# Lesson 12

Comparemos números de varios dígitos

## Warm-up: Cuál es diferente: Números amigables (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare features of 4 multi-digit numbers. Although students used place value terminology in the previous section, this activity lets the teacher hear the terminologies students use as they describe and compare each number. The observations and reasoning here prepare students to reason about the relative size of numbers later in this lesson.

### Launch

• Groups of 2
• Display image.
• “Escojan uno que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 23 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

1. 1,395
2. 3,095
3. 9,530
4. 30,195

### Activity Synthesis

• “La mayoría de estos números tienen los mismos dígitos (0, 1, 3, 5 y 9). ¿Todos son del mismo tamaño?” // “These numbers have mostly the same digits—0, 1, 3, 5, and 9. Are they all the same size?” (No)
• “¿Cuál de estos es el mayor? ¿Cómo lo saben?” // “Which of these is the greatest? How do you know?” (30,195, because it has five digits and is in the ten-thousands. The others have fewer digits and are in the thousands.)
• “Todos los demás números tienen cuatro dígitos. ¿Cómo los podríamos comparar?” //“The other numbers are all four-digit numbers. How might we compare them?” (Compare the digits in the thousands place and see which one is greater.)

## Activity 1: ¿Cuál es mayor? (15 minutes)

### Narrative

In this activity, students compare pairs of numbers with the same number of digits and the same set of digits (for example, 278 and 872, or 1,356 and 3,156). The goal is to elicit the idea that both the placement and the size of the digits matter in determining the value of a number, and that the digits in certain places matter more than others for making comparisons (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 3–4
• Give each group a set of cards 0–10. Ask students to remove the cards that show 10.
• “¿Qué números de tres dígitos podemos formar con 5, 7 y 3?” // “What three-digit numbers can we make with 5, 7, and 3?” (357, 375, 537, 573, 735, 753)

### Activity

• “Con su grupo, formen parejas de números usando los dígitos que están en las tarjetas. Primero usen 3 tarjetas, luego 4, luego 5 y por último 6. Comparen los números de cada pareja de números” // “Work with your group to make pairs of numbers using the digits on the cards. First use 3 cards, then 4, then 5, and lastly 6. Compare each pair of numbers.”
• “Piensen en cómo hacer para comparar los dos números de cada pareja” // “Think about how you go about comparing each pair of numbers.”
• 5 minutes: group work time
• Monitor for students who:
• notice that not all digits need to be compared in order to tell which number in a pair is greater
• use place-value language in describing the numbers
• make two numbers with the same first digit or the same first and second digits

### Student Facing

Tu profesor te va a dar varias tarjetas. Cada una tiene un solo dígito del 0 al 9.

1. Usa las tarjetas del 2, 7 y 8 para formar dos números diferentes de tres dígitos. Usa el símbolo < o > para compararlos.

$$\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}$$

2. Ahora agrega la tarjeta del dígito 1 a tus tarjetas y forma dos números diferentes de cuatro dígitos. Compara los números.

$$\boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}$$

3. Mezcla las tarjetas. Repite los pasos anteriores usando otras tarjetas.

1. Números de cuatro dígitos

$$\boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}$$

2. Números de cinco dígitos

$$\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}}\ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \$$

3. Números de seis dígitos

$$\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}$$

4. En cada pareja, ¿cómo decidiste cuál número es mayor?

### Student Response

When answering the question about how they compared the numbers, students may say that they “just know” which one was greater. Encourage them to think about the features of the numbers that gave them an immediate clue about the size of the numbers, or the features that they didn’t find as useful.

### Activity Synthesis

• Select students to display their number statements and read them. Ask if the class agrees with their comparison.
• “¿Cómo decidieron cuál número es mayor? ¿Compararon todos los dígitos?” // “How did you decide which number is greater? Did you compare every digit?”
• Select students who wrote numbers with the same first digit (or the same first two digits) to share their number statements. Ask them to explain how they compared the numbers.
• If no students mentioned that the digits in some places matter more than those in others, ask them about it.
• “¿Tuvieron en cuenta solo algunos dígitos, y otros no?” // “Did you pay attention only to some digits but not others?”
• “¿A cuáles les dieron prioridad? ¿Tendieron a ignorar a algunos?” // “Which ones did you prioritize? Were there any you tended to ignore?”

