# Lesson 20

Sumemos y restemos hasta 1,000,000

## Warm-up: Observa y pregúntate: Restemos decenas de mil (10 minutes)

### Narrative

This warm-up prompts students to analyze an example of subtraction using both the standard algorithm and expanded form. The numbers require decomposing multiple units, which are shown in both strategies. The observations here prepare students to later reason with similar subtraction problems in which more than one decomposition is needed.

### Launch

• Groups of 2
• Display subtraction calculations.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

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

### Student Facing

¿Qué observas? ¿Qué te preguntas?

​​​​​​

### Activity Synthesis

• “¿En qué se parecen los dos ejemplos? ¿En qué son diferentes?” // “How are the two examples alike? How are they different?” (In the second one the numbers are written in expanded form, including the numbers that show regrouping.)
• “Ya hemos visto problemas de resta en donde se descomponen unidades en base diez. ¿En qué son diferentes estos nuevos problemas?” // “We’ve seen subtraction problems with decomposed units before. How are these different?” (In the problems we have seen so far, only one place needed to be decomposed in order to subtract. In these examples, more than one place needs to be decomposed.)

## Activity 1: Sumemos y restemos números grandes (15 minutes)

### Narrative

The purpose of this activity is for students to add and subtract multi-digit numbers through the hundred-thousands place. To find the value of some differences, students will decompose more than one unit.

The last question includes problems with a missing addend and a missing subtrahend. Besides making use of the structure of the standard algorithm (MP7), students will need to rely on what they know about the relationship between addition and subtraction to find the missing numbers.

Action and Expression: Develop Expression and Communication. Develop fluency with the standard algorithm by offering and gradually releasing scaffolds. Some students may continue to benefit from access to tools such as base-ten blocks, a blank place value chart, and place value cards. Use place value language to make explicit connections between these representations and the standard algorithm. Invite students to record what they are doing with these tools using the notation of the standard algorithm.
Supports accessibility for: Conceptual Processing, Language, Memory

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• 5–7 minutes: independent work time
• “Revisen sus respuestas del primer problema con su compañero y hagan los ajustes necesarios. Después trabajen juntos en el segundo problema” // “Check your responses to the first problem with your partner and make any adjustments you need to make. Then work on the second problem together.”
• 2–3 minutes: partner discussion
• As students work on the second problem monitor for students who:
• Find the missing addend by adding up from 67,182 to 129,400, or by subtracting $$129,\!400 - 67,\!182$$.
• Find the missing subtrahend by adding up from 193,710 to 234,650, or by subtracting $$234,\!650 - 193,\!710$$

### Student Facing

1. Usa el algoritmo estándar para encontrar cada suma y cada diferencia. Si tienes dificultades, trata de escribir los números en forma desarrollada.

1. $$7,\!106 + 2,\!835$$
2. $$8,\!179 - 3,\!599$$
3. $$142,\!571 + 10,\!909$$
4. $$268,\!322 - 72,\!145$$
2. En cada caso, encuentra el número que falta para que el cálculo sea correcto.

### Student Response

Students may not recognize the relationship between addition and subtraction, or may not know how to begin the problem involving a missing addend and missing subtrahend. Consider asking: “¿Cómo podríamos escribir este problema como una ecuación que tenga un símbolo que represente el número desconocido?” // “How might we write this problem as an equation with a symbol for the unknown?”

### Activity Synthesis

• “¿Cómo pueden saber si sus respuestas son correctas? ¿Cómo las pueden revisar?” // “How can you tell if your answers are correct? How can you check them?” (One way to check the result of subtraction is by adding it back to the number being subtracted. One way to check the result of addition is by subtracting one addend from the sum. Another way of checking is by performing the calculations with the numbers written in expanded form.)

## Activity 2: Descubramos errores (20 minutes)

### Narrative

In this activity, students analyze addition and subtraction calculations, identify errors, and explain what makes certain ways of calculating problematic. The work here gives students opportunities to construct logical arguments and critique the reasoning of others (MP3).

MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “Te escuché decir . . .” // “I heard you say . . . .” Original speakers can agree or clarify for their partner.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display the first problem.
• 2 minutes: quiet think time
• 2–3 minutes: partner discussion

### Activity

• 5 minutes: independent work time to complete the second problem.
• 2 minutes: partner discussion
• Monitor for students who describe the errors using place value language and an understanding of how and when to decompose a unit.

### Student Facing

1. Kiran estaba tratando de encontrar la suma de 204,500 y 695. No estaba seguro de cómo escribir los cálculos, así que escribió dos ideas. ¿Cuál es correcta? Prepárate para compartir con tu compañero cómo pensaste.

2. Lin cometió algunos errores al restarle 4,325 a 61,870. Identifica todos los errores que puedas. Después, muestra la forma correcta de restar.

### Student Response

If students have trouble making sense of Lin's work, encourage them to find the difference and then compare their work to Lin's.

### Activity Synthesis

• Select 2–3 students to share their responses and reasoning.
• Record responses as students share.
• “¿Qué errores se cometen a menudo cuando se restan dos números usando el algoritmo estándar?” // “What are some errors that are commonly done when subtracting two numbers using the standard algorithm?” (Subtracting without lining up the numbers by place value. Using the notation of regrouping without decomposing a unit. Crossing out digits without including them in the regrouping. Subtracting from left to right without decomposing units.)
• “¿Qué ideas tienen para evitar esos errores?” // “What ideas do you have for avoiding those errors?” (Use grid paper to line up the digits by place value. Decompose the unit before regrouping. Check answers using addition.)

## Lesson Synthesis

### Lesson Synthesis

Display these subtraction problems.

Ask students to decide if they agree with the following statements, without doing any calculations. Students should be prepared to defend their decisions.

• “Solamente es posible hacer el problema A. El problema B no se puede completar porque los ochos del segundo número son mayores que la mayoría de los dígitos del primer número” // “Only problem A can be done. Problem B cannot be completed because the 8s in the second number are greater than most digits in the first number.” (Disagree)
• “En el problema A no se necesita descomponer” // “Problem A doesn’t require any decomposing.” (Agree)
• “En el problema B se necesita descomponer unidades en base diez de cuatro posiciones” // “Problem B requires decomposing units in four places.” (Disagree)
• “El resultado de la resta del problema A es mayor que cien mil” // “The result of the subtraction in A is in the hundred-thousands.” (Agree)
• “El resultado de la resta del problema B también es mayor que cien mil” // “The result of the subtraction in B is also in the hundred-thousands” (Disagree)

If time permits, ask students to find each difference (Problem A is 143,210, and Problem B is 98,766) and to show a way to check their answers.