Lesson 9

This warm-up prompts students to carefully analyze and compare four equivalent expressions. In making comparisons, students have a reason to attend closely to the features and the value of the expressions. The activity also enables the teacher to gain insight into students’ understanding of properties of operations and how they talk about them.

• Groups of 2
• Display the expressions.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “¿En qué se parecen las expresiones?” // “How are the expressions alike?” (They all have a value of 350.)
• “¿Supieron inmediatamente que algunas expresiones eran equivalentes? ¿Cuáles? ¿Cómo lo supieron?” // “Were there expressions that you knew right away were equivalent? Which ones? How did you know?”

The purpose of this activity is to analyze an algorithm that uses partial products. Students are not required to use a specific notation, but analyzing each algorithm deepens their understanding of the structure of place value in multiplication.

When students interpret and make sense of Noah's work, they construct viable arguments and critique the reasoning of others (MP3).

• Groups of 2
• Display the diagram in the activity.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: partner discussion
• Share and record responses. 

• 4 minutes: independent work time on the first problem about Noah’s diagram
• 4 minutes: partner discussion
• 5 minutes: group work time on the rest of the activity
• Monitor for students who include the place value of each digit in 124 in explaining what is happening in the algorithm. 

• Select students to share their interpretations of the steps in the second problem.
• Highlight these key ideas along the way:
  • Multiply the 7 by the 4 ones in 124 and write the product below the line.
  • Multiply the 7 by 2 tens in 124 and record this product below the first product.
  • Multiply the 7 by the 1 hundred and record this product below the previous product.
  • Add all the partial products to calculate the final product.

    

    

• If needed, review these steps for \(8 \times 217\).
• Reiterate that we typically begin by multiplying the single-digit factor by the digit in the ones place, and then repeat with then the digit in the tens place, and so on.
• “La estrategia de registro que Noah aprendió es un algoritmo en el que se usan productos parciales. En la unidad anterior, usamos un algoritmo para sumar o restar números grandes” // “The recording strategy that Noah learned is an algorithm that uses partial products. We used an algorithm to add and subtract large numbers in our last unit.”
• “¿Cómo sabemos si ya terminamos de encontrar todos los productos parciales?” // “How do we know if we have finished finding all the partial products?” (We keep track of all the partial products or smaller parts of the whole product as we go. We went in order from the smallest place value to the largest place value.)

In this activity, students continue to analyze an algorithm that uses partial products and learn that there are different ways to write the partial products. While students are not required to use a specific algorithm for multiplication, analyzing variations in the partial products notation deepens their understanding of how to use base-ten structure to multiply efficiently (MP7). 

• Groups of 2
• “¿Qué estrategia usarían para encontrar el valor de \(8 \times 3,\!419\)?” // “What strategy would you use to find the value of \(8 \times 3,\!419\)?”
• 1 minute: partner discussion
• Share responses.
• Display Noah’s and Mai’s computations.
• “Tómense un momento en silencio para entender el trabajo de Noah y de Mai. ¿Cómo creen que ellos obtuvieron los últimos cuatro números?” // “Take a quiet moment to make sense of Noah and Mai’s work. How do you think they arrived at the last four numbers?”
• 1 minute: quiet think time
• 1 minute: partner discussion

• “¿Cuál representación se parece más a la forma en la que ustedes pensaron?” // “Which representation is more like how you thought about it?”

• “Trabajen con su compañero en las dos primeras preguntas” // “Work with your partner on the first two questions.”
• 5 minutes: partner work time on the first two problems
• Pause for a brief discussion.
• “¿En qué se parecen las notaciones de Mai y de Noah? ¿En qué son diferentes?” // “How are Mai’s and Noah’s notation the same and how are they different?” (Both of these methods calculate partial products, but Noah’s goes in order from multiplying the ones, tens, hundreds, then the thousands. Mai’s goes in order the other way from the thousands, hundreds, tens, and then the ones.) 
• “¿Ambos métodos son correctos? ¿Cómo lo saben?” // “Are both methods correct? How do you know?” (Yes, because they both result in the same product when you calculate it.) 
• “Trabajen individualmente en el último grupo de problemas antes de discutir con su compañero” // “Work on the last set of problems independently before discussing with your partner.”
• 5 minutes: independent work time on the last set of problems

• See lesson synthesis.