Lesson 5

The purpose of an Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. 

• 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
• 1 minute: partner discussion
• Record responses.

• “¿Cómo saben que 100 es una estimación muy baja?” // “How do you know 100 is too low?” (Because \(100 \times 37 = 3,\!700\) and that’s less than 9,953.)
• “¿Cómo pueden usar el valor del producto \(100 \times 37\) para estimar el valor de \(9,\!953 \div 37\)?” // “How can you use the value of the product \(100 \times 37\) to estimate the value of \(9,\!953 \div 37\)?” (I know that 9,953 is more than \(2 \times 3,\!700\) but less than \(3 \times 3,\!700\) so the quotient is more than 200 but less than 300.)

The purpose of this activity is for students to revisit the partial quotients method to find whole number quotients. Students compare their strategy with Elena's strategy and reason about the similarities and differences using their understanding of place value. They may use estimation to identify that Elena's answer is not reasonable while they may also use parts of their own calculation to identify Elena's error (MP3).

• Groups of 2
• “Completen el primer problema” // “Complete the first problem.”
• 1–2 minutes: independent work time

• “Con su compañero, completen el segundo, el tercer y el cuarto problema” // “Work with your partner to complete the second, third, and fourth problems.”
• 5–7 minutes: partner work time
• “Ahora van a revisar el trabajo que hicieron en el primer problema” // “Now you will have a chance to revisit your work from the first problem.”
• 1–2 minutes: independent work time
• Monitor for students who:
  • revised their original solution
  • used different partial quotients

• Invite students to share different partial products.
• Keep student work displayed or display provided solutions.
• “¿En qué se parecen las soluciones? ¿En qué son diferentes?” // “How are the solutions the same? How are they different?” (They both subtract some groups of 100, 10, and 1. They both show that the value of the quotient is 521. The first strategy takes out all the hundreds at once while the second strategy takes out 400 and then 100 more.)
• “¿Por qué los múltiplos de 100 son una buena opción para restar?” // “Why are multiples of 100 good to subtract?” (I can calculate them quickly and they make the number I’m dividing get smaller quickly.)

The purpose of this activity is for students to practice using partial quotients. Students compare their strategy with the strategies of their classmates. They reason about the similarities and differences using their understanding of place value, balancing the complexity of calculations versus subtracting a large amount quickly.

• Groups of 2, then 4
• “Ustedes y su compañero van a encontrar cada uno un cociente. Cuando terminen, discutan su trabajo con su compañero” // “You and your partner will each find a quotient independently. After you’re done, discuss your work with your partner.”

• 3–5 minutes: independent work time
• 1–3 minutes: partner discussion
• “Ahora busquen otro grupo de 2 estudiantes y comparen su trabajo. ¿En qué se parece? ¿En qué es diferente?” // “Now, find another group of 2 and compare your work. How is it the same? How is it different?”

• Ask selected students who used different partial quotients to share or display or write work as shown below for all to see.
• Display:

  

  

  

  

• “¿En qué se parecen los cálculos? ¿En qué son diferentes?” // “How are the calculations the same? How are they different?” (They both show that the quotient is 71. One of them finishes very quickly because it starts by taking out 70 groups of 32. The other one takes longer because it subtracts smaller groups of 32.)
• “¿Cuál forma de calcular prefieren?” // “Which calculation do you prefer?” (The one that takes out all of the thirty-twos at once because it is fast. The one that takes more steps because each calculation is easier to do in my head and I know that I can subtract each multiple of 32, that is I am not trying to take out too much.)