## Activity 2: Números incompletos (10 minutes)

### Narrative

In the previous activity, students noticed that the digits in certain places within numbers matter more than others when comparing numbers. In this activity, students deepen that understanding by comparing pairs of numbers with a missing digit. The missing digit is the same for each pair but may not be in the same place in the two numbers. The reasoning here prompts students to pay closer attention to place value. It also reinforces the idea that the digit with the greatest place value affects the size of the number the most, followed by the digits with the second greatest place value, and so on (MP7).

MLR2 Collect and Display. Circulate, listen for and collect the language students use as they compare the pairs of numbers. On a visible display, record words and phrases such as: hundreds place, tens place, place value, bigger, smaller, greater than, less than. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Action and Expression: Develop Expression and Communication. Give students access to digit cards, base-ten blocks, and a visual display reminding them of place-value language. Invite students to make and explain an educated guess about which number is greater using place-value language, then to use the cards or blocks to test their theory.
Supports accessibility for: Conceptual Processing, Organization

### Launch

• Groups of 2
• “Veamos otras parejas de números de varios dígitos. Esta vez, a los números les falta un dígito. ¿Aun así se pueden comparar?” // “Let’s look at some other pairs of multi-digit numbers, but this time the numbers are missing a digit. Can they still be compared?”

### Activity

• “Tómense unos minutos en silencio para pensar en el primer problema. Luego, compartan con su compañero cómo pensaron” // “Take a few quiet minutes to think about the first problem. Then, share your thinking with your partner.”
• 1 minute: independent work time
• 1–2 minutes: partner discussion
• Pause for a whole-class discussion. Invite students to share their responses, hearing first from those who agree with Han, and then those who agree with Clare.
• “Si el dígito que falta no es el mismo dígito, ¿podemos comparar los dos números?” // “If the missing digit is not the same digit, can we compare the two numbers?” (No) “¿Por qué no?” // “Why not?” (Because we don’t know which number has the greater missing digit, so there’s no way to compare.)
• If not mentioned by students, point out the features of the pair of numbers being compared: Both are three-digit numbers, both are missing the first digit, and 62 is greater than 17.
• “Comparemos otros números a los que les falta un dígito” // “Let’s compare some other numbers that have a missing digit.”
• 5 minutes: partner work time
• Monitor for students who use place-value language to explain their thinking.

### Student Facing

1. Estos son dos números. El dígito que falta es el mismo en ambos números.

$$\large \boxed{\phantom{0}}\ \boxed{1} \ \boxed{7} \qquad \qquad \boxed{\phantom{0}}\ \boxed{6} \ \boxed{2}$$

• Han dice que los números no se pueden comparar porque están incompletos.

• Clare dice que el segundo número es mayor sin importar cuál es el dígito que falta.

¿Estás de acuerdo con alguno de ellos? Explica cómo razonaste.

2. Estas son algunas parejas de números. En cada pareja, el dígito que falta es el mismo en ambos números. ¿Puedes saber cuál número es mayor? Prepárate para explicar cómo razonaste.

1. $$\large \boxed{4} \ \boxed{9} \ \boxed{\phantom{0}}$$
$$\large \boxed{3} \ \boxed{\phantom{0}}\ \boxed{9}$$
2. $$\large \boxed{1} \ ,\boxed{\phantom{0}}\ \boxed{7} \ \boxed{2}$$
$$\large \boxed{1} \ , \boxed{\phantom{0}}\ \boxed{8} \ \boxed{5}$$
3. $$\large \boxed{8} \ ,\boxed{\phantom{0}}\ \boxed{1} \ \boxed{6}$$
$$\large \boxed{5} \ , \boxed{8} \ \boxed{\phantom{0}}\ \boxed{2}$$
4. $$\large \boxed{2} \ \boxed{7} \ ,\boxed{\phantom{0}}\ \boxed{9} \ \boxed{5}$$
$$\large \boxed{2} \ \boxed{\phantom{0}} \ , \ \boxed{7} \ \boxed{4} \ \boxed{5}$$
5. $$\large \boxed{\phantom{0}}\ \boxed{9} \ \boxed{0} \ ,\boxed{1} \ \boxed{6} \ \boxed{5}$$
$$\large \boxed{9} \ \boxed{\phantom{0}}\ \boxed{0} \ ,\boxed{0} \ \boxed{6} \ \boxed{4}$$

