# Lesson 8

Diferentes maneras de descomponer

## Warm-up: Conversación numérica: Múltiplos de 10 (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit the ways students look to add or subtract based on place value. When students describe ways to add or subtract by adding or subtracting tens and tens, they make use of the base-ten structure of the numbers. When they describe ways to use the value of the sums to find the value of the differences, they look for and make use of the structure of expression and the relationship between addition and subtraction (MP7). Both of these understandings help students develop fluency with addition and subtraction within 100.

### 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

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$18 + 10 + 10$$
• $$18 + 20 + 10$$
• $$38 - 20$$
• $$48 - 30$$

### Activity Synthesis

• “¿Cómo se relacionan las expresiones de suma con las expresiones de resta?” // “How are the addition expressions related to the subtraction expressions?” (The second expression is the opposite of the last expression. They are in the same fact family. The first expression helped me solve the third expression because I know $$18 + 20 = 38$$, so $$38 - 20$$ must be 18.)

## Activity 1: ¿No estás olvidando algo? (15 minutes)

### Narrative

The purpose of this activity is for students to analyze two different subtraction methods that are based on place value and connect the methods to equations. In previous lessons, students analyzed base-ten drawings like Lin’s where a student recognizes a ten needs to be decomposed before they draw the blocks. Clare’s drawing represents a student who mentally subtracts tens from tens before drawing and then considers decomposing units. Students discuss how each method works and deepen their understanding of the properties of operations (MP7).

Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, encourage students to take turns sharing their ideas about Lin and Clare’s methods and give feedback based on their responses.
Supports accessibility for: Social-Emotional Functioning

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students access to base-ten blocks.
• “Lin y Clare hicieron diagramas en base diez para encontrar el valor de $$71-56$$” // “Lin and Clare made base-ten diagrams to find the value of $$71-56.$$
• “¿Qué observan sobre su trabajo? ¿Qué se preguntan?” // “What do you notice about their work? What do you wonder?” (Lin drew 6 tens and decomposed one ten. Lin crossed out 5 tens and 6 ones. Clare just drew 2 tens and decomposed 1 of them. Clare crossed out 6 ones and the ten she decomposed. Did Clare find $$71-56$$? Why did Clare only draw 2 tens?)
• 12 minutes: partner discussion
• Share responses.
• “¿Creen que Clare encontró el valor de $$71-56$$? ¿Por qué sí o por qué no?” // “Do you think Clare found the value of $$71-56$$? Why or why not?” (Yes, her diagram shows 15 left. No, she showed $$21-6$$. She didn’t represent all the tens.)
• 1 minute: partner discussion
• Share responses.

### Activity

• “Lin y Clare usaron ecuaciones para mostrar cómo pensaron. Con su compañero, emparejen un grupo de ecuaciones con el trabajo de Lin y otro con el trabajo de Clare. Después, discutan en qué se parecen los métodos y en qué son diferentes” // “Lin and Clare used equations to show their thinking. Work with your partner to match the equations to Lin’s work and Clare’s work. Then discuss how the methods are the same and how they are different.”
• 5 minutes: partner work time

### Student Facing

Lin y Clare hicieron diagramas en base diez para encontrar el valor de $$71 - 56$$.

Lin

Clare

1. ¿Qué observas sobre su trabajo? ¿Qué te preguntas?
2. Lin y Clare escribieron, cada una, varias ecuaciones para mostrar cómo pensaron. Explica cómo sabes cuál grupo de ecuaciones corresponde al trabajo de Lin y cuál corresponde al trabajo de Clare.

A

B

$$71 - 50 = 21$$
$$21 = 10 + 11$$
$$11 - 6 = 5$$
$$10 + 5 = 15$$

$$71 = 60 + 11$$
$$11 - 6 = 5$$
$$60 - 50 = 10$$
$$10 + 5 = 15$$

3. ¿En qué se parecen los métodos de Lin y Clare? ¿En qué son diferentes?

