# Lesson 19

Compongamos y descompongamos para sumar y restar

## Warm-up: Conversación numérica: Restemos fracciones (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for subtracting fractions and mixed numbers, particularly cases where it is necessary to rewrite a whole number as a fraction, or decompose it into a different whole number and fraction, in order to perform subtraction. These understandings will be helpful later in this lesson when students subtract multi-digit numbers that involve decomposing units.

### 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.”

### Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$2\frac{3}{4} - 1\frac{1}{4}$$
• $$1\frac{1}{4} - \frac{3}{4}$$
• $$5\frac{1}{8} - 2\frac{3}{8}$$
• $$3\frac{2}{10} - 2\frac{7}{10}$$

### Activity Synthesis

• Highlight strategies in which students decomposed the first mixed number in each expression.
• “¿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 la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
• “¿Alguien quiere agregar algo a la estrategia de ____?” // “Does anyone want to add on to____’s strategy?”

## Activity 1: Encontremos y revisemos algunas sumas (15 minutes)

### Narrative

The purpose of this activity is to use the standard algorithm to add multi-digit numbers, taking care to compose a new unit and record it accurately. They also analyze common errors and critique the given reasoning when composing a new unit (MP3).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Completen los primeros 2 problemas y después hablen con su compañero sobre los patrones que observen” // “Complete the first 2 problem and then talk to your partner about any patterns you notice.”

### Activity

• 3 minutes: independent work time
• 2 minutes: partner discussion
• Share and record responses.
• If needed, use the expanded form to help students make connections between the way composing a larger unit is recorded when using the standard algorithm and what is happening in each place. In the bottom line of the example here, we see 10 in the ones place and 100 in the tens place. Both partial sums do not match the assigned values of their places.
• 3–4 minutes: independent work time for the last problem.

### Student Facing

1. Encuentra el valor de cada suma.

2. Usa la forma desarrollada de 8,299 y de 1,111 para revisar el valor que encontraste de la última suma.
3. Cada uno de estos cálculos tiene al menos un error. Encuentra los errores y muestra el cálculo correcto.

### Student Response

Students may need support identifying errors in the last problem. Consider asking: “¿Cómo se podría usar la resta como ayuda para identificar el error?” // “How might subtraction be used to help identify the error?”

### Activity Synthesis

• Ask students to share the errors they identified in the last set of questions.
• “¿Qué errores comunes se cometen al sumar números grandes?” // “What are some common errors when adding large numbers?”

## Activity 2: Las reliquias de la familia de Priya (20 minutes)

### Narrative

This activity revisits the idea of decomposing a unit in one place into 10 units of the place value to its right when subtracting multi-digit numbers using the standard algorithm. Students recall how this is done as they subtract numbers in which decomposition is necessary.

To support students with understanding the context, the activity launch introduces a saree (traditional wedding attire for women in India) and the idea of family heirlooms, or gifts passed down from generation to generation.

When students create a subtraction problem that does not require decomposition of a unit when using the standard algorithm, they make use of structure and their understanding of subtraction as they choose the digits for the numbers in their difference (MP7).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Representation: Develop Language and Symbols. Activate or supply background knowledge. To help students recall the term decompose, represent a four-digit number (for example 2,467) with both base-ten blocks and digits in a place value chart. Ask, “¿Qué significa descomponer una unidad en base diez?” // “what does it mean to decompose a unit?” Show an example of decomposition, such as exchanging one long rectangle for ten small cubes. Notate this in the place value chart by crossing out the 6 and 7, and writing 5 and 17 above them. Reset the place value blocks and show additional examples as needed.
Supports accessibility for: Conceptual Processing, Memory, Language

### Launch

• Groups of 2
• “¿Qué observan y qué se preguntan acerca de estas imágenes?” // “What do you notice and wonder about these pictures?”
• Collect student ideas.
• “En esta actividad, Priya investiga su historia familiar. Veamos qué descubre” // “In this activity, Priya is researching her family history. Let’s see what she discovers.”
• “Las mujeres de la imagen están vestidas con un traje tradicional de la India llamado ‘sari’. Los saris están hechos de telas coloridas y casi siempre tienen bordados elaborados o diseños con estampados” // “The women in the picture are dressed in a traditional Indian garment called a ‘saree’. Sarees are made of colorful fabric and often have intricate embroidery or patterned print designs.”
• “Las pulseras que hay en la imagen también son de la India. A veces las joyas de este tipo son una reliquia o se usan como regalos que pasan de una generación a la siguiente” // “The bracelets in the picture are also from India. Sometimes jewelry like this is used as heirlooms, or gifts that are passed down from one generation to the next.”

