The purpose of this activity is for students to 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.

• 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. )
  • “¿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.”

The purpose of this activity is to introduce students to an algorithm that uses partial quotients, a vertical method of recording partial quotients. Students compare and contrast this approach with other ways of dividing numbers using partial quotients. They also use partial quotients to divide multi-digit numbers.

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

• Invite students to share their responses. Highlight the different ways to decompose 428 or the different partial quotients that could be used to find .
• 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.”