# Lesson 17

Comparemos fracciones

## Warm-up: Exploración de estimación: La longitud de una mariquita (10 minutes)

### Narrative

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In this warm-up, students apply what they know about fractions to estimate the length of an insect that is less than 1 inch.

### Launch

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

### Student Facing

¿Cuál es la longitud de esta mariquita?

Escribe una estimación que sea:

muy baja razonable muy alta
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Student Response

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### Activity Synthesis

• Consider asking:
• “¿La estimación de alguno es menor que _____? ¿La estimación de alguno es mayor que _____?” // “Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____?”
• “Teniendo en cuenta esta discusión, ¿alguien quiere ajustar su estimación?” // “Based on this discussion, does anyone want to revise their estimate?”

## Activity 1: Problemas de comparación (15 minutes)

### Narrative

The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and record the results of the comparisons with the symbols >, =, or < (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.

Students are likely to generate different comparison statements for the same situation. For example, they may write $$\frac{5}{8} > \frac{3}{8}$$ or $$\frac{3}{8}<\frac{5}{8}$$ to represent $$\frac{5}{8}$$ being the greater fraction. During synthesis, discuss how both statements capture the comparison and are valid.

MLR8 Discussion Supports. Synthesis: As students share the similarities and differences between the representations and comparison statements, use gestures to emphasize what is being described. For example, show with your fingers the partitions such as fourths or eighths that are the same in the representations that are being compared.
Advances: Listening, Representing
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share a connection between activity content and their own lives. Ask “¿Cómo puedo usar esto en mi propia vida?” // “How can I use this in my own life?”
Supports accessibility for: Conceptual Processing

### Launch

• Groups of 2
• “Usemos lo que aprendimos acerca de cómo comparar fracciones y cómo reconocer fracciones equivalentes para resolver problemas sobre longitudes” //  “Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”

### Activity

• “Resuelvan los problemas individualmente. En cada uno, asegúrense de mostrar cómo pensaron y de escribir una afirmación de comparación” // “Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
• 6–8 minutes: independent work time
• “Comparen sus respuestas y su razonamiento con su compañero” // “Share your responses and reasoning with your partner.”
• 2–3 minutes: partner discussion
• Monitor for:
• different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words
• different statements written for the same problem, such as $$4=\frac{12}{3}$$ and $$\frac{12}{3}=4$$, or $$\frac{2}{3} > \frac{2}{8}$$ and $$\frac{2}{8} < \frac{2}{3}$$

### Student Facing

En cada problema:

• Resuelve la pregunta y explica o muestra cómo razonaste.
• Representa tu respuesta con una afirmación en la que uses los símbolos >, <, o =.
1. Un escarabajo avanzó lentamente $$\frac{2}{8}$$ de la longitud de un tronco. Una oruga avanzó lentamente $$\frac{2}{3}$$ de la longitud del mismo tronco. ¿Cuál insecto avanzó más?

2. Un saltamontes tiene 4 centímetros de largo. Una oruga tiene $$\frac{12}{3}$$ centímetros de largo. ¿Cuál insecto es más largo?

3. Una mariquita avanzó lentamente $$\frac{3}{8}$$ de la longitud de una rama. Una hormiga avanzó lentamente $$\frac{5}{8}$$ de la longitud de la misma rama. ¿Cuál insecto avanzó más?

4. Un saltamontes saltó $$\frac{5}{8}$$ del ancho de la acera. Una rana saltó $$\frac{5}{6}$$ del ancho de la misma acera. ¿Cuál de los dos saltó una mayor distancia?

### Student Response

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### Activity Synthesis

• Select 2–3 students to share their reasoning strategies or representations for at least one of the situations.
• “¿Cómo usaron lo que han aprendido en lecciones anteriores para comparar las fracciones?” // “How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
• Select students who wrote different but equally valid comparison statements (for instance, $$\frac{2}{8}<\frac{2}{3}$$ and $$\frac{2}{3} > \frac{2}{8}$$) to share.
• Discuss how to read each statement and ask students whether both accurately represent the comparison.
• Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use.

## Activity 2: ¿Qué fracción tiene sentido? (15 minutes)

### Narrative

The purpose of this activity is for students to generalize what they have learned about comparing fractions to complete comparison statements and to generate new ones, using the symbols <, >, or =. Students first consider all numbers that could make an incomplete comparison statement true. Then, they find a fraction less than, greater than, and equivalent to a given fraction and write statements to record the comparisons. As in the previous activity, students see that there are different ways to record the same comparison of two numbers.

### Launch

• Groups of 2
• “Ahora que hemos practicado cómo comparar fracciones, inventemos fracciones que sean mayores o menores que una fracción dada o equivalentes a ella” // “Now that we have practiced comparing fractions, let's come up with fractions that are greater than, less than, or equivalent to a given fraction.”

### Activity

• “Noah estaba trabajando con fracciones cuando se le derramó un jugo. Ahora no logra saber cuáles eran algunos números. Ayúdenlo a averiguar qué estaba escrito antes de que se derramara el jugo” //  “Noah was working with fractions when some juice spilled. Now he can’t tell what some numbers were. Help him figure out what was written before the juice was spilled.”
• 5 minutes: partner work time
• Pause for a discussion and invite students to share the numbers that they think make sense in the first statement ($$\frac{2}{8}< \frac{?}{8}$$).
• Display or write the comparison statements using students’ numbers.
• “¿Todas estas afirmaciones tienen sentido? ¿Cómo lo saben?” // “Do all of these statements make sense? How do you know?”
• “¿Podríamos escribir más afirmaciones?” // “Are there any more statements that we could write?”
• If time permits, repeat with the next two parts.
• “Ahora trabajen individualmente en el último grupo de problemas” // “Now work independently on the last set of problems.”
• 5 minutes: independent work time

### Student Facing

1. ¡Oh, no! Se derramó jugo sobre las fracciones de Noah. Ayúdalo a averiguar qué estaba escrito antes de que el jugo se derramara.