### Activity

• Groups of 2
• 5–6 minutes: independent work time
• 3 minutes: partner work time
• As students work, listen for student discourse that includes language about the place value of the digits and when to decompose a unit.

### Student Facing

Para su boda en 1996, la mamá de Priya usó una pulsera especial. La pulsera era una reliquia y fue hecha en 1947.

Priya decidió restar para averiguar cuántos años tenía la pulsera cuando sus padres se casaron.

Priya se enteró de que su abuela también usó la pulsera en su boda, 24 años antes.

Priya decidió restar para averiguar cuándo se casaron sus abuelos.​​​​​

1. ¿Son correctos ambos cálculos? ¿Por qué un cálculo tiene números tachados y números nuevos, y el otro no? Explica cómo razonaste.

2. Cuando la abuela de Priya se casó en 1972, usó un juego de collar y aretes que era una reliquia de 63 años de antigüedad.

1. Si Priya usa el algoritmo estándar para restar $$1972 - 63$$, ¿va a necesitar descomponer una unidad en base diez? Explica cómo razonaste.
2. Usa el algoritmo estándar para restar $$1972 - 63$$ y encontrar el año en el que fue hecho el collar.
3. Inventa un problema de resta en el que no sea necesario descomponer una unidad en base diez para hacer la resta. Después, resuelve el problema.

### Student Response

Students may not remember when to decompose a unit or how to record regrouping. Urge students to begin to subtract by place value, either by using expanded form or lining up the digits. Ask: “¿Qué dificultad surge cuando restas las unidades en la expresión $$1972 - 63$$?” // “What issue comes up when you subtract the ones in $$1972 - 63$$?” Allow students to explain that they don't have enough ones in the ones place to subtract 3, but they can decompose a ten to get 10 ones and add it to the 2 already there. Then consider asking:

• “¿Cómo vas a registrar todas las unidades que tienes después de que descompongas una decena?” // “How will you record all of the ones you have after you decompose a ten?”
• “¿Cómo sabes si necesitas descomponer una unidad en base diez cuando le estás restando un número a otro?” // “How will you know if you need to decompose a unit when subtracting one number from another?”

### Activity Synthesis

MLR7 Compare and Connect
• “Hagamos un recorrido por el salón para ver qué problemas inventaron” // “Let’s do a gallery walk to see what problems you created.”
• “Mientras caminan por el salón, discutan con un compañero lo que observan sobre el valor de los dígitos de los números que se escogieron” // “As you walk, discuss with a partner what you notice about the value of the digits in the numbers that were chosen.”
• 3 minutes: gallery walk
• Collect 1–2 responses from student discussions during the gallery walk.
• Share 1–2 of the responses you collected.
• 1 minute: partner discussion
• “¿En qué se parecen todos los problemas que inventaron?” // “What is the same about each of the problems you created?” (In each problem each digit is greater in the first number than in the second number)
• 2 minutes: whole-group discussion

## Lesson Synthesis

### Lesson Synthesis

Write $$1972 - 63$$ for all students to see.

“Cuando vemos un problema, ¿cómo sabemos si va a ser necesario descomponer una unidad en base diez?” // “When we look at a problem, how do we know if we will need to decompose a unit?” (If the digit we are subtracting is larger than the digit we are subtracting from, we will need to decompose a unit and regroup.)

Display for all to see:

“Cuando vemos un problema de suma, ¿cómo sabemos cuándo va a ser necesario componer una nueva unidad en base diez?” // “When we look at an addition problem, how do we know when we will need to compose a new unit?” (If the sum of the digits in one place is greater than 9, we will compose a new unit, and record 1 more for the place to the left.)