In a previous lesson, students saw that there are many ways to find products of multi-digit numbers. In this activity, students analyze and connect different ways to divide a multi-digit whole number by a single-digit whole number, and complete calculations to find the value of the quotient. In the Activity Synthesis, students compare the different methods and explain their preference.

• “¿En qué se parecen las cuatro estrategias? ¿En qué son diferentes?” // “How are the four strategies the same? How are they different?” (The first three are the same because they involve partial quotients. Each one records the partial quotients in different ways. The last one involves estimation.)
• Consider asking:
  • “¿Cuáles métodos les parecen más fáciles de entender? ¿Cuáles les parecen complicados?” // “Which method or methods do you find easy to follow? Which did you find hard to follow?”
  • “¿En cuál método se usan más pasos? ¿En cuál método se usan menos pasos?” // “Which method uses the most steps? Which uses the fewest steps?”
  • “¿Cuáles métodos sirven para encontrar el valor de cualquier cociente? ¿Cuál puede servir para esta expresión, pero puede ser menos útil para otras?” // “Which methods would work to find the value of any quotient? Which might work for this expression, but might be less useful for others?”

This activity serves two goals. First, it prompts students to consider whether the order in which parts of the dividend are divided makes a difference in the process or in the result. Second, it deepens students’ understanding of the structure of algorithms that use partial quotients.

Students first explain why different initial steps could be equally productive for starting a division process. Next, they analyze and complete some partial-quotients calculations with unknown numbers. The unknown numbers could be partial quotients, parts of the dividend being removed, or the results of subtraction. To find the unknown numbers, students need to recognize and make use of the structure of the algorithm (MP7). Lastly, students use the algorithm to find a quotient, being mindful of their starting move and of the efficiency of their process.

MLR2 Collect and Display. Collect the language students use to explain how they found the quotient. Display words and phrases such as “cociente” // “quotient,” “cociente parcial” // “partial quotient,” and “dividendo” // “dividend.” During the Activity Synthesis, invite students to suggest ways to update the display: “¿Qué otras palabras o frases deberíamos incluir?” // “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading

• See the Lesson Synthesis.