# Lesson 12

Representemos problemas en la cuadrícula de coordenadas

## Warm-up: Verdadero o falso: Suma y multiplicación (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate understandings they have of the properties of operations. These understandings will be helpful later when students will need to be able to use addition and multiplication to solve problems involving money. Each expression here is chosen to represent the total value of a set of coins (nickels, dimes, and quarters).

### 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 statement.

### Student Facing

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

• $$(2 \times 10) + (3 \times 5) = (3 \times 10) + (1 \times 5)$$
• $$(3 \times 25) + (5 \times 5) = 8 \times 25$$
• $$(4 \times 25) + (10 \times 5) = (2 \times 25) + (10 \times 10)$$

### Student Response

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### Activity Synthesis

• “¿Cuál afirmación fue su favorita y por qué?” // “Which statement was your favorite to think about and why?” (I liked the first one because I could calculate all the values mentally.)

## Activity 1: Cara o cruz (15 minutes)

### Narrative

The purpose of this activity is for students to plot and interpret points that represent the result of flipping a coin 10 times (MP2). Students also interpret points that are already on the graph, representing the number of heads and tails two other students got when they flipped a coin.

Students may wonder what to do if they get the same result twice or the same result as their partner since that point is already plotted on the graph. They may:

• put a letter for their name next to the point
• put a number to indicate that they got that result on their first and second coin tosses

Students may notice that the points all lie on a line (MP7). It is not necessary for students to understand why the points form a line. Focus students’ attention on the meaning of each point.

This activity uses MLR6 Three Reads. Advances: Reading, Listening, Representing

Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Attention, Organization

### Required Materials

Materials to Gather

### Required Preparation

• Gather pennies, nickels, dimes, and quarters to show students during the launch.

### Launch

• Groups of 2

MLR6 Three Reads

• Display only the problem stem, without revealing the grid or question(s):
• “Vamos a leer este problema 3 veces” // “We are going to read this problem 3 times.”
• 1st Read: “Han y Jada lanzaron una moneda de un centavo varias veces y contaron cuántas veces salía cara y cuántas veces salía cruz” // “Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails.”
• “¿De qué se trata esta situación?” // “What is this situation about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• 2nd Read: “Han y Jada lanzaron una moneda de un centavo varias veces y contaron cuántas veces salía cara y cuántas veces salía cruz” // “Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails.”
• “Nombren las cantidades. ¿Qué podemos contar o medir en esta situación?” // “Name the quantities. What can we count or measure in this situation?”
• 30 seconds: quiet think time
• 2 minutes: partner discussion
• Share and record all quantities.
• Reveal the question(s).
• 3rd Read: Read the entire problem, including question(s) aloud.
• “¿Qué estrategias podemos usar para resolver este problema?” // “What are some strategies we can use to solve this problem?”

### Activity

• 2 minutes: independent think time
• 8 minutes: partner work time

### Student Facing

Han y Jada lanzaron una moneda de un centavo varias veces y contaron cuántas veces salía cara y cuántas veces salía cruz. Sus resultados se muestran en esta gráfica.

1. ¿Cuántas caras le salieron a Jada? ¿Cuántas cruces le salieron a Jada? Explica o muestra cómo lo sabes.
2. ¿Cuántas caras le salieron a Han? ¿Cuántas cruces le salieron a Han? Explica o muestra cómo lo sabes.
3. Lanza la moneda 10 veces y anota cuántas caras y cuántas cruces te salieron. Ubica en la cuadrícula de coordenadas el punto que representa lo que salió en tus lanzamientos.
4. Muéstrale a tu compañero el punto que ubicaste en la cuadrícula de coordenadas. Mira la cuadrícula de tu compañero. ¿Cuántas caras le salieron a tu compañero?, ¿cuántas cruces le salieron a tu compañero? Explica o muestra cómo razonaste.
5. ¿Alguno de los puntos que ubicaron está en el eje horizontal? ¿Qué significaría un punto que estuviera en el eje horizontal en esta situación?
6. Si te queda tiempo, lanza la moneda 10 veces más y marca tus resultados y los de tu compañero en la cuadrícula de coordenadas.

### Student Response

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### Advancing Student Thinking

If students say they aren’t sure where to plot a point to represent their coin tosses, refer to the point that represents Jada’s data and ask, “¿Qué representa este punto?” // “What does this point represent?”

