Lesson 8

Ecuaciones en una recta numérica

Warm-up: Conteo grupal: Hacia atrás de 10 en 10 (10 minutes)

Narrative

The purpose of this Choral Count is to invite students to practice counting back by 10 from any number and notice patterns in the count. When students recognize that the digit in the ones place remains the same, while the digit in the tens place decreases by 1 each time, they look for and make use of the base ten-structure and express regularity in repeated reasoning (MP7, MP8). These understandings help students develop fluency and will be helpful later in this lesson when students show their thinking on the number line.

Launch

• “Contemos hacia atrás de 10 en 10, empezando en 99” // “Count back by 10, starting at 99.”
• Record as students count.
• Stop counting and recording at 49.
• “Contemos hacia atrás de 10 en 10, empezando en 98” // “Count back by 10, starting at 98.”
• Record as students count directly below the first count.
• Stop counting and recording at 48.
• “Contemos hacia atrás de 10 en 10, empezando en 95” // “Count back by 10, starting at 95.”
• Record as students count directly below the second count.
• Stop counting and recording at 45.

Activity

• “¿Qué patrones observan?” // “What patterns do you see?” (The ones digit stays the same, but the tens digit goes down by one.)
• 1–2 minutes: quiet think time
• Record responses.

Activity Synthesis

• “¿Alguien puede describir el patrón con otras palabras?” // “Who can restate the pattern in different words?”
• “¿Alguien quiere compartir otra observación sobre por qué ocurre ese patrón aquí?” // “Does anyone want to add an observation on why that pattern is happening here?”

Activity 1: Representemos ecuaciones (15 minutes)

Narrative

The purpose of this activity is for students to represent addition and subtraction equations on a number line. Students consider where to begin and in which direction to draw their arrows in order to accurately represent the operation in the given equation. Throughout the activity, encourage students to explain how they know their representation matches the equation. Listen for the way they explain how they know where to start and end their arrow, which direction they draw their arrow, and how they connect the length the jump represents to the equation (MP3).

MLR8 Discussion Supports. Display sentence frames to support partner discussion: “Nuestras rectas numéricas se parecen porque…” // “Our number lines are the same because…”. “Nuestras rectas numéricas son diferentes porque…” // “Our number lines are different because…”.

Launch

• Groups of 2
• “Hemos visto ecuaciones de suma y ecuaciones de resta representadas en una recta numérica. ¿Cómo pueden saber si una recta numérica está representando una suma o está representando una resta?” // “We have seen addition and subtraction equations represented on a number line. How can you tell whether a number line is representing addition or subtraction?” (Look at the direction of the arrow.)
• 30 seconds: quiet think time
• Share responses.

Activity

• “Ahora van a representar algunas ecuaciones en rectas numéricas. Asegúrense de que los demás, al ver su recta numérica, van a poder saber si ustedes están representando una suma o si están representando una resta” // “Now you are going to represent some equations on number lines. Be sure others will be able to tell from looking at your number line whether you are representing addition or subtraction.”
• 8 minutes: independent work time
• Monitor for students who:
• Mark the first number and last number in each equation with a point first. Then draw their arrow to match the operation.
• Locate the first number in the equation, count a number of spaces forward or backward to match the operation, and draw an arrow to connect the two numbers.
• “Comparen con su compañero sus rectas numéricas. Expliquen cómo saben que su representación corresponde a la ecuación” // “Compare your number lines with your partner. Explain how you know your representation matches the equation.”
• 3 minutes: partner discussion

Student Facing

Representa cada ecuación en la recta numérica.

1. $$15 + 7 = 22$$

2. $$18 - 6 = 12$$

3. $$46 + 7 = 53$$

4. $$33 - 9 = 24$$

Student Response

If a student's number line representation doesn't match the equation, consider asking:

• “¿Podrías explicar tu representación?” // “Will you please explain your representation?”
• “¿Cómo decidiste dónde empezar o dónde terminar en la recta numérica?” // “How did you decide where to start or end on the number line?”
• “¿Cómo decidiste en qué dirección dibujar tu flecha?” // “How did you decide which direction to draw your arrow?”

