# Lesson 21

Los ceros en el algoritmo estándar

## Warm-up: Cuál es diferente: Números con 0, 2 y 5 (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare features of multi-digit numbers. In making comparisons, students have a reason to use language precisely (MP6), especially place value names. The activity also enables the teacher to hear how students talk about the meanings of non-zero digits in different places of a multi-digit number.

Students observations will support their reasoning in the next activity when they subtract a number with non-zero digits from the four numbers listed.

### Launch

• Groups of 2
• Display numbers.
• “Escojan uno que sea diferente. Prepárense para explicar 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.”
• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

1. 2,050
2. 2,055
3. 205.2
4. 20,005

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

• “¿Qué tal si le restáramos 44 a estos números enteros?” // “What if we subtracted 44 from each whole number?” Record “– 44” under each whole number.
• “¿A cuál número sería más fácil restarle 44?” // “Which number would it be easiest to subtract 44?” $$(2,\!055 - 44$$ because you don’t have to decompose units.)
• “¿Cómo podríamos restarle 44 a los otros números enteros?” // “How could we subtract 44 from the other whole numbers?” (We would have to decompose other place values.)

## Activity 1: ¿Qué hacemos si no hay nada para descomponer? (20 minutes)

### Narrative

The purpose of this activity is to examine subtraction cases in which non-zero digits are subtracted from zero digits. In some cases, students could simply look at the digit to the left of a 0 and decompose 1 unit of that number. But in other cases, the digit to the left is another 0 (or more than one 0), which means looking further to the left until reaching a non-zero digit. Students learn to decompose that unit first, and then move to the right, decomposing units of smaller place values until reaching the original digits being subtracted. The problems are sequenced from fewer zeros to more zeros to allow students to see how to successively decompose units.

Recording all of the decompositions can be challenging. For the last problem, two sample responses are given to show two different ways of recording the decompositions. The important point to understand is that because there are no tens, hundreds, or thousands to decompose, a ten-thousand must be decomposed to make 10 thousands. Then, one of the thousands is decomposed to make 10 hundreds, and so on, until reaching the ones place. Those successive decompositions can be lined up horizontally, but this can make it hard to see what happened first. A second way shows more clearly the order in which the decompositions happen, but it may be challenging to see which place the successive decomposed units are in.

To add movement to this activity, the second problem could be done as a gallery walk where each group completes one problem and then walk around the room to look for similarities and differences in others’ posters.

MLR8 Discussion Supports. Display sentence frames to support partner discussion: “Observé _____, entonces yo . . .” // “I noticed _____ so I . . .”, and “Me pregunto si . . .” // “I wonder if . . . .”
Advances: Conversing, Representing
Representation: Develop Language and Symbols. Maintain a visible display to record language that students can use to explain their work. Include important vocabulary, such as the name of each place value category, decompose, and exchange. Also include sentence frames, such as: “Observé la posición de las _____ y vi que no tenía suficientes ____ para restar” // “I looked at the _____ place and saw that I did not have enough ____ to subtract.” and “Cambié una ______ por 10 _____” // “I exchanged one ______ for 10 _____.”
Supports accessibility for: Conceptual Processing, Language, Organization

### Launch

• Groups of 2–4
• 5 minute: independent think time
• 3 minute: partner share

### Activity

• “Tómense un momento para pensar en el primer problema y luego discútanlo con su compañero” // “Take some time to think about the first problem and then discuss it with your partner.”
• 2 minutes: quiet think time
• 3 minutes: partner discussion
• Pause for clarifying questions, as needed.
• “Ahora, con su compañero, completen los siguientes 2 problemas” // “Now, work with your partner to complete the next 2 problems.”
• 5 minutes: partner work time

### Student Facing

Estos son algunos números que viste antes. Cada número tiene al menos un 0. A todos los números se les resta 1,436.

