# Lesson 10

Interpretemos relaciones

## Warm-up: Verdadero o falso: Multiplicar y dividir (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate understandings they have for the relationship between multiplication and division. Students will use this understanding in the lesson when they recognize multiplicative relationships between patterns.

### Launch

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each problem.

### Student Facing

En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar cómo razonaste.

• $$276 \div 3 = \frac{1}{3} \times 276$$
• $$276 \div 3 = \frac{276}{6}$$
• $$(276 \div 3)\times2=\frac{2}{3}\times 276$$

### Activity Synthesis

• “Para justificar su razonamiento, ¿cómo les puede ayudar la relación que hay entre la multiplicación y la división?” // “How can the relationship between multiplication and division help you justify your reasoning?” (I used the fact that dividing by 3 is the same as multiplying by $$\frac{1}{3}$$.)

## Activity 1: Completemos y relacionemos: 3 patrones (20 minutes)

### Narrative

The purpose of this activity is for students to practice interpreting relationships between patterns generated from two different rules. Students may need to generate patterns beyond the boxes provided. Encourage them to continue the pattern as needed. Students may describe the patterns and relationships in different, but accurate ways. Encourage them to notice as many relationships as they can and describe the relationships in whatever way makes sense to them. This is the first time students see two patterns where the numbers in one pattern are not multiples of the numbers in the other pattern.

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

• Groups of 2

### Activity

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

MLR7 Compare and Connect

• “Creen una presentación visual que muestre lo que pensaron sobre las relaciones que hay entre cada grupo de patrones. Incluyan detalles, como notas, diagramas y dibujos, para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about the relationships between each set of patterns. Include details such as notes, diagrams, and drawings to help others understand your thinking.”
• 2–5 minutes: independent or group work
• 3–5 minutes: gallery walk

### Student Facing

1. Completa los patrones para cada grupo de reglas.
2. ¿Qué relaciones hay entre los patrones de cada grupo de reglas? Prepárate para explicar cómo pensaste.

Grupo A

Regla 1: empezar en 0 y siempre sumar 3.

Regla 2: empezar en 0 y siempre sumar 6.

Grupo B

Regla 1: empezar en 4 y siempre sumar 3.

Regla 2: empezar en 9 y siempre sumar 6.

Grupo C

Regla 1: empezar en 0 y siempre sumar 5.

Regla 2: empezar en 0 y siempre sumar 3.

### Activity Synthesis

• “¿En qué se parecen y en qué son diferentes las formas en las que representamos las relaciones que hay entre los patrones?” // “What is the same and what is different in the way we represented the relationships between the patterns?” (Some of us used numbers and symbols, some of us wrote sentences.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “¿En qué se parecen y en qué son diferentes el grupo A y el grupo B?” // “What is the same and what is different about Set A and Set B?” (They both add the same amount, but one set starts at 0. Set B has a lot more odd numbers in it. In set A, each number in rule 2 is double the number in rule 1, but in set B, each number in rule 2 is 1 more than double the number in rule 1.)
• “¿Para cuál pareja de reglas fue más retador observar y describir las relaciones?” // “For which pair of rules was it most challenging to notice and describe relationships?” (Set B because there was no multiple that worked to get from one pattern to the other. Set C because there weren’t doubles or halves.)

## Activity 2: Generemos patrones (15 minutes)

### Narrative

The purpose of this activity is for students to interpret more complex relationships in corresponding terms from patterns generated from two different rules. Both sets of rules generate patterns that have the same relationships between corresponding terms. Each of the terms in rule 2 is $$\frac{3}{2}$$ times greater than each of the corresponding terms in rule 1 and each of the terms in rule 1 is $$\frac{2}{3}$$ the corresponding term in rule 2. Some students may state these relationships in different ways. The relationships between these patterns build directly on the third pair of rules from the previous activity where the terms for rule 1 were $$\frac{5}{3}$$ the corresponding terms in rule 2.

Representation: Internalize Comprehension. Synthesis: Use multiple examples and non-examples to emphasize the relationship between corresponding terms in patterns.
Supports accessibility for: Attention, Memory, Conceptual Processing

### 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

• 5–7 minutes: independent work time
• 3–5 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 your discussion.”
• 1–2 minutes: independent work time
• Monitor for students who:
• recognize that each term in pattern 2 is $$1\frac{1}{2}$$ times the corresponding term in pattern 1
• recognize that each term in pattern 1 is $$\frac{2}{3}$$ the corresponding term in pattern 2

### Student Facing

Compañero A

1. Genera patrones para las dos reglas.

Regla 1: empezar en 0 y siempre sumar 4.

Regla 2: empezar en 0 y siempre sumar 6.

2. Compara tus patrones. ¿Qué relaciones observas?
3. ¿Qué número saldrá en el patrón 2 cuando el número 40 salga en los cuadros del patrón 1?
4. ¿Qué número saldrá en el patrón 1 cuando el número 120 salga en los cuadros del patrón 2?

Compañero B

1. Genera patrones para las dos reglas.

Regla 1: empezar en 0 y siempre sumar 2.

Regla 2: empezar en 0 y siempre sumar 3.

2. Compara tus patrones. ¿Qué relaciones observas?
3. ¿Qué número saldrá en el patrón 2 cuando el número 30 salga en los cuadros del patrón 1?
4. ¿Qué número saldrá en el patrón 1 cuando el número 60 salga en los cuadros del patrón 2?

### Activity Synthesis

• Ask previously selected students to share their thinking.
• Display or write the numbers in the patterns for partner A.
• “¿Cómo podemos representar la relación que hay entre los números de los patrones usando ecuaciones de multiplicación?” // “How can we represent the relationship between the numbers in the patterns with multiplication equations?” (Each number in rule 2 is $$\frac{3}{2}$$ the corresponding number in rule 1. Each number in rule 1 is $$\frac{2}{3}$$ the corresponding number in rule 2:  $$6=\frac{3}{2}\times 4$$, $$4= \frac{2}{3}\times 6$$.)
• Display or write the numbers in the patterns for partner B.
• “¿Cómo podemos representar las relaciones que hay entre los números de los patrones usando ecuaciones de multiplicación?” // “How can we represent the relationships between the numbers in the patterns with multiplication equations?” ($$3=1\frac{1}{2}\times 2$$ , $$2=\frac{2}{3}\times 3$$.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy observamos y explicamos relaciones entre patrones. Algunas de las relaciones involucraban fracciones” // “Today we noticed and explained relationships between patterns. Some of the relationships involved fractions.”

“¿Qué relaciones encontraron entre los patrones que estudiamos hoy?” // “What relationships did you find between the patterns we studied today?” (Sometimes I could multiply each term in one pattern by the same number to get the corresponding number in the other pattern. Sometimes that number was a fraction.)

Consider asking students to record their response in a math journal and then share their response with a partner.