su compañero cómo pensaron” // “Take a few quiet minutes to work on the first problem. Then, share your thinking with your partner.”
• 3 minutes: independent work time
• 3 minutes: partner discussion
• Invite students to share their responses: “¿Qué expresiones de multiplicación pueden estar representadas por los diagramas?” // “What multiplication expression can be represented by each diagram?”
• “Completen el resto de la actividad” // “Complete the rest of the activity.”
• 3 minutes: independent or partner work time
• Monitor for students who decompose both factors into tens and ones and those who choose to keep one of the factors intact.

• Select two students to share their diagrams, solutions, and reasoning for the last problem. 
• “¿Cómo descompusieron los factores y el diagrama?” // “How did you decompose the factors and the diagram?” (I decomposed the 13 into 10 and 3 and the 21 into 20 and 1.) 
• “¿Qué expresiones podríamos escribir para mostrar el producto parcial que está representado por cada parte del diagrama?” // “What expression could we write to show the partial product represented by each part of the diagram?” 
• “¿Cómo nos ayudan estos productos parciales a encontrar el valor de \(13 \times 21\)?” // “How do these partial products help us find the value of \(13 \times 21\)?” (Adding them gives us the value of \(13 \times 21\): \(200 + 10 + 3 + 60 = 273\))

In this activity, students analyze two ways of decomposing a factor: by place value and not by place value. As they write the corresponding partial products, they see more clearly why it is helpful to decompose each of the factors by place value (MP7). Students may notice that when the factors are decomposed by place value, they end up finding multiples of 10 and multiplying a number by single-digit factors—both of which they can do with some degree of fluency.

This activity uses MLR5 Co-craft Questions. Advances: writing, reading, representing

• Groups of 2

• Display only the two diagrams, without revealing the opening sentence or the question(s). 
• “Escriban una lista de preguntas matemáticas que se pueden hacer sobre esta situación” // “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “¿En qué se parecen estas preguntas? ¿En qué son diferentes?” // “What do these questions have in common? How are they different?”
• Reveal the task (students open books), and invite additional connections. 
• “Piensen en silencio en la primera pregunta durante dos minutos. Luego, compartan con su compañero cómo pensaron” // “Take two quiet minutes to think about the first question. Then, share your thinking with your partner.”
• 2 minutes: independent work time on the first problem
• 1 minute: partner discussion
• “Ahora completen el resto de la actividad” // “Now complete the rest of the activity.”
• Monitor for students who: 
  • decompose the side lengths of their diagrams by place value 
  • write expressions to show the sum of the partial products (This is not required, but it is helpful for the synthesis.)
• If extra time is available, add more two-digit by two-digit expressions to the last problem:
  • \(83 \times 39\)
  • \(64 \times 92\)    

• “¿Cómo se pueden usar ambos diagramas para encontrar el valor de \(49 \times 57\)?” // “How can both diagrams be used to find the value of \(49 \times 57\)?” (We can partition each side of a rectangle in many ways without changing the total side lengths.)
• “¿Por qué fue más útil la descomposición del diagrama A que la del diagrama B?” // “Why was the partitioning in diagram A more helpful than in diagram B?” (\(40 \times 50\) is easier than \(37 \times 50\) or \(12 \times 50\) to find mentally, because we are multiplying multiples of 10.)