Lesson 18

This Number Talk encourages students to look for and make use of the structure of numbers in base-ten to mentally solve division problems. The reasoning elicited here will be helpful later in the lesson when students divide large numbers using increasingly more abstract strategies.

• Groups of 2
• 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 ayudó cada expresión a encontrar la siguiente?” // “How did each expression help you find the next one?”
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _______ de otra forma?” // “Who can restate _______ 's reasoning in a different way?”
  • “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
  • “¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
  • “¿Alguien quiere agregar algo a la estrategia de ____?” // “Does anyone want to add on to____’s strategy?”

In this activity, students encounter a way to divide a multi-digit number by using partial quotients and writing equations for them. They analyze and interpret the equations and consider how it is like and unlike finding quotients using base-ten representations. In the next activity, students will be introduced to a way to record partial quotients vertically.

• Groups of 4.
• Give students access to base-ten blocks.

• Pause after the first question and discuss students’ responses. Record and display responses for all to see.

• Invite students to share their interpretations of Priya’s work and compare it to their reasoning in the first question.
  • “¿Cómo descompuso Priya el número 465?” // “How did Priya decompose the number 465?” (By place value. \(400 + 60 + 5\))
  • “¿Qué hizo Priya después de escribir las tres primeras ecuaciones?” // “What does Priya do after writing the first three equations?” (She adds up the quotients.)
• “Podemos encontrar un cociente por partes, dividiendo una porción del dividendo a la vez, hasta que no haya más partes para dividir (o hasta que no haya suficiente cantidad para dividir)” // “We can find a quotient in parts—dividing a portion of the dividend at a time—until there is no more (or until there is not enough) of the dividend to divide.”
• “Cada cociente se llama un cociente parcial” // “Each quotient is called a partial quotient.”

In this activity, students are introduced to an algorithm that uses partial quotients, a vertical method of recording partial quotients. They compare and contrast this approach with other ways of dividing numbers using partial quotients and try using it to divide multi-digit numbers.

When students analyze Priya and Tyler's work and explain their reasoning, they critique the reasoning of others (MP3).

• Groups of 4
• Display Priya and Tyler’s methods.

• “Tyler usó otro método para registrar \(465 \div 5\). Analicen lo que está pasando en su método. Piensen en las semejanzas y las diferencias entre los dos métodos” // “Tyler used a different method to record \(465 \div 5\). Analyze what is happening in his method. Think about how the two methods are alike and different.”
• 3 minutes: independent work time on the first two questions
• 3 minutes: small-group discussion
• Invite students to share their analyses on the two methods.
• If not mentioned by students, highlight that both Priya and Tyler divided in parts, but reasoned and recorded differently.
  • Priya recorded the partial quotients with division equations.
  • The partial quotients in Tyler's work are recorded as factors being multiplied by 5, and also listed above the dividend.
  • Tyler kept dividing in parts and subtracting until there's nothing left of the dividend to divide.
• Clarify the meaning of the numbers in Tyler’s method before students work on the last question.
• 3 minutes: independent work time to find the value of \(428 \div 4\).

• Invite students to share their responses. Highlight the different ways to decompose 428 or the different partial quotients that could be used to find \(428 \div 4\).
• Make sure students see that Priya’s equations and Tyler’s method are simply two ways to record partial quotients, but they are not fundamentally different.
• “El método de Tyler, en el que registra de forma vertical, es otro tipo de algoritmo” // “Tyler’s vertical recording method is another type of algorithm.”