### Student Response

When the missing digit in a pair of numbers is in different places  (such as 27,__95 and 2__,745), students may generalize about how the numbers compare after trying one possibility for the missing digit, not realizing that a different possibility may change the relative size. Ask them to check their conclusion with other numbers for the missing digits.

### Activity Synthesis

• Invite selected students to share their explanations.

## Activity 3: ¿Es posible? [OPTIONAL] (20 minutes)

### Narrative

This optional activity gives students additional opportunities to compare multi-digit numbers by reasoning about the value of the digits in different places. It also prompts students to generalize the relative size of two numbers based on their understanding of place value. Students practice constructing logical arguments (MP3) as they explain whether it is possible for $$4 \underline{\phantom{5}},\!300$$ to be less than $$3 \underline{\phantom{5}},\!400$$, or for $$\underline{\phantom{5}}4,\!300$$ to be less than $$\underline{\phantom{5}}3,\!400$$.

### Required Preparation

• Each group of 2 needs a set of cards from the previous activity.

### Launch

• Groups of 2
• Give each group a card with a digit between 0 and 9 for each group

### Activity

• “Usen el dígito que está en la tarjeta para completar las afirmaciones de comparación del primer problema y decidan si son verdaderas” // “Use the digit on the card to complete the comparison statements in the first problem and decide if they are true.”
• 4–5 minutes: group work time
• Pause to collect responses from all the groups.
• “¿La primera afirmación de comparación es verdadera para todos los dígitos?” // “Is the first comparison statement true for all digits?” (Yes, because if the digits are the same, we can just compare 999 to 500.)
• Repeat the question with all statements. Record responses in a chart such as shown:
$$\underline{\phantom{00}} \ , 999 > \underline{\phantom{00}} \ , 500$$ todos los dígitos
$$15,2 \underline{\phantom{00}}0 > 15, \underline{\phantom{00}}02$$ 0, 1, 2 3, 4, 5, 6,
7, 8, 9
$$4 \underline{\phantom{00}},700 < 7 \underline{\phantom{00}} ,400$$ todos los dígitos
$$1 \underline{\phantom{00}}5,000 > 5 \underline{\phantom{00}}1,000$$ todos los dígitos
//
statement true false
$$\underline{\phantom{00}} \ , 999 > \underline{\phantom{00}} \ , 500$$ all digits
$$15,2 \underline{\phantom{00}}0 > 15, \underline{\phantom{00}}02$$ 0, 1, 2 3, 4, 5, 6,
7, 8, 9
$$4 \underline{\phantom{00}},700 < 7 \underline{\phantom{00}} ,400$$ all digits
$$1 \underline{\phantom{00}}5,000 > 5 \underline{\phantom{00}}1,000$$ all digits

• “¿Por qué son verdaderas las afirmaciones de las partes a y c sin importar qué dígito se use?” // “Why are the statements in parts a and c true no matter what digit is used?” (Sample response:
• In part a, both numbers are missing the first digit, in the thousands place. Since the missing digit is the same, we’re comparing the hundreds, and 999 is always greater than 500.
• In part c, the missing digit is in the thousands place of both numbers, but the digits in the ten-thousands place need to be compared first, and 4 ten-thousand is always less than 7 ten-thousand.)
• “¿Por qué es falsa la afirmación de la parte d sin importar qué dígito se use?” // “Why is the statement in part d false no matter what digit is used?” (Ten-thousand is never greater than 50 thousand.)
• “En silencio, trabajen unos minutos individualmente en los últimos dos problemas” // “Take a few quiet minutes to work on the last two problems independently.”
• 5 minutes: independent work time

### Student Facing

1. A los números de las siguientes parejas les falta el mismo dígito, pero en lugares diferentes.

Tu profesor te va a asignar un dígito. Úsalo para remplazar el dígito que falta y decide si cada afirmación de comparación es verdadera.