Activity Synthesis

• Display the equation and blank number line for $$15 + 7 = 22$$.
• Invite previously selected students to share how they represented the equation on the number line.
• “¿Cómo ven cada número de la ecuación en la representación de ____?” // “How do you see each number in the equation in ____’s representation?”
• “¿Cómo saben que la representación de ___ corresponde a la operación (suma o resta)?” // “How do you know ___’s representation matches the operation (addition or subtraction)?”
• As time permits, continue with the other equations.

Activity 2: Escribamos ecuaciones (20 minutes)

Narrative

The purpose of this activity is for students to write addition and subtraction equations to match number line representations. Students determine the operation represented by looking at the direction of the arrows. They identify the starting number, the length of the jump, and the ending number in order to write the equation. Each representation shows a relationship between the same three numbers. When students explain to each other why they think two representations are most alike, they look for and make sense of the structure of the number line and deepen their understanding of the relationship between addition and subtraction (MP3, MP7).

Representation: Access for Perception. Use a cut out of an animal that hops (bird, frog, rabbit, or another example) to demonstrate the jumps on the number line.
Supports accessibility for: Conceptual Processing, Attention

• Groups of 2

Activity

• “Ahora van a observar rectas numéricas y a escribir ecuaciones que correspondan al diagrama” // “Now you are going to look at number lines and write equations that match the diagram.“
• 5 minutes: independent work time
• “Comparen con su compañero sus ecuaciones. Tomen turnos para explicar de qué manera su ecuación corresponde al diagrama de recta numérica” // “Compare your equations with your partner. Take turns explaining how your equation matches the number line diagram.”
• “Luego, escojan juntos dos rectas numéricas que crean que son las que más se parecen. Prepárense para explicar su elección” // “Then work together to choose two number lines that you think are the most alike. Be prepared to explain your choice.”
• 5 minutes: partner discussion
• Monitor for students who:
• explain how their equation matches the point, direction of the arrow, final position of the arrow, and the length of the movement on the number line
• discuss how number lines are the same because they show the same operation
• discuss how number lines are the same because the length of the jump is the same

Student Facing

En cada caso, escribe una ecuación que corresponda al diagrama.

1. Ecuación:__________________________

2. Ecuación:__________________________

3. Ecuación:__________________________

4. Ecuación:__________________________

5. Compara con tu compañero tus ecuaciones.
6. Escoge las 2 rectas numéricas que creas que son las más parecidas. Explica tu elección a tu compañero.

Student Response

If students use language that is not precise mathematical language when they explain how they know their equations match the diagrams, consider asking:

• “¿Cómo ves cada número en el diagrama?” // “How do you see each number in the diagram?”
• “¿Cómo tu ecuación corresponde a la flecha en el diagrama?” // “How does your equation match the arrow in the diagram?”
• “¿Por qué tu ecuación empieza con __? ¿En qué parte del diagrama está ese número?” // “Why does your equation start with __? Where is that number in the diagram?”
• “¿Cuál número de tu ecuación representa la longitud entre __ y __?” // “Which number in your equation represents the length between __ and __?”

Activity Synthesis

• Invite previously identified students to share their equations for each number line.
• “¿Están de acuerdo o en desacuerdo? Expliquen” // “Do you agree or disagree? Explain.”
• “¿Alguien puede explicar de otra forma por qué la ecuación de ___ corresponde a la recta numérica?” // “Can anyone explain why ___’s equation matches the number line in another way?”
• Invite previously identified students to share the number lines they feel are most alike.
• If time, ask “¿En qué más se parecen las rectas numéricas? ¿En qué son diferentes?” // “In what other ways are the number lines alike? How are they different?”

Lesson Synthesis

Lesson Synthesis

“Hoy trabajamos con ecuaciones y representaciones en la recta numérica” // “Today we worked with equations and number line representations.”

“¿Cómo podrían explicarle a un amigo de qué forma puede usar la recta numérica para mostrar sumas y restas?” // “How could you explain to a friend how they could use the number line to show addition and subtraction?” (You can use an arrow to show moving from one spot to another on the number line. Moving right is like adding. Moving to the left is like subtracting. You can use a point to show where you start and the arrow shows where you end.)