Lesson 17

The purpose of this True or False is to elicit strategies students have for estimating. The reasoning students do here helps to deepen their understanding of how rounding can be used to estimate. It will also be helpful later when students are to determine a reasonable estimate.

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• “¿Cómo pueden explicar su respuesta sin encontrar el valor de ambos lados?” // “How can you explain your answer without finding the value of both sides?”
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de ___ de otra forma?” // “Who can restate ___ 's reasoning in a different way?”
  • “¿Alguien quiere agregar algo al razonamiento de _____?” // “Does anyone want to add on to _____ ’s reasoning?”
  • “¿Podemos hacer alguna generalización basándonos en las afirmaciones?” // “Can we make any generalizations based on the statements?”
• “¿En qué problemas fue útil redondear? ¿Para cuáles no necesitaron redondear?” // “For which problems was rounding helpful? For which ones did you not need to round?” (For the first problem, I didn’t need to round once I knew that the number was over 200. For the last problem, I rounded to add 17 and 51 just to see if it would go over 100.)

The purpose of this activity is for students to consider what it means for an answer to make sense. They see that rounding is a useful strategy to estimate the answer to a problem and determine if an answer makes sense.

The quantities chosen are close to multiples of 100 and 10 to encourage students to round as they decide if an answer makes sense. The first problem also says that “Priya makes an estimate” and “about 400 beads.” If students begin computing the exact numbers of beads, remind them of the situation and that they do not need to solve to determine if the answer makes sense.

As students work, prompt them to explain their strategies for making estimates and relate them to the idea of rounding (MP3). When students use language such as “about 600 beads” to convey that they are estimating, they practice communicating with precision (MP6).

• Groups of 2
• “¿Qué significa que 'una respuesta tenga sentido'? Hablen de esto con su compañero” // “What does it mean for an answer to make sense? Turn and talk with your partner.”
• 2 minutes: partner discussion
• Share responses.
• “En la clase de matemáticas, podemos pensar si una respuesta tiene sentido para nosotros con respecto a la situación y a los números del problema. Si parece que la respuesta podría ser correcta, decimos que la respuesta tiene sentido” // “In math class, we can think about whether an answer makes sense to us given the situation and the numbers in the problem. If the answer seems like it could be correct, we say it makes sense.”
• “Van a trabajar en algunos problemas sobre chaquiras. ¿Para qué cosas saben que se usan las chaquiras?” // “You’re going to work with some problems about beads. What are some ways of using beads that you know about?” (To make bracelets or necklaces. To decorate hair. To decorate clothing.)
• 30 seconds: quiet think time
• Share responses.
• Display the problem.
• “Piensen en la primera situación por 30 segundos” // “Take 30 seconds to think about the first situation.”
• 30 seconds: quiet think time

• “Con su compañero, piensen en la estimación de Priya” // “Work with your partner to consider Priya’s estimate.”
• 2–3 minutes: partner work time
• Consider asking:
  • “Ahí dice que Priya hace una estimación. ¿Cómo podrían decidir si la estimación tiene sentido sin encontrar la respuesta exacta?” // “It says Priya makes an estimate. How could you decide without solving the problem exactly?”
  • “¿Tienen alguna estrategia para estimar sin encontrar el valor exacto?” // “Do you have any strategies for estimating without solving?”
• Monitor for various student strategies, particularly a pair that uses rounding to determine if the answer makes sense.
• Select students to share strategies for evaluating Priya’s answer. Be sure to include a rounding strategy.
• For the rounding strategy, ask:
  • “¿Cómo se usó el redondeo en esta estrategia? ¿Pueden describirlo con sus propias palabras?” // “How did this strategy use rounding? Can you describe it in your own words?”
  • “¿Por qué es útil redondear acá?” // “Why is rounding helpful here?” (I can change the numbers to be easier to think about quickly.)
  • “Cuando estaban decidiendo si la respuesta de Priya tenía sentido, ¿redondearon al múltiplo de 10 o al múltiplo de 100 más cercano?” // “Did you round to the nearest multiple of 10 or 100 when you were deciding if Priya’s answer makes sense?”
• Consider asking:
  • “¿En qué se parecen (o se diferencian) lo que hicieron _____ y _____, y la estrategia de ustedes para decidir si la respuesta de Priya tiene sentido?” // “How is what _____ and _____ did similar to (or different from) your strategy for deciding if Priya’s answer makes sense?”
• If no students mention rounding in their explanation, ask: “¿Cómo podrían usar redondeo para decidir si la respuesta de Priya tiene sentido?” // “How could you use rounding to decide if Priya’s answer makes sense?”
• “Estimen la respuesta de los dos últimos problemas con su compañero” // “Work with your partner to estimate the answer for the last two problems.”
• 3–5 minutes: partner work time

