# Lesson 9

La diferencia entre números

## Warm-up: Conversación numérica: Agregar (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for different ways to find the value of a difference. Students often think of subtraction only as taking away. These equations were chosen to encourage students to show their understanding of subtraction as an unknown addend problem and express strategies based on counting on from the smaller number. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to interpret and represent subtraction strategies as counting on or counting back on the number line.

### Launch

• Display one problem.
• “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 strategies using a number line.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$20 - 2$$

• $$20 - 17$$

• $$49 - 3$$

• $$67 - 64$$

### Activity Synthesis

• “Observé que para encontrar el valor de $$20 - 17$$, algunos estudiantes contaron hacia adelante en lugar de restar o de contar hacia atrás” // “I noticed that some students counted on for $$20 - 17$$ instead of subtracting or counting back.”
• “¿Cómo deciden cuándo contar hacia atrás o restar, y cuándo contar hacia adelante?” // “How do you decide when to count back or subtract and when to count on?” (If the numbers are close together, I count on. 17 is only 3 away from 20.)
• “Vamos a seguir pensando sobre cuándo es útil sumar para encontrar la diferencia” // “We are going to continue to think about when it is helpful to add to find the difference.”

## Activity 1: Sumar o restar (15 minutes)

### Narrative

The purpose of this activity is for students to describe how representations of subtraction on the number line show the difference between two numbers in different ways. The number choices in the activity encourage students to use methods based on taking away or counting back that may be represented as a jump to the left from the larger number. Other number choices encourage students to consider methods that show their understanding of subtraction as an unknown addend problem. Students may start at the smaller addend and find the length of the jump to the right to reach the total. Students may also place both known numbers on the number line and find the length of the space between them by counting from the smaller to larger number or larger number to smaller number. Monitor for each strategy and the ways students represent them on the number line. All of these strategies show an important understanding of numbers and operations and how they can be represented on the number line (MP7).

The synthesis focuses on describing and comparing how the difference is represented on the number line. If a student uses a method that would be difficult to represent on the number line, pair the student with another student who uses a conceptually similar method. For example, if a student finds the value of $$57-24$$ as $$7-4=3$$, $$50-20=30$$, $$30+3=33$$, acknowledge the method would be hard to show on a single number line, and partner them with someone who shows jumps to the left based on place on the number line to compare methods.

### Required Materials

Materials to Gather

Materials to Copy

• Number Line to 100

### Required Preparation

• Place the number line recording sheets in sheet protectors. They will be used in the next activity and future lessons.

### Launch

• Groups of 2
• Give each student a copy of the number lines.

### Activity

• “De una manera que tenga sentido para ustedes, encuentren el número que hace que la ecuación sea verdadera” // “You are going to find the number that makes each equation true in a way that makes sense to you.”
• “Luego, usen la recta numérica para mostrar cómo pensaron” // “Then, use the number line to show your thinking.”
• 6 minutes: independent work time
• “Con un compañero, comparen sus métodos, sus soluciones y sus representaciones en la recta numérica” // “Compare your methods, solutions, and number line representations with a partner.”
• 4 minutes: partner discussion
• For $$75-68={?}$$, monitor for students who:
• represent taking away or counting back with arrow(s) moving to the left from 75
• represent an unknown addend problem with arrow(s) pointing to the right from 68
• locate 75 and 68 on the number line and count the length of the space between by counting from the smaller to larger number or larger to smaller number

### Student Facing

1. ¿Qué número hace que esta ecuación sea verdadera?_____

$$38 - 4 = {?}$$

Usa una recta numérica para representar cómo pensaste.

2. ¿Qué número hace que esta ecuación sea verdadera?_____

$$75 - 68 = {?}$$

Usa una recta numérica para representar cómo pensaste.

3. ¿Qué número hace que esta ecuación sea verdadera?_____

$$57 - 24 = {?}$$

Usa una recta numérica para representar cómo pensaste.

### Activity Synthesis

• Invite previously identified students to share their response and reasoning for $$75 - 68 = {?}$$ or draw the number lines and select the identified students to share their reasoning.
• “Ambas rectas numéricas muestran estrategias para encontrar $$75 - 68 = {?}$$” // “Both of these number lines show strategies for finding $$75 - 68 = {?}$$
• If it does not come up when students share, ask:
• “¿Cuál representación muestra la resta como quitar?” // “Which representation shows subtraction as taking away?”
• “¿Cuál representación muestra la resta como encontrar un sumando desconocido?” // “Which representation shows subtraction as finding an unknown addend?”
• “¿De qué manera cada una de las representaciones muestra la diferencia?” // “How do both representations show the difference?” (In the first number line, the arrow is pointing to the difference. In the second one, the length of the jump shows the difference.)
• As needed, gesture and restate student responses to emphasize how the difference is represented as a length in each representation.
• “Cuando en la recta numérica mostramos la resta como quitar, la diferencia es el número al que apunta la última flecha. La diferencia se representa como una longitud desde el 0 hasta donde la flecha apunta” // “When we show subtraction as taking away on the number line, the difference is the number the last arrow points to. It’s represented as a length from 0 to where the arrow is pointing.”
• “Cuando en la recta numérica mostramos la resta como un problema de sumando desconocido, la diferencia se presenta como la longitud del espacio entre los dos números” // “When we show subtraction as an unknown addend problem on the number line, the difference is shown as the length of the space between the two numbers.”