### Activity Synthesis

• Display the coordinate grid from the activity.
• As each student shares, ask them to explain where to plot their point on the displayed graph.
• “¿Cuáles son las coordenadas del punto de Jada?” // “What are the coordinates of Jada’s point?” ($$(6,3)$$)
• “¿Cuántas veces lanzó Jada la moneda? ¿Cómo lo saben?” // “How many times did Jada toss the coin? How do you know?” (9 times because she got 6 heads and 3 tails.)
• Highlight the point $$(10,0)$$ on the grid. “¿Qué quiere decir este punto?” // “What does this point mean?” (10 heads and no tails)
• “¿Alguien obtuvo este resultado?” // “Did anyone get this result?” (most likely no)
• “¿Ustedes creen que sacar siempre cara sucede muy a menudo?” // “Do you think all heads happens very often?” (This probably does not happen very often because that means that you can get only heads on every toss.)

## Activity 2: Valores de las monedas (15 minutes)

### Narrative

The purpose of this activity is for students to plot and interpret points on the coordinate grid. The context remains coins but there is a variety of coins and the vertical coordinate is determined by the value of the coins. Students plot points corresponding to different combinations of coins. They identify the coordinates of plotted points and interpret them in terms of the context of coins and their value (MP2). During the activity synthesis, students discuss how they decided where to plot points and how they interpreted points on the graph.

### Launch

• Groups of 2
• “¿Qué saben sobre las monedas?” // “What do you know about coins?” (They're round. I can buy things with them. There are different kinds and they have different values.)
• Record responses for all to see.
• Display a penny, dime, nickel, and quarter.
• If no student mentions it, say and record the value of each coin.

### Activity

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

### Student Facing

Esta gráfica muestra el número y el valor de las monedas que tenían algunos estudiantes.

1. Tyler tiene 1 moneda de diez centavos, 3 de cinco centavos y 2 de un centavo. ¿Cuál punto representa las monedas de Tyler? Marca el punto.
2. Lin tiene 3 monedas de veinticinco centavos, 1 de diez centavos y 1 de un centavo. ¿Cuál punto representa las monedas de Lin? Marca el punto.
3. Diego tiene 1 moneda de veinticinco centavos y 1 de diez centavos. Escribe las coordenadas del punto que representa las monedas de Diego. Explica o muestra cómo razonaste.
4. Clare tiene 5 monedas y no tiene monedas de veinticinco centavos. Escribe las coordenadas del punto que representa las monedas de Clare.
5. ¿Qué monedas podría tener Clare? Explica o muestra cómo razonaste.

### Student Response

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### Advancing Student Thinking

If students are not familiar with American coins or need support determining the total value of the coins, display only the problem stem, without revealing the question and ask, “¿Qué sabes sobre cada estudiante?” // “What do you know about each student?”

### Activity Synthesis

• “¿Cómo supieron cuál punto representa las monedas de Tyler?” // “How did you know which point represents Tyler’s coins?” (Tyler has 6 coins so I looked for the 6 below the horizontal line at the bottom of the graph. The value of the coins is 27 cents so I looked for a point between 20 and 30 in the vertical direction, closer to 30 than 20.)
• “¿Cómo supieron cuál punto representa las monedas de Clare?” // “How did you know which point represents Clare's coins?” (There are three points that represent 5 coins. Two of them have a vertical coordinate of more than 70. That can’t be Clare because she has no quarters. So Clare is the other point representing 5 coins.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy representamos problemas de la vida real y problemas matemáticos graficando e interpretando puntos del primer cuadrante de la cuadrícula de coordenadas” // “Today we represented real world and mathematical problems by graphing points in the first quadrant of the coordinate grid and interpreting the points.”

Display the image from the second activity.

“¿Cuál punto de la gráfica representa el número más pequeño de monedas? ¿Cómo lo saben?” // “Which point on the graph represents the smallest number of coins? How do you know?” (The point at the bottom right since it’ ;s just 1 coin. All the others represent more than one coin.)

“¿Qué moneda representa? ¿Cómo lo saben?” // “Which coin does it represent? How do you know?” (It’ s a nickel because it’ s less than 10 cents but more than 1 cent.)

“¿Cuál punto representa la mayor cantidad de dinero? ¿Cómo lo saben?” // “Which point represents the most money? How do you know?” (The one to the top right because it’ s almost 100 cents. Everything else is below 90.)

“¿Cuántas monedas están representadas por cada punto? ¿Cómo lo saben?” // “How many coins does that point represent? How do you know?” (9, because the horizontal coordinate is 9.)

## Cool-down: Monedas de medio dolar (5 minutes)

### Cool-Down

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