### Activity Synthesis

• Invite students to share the group of equations that match Lin's work and Clare’s work.
• “¿En qué se parecen estos métodos?” // “How are these methods the same?” (They both decomposed a ten. They both showed subtracting 6 in the diagram. They both subtracted 5 tens.)
• “¿En qué son diferentes los métodos?” // “How are the methods different?” (Clare took away 5 tens before she drew and before she decomposed. Then she took away the ones. Lin drew tens and decomposed first, then took away ones, and then tens. They did the same things, just in a different order and drew in different ways.)
• “¿Encontró Clare el valor de $$71-56$$? ¿Qué aprendieron sobre el método de ella?” // “Did Clare find the value of $$71-56$$? What did you learn about her method?” (Yes, she did because she did take away tens. I learned you can take away tens first and it doesn’t change the difference.)

## Activity 2: Diferentes maneras de descomponer (20 minutes)

### Narrative

The purpose of this activity is to analyze a subtraction method that is based on place value and connect it to equations. Students analyze the method and explain why they think it best matches one of the methods they saw in the previous activity. Then they practice subtraction using any method that makes sense to them. Monitor for different methods to share in the synthesis, including students who show ways to subtract ones from ones first and those that subtract tens from tens first.

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “cambiar una decena o intercambiar una decena” // “exchanged ten or traded ten” for “descomponer una decena” // “decomposed ten”.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students access to base-ten blocks.
• “Andre encontró el valor de ​​$$65-28$$. Tómense un minuto para examinar su trabajo” // “Andre found the value of $$65-28$$.  Take a minute to look at his work.”
• 1 minute: quiet think time
• “¿Creen que se parece más al método de Clare o al de Lin? Discutan con su pareja” // “Do you think it’s more like Clare or Lin’s method? Discuss with your partner.” (It’s more like Lin’s because he drew all the tens first. It’s more like Clare’s because he took away tens first, he just drew them out.)
• 2-3 minutes: partner discussion
• Share responses.

### Activity

• “Encuentren el valor de cada diferencia. Usen cualquier método que tenga sentido para ustedes. Después, compartan con su pareja cómo pensaron” // “Find the value of each difference. Use any method that makes sense to you. Then share your thinking with your partner.”
• 5 minutes: independent work time
• 2-3 minutes: partner work time

### Student Facing

Andre encontró el valor de $$65 - 28$$. Hizo un diagrama en base diez y escribió ecuaciones para mostrar cómo pensó.

$$65 - 28$$
$$65 - 20 = 45$$
$$45 = 30 + 15$$
$$15 - 8 = 7$$
$$30 + 7 = 37$$

1. ¿Crees que el método de Andre se parece más al método de Clare o al de Lin? Explica.
2. Encuentra el valor de cada diferencia. Muestra cómo pensaste.

1. $$34 - 18$$

2. $$82 - 37$$
3. $$71 - 53$$

### Advancing Student Thinking

• Students may use a method that shows they decompose and subtract by place, but they write a value that does not match the difference. Consider asking:
• “¿Qué hiciste primero para encontrar el valor de _____? ¿Puedes mostrármelo en tus dibujos/ecuaciones?” // “What did you do first to find the value of _____? Can you show me in your drawing/equations?”
• “¿Qué hiciste después?” // “What did you do next?”
• “¿Cómo puedes usar marcas o ecuaciones para ayudarte a llevar un registro de tus pasos?” // “How could you use labels or equations to help keep track of your steps?”

### Activity Synthesis

• Invite 12 students to share their method for each difference.
• Consider asking or inviting peers to ask questions of students who share:
• “¿Qué hicieron primero?” // “What did you do first?”
• “¿Por qué escogieron esta representación?” // “Why did you choose this representation?”
• “¿En qué se parece su método al método de _____?” // “How is your method like _____’s method?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy le dimos sentido a varios métodos para restar números de dos dígitos y comparamos estos métodos” // “Today we made sense of and compared different methods for subtracting two-digit numbers.”

Display Lin, Clare, and Andre’s methods or different examples of student work from the last activity.

“¿En qué se parecieron los métodos que vieron hoy? ¿En qué fueron diferentes?” // “How were the methods you saw today the same? How were they different?”

“¿Cuáles métodos entendieron mejor? Expliquen” // “Which methods make the most sense to you? Explain.”