Lesson 9

The purpose of this Estimation Exploration is for students to practice the skill of making a reasonable estimate for a point on a number line based on the location of other numbers represented. Students give a range of reasonable answers when given incomplete information. They have the opportunity to revise their thinking as additional information is provided. Revealing the actual answer is not necessary because leaving it open ended provides an opportunity to focus on reasonableness and not just one right answer.

• Group of 2
• Display the image.
• “¿Qué número podría estar representado por el punto en la recta numérica?” // “What number could be represented by the point on the number line?”
• “¿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 compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• “Al adivinar, dijimos muchos números diferentes porque no tenemos mucha información” // “We had a lot of different guesses, because we don’t have a lot of information.”
• Add 3 tick marks to the number line, so that it looks like this: 

• “Teniendo en cuenta esta nueva información, ¿quieren ajustar o cambiar sus estimaciones?” // “Based on this new information, do you want to revise or change your estimates?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.
• “¿Cómo cambió su estimación?” // “How did your estimation change?”
• 30 seconds: quiet think time
• Share responses.

• “¿Cómo les ayudaron las marcas a ajustar su estimación?” // “How did the tick marks help you revise your estimate?”

The purpose of this activity is for students to make sense of different methods they can use to compare three-digit numbers. They analyze the thinking of others and make connections across representations (MP2, MP3). Although students have compared numbers using different representations in prior units, this activity offers them the opportunity to consider using the number line as a tool to compare three-digit numbers. In the synthesis, students will discuss which representation makes it easier to see the comparison. While students could make a case that each of the representations was easier for them, the focus is on Jada’s way, the number line.

• Groups of 2

• “Les pidieron a Diego, Jada y Clare que compararan 371 y 317. Cada uno representó cómo pensó de una manera diferente” // “Diego, Jada, and Clare were asked to compare 371 and 317. They each represented their thinking differently.”
• “Tómense un momento para examinar sus métodos” // “Take some time to look over their methods.”
• 2 minutes: independent work time
• “Discutan con su pareja en qué se parecen y en qué son diferentes los métodos de ellos” // “Discuss with your partner how their methods are the same and different.”
• 4 minutes: partner discussion
• “Ahora prueben el método de Jada” // “Now try Jada’s way.”
• 6 minutes: partner work time

If students write comparison statements that are not true, ask them to read their statements and consider asking:

• “¿Como decidiste cuál número era mayor?” // “How did you decide which number was greater?”
• “¿De qué manera tu afirmación corresponde a la recta numérica?” // “How does your statement match the number line?”

• “Clare, Diego y Jada representaron cómo pensaron de diferentes formas. ¿Con el método de quién fue más fácil ver que 371 es mayor que 317? Expliquen” // “Clare, Diego, and Jada represented their thinking in different ways. Whose method made it easier to see that 371 is greater than 317? Explain.” (Jada’s because you just have to look at which number is farther to the right. You can see 371 is farther from 0. Diego’s because I use diagrams and I can see quickly that the hundreds are the same and there are more tens, but some things were the same.)
• As needed, “Jada y Diego escribieron \(371 > 317\), pero Clare escribió \(317 < 371\). ¿Con quién están de acuerdo? Expliquen” // “Jada and Diego wrote \(371 > 317\), but Clare wrote \(317 < 371\). Who do you agree with? Explain.” (They both mean the same thing. 371 is greater than 317, so 317 is less than 371.)

The purpose of this activity is for students to compare three-digit numbers based on different representations. Students continue to make connections and deepen their understanding of place value as they compare numbers using base-ten diagrams and number lines. They have the opportunity to reflect and share about the representation that makes the most sense to them and how they can use it to justify their thinking (MP3). This understanding will be helpful in upcoming lessons when students choose their own methods to compare three-digit numbers.

• Groups of 2

• “En la actividad anterior, vimos que a Jada le pareció útil usar la recta numérica para explicar que 371 es mayor que 317” // “In the last activity, we saw that Jada found it helpful to use the number line to explain that 371 is greater than 317.”
• “En esta actividad, van a comparar números de tres dígitos y a explicar cómo pensaron usando la recta numérica” // “In this activity, you will compare three-digit numbers and explain your thinking using the number line.”
• 6 minutes: independent work time
• “Comparen sus respuestas con las de un compañero y usen la recta numérica para explicar su razonamiento” // “Compare your answers with a partner and use the number line to explain your reasoning.”
• 4 minutes: partner discussion

• Display the images for 432 and 423.

• “¿En qué se parece o en qué es diferente ver estos números en la recta numérica a verlos en un diagrama en base diez?” // “What is the same or different about seeing these numbers on the number line compared to looking at a base-ten diagram?” (The one farthest to the right is greater. With the diagram you have to count to see which one has more. The number with more hundreds or tens is farther to the right on the number line.)