## Activity 2: Distintas formas de encontrar la diferencia (20 minutes)

### Narrative

The purpose of this activity is for students to compare methods for solving subtraction problems. Students compare representations of methods that show subtraction as taking away and subtraction as an unknown addend problem. Students discuss how some representations may better show the actions in a problem and others may show a different way to find the unknown value (MP2). In the synthesis, students consider how to select a strategy based on the numbers in the problem.

Action and Expression: Develop Expression and Communication. Provide access to poster paper and different colored markers, or whiteboards and different colored markers. Invite students to represent jumping forward on the top of their number line in one color, and jumping backward on the bottom of the number line in another color. Write the equation that matches the work shown.
Supports accessibility for: Organization

### Required Materials

Materials to Copy

• Number Line to 100

### Launch

• Groups of 2
• Give each student a copy of the number lines.
• Display only the problem stem for Elena’s string problem without revealing the question.
• “Vamos a leer este problema 3 veces” // “We are going to read this problem 3 times.”
• 1st Read: “Elena tenía una cuerda que era demasiado larga para su proyecto. La cuerda medía 65 pulgadas de largo. Elena le cortó 33 pulgadas” // “Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches.”
• “¿De qué se trata la historia?” // “What is this story about?”
• 1 minute: partner discussion.
• Listen for and clarify any questions about the context.
• 2nd Read: “Elena tenía una cuerda que era demasiado larga para su proyecto. La cuerda medía 65 pulgadas de largo. Elena le cortó 33 pulgadas” // “Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches.”
• “¿Qué podemos contar o medir en esta situación?” // “What can we count or measure in this situation?” (Elena’s string. The piece she cut off. The piece she has left.)
• 30 seconds: quiet think time
• 2 minutes: partner discussion
• Share and record all quantities.
• Reveal the question.
• “Elena tenía una cuerda que era demasiado larga para su proyecto. La cuerda medía 65 pulgadas de largo. Elena le cortó 33 pulgadas. ¿Cuál es la longitud de la cuerda ahora?” // “Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches. How long is the string now?”
• “Basándose en esta historia, ¿qué métodos podemos usar para resolver el problema?” // “Based on this story, what are some methods we can use to solve the problem?”
• 30 seconds: quiet think time
• 1–2 minutes: partner discussion

### Activity

• “Con un compañero, escojan dos rectas numéricas que muestren un método que se puede usar para encontrar la longitud de la cuerda de Elena” // “Work with your partner to choose two number lines that show a method that could be used to find the length of Elena’s string.”
• “Luego, resuelvan el problema de Han de manera individual. Cuando terminen, piensen sobre cómo pueden explicarle su método a los demás” // “Then solve Han’s problem on your own. When you finish, think of how you could explain your method to others.”
• 3 minutes: partner work time
• 3 minutes: independent work time
• “Ahora, encuentren a alguien que haya usado un método distinto al de ustedes para resolver el problema de Han. Compartan por turnos y luego intenten solos el método de su compañero” // “Now, find someone who used a different method than you to solve Han’s problem. Take turns sharing and then try their method on your own.”
• 5 minutes: group work time
• Monitor for students who:
• represent taking away or counting back with arrow(s) moving to the left from 87 to 2.
• represent an unknown addend problem with arrow(s) pointing to the right from 85 to 87.
• locate 85 and 87 on the number line and count the length of the space between by counting from the smaller to larger number or larger to smaller number

### Student Facing

1. Elena tenía una cuerda que era demasiado larga para su proyecto. La cuerda medía 65 pulgadas de largo. Elena le cortó 33 pulgadas. ¿Cuál es la longitud de la cuerda ahora?

Escoge 2 rectas numéricas que muestren formas de encontrar la longitud de la cuerda de Elena.

2. Han tenía 87 pulgadas de cuerda. Él le cortó 85 pulgadas. ¿Cuánta cuerda le queda?

1. Escribe una ecuación que represente el problema. Usa el signo ? para representar el número desconocido.

2. Encuentra el número que hace que la ecuación sea verdadera.

3. Representa en la recta numérica cómo pensaste.

3. Encuentra alguien que haya usado un método distinto.

Usa la recta numérica para mostrar su método.

### Activity Synthesis

• Invite students to share responses and reasoning for Elena’s string.
• Invite previously identified students to share their thinking and number lines for Han’s string.
• If it doesn’t come up, show how you could locate the 85 and 87 on the number line and find the difference.
• “¿Cuál método preferirían usar para resolver el problema de la cuerda de Elena? ¿Por qué?” // “Which method would you prefer to use to solve Elena’s string problem? Why?” (Taking away 33 because it matches the story. Adding on from 33 because I know I could add tens and ones and not have to regroup.)
• “¿Cuál método preferirían usar para el problema de la cuerda de Han? ¿Por qué?” // “Which method would you prefer to use for Han’s string problem? Why?” (Finding the unknown addend because I know it was small because the numbers are so close together.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy vimos distintas formas de resolver problemas de resta representados en la recta numérica. Pueden pensar en la resta como quitar o como encontrar un sumando desconocido” // “Today we saw different ways to solve subtraction problems represented on the number line. You can think about subtraction as taking away or as finding an unknown addend.”

“Descríbanle a su compañero las distintas formas como pueden representar y pensar en $$35 - 3 = ?$$. También, discutan cómo encontrarían el número que hace que la ecuación sea verdadera” // “Describe to your partner the different ways you can represent and think about $$35 - 3 = ?$$ and discuss how you would find the number that makes the equation true.”