Lesson 3
Interpretemos ecuaciones
Warm-up: ¿Qué sabes sobre $\frac{3}{2}$? (10 minutes)
Narrative
The purpose of this What Do You Know About ____? is for students to share what they know and how they can represent the number \(\frac{3}{2}\). This will be useful when students write equations to represent the relationship between a division expression and a fraction in a later activity. Record answers on a poster because students will revisit their answers to the warm-up during the lesson synthesis.
Launch
- Display the number.
- “¿Qué saben sobre \(\frac{3}{2}\)?” // “What do you know about \(\frac{3}{2}\)?”
- 1 minute: quiet think time
Activity
- Record responses.
- “¿Cómo podríamos representar el número \(\frac{3}{2}\)?” // “How could we represent the number \(\frac{3}{2}\) ?”
Student Facing
¿Qué sabes sobre \(\frac{3}{2}\)?
Student Response
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Activity Synthesis
- “¿Qué diagramas puedo dibujar para representar \(\frac{3}{2}\)?” // “What diagrams can I draw to represent \(\frac{3}{2}\)?” (Sample responses may include a number line, shaded rectangles, or a tape diagram.)
- Draw the diagrams.
- “¿En qué se parecen y en qué son diferentes los diagramas?” // “How are the diagrams the same and different?”
Activity 1: Bailarines deshidratados (20 minutes)
Narrative
The purpose of this activity is for students to write and interpret division expressions and equations that represent equal sharing situations. They explain the relationships between the dividend and the numerator and divisor and the denominator. Students may draw diagrams to help them make sense of these relationships (MP1).
The last problem provides an opportunity for students to think critically about a proposed solution to a problem (MP3). Different ways to think about the proposed solution include:
- estimation: with 3 friends sharing 2 liters, each friend gets less than 1 liter
- thinking about the meaning of the numerator (how many liters are being shared) and denominator (how many people are sharing the water)
This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.
Supports accessibility for: Conceptual Processing, Memory, Language
Launch
- Groups of 2
Activity
- 5 minutes: independent work time
- 5 minutes: partner discussion
- As students work, monitor for students who:
- draw a diagram to determine the amount of water each dancer drinks if 3 dancers share 2 liters of water.
- revise their solution for how much water each dancer gets after explaining why Mai’s answer doesn’t make sense.
Student Facing
- Tres bailarines comparten 2 litros de agua. ¿Cuánta agua recibe cada bailarín? Escribe una ecuación de división que represente la situación.
- Mai dijo que cada bailarín recibe \(\frac{3}{2}\) de un litro de agua porque 3 dividido en 2 grupos iguales es \(\frac{3}{2}\). ¿Estás de acuerdo con Mai? Muestra o explica tu razonamiento.
Student Response
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Advancing Student Thinking
If students write the numbers in the division equation in the wrong order, ask “¿Puedes describir cómo tu ecuación representa la situación?” // “Can you describe how your equation represents the situation?”
Activity Synthesis
- Invite previously selected students to share.
- If no student mentions it, ask students to explain which part of Mai’s solution doesn’t make sense and why. (It doesn’t make sense for each dancer to get \(\frac{3}{2}\) of a liter of water because \(\frac{3}{2}\) is equal to \(1\frac{1}{2}\) and if the dancers are only sharing 2 liters of water, they don’t have enough water for each person to get \(1\frac{1}{2}\) liters of water.)
MLR3 Clarify, Critique, Correct
- Display the following partially correct answer and explanation:
- Mai said that each dancer gets \(\frac{3}{2}\) liter water because 3 divided into 2 equal groups is \(\frac{3}{2}\).
- Read the explanation aloud.
- “¿Qué creen que Mai quiere decir?” // “What do you think Mai means?” (She thinks \(3\div2=\frac{3}{2}\) represents the situation.)
- “¿Hay algo que no esté claro?” // “Is anything unclear?” (The 3 represents the number of dancers, not the amount of water so Mai would be dividing dancers instead of liters of water.)
- 1 minute: quiet think time
- 2 minute: partner discussion
- “Con su pareja, escriban una explicación ajustada” // “With your partner, work together to write a revised explanation.”
- Display and review the following criteria:
- Specific words and phrases, such as dancers and liters of water
- Labeled equation or diagram
- 3–5 minute: partner work time
- Select 1–2 groups to share their revised explanation with the class. Record responses as students share.
- “¿En qué se parecen y en qué son diferentes las explicaciones?” // “What is the same and different about the explanations?” (Some people labeled the diagram to show what is being divided and some people labeled the numbers in the equation.)
Activity 2: Interpretemos expresiones (15 minutes)
Narrative
In previous activities, students interpreted diagrams and expressions that represented equal sharing situations. The purpose of this activity is for students to interpret division expressions without the support of diagrams in order to deepen their understanding about the relationship between fractions and division expressions. The activity is designed to highlight the relationship between the number of objects being shared and the numerator, on the one hand, and the number of people sharing and the denominator on the other (MP7).
Advances: Conversing, Reading
Launch
- Groups of 2
Activity
- 5-8 minutes: independent work time
- 5-8 minutes: partner work time
- Monitor for students who:
- notice and can explain the relationship between the numerator and the number of liters of water and the denominator and the number of dancers.
Student Facing
- Completa la tabla. Si te ayuda, dibuja un diagrama.
número de bailarines litros de agua expresión de división cantidad de agua que bebió cada bailarín, en litros 4 2 \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(3 \div 4\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(3 \div 5\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) 4 \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) 5 \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) - ¿Qué patrones observas en la tabla?
Student Response
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Advancing Student Thinking
If students fill in the table according to the order of the dividend and divisor, encourage them to draw a diagram to represent each situation in the table. Ask “¿De qué manera tus diagramas representan las expresiones de división?” // “How do your diagrams represent the division expressions?”
Activity Synthesis
- Display the table from the activity.
- “¿Cuáles son algunos de los números que usaron en las dos últimas filas?” // “What are some of the numbers you used for the last two rows?”
- Record the answers as additional rows to the table.
- “¿Cuáles son algunos de los patrones que observan en la tabla?” // “What are some patterns that you notice in the table?”
- Ask previously selected students to share their solutions.
Lesson Synthesis
Lesson Synthesis
Display poster of responses from the warm-up.
“A partir de lo que aprendimos hoy, ¿qué podemos agregar a nuestras respuestas del calentamiento?” // “What can we add to our answers from the warm-up question based on what we learned today?” (Sample responses: \(\frac{3}{2}\) can describe a division situation, \(\frac{3}{2}\) can represent 2 dancers sharing 3 liters of water, \(3 \div 2 = \frac{3}{2}\) or \(1\frac{1}{2}\))
If no student mentions it, say, “Si dos bailarines comparten 3 litros de agua, ¿cuánta agua recibe cada bailarín? Escriban una ecuación que represente la situación” // “If 2 dancers share 3 liters of water, how much water does each dancer get? Write an equation that represents the situation.” Invite students to share their equation.
Display equation: \(3 \div 2 = \frac{3}{2}\).
“¿De qué manera esta ecuación representa la situación?” // “How does this equation represent the situation?” (3 liters of water are being shared by 2 dancers. Each of the 2 dancers gets \(\frac{3}{2}\) liter of water.)
Cool-down: Comparte agua (5 minutes)
Cool-Down
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