# Lesson 9

Generemos patrones

## Warm-up: Conteo grupal: Dos patrones (10 minutes)

### Narrative

The purpose of this Choral Count is to invite students to notice patterns and relationships in two different counts. These understandings help students develop fluency with multiples and will be helpful when students identify relationships between corresponding terms in two patterns in the next several lessons.

### Launch

• “Cuenten de 6 en 6, empezando en 0” // “Count by 6, starting at 0.”
• Record as students count.
• Stop counting and recording at 60.

### Activity

• “¿Qué patrones ven?” // “What patterns do you see?”
• 1–2 minutes: quiet think time
• Record responses.
• “Ahora cuenten de 12 en 12, empezando en 0” // “Now count by 12s, starting at 0.”
• Record as students count.
• Stop counting and recording at 120.
• “¿Qué patrones ven?” // “What patterns do you see?”
• 1–2 minutes: quiet think time
• Record responses.

### Activity Synthesis

• “Si seguimos contando, ¿qué números van a estar en ambos conteos?” // “If we continue counting, what numbers will be in both counts?” (All of the numbers on the second list, the ones where we count by 12)

## Activity 1: ¿Cuál es el patrón? (20 minutes)

### Narrative

The purpose of this activity is for students to generate two different patterns, given two different rules, and recognize relationships between corresponding terms (MP7). Students may notice a variety of relationships between the two patterns and may describe them generally (all of the numbers in one pattern are contained in the other pattern) or more specifically (the numbers of the second pattern are every other number from the first pattern). To answer the questions about corresponding terms in the patterns, students may continue the patterns or may use the relationship between the patterns. The number 192 is deliberately chosen to encourage using the relationship between the patterns.

When students find and explain patterns related to the rules and relationships, they look for and express regularity in repeated reasoning (MP8).

Action and Expression: Internalize Executive Functions. Synthesis: Invite students to plan a strategy, including the tools they will use, for identifying apparent relationships between corresponding terms in patterns. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Memory

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 10–12 minutes: partner work time
MLR2 Collect and Display
• Circulate, listen for and collect the language students use to describe the relationships between Jada's and Priya’s rules. Listen for: double, half of, twice as much, times 2, divided by 2.
• Record students’ words and phrases on a visual display and update it throughout the lesson. Monitor for students who notice that:
• all of Priya’s numbers are in Jada’s pattern, but not all of Jada’s numbers are in Priya’s pattern
• each of Priya’s numbers is double the corresponding number in Jada’s pattern
• each of Jada’s numbers is half the corresponding number in Priya’s pattern

### Student Facing

1. Jada y Priya crean patrones usando unas reglas. Usa cada regla para completar sus patrones.

La regla de Jada: empezar en 0 y siempre sumar 4.

La regla de Priya: empezar en 0 y siempre sumar 8.
2. Si Jada y Priya continúan sus patrones, encuentra el número que saldrá en el patrón de Priya cuando el patrón de Jada tenga:

1. 40
2. 60
3. Si Jada y Priya continúan sus patrones, ¿qué número saldrá en el patrón de Jada cuando el patrón de Priya tenga 192?
4. ¿Qué relaciones observas entre los números del patrón de Priya y los números del patrón de Jada?

### Student Response

If students are not able to identify a relationship between the two different patterns, ask:

• “¿En qué se parecen los números de los patrones?” // “What is the same about the numbers in the patterns?”
• “¿En qué son diferentes los números de los patrones?” // “What is different about the numbers in the patterns?”

### Activity Synthesis

• “¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• “¿Todos los números de Priya están en el patrón de Jada? ¿Cómo lo saben?” // “Are all of Priya’s numbers in Jada’s pattern? How do you know?” (Yes. There are 2 fours in 8 so each of Priya’s numbers appears twice as far out in Jada’s list.)
• “¿Qué relaciones ven entre el patrón de Jada y el patrón de Priya?” // “What relationships do you see between Jada's pattern and Priya’s pattern?” (Each number in Priya’s pattern is double the corresponding number in Jada’s pattern. Each number in Jada’s pattern is half the corresponding number in Priya’s pattern.)

