# Lesson 10

El valor posicional y la recta numérica

## Warm-up: Observa y pregúntate: Base diez y la recta numérica (10 minutes)

### Narrative

The purpose of this warm-up is for students to connect a base-ten diagram to addition on the number line. This will support their work later in the lesson when they connect place value methods to representations of addition and subtraction on the number line.

### Launch

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

### Activity

• “Discutan con su compañero 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

• “¿Qué ecuación de suma puede estar representada aquí?” // “What addition equation could be represented here?” ($$28 + 10 + 5 = 43$$ or $$28 + 15=43$$)
• “Vamos a seguir pensando en qué se parecen y en qué se diferencian los diagramas en base diez y las rectas numéricas” // “We are going to keep thinking about what is the same and what is different between base-ten diagrams and number lines.”

## Activity 1: Comparemos representaciones (20 minutes)

### Narrative

The purpose of this activity is for students to connect a subtraction method based on place value to representations on the number line. Students compare representations of a subtraction method using base-ten diagrams, equations, and number lines (MP2). They notice that, just like with base-ten blocks, they can think about subtracting or counting by tens first or by ones first on the number line.

Representation: Develop Language and Symbols. Support understanding of the problem, by inviting students to act it out. For example, create a number line on the ground or across a large white board in the front of the classroom. Allow students to physically move on the number line.
Supports accessibility for: Conceptual Processing

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display image of Clare’s base-ten diagram.

### Activity

• “Clare restó y representó en un diagrama en base diez cómo pensó” // “Clare subtracted and represented her thinking with a base-ten diagram.”
• “¿Qué nos dice este diagrama?” // “What does this diagram tell us?” (She started with 46. She took away 35. She has 1 ten and 1 one left.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses
• “Van a escribir una ecuación para representar el trabajo de Clare. Luego, representarán en la recta numérica el método de Clare” // “You are going to write an equation to represent Clare’s work. Then you will represent Clare's method on a number line.”
• “Con su compañero, decidan cómo representar de la mejor manera lo que ustedes piensan que Clare hizo. Pueden discutir sobre dónde empezar, cuántos saltos deben dibujar, qué tan largo debe ser cada salto y a dónde llegar” // “Work with your partner to decide how to best represent what you think Clare did. You may discuss where to start, how many jumps you should draw, how long each jump should be, and where to land.”
• 5 minutes: partner work time
• “Ahora, intenten uno solos” // “Now try one on your own.”
• 8 minutes: independent work time
• Monitor for students who:
• start at 58 and jump 20 and then 4
• start at 58 and jump 4 and then 20
• start at 58 and jump 10, 10, and then 4
• start at 58 and show a jump for each ten and each one (6 total jumps)

### Student Facing

Clare restó y representó su trabajo con un diagrama en base diez.
1. Escribe una ecuación para representar el trabajo de Clare.

2. Usa una recta numérica para representar el método de Clare.

3. Encuentra el valor de $$58 - 24$$.

Usa un diagrama en base diez para mostrar cómo pensaste.

4. Usa la recta numérica para representar cómo encontraste el valor de $$58-24$$.

### Student Response

If students represent their thinking using a base-ten diagram, but the number line doesn't match their thinking, consider asking::

• “¿Puedes decirme más acerca de tu recta numérica? ¿Cómo supiste dónde empezar?” // “Can you tell me more about your number line? How did you decide where to start?”
• “¿Cómo se relaciona la recta numérica con tu diagrama en base diez?” // “How does the number line connect to your base-ten diagram?”

### Activity Synthesis

• Invite previously identified students to share.
• Display or record their methods.
• “¿Cómo decidieron dónde empezar, cuántos saltos dar y la longitud de cada salto?” // “How did you decide where to start, how many jumps to make, and the length of each jump?”
• “¿En qué se parecen estos métodos? ¿En qué son diferentes?” // “How are these methods the same? How are they different?” (They started at 54. Some show subtracting tens first, some show ones first. Some show subtracting the value of all the tens or all the ones. Some show subtracting each ten or each one.)
• “¿De qué manera la recta numérica les ayuda a ver en qué se parecen estos métodos?” // “How does the number line help you see how these methods are the same?” (It helps you see that it doesn't matter if you subtract tens first or ones first. They both show subtracting 24.)

