Lesson 13

The purpose of this True or False is to activate what students know about multiplying a fraction by a whole number (\(n \times \frac{a}{b}\), in particular fractions with denominators 4, 8, and 12) and about fractions that are equivalent to whole numbers. The reasoning students do here will be helpful later when students solve problems involving fractional units of measurement in pounds, ounces, hours, and minutes.

The whole numbers and the denominators in the equations are multiples or factors of one another, so students have an opportunity to look for and make use of structure (MP7) to determine whether the equations are true.

• 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

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

• “¿Pueden saber si una ecuación es verdadera o no solo mirando los tamaños de los números enteros y las fracciones, sin realizar ningún cálculo? Por ejemplo, sin multiplicar \(60 \times \frac{1}{12}\), ¿podemos decir que \(60 \times \frac{1}{12}\) no puede ser 10? ¿Cómo?” // “Can you tell whether an equation is true by looking at the sizes of the whole numbers and fractions, without performing computation? For instance, without multiplying \(60 \times \frac{1}{12}\), can we say that \(60 \times \frac{1}{12}\) can’t be 10? How?” (Yes, we know that 12 groups of \(\frac{1}{12}\) make 1, which means 120 groups of \(\frac{1}{12}\), not 60 groups, are needed to make 10.)

The purpose of this activity is to introduce students to the structure of the MLR4 Information Gap routine. This routine facilitates meaningful interactions by positioning some students as holders of information that is needed by other students.

Tell students that first, a demonstration will be conducted with the whole class, in which they are playing the role of the person with the problem card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question.

For each question that is asked, students are expected to explain what they will do with the information, by responding to the question, “¿Por qué necesitas saber _____?” // “Why do you need to know _____ [that piece of information]?” If the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, “No tengo esa información” // “I don’t have that information.”

Once the students have enough information to solve the problem, they solve the problem independently.

The info gap routine requires students to make sense of problems by determining what information is necessary and then ask for information they need to solve them. This may take several rounds of discussion if their first requests do not yield the information they need (MP1).

• Groups of 2

MLR4 Information Gap

• Display problem card, as shown in the activity statement.
• Read the problem aloud. 
• Listen for and clarify any questions about the context. 
• “Hace falta parte de la información que necesitan para resolver este problema y yo la tengo aquí. ¿Qué información específica necesitan?” //  “Some of the information you need to solve this problem is missing, and I have it here. What specific information do you need?”
• 1–2 minutes: quiet think time
• “Con su compañero, decidan qué información necesitan para resolver el problema y hagan una lista de preguntas que pueden hacer para averiguarla” // “With your partner, decide what information you need to solve the problem, and create a list of questions you can ask to find out.”
• 2–3 minutes: partner discussion
• Invite students to share 1 question at a time. 
• Record each question on a display, and respond with, “¿Por qué necesitan saber _____?” // “Why do you need to know _____ [the information requested]?” Students should provide a justification for how they will use the information before the information is revealed. 
• Answer questions using only information on the data card in the narrative (do not reveal).
• Record information that is shared on the display. Give students time to decide whether they have enough information to solve the problem.
• Repeat until students decide they have enough information to solve. 
• 2–4 minutes: independent work time 

• Invite 1–2 students to share how they solved the problem.
• “¿Qué cantidades era importante saber para resolver este problema?” // “What were the important quantities to know to solve this problem?”
• “¿Cuáles preguntas de las que hicieron les ayudaron a encontrar esas cantidades?” // “Which questions that you asked helped you find out those quantities?”

This Info Gap activity prompts students to compare lengths of time given in different units. To make comparisons, students need to convert one unit into another or otherwise reason about equivalent amounts. They also need to relate quantities in multiplicative terms—to think of a quantity as a certain number of times as much as another quantity.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6). Here is an image of the cards for reference:

• Create a set of cards from the blackline master for each group of 2. 

• Groups of 2 

MLR4 Information Gap

• Display the task statement, which shows a diagram of the Info Gap structure.
• 1–2 minutes: quiet think time
• Read the steps of the routine aloud. 
• “Les voy a dar una tarjeta de problema o una tarjeta de datos. Lean su tarjeta en silencio. No se la muestren ni se la lean a su compañero” // “I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.” 
• Distribute the cards.
• 1–2 minutes: quiet think time
• Remind students that after the person with the problem card asks for a piece of information, the person with the data card should respond with, “¿Por qué necesitas saber _____?” // “Why do you need to know _____ [the information requested]?”

• 3–5 minutes: partner work time
• After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.

• “¿Qué cantidades era importante saber para resolver el primer problema? ¿Y para resolver el segundo problema?” // “What were the important quantities to know to solve the first problem? What about in the second problem?”
• “¿Alguien resolvió el problema de una forma diferente a la de su compañero?” // “Did anyone solve the problem in a different way than their partner?”
• “¿Cómo compararon 9 horas con 90 minutos?” // “How did you compare 9 hours and 90 minutes?”
• “¿Cómo averiguaron si Noah pasa más o menos de 4 horas con su familia en un día del fin de semana?” // “How did you find out if Noah spends more than or less than 4 hours with his family on a weekend day?”

This optional activity invites students to apply their knowledge of pounds and ounces and multiplicative reasoning to solve a puzzle about the quantities of ingredients on a shopping list. To solve the puzzle, students need to express pounds as ounces and reason deductively.

As they work to eliminate possibilities, draw conclusions, and explain their thinking to others, students practice constructing logical arguments (MP3).

• Groups of 2–4
• Read aloud the opening paragraph and the list of ingredients. Invite students to ask any clarifying questions they might have about what was just read.
• Ask students to take turns reading each of the clues and clarify any terms or statements, if needed.

• “Tómense unos minutos en silencio para leer las pistas de nuevo y trabajar en el acertijo. Después, discutan con su grupo cómo pensaron” // “Take a few quiet minutes to read the clues again and to work on the puzzle. Then, discuss your thinking with your group.”
• 5–7 minutes: independent work time
• 5 minutes: group discussion

• Discuss the order in which students completed the missing values. Ask questions such as:
  • “Cuando averiguaban la cantidad de cada ingrediente, ¿por cuál empezaron? ¿Hubo alguna razón por la cual empezaron con ese ingrediente?” // “Which was the first ingredient whose amount you figured out? Was there a reason you started with that item?” (Rice noodles, because it is lightest and \(\frac{1}{2}\) pound is very light.)
  • “¿De cuál ingrediente averiguaron su cantidad después?” // “Which ingredient and amount did you figure out next?” (Brown rice, because it is 20 times the weight of rice noodles.)
  • “¿Hubo algún momento en el que vieron varias posibilidades? ¿Cómo decidieron qué hacer?” // “Was there a point at which you saw multiple possibilities? How did you decide what to do?” (Any of the items could have been \(\frac{1}{2}\) pound, but if the heaviest amount was \(\frac{1}{2}\) pound, the other ingredients would be very small and not make much sense.)
  • “¿Cómo averiguaron el peso del artículo más pesado? ¿Cómo supieron cuál era?” // “How did you find the weight of the heaviest item? How did you know which one it was?” (I found \(20 \times \frac{1}{2}\).)