## Activity 2: Más patrones (15 minutes)

### Narrative

The purpose of this activity is for students to practice generating two different patterns from two different rules and observe and quantify relationships between corresponding terms. Both sets of rules generate patterns that have the same relationships between corresponding terms. Each of Priya’s terms is 3 times greater than each of Jada’s corresponding terms and each of Jada’s terms is $$\frac{1}{3}$$ of Priya’s corresponding terms.

When students find and explain patterns related rules and relationships, they look for and express regularity in repeated reasoning (MP8).

### Launch

• Groups of 2
• “Ustedes y su compañero van a trabajar individualmente en algunos problemas sobre patrones. Cuando acaben, discutan su trabajo con su compañero” // “You and your partner will each complete some problems about patterns independently. After you’re done, discuss your work with your partner.”

### Activity

• 5 minutes: independent work time
• 3 minutes: partner discussion
• “Revisen su trabajo y hagan los ajustes necesarios basándose en lo que aprendieron durante la discusión con su compañero” // “Look back at your work and make any revisions based on what you learned from discussing with your partner.”
• 1–2 minutes: independent work time
• Monitor for students who:
• revise their thinking based on partner discussion
• use multiplication expressions to represent the relationships between the patterns

### Student Facing

Compañero A

1. Usa cada regla para crear un patrón.

La regla de Jada: comenzar en 0 y siempre sumar 2.

La regla de Priya: comenzar en 0 y siempre sumar 6.

2. Si los patrones continúan,

1. ¿qué número saldrá en el patrón de Priya cuando el patrón de Jada tenga 34? Explica o muestra cómo razonaste.
2. ¿qué número saldrá en el patrón de Jada cuando el patrón de Priya tenga 120? Explica o muestra cómo razonaste.
3. ¿Qué relaciones observas entre los números del patrón de Priya y los números del patrón de Jada?

Compañero B

4. Usa cada regla para crear un patrón.

La regla de Jada: empezar en 0 y siempre sumar 3.

La regla de Priya: empezar en 0 y siempre sumar 9.
5. Si los patrones continúan,

1. ¿qué número saldrá en el patrón de Priya cuando el patrón de Jada tenga 54? Explica o muestra cómo razonaste.
2. ¿qué número saldrá en el patrón de Jada cuando el patrón de Priya tenga 198? Explica o muestra cómo razonaste.
6. ¿Qué relaciones observas entre los números del patrón de Priya y los números del patrón de Jada?

### Activity Synthesis

• Ask previously identified students to share their thinking.
• Display completed patterns from both sets of rules.
• “¿En qué se parecen los patrones generados por las parejas de reglas? ¿En qué son diferentes?” // “How are the patterns from the pairs of rules the same? How are they different?” (Some of the numbers are in all four patterns. Some numbers aren’t in any pattern like 5 and 11. In both sets of rules, Priya’s numbers are 3 times greater than Jada’s numbers and Jada’s numbers are $$\frac{1}{3}$$ Priya’s numbers.)
• “¿Cómo describirían la relación que hay entre el patrón de Priya y el patrón de Jada?” // “How would you describe the relationship between Priya's pattern and Jada's pattern?” (Priya's numbers are all 3 times as much as Jada's corresponding numbers.)
• “¿Por qué número puedo multiplicar los números del patrón de Priya para obtener los números correspondientes en el patrón de Jada?” // “What can I multiply the numbers in Priya’s pattern by to get the corresponding numbers in Jada’s pattern?” ($$\frac{1}{3}$$, each number in Jada's pattern is $$\frac{1}{3}$$ the corresponding number in Priya's pattern.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy generamos algunos patrones e identificamos relaciones entre dos patrones diferentes” // “Today we generated patterns and recognized relationships between two different patterns.”

Display or rewrite the numbers from the 2 choral counts in the warm-up.

“¿Qué relaciones observan entre los patrones de los conteos?” // “What relationships do you notice between the patterns in the counts?” (Each number in the second count is twice the matching number in the first count. Each number in the first count is one half the matching number in the second count. All the numbers on the second list are also on the first list. The second list is every other number of the first list.)

“Explíquenle a su compañero por qué estas relaciones van a continuar si los patrones continúan” // “Explain to your partner why these relationships will continue if the patterns continue.” (The second pattern adds 12 each time and that's twice what the first pattern adds each time.)