## Activity 2: En la recta numérica (15 minutes)

### Narrative

The purpose of this activity is for students to represent addition and subtraction within 100 on a number line. Students make connections to strategies based on counting on or back by place. The numbers in each subtraction equation are designed to elicit methods that do not require students to explicitly decompose a ten. For example, when finding the value of $$50 - 32$$, students may first add on to make a ten ($$32 + 8 = 40$$), then add on more tens to reach the total ($$40 + 10 = 50$$). Others may see they can count back 2, then subtract the tens. In the synthesis, students share their thinking and discuss how the number line helps see how they can use what they know about the structure of counting sequence and what they know about tens and ones to add and subtract (MP7).

MLR8 Discussion Supports. Synthesis: For each comparison that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Ask, “¿Alguien puede usar el lenguaje de valor posicional para expresar de otra forma lo que ____ dijo?” // “Who can restate what ____ shared using the place value language?”

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display images of Diego’s number line.
• “Diego encontró el valor de $$33 + 45$$. Él usó una recta numérica para representar cómo pensó” // “Diego found the value of $$33 + 45$$. He used a number line to represent his thinking.”
• “¿En qué parte de su recta numérica ven 33 y en qué parte ven 45?” // “Where do you see 33 and 45 on his number line?” (On the number line there is a point on 33 and 4 jumps of 10 and 1 jump of 5.)
• 30 seconds: quiet think time
• Share responses

### Activity

• “Van a encontrar el valor de expresiones y a usar rectas numéricas para representar cómo pensaron” // “You will be finding the value of expressions and representing your thinking on the number line.”
• “Si les ayuda, dibujen diagramas en base diez” // “Draw base-ten diagrams if it helps.”
• 810 minutes: independent work time

### Student Facing

Diego está encontrando el valor de $$33 + 45$$. Él dice que puede contar hacia adelante de decena en decena y luego de unidad en unidad. Él usó una recta numérica para mostrar lo que quiere decir.

1. Escribe una ecuación para mostrar la suma en el trabajo de Diego.

2. Encuentra el valor de $$23 + 24$$.

Usa una recta numérica para representar cómo pensaste.

3. Encuentra el valor de $$50 - 32$$.

Usa una recta numérica para representar cómo pensaste.

4. Encuentra el valor de $$40 - 26$$.

Usa una recta numérica para representar cómo pensaste.

### Student Response

If students use base-ten diagrams or blocks to show decomposing a ten to subtract, validate their reasoning and consider asking:

• “Después de descomponer una decena, ¿restaste decenas o unidades primero?” // “After you decomposed a ten, did you subtract tens or ones first?”
• “Ubica 50 (o 40) en la recta numérica. Si usaras la recta numérica para mostrar cómo contar hacia atrás, ¿contarías hacia atrás, primero de decena en decena o de unidad en unidad? ¿Por qué?” // “Locate 50 (or 40) on the number line. If you used the number line to show counting back, would you count back by tens first or ones first? Why?”
• “¿Qué debes hacer ahora?” // “What should you do next?”
• “¿En qué se parece este método a lo que hiciste con el bloque (o el diagrama en base diez)? ¿En qué es diferente?” // “How is this method like what you did with the block (or base-ten diagram)? How is it different?”

### Activity Synthesis

• Invite 23 previously selected students to share.
• “¿Cómo decidieron dónde empezar?” // “How did you decide where to start?”
• “¿Cómo decidieron cuánto sumar/restar primero?” // “How did you decide how much to add/subtract first?”
• “¿De qué manera su recta numérica muestra el valor de la diferencia?” // “How does your number line show the value of the difference?”
• “¿Qué otras preguntas tienen sobre la recta numérica de _____?” // “What other questions do you have about _____'s number line?”
• “¿De qué manera la recta numérica les ayudó a entender el método de _____?” // “How does the number line help you make sense of _____'s method?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy aprendimos que algunos métodos que usamos para sumar o para restar se pueden representar en la recta numérica. Vimos que se puede sumar o restar las decenas primero y luego las unidades, o las unidades primero y luego las decenas. Vimos métodos para restar en los que se contaba hacia atrás, de decena en decena y de unidad en unidad, a partir del número mayor. También vimos otros métodos en los que se contaba hacia adelante, de decena en decena o de unidad en unidad, a partir del número menor” // “Today we learned that some of the methods we use to add or subtract can be represented on the number line. We saw you can add or subtract the tens first and then the ones or the ones first and then the tens. We saw methods for subtraction that counted back by tens and ones from the larger number and those that showed counting on by tens or ones from the smaller number.”

“¿Prefirieron usar diagramas en base diez, la recta numérica u otra manera para mostrar cómo pensaron? ¿Fue lo mismo para la suma que para la resta? Expliquen” // “Did you prefer showing your thinking with base-ten diagrams, the number line, or another way? Was it the same for addition and subtraction? Explain.”