The purpose of this activity is for students to relate their understanding of fractions as representing division to think about decomposing a quotient into partial quotients in a way that simplifies the calculation. To find the value of , students may use:

• Their understanding of division.
• Multiplication and find how many groups of 6 are in 78.
• The fraction expressions from the first part of the problem.

• Invite students to share the expressions that match .
• Display:
• “¿Cómo sabemos que esto es verdadero?” // “How do we know this is true?” (A fraction shows that we are dividing the numerator by the denominator.)
• Display:
• “¿Cómo pueden usar esta ecuación para encontrar el valor de ?” // “How can you use this equation to find the value of ?” (I know is 10 and is 3, so is 13.)
• “En la siguiente actividad, vamos a usar expresiones que tienen fracciones para encontrar los valores de otros cocientes” // “In the next activity, we will use expressions with fractions to find values of other quotients.”

The purpose of this activity is for students to find the whole-number values of quotients, using sums of fractions, and to think about which sums were most helpful. They may notice that it is helpful to decompose the dividend into a multiple of the divisor, and that multiples of 10 are particularly helpful. This is closely related to how students found quotients, using partial products, which requires strategically choosing the number of groups of the divisor to subtract.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.

• “Compartan con su compañero sus respuestas acerca de por qué algunas expresiones ayudaron y otras no. 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 with your partner your response as to why some expressions were helpful and others were not. 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.
  • “¿Me 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?”
  • “La parte que mejor entendí fue . . .” // “The part I understood best was . . . .”
• “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