1. $$\large \boxed{\phantom{0}} \ , \boxed{9} \ \boxed{9} \ \boxed{9} > \boxed{\phantom{0}} \ , \boxed{5} \ \boxed{0} \ \boxed{0}$$
2. $$\large \boxed{1} \ \boxed{5} \ , \boxed{2} \ \boxed{\phantom{0}} \ \boxed{0} > \boxed{1} \ \boxed{5} \ , \boxed{\phantom{0}} \ \boxed{0} \ \boxed{2}$$
3. $$\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{7} \ \boxed{0} \ \boxed{0} < \boxed{7} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}$$
4. $$\large \boxed{1} \ \boxed{\phantom{0}} \ \boxed{5} \ , \boxed{0} \ \boxed{0} \ \boxed{0} > \boxed{5} \ \boxed{\phantom{0}} \ \boxed{1} \ , \boxed{0} \ \boxed{0} \ \boxed{0}$$
2. Estos son dos números. A ambos les falta el mismo dígito.

$$\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0} \qquad \qquad \boxed{3} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}$$

Escoge un dígito para completar los números y muestra en qué lugar de la recta numérica estarían.

3. ¿Es posible completar los dos espacios en blanco con el mismo dígito y que las afirmaciones sean verdaderas? Si crees que sí, da por lo menos un ejemplo del dígito que podrías usar. Si no, explica por qué no es posible.

1. $$\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0}$$ es menor que $$\large {\boxed{3}} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}$$ .
2. $$\large \boxed{\phantom{0}} \ \boxed{4} \ , \boxed{3} \ \boxed{0} \ \boxed{0}$$ es mayor que $$\large \boxed{\phantom{0}} \ \boxed{3} \ , \boxed{4} \ \boxed{0} \ \boxed{0}$$ .

### Activity Synthesis

• Select students to share their responses to the last two problems.
• Highlight explanations that make it clear that:
• $$4 \underline{\phantom{5}},\!300$$ will always be greater than $$3 \underline{\phantom{5}},\!400$$ because 4 ten-thousand is always greater than 3 ten-thousands.
• $$\underline{\phantom{5}}4,\!300$$ will always be greater than $$\underline{\phantom{5}}3,\!400$$ because, given the first digit is the same, the digits to compare are the thousands, and 4 thousands is always greater than 3 thousands.

## Lesson Synthesis

### Lesson Synthesis

“Hoy comparamos varios números grandes. Al principio, veíamos todos los dígitos de los números que comparábamos. Más adelante en la lección, hacía falta un dígito de cada número, y aun así, en muchos casos fuimos capaces de comparar el tamaño de los números” // “Today we compared many large numbers. At first, all the digits of the numbers being compared were known. Later in the lesson, one digit of each number was missing, but in many cases we were still able to compare the size of the numbers.”

“Supongamos que un compañero dice que no podemos comparar $$380,\!\underline{\phantom{5}}51$$ y $$384,\!\underline{\phantom{5}}89$$ porque hace falta un dígito de cada uno. ¿Cómo podrían convencerlo de que sí se puede hacer? Escriban lo que le dirían a ese compañero” // “Suppose a classmate says that we can’t compare $$380,\!\underline{\phantom{5}}51$$ and $$384,\!\underline{\phantom{5}}89$$ because a digit is missing from each. How might you convince them that it can be done? Write down what you might say to that classmate.”

Invite students to share their explanations. Highlight those that make it clear that both numbers have 3 hundred-thousands and 8 ten-thousands, but one has 0 thousand and the other has 4 thousands. This tells us that the second number is greater, regardless of what digit is missing in the places to the right of the thousands place.