Lesson 16

The purpose of this Estimation Exploration is for students to apply their understanding of dividing a whole number by a fraction from previous lessons. The dividend in this expression is much larger than those that students have previously worked with to encourage students to use multiplication to estimate.

• Groups of 2
• Display the expression.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Cómo saben que el valor de \(98 \div \frac{1}{5}\) es menor que 500?” // “How do you know the value of \(98 \div \frac{1}{5}\) is less than 500?" (It's less than \(100 \div \frac{1}{5}\) and that's 500)

The purpose of this activity is for students to reason about the size of quotients, involving a unit fraction and a whole number, by carefully analyzing the relative sizes of the dividend and divisor rather than finding the value of the expressions. As students work, listen for the language they use to explain why they think the value of an expression is greater than or less than 1. Highlight the language during the synthesis. When students explain to each other how they decided whether a quotient is greater than 1 or less than 1 they construct viable arguments (MP3).

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

• Groups of 2

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time

• “Compartan con su compañero su explicación sobre cómo decidir si un cociente es menor que 1 o mayor que 1. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your explanation for determining whether a quotient is less than or greater than 1 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.
• If needed, display question starters and prompts for feedback.
  • “¿Puedes darme un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
  • “¿Cómo puedes usar las palabras divisor y dividendo en tu explicación?” // “How can you use the words divisor and dividend 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

The purpose of this activity is for students to order the quotients from the previous activity from least to greatest, without calculating. The quotients of a whole number by a unit fraction have the same dividend so students reason that the expression with the smallest unit fraction divisor represents the largest quotient. In the same way, the quotients of a unit fraction by a whole number all have the same divisor so the expression with the largest unit fraction dividend is the largest.

• Groups of 2

• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
  • explain that the greatest quotient is \(25 \div \frac {1}{8}\) because it represents the largest number of pieces
  • explain that the smallest quotient is \(\frac {1}{8} \div 25\) because it represents the smallest sized piece
  • change their response after the partner discussion

Students may confuse the strategies for dividing a whole number by a unit fraction with dividing a unit fraction by a whole number. Ask:

• “¿El cociente va a ser mayor o menor que el dividendo?” // “Will the quotient be greater than or less than the dividend?”

• Ask previously selected students to explain their reasoning.
• “¿Qué fue un reto en esta actividad?” // “What was challenging about this activity?” (It was hard to explain without finding any values.)
• “¿Cómo les ayudó razonar sobre el orden de las expresiones a encontrar el valor de 2 de las expresiones?” // “How did reasoning about the order of the expressions help you find the value of 2 of the expressions?” (I knew the value was going to be a unit fraction [or whole number].)