1. Dale sentido a los problemas y explícaselos a un compañero.

2. Usa la estrategia que se muestra en el primer problema para encontrar estas dos diferencias:

3. Encuentra el valor de cada diferencia. Prepárate para explicar cómo razonaste. Si tienes dificultades, trata de restar usando la forma desarrollada.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Advancing Student Thinking

Students may lose track of the units when multiple rounds of decompositions are required. Offer base-ten blocks and prompt students to show multiple decompositions when finding the value of $$100 - 1$$ using a large square base-ten block that represents 100. Consider asking:

• “¿Cómo puedes mostrar la resta de 1?” // “How can you show subtraction of 1?” (Exchange it for 100 ones.)
• “¿Qué haríamos si solo pudiéramos intercambiar 1 unidad en base diez por 10 unidades en base diez a la vez?” // “What if we could only exchange 1 unit for 10 units at a time?” (Exchange the hundred for 10 tens, then exchange 1 ten for 10 ones, so that we’d have 9 tens and 10 ones. Then, we can remove 1.)
• “Usando el algoritmo estándar, ¿cómo mostrarías la forma de restarle 1 a 100?” // “How would you show the subtraction of 1 from 100 using the standard algorithm?”

### Activity Synthesis

• Ask students to share responses and to demonstrate consecutive decomposing when multiple zeros are involved.
• If needed, use expanded form to represent the decompositions students are explaining.

## Activity 2: ¿Cuál es tu edad? (15 minutes)

### Narrative

In this activity, students solve contextual problems that involve subtracting numbers with non-zero digits from numbers with one or more zero digits. Students may choose other ways to find the difference (for example, adding up and using a number line to keep track), but are asked to use the standard algorithm at least once.

In the launch, students subtract their age from the current year. This provides an opportunity for students to notice the relationship between this difference and their current age (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students access to grid paper.
• “Escriban el año en el que nacieron y réstenle ese año al año en el que estamos” // “Write the year that you were born down and subtract that year from the current year.”
• “Compartan con un compañero lo que descubrieron” // “Share with a neighbor what you found out.” (Students will notice that the number that results is their current age or the age they will be on their upcoming birthday.)
• “Jada usó este método para encontrar la edad de algunos de sus familiares” // “Jada used this method to find the age of some of her relatives.”

### Activity

• 10 minutes: partner work time

### Student Facing

Para un proyecto de historia familiar, Jada anotó el año en el que nacieron algunos de sus abuelos maternos.
miembro de la familia año de nacimiento
abuela 1952
abuelo 1948
bisabuela 1930
bisabuelo 1926
Este año, ¿cuál es la edad de cada uno de esos miembros de la familia? Muestra cómo razonaste. Usa el algoritmo estándar por lo menos una vez.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

• Ask students to share responses.
• “¿Cuáles años les parecieron más fáciles de restar? ¿Cuáles les parecieron más difíciles? ¿Por qué?” // “Which years did you find easiest to subtract? Which were more difficult? Why?” (Subtracting a year without decomposing any units was easier than subtracting a year that required decomposing.)
• Prompt students to check and compare ages found using the standard algorithm and those found using other ways of reasoning.

## Lesson Synthesis

### Lesson Synthesis

Display these expressions. “Estas son tres expresiones” // “Here are three expressions.”

“¿En qué se parecen las expresiones?” // “How are the expressions alike?” (They all involve 2,222 that is subtracted from a six-digit number with three 5s and three 0s. Finding each difference requires multiple regroupings.)

“¿En qué son diferentes?” // “How are they different?” (The fives and zeros are in different positions in each number. In the first expression, only one unit needs to be decomposed before 2 ones could be subtracted from it. In the second expression, two units need to be decomposed. In the third expression, three units need to be decomposed before 2 ones could be subtracted.)

“Un amigo no está seguro de cómo resolver la segunda expresión, $$505,\!500 - 2,\!222$$. Explícale a un compañero cómo usarías el algoritmo estándar para encontrar el valor de la diferencia” // “A friend is unsure how to solve the second expression, $$505,\!500 - 2,\!222$$. Explain to a partner how you would use the standard algorithm to find the value of the difference.”

2 minutes: partner discussion

2 minutes: whole-group discussion

## Cool-down: Encuentra algunas diferencias (5 minutes)

### Cool-Down

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs.