Lesson 12

The purpose of this warm-up is for students to analyze the structure of an algorithm that uses partial quotients, which will be useful when students use this method to divide multi-digit numbers in a later activity. Students have seen algorithms that use partial quotients in grade 4. The new aspect to these calculations in grade 5 is that the dividend is now a 2-digit number. While students may notice and wonder many things about this image, the relationship between multiplication and division and the purpose of subtraction are the important discussion points.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Este es un algoritmo que se usó para dividir números enteros. El proceso no está completo. ¿Qué harían después?” // “This is an algorithm used to divide whole numbers. The algorithm is not complete. What might you do next?” (Find \(5 \times 16\) and subtract it from 128.)

The purpose of this activity is for students to interpret a partial quotients calculation with a two-digit divisor. Before interpreting the partial products calculation, students find the value of the quotient in a way that makes sense to them. This will help them understand the partial quotients calculation by familiarizing themselves with the numbers and likely some of the steps in the calculation. In explaining both their answers and strategies and Elena's, students need to be precise in their word choice and use of language (MP6) and they also have an opportunity to improve their argument and critique the reasoning of others (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Writing, Speaking, Listening.

• Groups of 2
• “Hagan una pausa después de haber encontrado el valor de \(448 \div 16\)” // “Pause your work after you find the value of \(448 \div 16\).”

• 3–5 minutes: independent work time
• “Discutan con su compañero cómo encontraron el valor de \(448 \div 16\)” // “Discuss how you found the value of \(448 \div 16\) with your partner.”
• 1–2 minutes: partner discussion
• “Describan los pasos que Elena siguió para encontrar el valor de \(448 \div 16\)” // “Describe the steps Elena took to find the value of \(448 \div 16\).”
• 3–5 minutes: independent work time

If students do not find the correct value of \(448 \div 16\), ask, “¿En qué se parece tu trabajo al de Elena? ¿En qué se diferencia?” // “ How is your work similar to and different from Elena’s work?”

• “Compartan con su compañero su descripción de la forma en que Elena encontró el valor de \(448 \div 16\). Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your description of how Elena found the value of \(448 \div 16\) with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion
• Repeat with 2–3 different partners.
• (Optional) If needed, display question starters and prompts for feedback.
  • “¿Puedes dar un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
  • “¿Puedes usar la palabra _____ en tu explicación?” // “Can you use the word _____ in your explanation?”
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• “¿En qué se parece la estrategia de Elena a su estrategia? ¿En qué es diferente?” // “How was Elena’s strategy the same as and different from your strategy?” (She used the same calculations, but organized her work differently.)
• “¿Cómo le ayuda el método que usó Elena a organizar su trabajo?” // “How does the method Elena used help her organize her work?” (She records the multiples of 16 in one place and subtracts them in another.)

The purpose of this activity is for students to deepen their understanding of an algorithm that uses partial quotients and practice using it. Students use an algorithm that uses partial quotients to find quotients with a three-digit dividend and a two-digit divisor. Different levels of scaffolding are provided as some of the calculations are partly completed. If students struggle to decide what multiple of the divisor to subtract, encourage them to pick a multiple they can calculate easily and that is less than or equal to what remains of the dividend. 

• Groups of 2

• 8–10 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
  • multiply by 10 to find the value of \(364 \div 14\).
  • multiply by multiples of 10 to find the value of \(364 \div 14\).

If students do not complete the steps correctly, refer to the partially completed problems and ask them to describe the steps that are shown.

• Ask students to share the steps in the algorithm for the first problem and complete the problem.
• Ask previously identified students to share their work for \(364 \div 14\).
• “¿Tienen alguna pregunta sobre el trabajo de sus compañeros?” // “Do you have any questions about your classmates' work?”
• If needed, clarify any of the steps for using the algorithm.