# Lesson 11

Analicemos algoritmos de resta

## Warm-up: Conversación numérica: Restas hasta 1,000 (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for subtracting multi-digit numbers. These understandings help students develop fluency and will be helpful later in a subsequent lesson when students are to use strategies flexibly to subtract within 1,000.

### Launch

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$400 - 200$$
• $$450 - 200$$
• $$450 - 205$$
• $$450 - 215$$

### Activity Synthesis

• “Al restar estos números, ¿cómo les ayudó el valor posicional?” // “How did place value help as you subtracted these numbers?” (I subtracted hundreds from hundreds, tens from tens, and ones from ones. I was able to think about each place value position separately, which helped me find the difference.)
• “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____'s reasoning in a different way?”
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
• “¿Alguien pensó en el problema de otra forma?” // “Did anyone approach the problem in a different way?”
• “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

## Activity 1: Comparemos dos algoritmos de resta (20 minutes)

### Narrative

The purpose of this activity is for students to consider two subtraction algorithms. In the first algorithm, students first look for any place value units where they need to decompose to get more units, then subtract right to left. In the second algorithm, subtraction occurs right to left, and units are decomposed as the need arises. Students try each algorithm and consider potential advantages and disadvantages of each algorithm.

In the synthesis, students carefully analyze and discuss the two algorithms, explaining the motivation behind them and how they are the same and different (MP3, MP6).

MLR8 Discussion Supports.
Synthesis: For each idea that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Engagement: Develop Effort and Persistence. Some students may benefit from feedback that emphasizes effort, and time on task. For example, check in with students after completing the problem using the first algorithm.
Supports accessibility for: Attention

### Launch

• Groups of 2
• Display the image.
• “Estos son los primeros pasos de dos algoritmos de resta. Tómense un minuto para pensar en qué son diferentes” // “The first steps of two subtraction algorithms are shown. Take a minute to think about how they are different.”
• 1 minute: quiet think time

### Activity

• “Con su pareja, discutan las diferencias en los pasos de cada algoritmo” // “Discuss how the steps are different in each algorithm with your partner.”
• 2 minutes: partner discussion
• Share and record responses.
• “Con su pareja, terminen cada algoritmo” // “Work with your partner to finish each algorithm.”
• 2–3 minutes: partner work time
• “Ahora, con su pareja, usen los dos algoritmos para restarle 541 a 824” // “Now work with your partner to use both algorithms to subtract 541 from 824.”
• 5–7 minutes: partner work time

### Student Facing

1. Estos son los primeros pasos de los dos algoritmos.

Paso 1 del algoritmo A

Paso 1 del algoritmo B

¿En qué son diferentes los pasos?

2. Usa cada algoritmo para encontrar el valor de $$824 - 541$$.

### Activity Synthesis

• Select students to share how they used both algorithms to find the value of $$824 - 541$$.
• Keep algorithms from the first problem displayed.
• “Aunque puede que los algoritmos parezcan ser el mismo después de unos pasos, estos empezaron de manera diferente. Piensen sobre las ventajas y desventajas de usar cada algoritmo” // “Even though the algorithms may look the same after a few steps, they started out differently. Think about advantages and disadvantages of using each algorithm.”
• 1 minute: quiet think time
• “Ahora discutan con su pareja sobre las ventajas y desventajas de cada algoritmo” // “Now, discuss the advantages and disadvantages of each algorithm with your partner.”
• 1 minute: partner discussion
• Invite students to share advantages and disadvantages they come up with. (In algorithm 1, I look for decompositions first, so I probably won’t mix up the order of subtracting. In algorithm 1, I could subtract left to right or right to left. In algorithm 2, I can start subtracting right away. That means I don’t have to worry about decomposing until I know I need to do it.)

## Activity 2: ¿Usamos un algoritmo? (15 minutes)

### Narrative

The purpose of this activity is for students to make sense of an algorithm in which a number with non-zero digits is subtracted from a number with a zero in the tens place. In the given problem, it is necessary to decompose a larger unit to have enough ones to subtract. There are no tens to decompose, however, prompt students to consider whether subtraction is possible, and if so, how it could be done.

When students make sense of Elena’s reasoning, they construct viable arguments and critique the reasoning of others (MP3).

### Launch

• Groups of 2
• “Tómense un minuto para revisar el trabajo de Noah y lo que Elena dice sobre él” // “Take a minute and look over Noah’s work and what Elena says about it.”
• 1 minute: quiet think time

### Activity

• “Ahora completen la actividad con su pareja” // “Now, work with your partner to complete the activity.”
• 5–7 minutes: partner work time
• Monitor for a student who shows how the problem could be completed with decomposing a hundred into tens, then decomposing a ten into more ones.

### Student Facing

Noah quería encontrar el valor de $$301 - 167$$ y escribió:

Elena dijo que no podemos restar de esta forma porque necesitaríamos más unidades para restar 7 unidades, pero hay un cero en la posición de las decenas de 301.

1. ¿Estás de acuerdo con la afirmación de Elena? Explica tu razonamiento.
2. Muestra cómo usarías un algoritmo (el de Noah u otro algoritmo) para encontrar la diferencia entre 301 y 167.

### Activity Synthesis

• Select students to share their responses.
• Display student work that shows a hundred decomposed into tens, then a ten decomposed into ones (or show the example in Student Responses).
• “¿Cómo muestra esto que podemos tener suficientes unidades para restar, aunque 301 tenga un 0 en la posición de las decenas?” // “How does the work here show that we could have enough ones to subtract even though there is 0 in the tens place of 301?” (Crossing out the 3 in the hundreds and writing 10 in the tens place shows a hundred decomposed to get tens. Crossing out the 10 and writing a 9 in the tens place and writing 11 in the ones place shows a ten decomposed to get ones.)

## Lesson Synthesis

### Lesson Synthesis

• “Hemos aprendido varios algoritmos de resta. ¿Cuál algoritmo de resta es su favorito y por qué?” // “We've learned different algorithms for subtracting. Which subtraction algorithm is your favorite and why?” (The expanded form algorithms because we can really see all the parts of the number. The algorithm where we decompose the units as we go because I don’t like to do them all at once. The algorithms that use 1 digit for each place value because they don’t take as long to write.)