# Lesson 11

Patrones y pares ordenados

## Warm-up: Observa y pregúntate: La cuadrícula de coordenadas (10 minutes)

### Narrative

The purpose of this warm-up is for students to discuss the patterns they see in points plotted on a coordinate grid, which will be useful when students graph ordered pairs consisting of corresponding terms from two patterns in a later activity. While students may notice and wonder many things about this image, the location of the points and their coordinates are the important discussion points.

### 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 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

• “¿Cómo podemos usar coordenadas para describir la ubicación de cada punto?” // “How can we use coordinates to describe the location of each point?” (The point $$D$$ is at $$(8,4)$$ since its horizontal coordinate is 8 and its vertical coordinate is 4. The other points are harder to tell though the vertical coordinate of $$B$$ is 4.)

## Activity 1: Patrones en la cuadrícula de coordenadas (parte 1) (20 minutes)

### Narrative

The purpose of this activity is for students to generate two patterns from rules and then graph them on the coordinate grid. Students first identify a point on the coordinate grid with one of the pairs of numbers from the patterns and then plot the rest of the points. Students may notice that the points on the graph are regularly spaced. They are invited to share this and other observations in the synthesis.

MLR8 Discussion Supports. Display sentence frames to support partner discussion: “Primero, yo _____ porque . . .” // “First, I _____ because . . .” and “Observé _____, entonces yo . . .” // “I noticed _____ so I . . . .”
Advances: Speaking, Writing, Conversing, Representing
Action and Expression: Develop Expression and Communication. Synthesis: Develop fluency with connecting rules, tables, and a coordinate grid to the same pattern. Provide access to blank or partially completed tables.
Supports accessibility for: Conceptual Processing, Attention

### Launch

• Groups of 2
• “Ustedes y su compañero van a comenzar a trabajar individualmente en algunos problemas sobre patrones y la cuadrícula de coordenadas. Después de un par de minutos, completen los problemas con su compañero” // “You and your partner will each start some problems about patterns and the coordinate grid independently. After a couple minutes, work with your partner to complete the problems.”

### Activity

• 2 minutes: independent time
• 5 minutes: partner work time

### Student Facing

Compañero A

Regla 1: empezar en 0 y siempre sumar 8.

Regla 2: empezar en 0 y siempre sumar 2.

1. Completa la tabla usando las reglas.
A B C D E F
regla 1
regla 2

2. ¿Cuál columna de la tabla representa el punto que está en la cuadrícula de coordenadas? Marca el punto con la letra que le corresponde.

3. Ubica y marca el resto de los puntos.

Compañero B

Regla 1: empezar en 0 y siempre sumar 10.

Regla 2: empezar en 0 y siempre sumar 40.

1. Completa la tabla usando las reglas.
A B C D E F
regla 1
regla 2

2. ¿Cuál columna de la tabla representa el punto que está en la cuadrícula de coordenadas? Marca el punto con la letra que le corresponde.

3. Ubica y marca el resto de los puntos.

### Activity Synthesis

• “¿Cómo decidieron en qué lugar de la cuadrícula ubicar cada punto?” // “How did you decide where to place the points on the grid?” (I used the top row to decide how far over to go on the horizontal axis and the second rule to decide how far up to go on the vertical axis.)
• Invite students to share completed graphs for parts A and B.
• “¿En qué se parecen las dos gráficas?” // “How are the two graphs the same?” (There is a point at the bottom left, $$(0,0)$$, on each. The points are regularly spaced and go up and to the right. They all lie at the intersection of gridlines.)
• “¿En qué son diferentes las dos gráficas?” // “How are the two graphs different?” (The numbers on the axes are different. The ones for partner B get bigger really quickly.)

## Activity 2: Patrones en la cuadrícula de coordenadas (parte 2) (15 minutes)

### Narrative

The purpose of this activity is for students to generate numerical patterns given two rules, form ordered pairs consisting of the corresponding terms, and graph the ordered pairs on the coordinate grid. The structure of the activity is the same as the previous activity but this time the multiplicative factor relating the two rules is a fraction. Monitor for students who express the relationship (MP8) between the two patterns by saying that
• the numbers in the second pattern are double the numbers in first pattern and half more
• the numbers in the second pattern are $$2\frac{1}{2}$$ times the numbers in the first pattern

• Groups of 2.

### Activity

• 5 minutes: independent time
• 5 minutes: partner work time
• Monitor for students who:
• notice the additive relationship for each rule
• notice the multiplicative relationship between rule 1 and rule 2

### Student Facing

1. Completa la tabla usando las reglas.
• Regla 1: empezar en 0 y siempre sumar 2.
• Regla 2: empezar en 0 y siempre sumar 5.
A B C D E F
Regla 1
Regla 2
2. ¿Qué relaciones observas entre los términos correspondientes en los dos patrones?
3. Ubica y marca los puntos de la tabla.

4. ¿Qué te dice el punto $$C$$ sobre la regla 1 y la regla 2?

### Advancing Student Thinking

If students don’t plot and label the points from the table correctly, plot points A and B and ask, “¿Cómo estos puntos representan las reglas?” // “How do each of these points represent the rules?”

### Activity Synthesis

• Invite previously selected students to share.
• “¿Qué representa el punto $$D$$ en términos de las dos reglas? ¿Cómo lo saben?” // “What does point $$D$$ represent in terms of the two rules? How do you know?” (When rule 1 is 6, rule 2 is 15. The coordinates are $$(6,15)$$ and the horizontal coordinate is rule 1 and the vertical coordinate is rule 2.)
• Display: $$(10,20)$$
• “El número 10 está en el patrón de la regla 1 y el número 20 está en el patrón de la regla 2. ¿El punto que tiene coordenadas $$(10, 20)$$ está en su gráfica?” // “10 is a number in rule 1 and 20 is a number in rule 2. Is the point with coordinates $$(10, 20)$$ on your graph?” (No, the points that represent the two rules are $$(10, 25)$$ or $$(8, 20)$$. The 10 from rule 1 and the 20 from rule 2 don’t match up with each other.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy ubicamos puntos, creados a partir de dos patrones, en una cuadrícula de coordenadas y descubrimos relaciones” // “Today, we plotted points from two patterns on a coordinate grid and noticed patterns.”

Display the image from the student solution in the second activity.

“¿Qué les dice la gráfica sobre las dos reglas?” // “What does the graph tell you about the two rules?” (They both start at 0. That’s what the point $$(0,0)$$ means. Then the first rule has 2 and the second rule has 5.)

“¿En qué se parecen mirar relaciones entre patrones en una tabla y mirar relaciones entre patrones en una cuadrícula de coordenadas? ¿En qué son diferentes?” // “How is looking at relationships between patterns in a table the same as looking at relationships between patterns on a coordinate grid? How is it different?” (In the table I can see each rule by going across or I can see the relationship between rules by looking at columns. The points on the coordinate grid help me visualize how the two patterns are changing relative to one another but they don’t help me see the pattern for each rule.)