Lesson 13

Ordenemos números de varios dígitos

Warm-up: Verdadero o falso: Descomposición de números (10 minutes)

Narrative

The purpose of this True or False is to elicit the insights students have about the composition of multi-digit numbers in terms of place value. It also reinforces the idea that the same digit has different values depending on its place in a number—that is, digits cannot be viewed in isolation of their positions. The reasoning students do here will be helpful later when students compare and order numbers within 1,000,000.

Launch

• 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

Activity

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

Student Facing

En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar cómo razonaste.

• $$1,\!923 = 1 + 90 + 200 + 3,\!000$$
• $$1,\!923 = 1,\!000 + 90 + 20 + 3$$
• $$19,\!203 = 10,\!000 + 9,\!000 + 200 + 3$$
• $$190,\!023 = 10,\!000 + 90,\!000 + 20 + 3$$

Activity Synthesis

• “¿Cómo corregirían las afirmaciones falsas para que sean verdaderas?” // “How would you correct the false statements so that they become true?”
• $$3,\!291 = 1 + 90 + 200 + 3,\!000$$
• $$190,\!023 = 100,\!000 + 90,\!000 + 20 +3$$

Activity 1: Formas de comparar (25 minutes)

Narrative

This activity prompts students to examine more closely how multi-digit numbers can be compared, and to use their insights to order several numbers. Students solidify their awareness that looking only at the first digit is not a definitive way of comparing numbers. They also practice constructing a logical argument and critiquing the reasoning of others (MP3) when they explain why the strategy of analyzing only one digit is not reliable. When students refine Tyler's statement about comparing numbers to include making sure to compare digits with the same place value, they attend to precision in the language they use (MP6).

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

Launch

• Groups of 2
• Read the first problem as a class.
• Ask 1–2 students to restate Tyler’s claim in their own words.

Activity

• “En silencio, trabajen unos minutos en los dos primeros problemas sobre la estrategia de Tyler. Luego, compartan sus respuestas con su compañero” // “Take a few quiet minutes to work on the first two problems about Tyler’s strategy. Then, share your responses with your partner.”
• 4–5 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who use place-value understanding to explain why Tyler’s strategy is not reliable.
• “Trabajen individualmente en los dos últimos problemas” // “Work on the last two problems independently.”
• 4–5 minutes: independent work time

Student Facing

1. Tyler compara números grandes mirando el primer dígito de cada número, es decir, el de más a la izquierda.

Él dice: “Entre mayor sea el primer dígito, mayor es el número. Si el primer dígito es el mismo, entonces comparamos el segundo dígito”.

En cada una de estas parejas de números, ¿el número que tiene el mayor primer dígito es también el mayor número?

1. 985,248 y 320,097

2. 72,050 y 64,830

3. 320,097 y 58,978

4. 54,000 y 587,000

5. 58,978 y 547,612

6. 146,001 y 1,483

2. ¿La estrategia de Tyler funciona para comparar cualquier pareja de números? Explica cómo razonaste.

3. ¿Cómo compararías números grandes? Describe tu estrategia para comparar 54,000 y 587,000.
4. Usa tu estrategia para ordenar estos números de menor a mayor.

• 87,696
• 847,040
• 84,381
• 63,591
• 630,951
• 63,951
• 631,051

Student Response

Students may think that a number with larger digits is always greater than one with smaller digits, regardless of their place value and the number of the digits in each number. For instance, they consider 58,978 to be greater than 547,612 because the former has the digits 5, 7, 8, and 9, and the latter has smaller digits: 1, 2, 4, 5, 6, and 7. Urge students to read each number aloud and then ask which is greater: a number in the 58 thousands and one in the 547 thousands. Alternatively, ask students to identify in each 58,978 and 547,612 the digit in the place with the greatest value. Urge them to consider which is greater: 5 ten-thousand or 5 hundred-thousand.

Activity Synthesis

MLR1 Stronger and Clearer Each Time
• “Compartan con su compañero su estrategia para comparar números de varios dígitos. 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 explicación” // “Share your strategy for comparing multi-digit numbers 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 explanation.”
• 3–4 minutes: structured partner discussion
• Repeat with 1–2 other partners.
• “Ajusten su descripción inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial description based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• Invite students to briefly share their ordered sets of numbers from the last problem and their reasoning. Record and display their responses.
• If not mentioned in students’ explanations, point out that in the last set of numbers, the third digit (in the thousands place) in each 630,951 and 631,051 is what tells us how the two numbers compare. The third digit (in the hundreds place) also tells us how 63,591 and 63,951 compare.

Activity 2: Puntajes en un videojuego (10 minutes)

Narrative

In this activity, students apply their understanding of place value to order multi-digit whole numbers and solve problems in context. They also reason about the range of numbers whose values are between two given numbers.

Engagement: Provide Access by Recruiting Interest. Synthesis: Invite students to generate a list of additional examples of times they would need or want to put large numbers in order.
Supports accessibility for: Memory

Launch

• “¿A quién le gusta jugar videojuegos? ¿Qué juegos les gusta jugar?” // “Who enjoys playing video games? What games do you enjoy playing?”
• “¿Alguien ha jugado un juego en el que los puntajes de los jugadores se acumulan o se suman a lo largo de varias rondas?” // “Who has played a game where the scores of the players get accumulated or added up over multiple rounds?”
• “Usemos lo que sabemos sobre los números grandes para comparar algunos puntajes de un videojuego y ordenar jugadores según su puntaje” // “Let’s use what we know about big numbers to compare some video game scores and rank some players.”

Activity

• “En silencio, trabajen en la actividad durante unos minutos. Luego, discutan sus respuestas con su compañero” // “Take a few quiet minutes to work on the activity. Then, discuss your responses with your partner.”
• 6–7 minutes: independent work time

Student Facing

Un fin de semana, Mai y sus amigos participaron en un torneo de videojuegos.

Estos fueron los puntajes al final del torneo:

Mai 93,005
Priya 101,012
Kiran 90,298
Noah 90,056
Clare 98,032
Elena 89,100
Andre --
1. Ordena los puntajes de mayor a menor. ¿Quién está en el primer lugar?

2. El puntaje de Andre se borró accidentalmente, pero todos estuvieron de acuerdo en que él está en el segundo lugar. ¿El puntaje de Andre podría ser un número de seis dígitos?

Describe cuál podría ser el puntaje de Andre y da un par de ejemplos.

Activity Synthesis

• Invite students to share the ranking of the players and their reasoning. Record their responses.
• Invite students to share examples of what Andre's score could be.
• If not mentioned in students’ explanations, point out that 101,012 (the highest score) is 1,012 greater than the first six-digit number, 100,000. This means there are many six-digit numbers that could be the second-highest score.

Lesson Synthesis

Lesson Synthesis

“Hoy comparamos y ordenamos números hasta 1,000,000” // “Today we compared and ordered numbers within 1,000,000.”

“¿Es verdad que los números enteros que tienen más dígitos siempre son mayores que los que tienen menos dígitos? ¿Por qué sí o por qué no? ¿Pueden dar un ejemplo?” // “Is it true that whole numbers with more digits are always greater than those with fewer digits? Why or why not? Can you give an example?” (Yes. More digits means greater place values. A three-digit number has hundreds for its largest place value. A four-digit number has thousands.)

“Escriban dos números grandes que nos muestren que es posible saber cuál número es mayor al comparar los primeros dígitos, es decir, los que están más a la izquierda. Luego, compartan los números con su compañero” // “Write down two large numbers that show that it is possible to tell which number is greater by comparing the first or leftmost digits. Then, share the numbers with your partner.” (Sample response: 6,315 and 4,315)

“Escriban otros dos números que nos muestren que no siempre podemos basarnos en los primeros dígitos para saber cuál número es mayor. Muéstrenle los números a su compañero” // “Write down two other numbers that show that we can’t rely on the first or leftmost digits to tell us which number is greater. Share them with your partner.” (6,315 and 42,315)