• “En general, ¿cómo nos ayuda redondear cuando queremos estimar cantidades?” // “In general, how can rounding help us estimate?” (Sample responses: Rounding makes the numbers easier to think about quickly. Rounding to the nearest 10 or 100 means dealing only with hundreds, or with only hundreds and tens, rather than hundreds, tens, and ones.)
• “Cuando queremos decidir si una respuesta tiene sentido, no estamos resolviendo el problema para obtener una respuesta exacta. Estamos estimando la respuesta. Redondear es una estrategia que puede ser útil para estimar una respuesta” // “When we are deciding whether an answer makes sense, we are not solving the problem to get an exact answer. We are estimating the answer. Rounding is a strategy that can be useful when we estimate an answer.”

The purpose of this activity is for students to solve two-step word problems involving addition and subtraction. After students solve the problems, they trade answers with a partner to decide if their answer makes sense.

When students assess the reasonableness of each other’s answers and communicate their assessment, they construct logical arguments and critique the reasoning of others (MP3).

• Groups of 2
• “Ahora van a encontrar la respuesta exacta de algunos problemas. Mientras lo hacen, piensen cómo les puede ayudar hacer una estimación a decidir si una respuesta tiene sentido” // “Now you are going to solve for the exact answer to some problems. As you do so, think about how estimating could help you decide if an answer makes sense.”

• “Con su compañero, decidan quién va a resolver cada problema. Luego, resuelvan su problema individualmente” // “Work with your partner and decide who will solve each problem. Then, work independently to solve your problem.”
• 3–5 minutes: independent work time
• “Ahora intercambien su trabajo con el de su compañero y decidan si la respuesta del problema que resolvió tiene sentido” // “Now, trade work with your partner and decide whether their answer for the problem they solved makes sense.”
• “Anoten sus ideas en la hoja de su compañero para que las tenga en cuenta si quiere ajustar su respuesta” // “Record your thoughts on your partner’s paper for them to refer back to if they want to adjust their answer.”
• 3–5 minutes: independent work time
• “Por turnos, compartan sus ideas sobre el trabajo de su compañero. Den la oportunidad a su compañero de compartir cómo resolvió su problema” // “Take turns sharing your thoughts on your partner’s work. Give your partner a chance to share how they solved their problem.”
• 5 minutes: partner work time

If students don't find a solution to the problems, consider asking:

• “¿De qué se trata este problema? ¿Qué cosas de esta situación se pueden contar o medir?” // “What is this problem about? What can be counted or measured in this situation?”
• “¿Cómo podrías representar el problema?” // “How could you represent the problem?”

• “¿Cómo les ayudó tener a alguien que examinara su trabajo en cada problema?” // “What was helpful about having someone look at your work for each problem?” (They thought my answer made sense, which helped me know that I had answered correctly. They caught a mistake that I made that I didn’t notice.)
• “¿En qué les ayudó examinar el trabajo de otra persona en cada problema?” // “What was helpful about looking at someone else’s work for each problem?” (I was able to see a different strategy that they used for the problem. I was able to help them catch a mistake they made.)