# Lesson 14

Problemas de comparación de fracciones

## Warm-up: Conversación numérica: Múltiplos de diez (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for adding and subtracting multi-digit numbers. These understandings help students develop fluency and will be helpful in later units as students will need to be able to add and subtract multi-digit numbers fluently using the standard algorithm.

When students make adjustments and create multiples of ten for mental addition they are looking for and making use of the base ten structure of numbers (MP7).

### Launch

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$119 + 119$$
• $$139 + 139$$
• $$159 + 159$$
• $$199 + 199$$

### Activity Synthesis

How did you use multiples of ten, for example 20, 40, and 60 to help add these numbers mentally? (I changed the addends by adding one more to each addend and the subtracting the extra two from the final sum.)

• “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
• “¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
• “¿Alguien quiere agregar algo a la estrategia de ______?” // “Does anyone want to add on to _____’s strategy?”

## Activity 1: Fracciones desconocidas (20 minutes)

### Narrative

In this activity, students are given several sets of fractions and some clues about the size of a particular fraction in each set. To identify a fraction that meets certain size requirements or falls within a specified range, students need to use multiple comparison strategies they have learned. For example, they can use comparisons to benchmarks such as $$\frac{1}{2}$$ and 1 to eliminate some fractions, and then use equivalent fractions to compare the remaining ones.

### Launch

• Groups of 3–4
• “Hay seis conjuntos de fracciones en esta actividad. Cada conjunto viene con unas pistas. Su tarea en cada conjunto es encontrar una fracción que cumpla con las tres pistas” // “There are six sets of fractions in the activity. Each set comes with some clues. Your task is to find one fraction that meets all three clues in each set.”

### Activity

• “Trabajen en grupo para encontrar las fracciones desconocidas” // “Work with your group to find the mystery fractions.”
• “Cada miembro del grupo debe empezar con un conjunto de fracciones distinto y debe encontrar las fracción desconocida de al menos dos conjuntos antes de discutir sus respuestas” // “Each group member should start with a different set, and should find at least two mystery fractions before discussing their responses with the group.“
• 5–6 minutes: independent work time
• 7–8 minutes: group work time

### Student Facing

A cada uno de seis amigos se les dio una lista con 5 fracciones. Cada uno eligió en secreto una fracción y escribió pistas sobre su elección. Usa sus pistas para identificar las fracciones que eligió cada uno.
 Andre: $$\ \frac{8}{12} \quad \frac{3}{6} \quad \frac{3}{4} \quad \frac{3}{2} \quad \frac{2}{12}$$ menor que 1 mayor que ​​​​$$\frac{1}{3}$$ menor que $$\frac{2}{3}$$
 Tyler: $$\ \ \frac{2}{6} \quad \frac{2}{2} \quad \frac{2}{4} \quad \frac{2}{3} \quad \frac{2}{5} \quad$$ mayor que $$\frac{1}{3}$$ menor que 1 menor que $$\frac{1}{2}$$
 Clare: $$\ \ \frac{4}{3} \quad \frac{4}{2} \quad \frac{3}{4} \quad \frac{1}{4} \quad \frac{2}{10} \ \$$ mayor que $$\frac{2}{8}$$ menor que $$\frac{11}{6}$$ mayor que 1
 Diego: $$\ \frac{2}{8} \quad \frac{6}{12} \quad \frac{6}{8} \quad \frac{12}{10} \quad \frac{11}{12}$$ mayor que $$\frac{1}{2}$$ menor que 1 mayor que​​ $$\frac{3}{4}$$
 Elena: $$\ \frac{2}{12} \quad \frac{50}{100} \quad \frac{4}{10} \quad \frac{3}{5} \quad \frac{7}{5}$$ mayor que $$\frac{2}{10}$$ menor que 1 mayor que $$\frac{3}{6}$$
 Noah: $$\ \frac{18}{10} \quad \frac{7}{8} \quad \frac{2}{5} \quad \frac{18}{5} \quad \frac{150}{100}$$ mayor que $$\frac{1}{2}$$ menor que $$\frac{25}{10}$$ mayor que $$\frac{8}{5}$$

### Activity Synthesis

• “¿Cuáles pistas les ayudaron a eliminar fracciones más rápido?” // “Which clues helped you eliminate fractions the fastest?” (clues about size relative to 1)
• “¿Qué estrategias usaron para comparar fracciones?” // “What strategies did you use to compare fractions?” (compare fractions to $$\frac{1}{2}$$, 1, or another benchmark, write equivalent fractions to compare two fractions, compare fractions with the same numerator or denominator)
• “¿En algún caso tuvieron que usar más de una estrategia para comparar fracciones?” // “Did you ever have to use more than one strategy to compare fractions?” (Yes, two or three were often needed to find the mystery fraction.)

## Activity 2: Distancias a pie (10 minutes)

### Narrative

This activity has two purposes: to give students an opportunity to solve fraction comparison problems in context, and to reinforce the idea that two fractions can be compared only if they refer to the same whole. To serve the former, students compare fractional distance measurements. To serve the latter, they investigate fractional measurements in two different units of distance: Chinese “li” and kilometer.

When comparing the distances in the first question, students can rely on a number of familiar strategies. Two of the fractional values are close to 2. Some students are likely to use that benchmark for efficient comparison. For example, they may note that the school and the market are both a little over 1 li from home, the library is more than 2 li, and the badminton club is a little under 2 li.