En cada caso, encuentra todos los números que puedas que hacen que la afirmación sea verdadera. Explica o muestra tu razonamiento.

1.

2.

3.

2. En cada caso, encuentra una fracción que sea menor, una que sea mayor y una que sea equivalente a la fracción. Después, escribe una afirmación en la que uses los símbolos >, <, o = para dejar registro de cada comparación.

1. Menos de $$\frac{4}{6}$$: __________

Afirmación:

Más de $$\frac{4}{6}$$: __________

Afirmación:

Equivalente a $$\frac{4}{6}$$: __________

Afirmación:

2. Menos de $$\frac{3}{4}$$: __________

Afirmación:

Más de $$\frac{3}{4}$$: __________

Afirmación:

Equivalente a $$\frac{3}{4}$$: __________

Afirmación:

### Student Response

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### Advancing Student Thinking

If students don’t find more than one number that would make the statements in the first problem true, consider asking:

• “Ya encontraste un número que hace que la afirmación sea verdadera. ¿Cómo encontraste ese número y cómo supiste que hacía que la afirmación fuera verdadera?” // “You found one number that made the statement true. How did you find that number and know that it made the statement true?”
• “¿Cómo puedes usar una estrategia similar para encontrar otro número que haga que la afirmación sea verdadera?” // “How could you use a similar strategy to find another number that would make the statement true?”

### Activity Synthesis

• Invite students to share their responses to the last set of problems.
• “¿Cómo encontraron una fracción que fuera menor que la fracción dada (o mayor que la fracción dada o equivalente a ella)?” // “How did you find a fraction that was less than (or greater than or equivalent to) the given fraction?”

## Activity 3: Ubiquemos y marquemos (versión extrema) [OPTIONAL] (10 minutes)

### Narrative

The purpose of this activity is for students to use their knowledge of fractions to locate fractions with different denominators on the number line. Students may use a variety of reasoning to locate the fractions, including their knowledge of equivalence, strategies about the same numerator or denominator, or benchmark numbers they are familiar with. The synthesis focuses on the variety of strategies that make sense, and students should be encouraged to use different strategies for different fractions as needed.

Although students have represented fractions on number lines (including those with two different denominators, when reasoning about equivalence), this activity is optional because representing multiple fractions of different denominators on the same number line involves a deeper understanding than required by the standards.

• Groups of 2

### Activity

• “Individualmente, comiencen a ubicar estas fracciones en la recta numérica” // “Work independently to start placing these fractions on the number line.”
• 3–5 minutes: independent work time
• “Compartan sus estrategias con su compañero y ubiquen juntos las fracciones que les quedan” //  “Share your strategies with your partner and place any fractions you have left together.”
• 5–7 minutes: partner work time
• Monitor for students who:
• place each fraction separately by partitioning for that single fraction
• compare the fraction they are placing to others they’ve already placed
• use equivalent fractions
• use strategies about same numerators or denominators
• use benchmarks like whole numbers or halves

### Student Facing

Ubica y marca cada fracción en la recta numérica. Prepárate para compartir cómo razonaste.

$$\frac{1}{2},\frac{3}{8},\frac{13}{8},\frac{2}{4},\frac{3}{4},\frac{9}{8},\frac{5}{4},\frac{12}{6},\frac{5}{2},\frac{9}{3},\frac{20}{8}$$

### Student Response

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### Activity Synthesis

• Invite students to share a variety of strategies for placing fractions on the number line.
• Consider asking:
• “¿Cuáles fracciones fueron más fáciles de ubicar en la recta numérica?” // “Which fractions were easier to place on the number line?”
• “¿Cuáles fueron más difíciles de ubicar?” // “Which fractions were more difficult?”
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”

## Lesson Synthesis

### Lesson Synthesis

“Hemos comparado muchas fracciones diferentes. Fracciones que tenían el mismo denominador, fracciones que tenían el mismo numerador y, en esta lección, volvimos a ver fracciones que eran equivalentes” // “We have compared a lot of different fractions. Fractions with the same denominator, fractions with the same numerator, and in this lesson, we saw fractions that were equivalent again.”

“¿Cuáles serían algunas de las cosas más importantes que le dirían a un amigo que quisiera aprender a comparar dos fracciones?” //  “What do you think would be some of the most important things to tell a friend who wanted to learn about comparing two fractions?” (I would tell my friend to think about whether they can draw a representation, like a number line or a diagram to see which fraction is greater. I think they need to know whether the fractions have the same numerator or the same denominator. They can check to see if the fractions are the same size or are the same location because that means they are equivalent.)

Consider asking: “¿Su estrategia para comparar fracciones cambia dependiendo de las fracciones?” // “Does your strategy for comparing fractions change depending on the fractions?”

## Cool-down: Todo tipo de comparaciones (5 minutes)

### Cool-Down

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## Student Section Summary

### Student Facing

En esta sección, comparamos fracciones que tenían el mismo numerador o el mismo denominador, y usamos los símbolos >, =, o < para escribir nuestros resultados. Usamos diagramas y rectas numéricas para representar cómo pensábamos.

$$\frac{4}{6} < \frac{5}{6}$$

$$\frac{5}{6} > \frac{5}{8}$$