Focus the synthesis on the last two questions about interpreting fractional measurements in two different units.

MLR8 Discussion Supports. Synthesis: Display sentence frames to support whole-class discussion: “Estoy de acuerdo porque . . .” // “I agree because . . .” and “No estoy de acuerdo porque . . .” // “I disagree because . . . .”
Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Share information about the unit “li” and help students understand the size of one li by referencing a common point of interest. Invite students to share this knowledge with family members and other teachers.
Supports accessibility for: Visual-Spatial Processing, Social-Emotional Functioning

### Launch

• Groups of 3–4
• “¿Cuáles son algunas unidades que usamos para medir distancia? Digamos todas las que se nos ocurran” // “What are some units that we use for measuring distance? Let’s name as many as we can think of.” (Sample responses: inches, feet, miles, meters, kilometers)
• Share and record responses.
• “Hoy vamos a pensar en distancias medidas en ‘li’, una unidad que se usa con frecuencia en China” // “Today we’ll look at distances measured in ‘li,’ a unit commonly used in China.”

### Activity

• “Tómense unos minutos para trabajar en silencio en las preguntas. Prepárense para explicar su razonamiento” // “Take a few quiet minutes to work on the questions. Be prepared to explain your reasoning.”
• “Después, discutan sus respuestas con su grupo y trabajen juntos para terminar la actividad” // “Afterward, discuss your responses with your group and work together to complete the activity.”
• 4 minutes: independent work time
• 4 minutes: group work time
• Monitor for students who:
• attend to the location of 1 whole when representing $$\frac{4}{5}$$ and $$\frac{7}{5}$$ on the two number lines (rather than partitioning both number lines into the same 5 parts)
• recognize that $$\frac{4}{5}$$ and $$\frac{7}{5}$$ in the two cases refer to different wholes and can articulate their reasoning

### Student Facing

En China y en algunos países del este de Asia se usa la unidad “li” para medir distancias.

Estas son las distancias que camina un estudiante en China entre su casa y algunos de los lugares que visita con frecuencia.

• escuela: $$\frac{7}{5}$$ li
• biblioteca: $$\frac{23}{10}$$ li
• mercado: $$\frac{7}{4}$$ li
• club de bádminton: $$\frac{23}{12}$$ li
1. Cuál queda a menor distancia desde la casa del estudiante:

1. ¿Su escuela o la biblioteca?
2. ¿El mercado o el club de bádminton?
3. ¿La biblioteca o el mercado?
2. Un estudiante en los Estados Unidos camina $$\frac{4}{5}$$ kilómetros (km) de la casa a la escuela. Estas rectas numéricas muestran cómo se relaciona 1 kilómetro con 1 li.

¿Cuál estudiante camina una mayor distancia a la escuela? Usa las rectas numéricas para mostrar tu razonamiento.

3. Explica por qué no podemos simplemente comparar las fracciones $$\frac{4}{5}$$ y $$\frac{7}{5}$$ para ver cuál estudiante camina una mayor distancia.

### Activity Synthesis

• Ask previously selected students to share their responses to the last two questions. Display their number lines, or display the number lines from the activity for them to annotate while they explain.
• Emphasize that just as 1 km is not the same distance as 1 li, $$\frac{4}{5}$$ km is not the same distance as $$\frac{4}{5}$$ li. We can’t compare two fractions that refer to different wholes.

## Lesson Synthesis

### Lesson Synthesis

“Hoy combinamos varias estrategias que nos ayudaron a comparar fracciones. También resolvimos problemas de comparación de fracciones en una situación sobre distancias” // “Today we used a combination of strategies to help us compare fractions. We also solved fraction comparison problems in a situation about distance.”

Keep students in groups of 3–4. Give tools for creating a visual display to each group.

Assign each group one set of fractions in Activity 1 or the first set of questions in Activity 2.

• For the former: “Hagan una presentación visual que explique cómo encontraron la fracción desconocida para el conjunto de fracciones que les asignaron en la actividad 1. Su presentación debe incluir las cinco fracciones, las tres pistas y una explicación de cómo la fracción elegida cumple con todas las pistas” // “Create a visual display that explains how you found the mystery fraction for your assigned set of fractions from Activity 1. Your display should list the five fractions, the three clues, and how the chosen fraction satisfies all the clues.”
• For the latter: “Hagan un presentación visual que muestre sus respuestas al primer grupo de preguntas de la actividad 2. Su presentación debe mostrar las cuatro distancias y cómo las compararon” // “Create a visual display that shows your responses to the first set of questions in Activity 2. Your display should show the four walking distances and how you compared them.”

“Incluyan diagramas, notas y cualquier descripción que pueda ayudarle a los demás a entender cómo pensaron” // “Include diagrams, notes, and any descriptions that might help others understand your thinking.”

Ask students to display their work around the room.

“Visiten las presentaciones de al menos otros 2 grupos” // “Visit the display of at least 2 other groups.”

“En cada presentación, revisen si las estrategias de razonamiento tienen sentido para ustedes. Piensen en qué se parece y en qué es diferente el razonamiento en las presentaciones” // “At each display, check to see if the reasoning strategies make sense to you. Think about how the reasoning in different displays is alike and how it is different.”

“¿En qué se parecen los diagramas, explicaciones o cálculos que vieron? ¿En qué son diferentes?” // “How are the diagrams, explanations, or calculations that you saw alike? How are they different?”

Share and record responses. Reference the displays that students created to show similarities and differences in their reasoning strategies.