# Lesson 2

Sumas y diferencias de fracciones

## Warm-up: Conversación numérica: Enteros y unidades (10 minutes)

### Narrative

This Number Talk encourages students to think about sums that make 10, 100, and one whole and look for ways to use these sums to mentally find the value of different expressions with whole numbers and fractions. The understandings elicited here will be helpful throughout this unit as students add and subtract whole numbers fluently and add and subtract fractions with the same denominator.

When students identify ways to make 1, 10, or 100, they look for and make use of the properties of operations and the structure of whole numbers and fractions (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.”

### Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$38 + 62$$
• $$38\frac{2}{6} + 62\frac{3}{6}$$
• $$38\frac{2}{6} + 62\frac{3}{6} + 17\frac{1}{6}$$
• $$138\frac{2}{6} + 162\frac{3}{6} + 17\frac{2}{6}$$

### Activity Synthesis

• “¿En qué se parecían estas expresiones? ¿En qué eran diferentes?” // “How were these expressions the same? How were they different?” (They each have $$38 + 62$$, so you can look for ways to make 100 in each. The last three expressions all have a mixed number with sixths. Some expressions did not have fractions. One expression had fractions that had a sum that was equivalent to 1 whole.)
• As needed, “En cada expresión, ¿cómo podemos buscar maneras de formar 100? ¿Cómo podemos buscar maneras de formar 1?” // “How could we look for ways to make 100 in each expression? How could we look for ways to make 1?”

## Activity 1: Pajillas para una montaña rusa (15 minutes)

### Narrative

The purpose of this activity is to represent and solve a measurement problem with fractions. Students may approach this activity in multiple ways and are invited to apply what they know about operations with fractions, comparing fractions, and fraction equivalence to make sense of and solve the problems (MP2). Throughout the activity, listen for the ways students use what they know about comparing fractions and fraction equivalence as they make sense of the problem. Although students may consider ways to subtract fractions with unlike denominators, this is not a requirement for grade 4. Focus the conversation during the activity and the synthesis on how students can solve the problem by reasoning about equivalent fractions and the representation they use to make sense of the problem.

MLR8 Discussion Supports. Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, next to each representation, write how the diagram relates to the situation.

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work
• Monitor for different representations of the situation (for example, equations, drawings, and number lines).

### Student Facing

En clase de Ciencias, Noah, Tyler y Jada construyen un modelo a escala de una montaña rusa con pajillas de papel de 1 pie de largo.

• Noah necesita un pedazo que mida $$\frac{7}{12}$$ de pie.
• Tyler necesita uno que mida $$\frac{1}{4}$$ de pie.
• Jada necesita uno que sea más corto que los otros dos.

Jada dice: “Podemos usar una sola pajilla para obtener todos los pedazos”.

1. Dibuja un diagrama que represente la situación. Explícale a tu compañero cómo le corresponde el diagrama a la situación. Después, encuentra la longitud del pedazo de pajilla que puede quedar para Jada.

2. ¿Noah usó más de $$\frac{1}{2}$$ pie o menos de $$\frac{1}{2}$$ pie de la pajilla? Explica o muestra cómo razonaste.
3. Tyler dice: “Si Jada usa un pedazo que mide $$\frac{1}{6}$$ de pie, sobrará un pedazo de pajilla que mide $$\frac{1}{12}$$ de pie”.

¿Estás de acuerdo o en desacuerdo con Tyler? Explica tu razonamiento.

### Student Response

If students create representations that do not match the quantities in the problem, consider asking:

• “¿Cómo se ve la longitud de la pajilla en tu representación? ¿Cómo se ven los pedazos de Noah y de Tyler?” // “How does your representation show the length of the straw? How does it show Noah and Tyler’s pieces?”
• “¿Cómo se ve la acción de la situación en tu representación?” // “How does your representation show the action in the situation?”

### Activity Synthesis

• Invite previously selected students to share their arguments and allow peers to extend and add ideas to the conversation.
• “¿De qué manera cada representación corresponde a esta situación de la pajilla?” // “How does each representation match this situation about the straw?” (They each show the total length of the straw and ways it could be broken into smaller parts. They each show the length of Noah and Tyler’s pieces. They show how long Jada’s piece could be.)

## Activity 2: ¿Lo suficientemente alto para montar? (20 minutes)

### Narrative

In this activity, students practice solving word problems that involve adding and subtracting mixed numbers. Students interpret fractions in the context of comparing heights and use what they know about decomposing whole numbers and equivalent fractions to make sense of and solve each problem (MP2). Look for the different ways students use what they know about the structure of whole numbers and fractions as they reason about how to solve each problem and share their thinking with others.

Action and Expression: Develop Expression and Communication. Provide access to a variety of tools, such as fraction strips and meter sticks (or if possible, tape measures).
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention

### Launch

• Groups of 2
• “¿Alguna vez han montado en una atracción de un parque de diversiones o de una feria? ¿Alguna vez no pudieron montar en una atracción porque eran muy jóvenes o porque no eran lo suficientemente altos?” // “Have you ever ridden a ride at an amusement park or a fair? Have you ever not been able to ride a ride because you were too young or not tall enough?”
• As needed, explain that some rides require the riders to be a certain height for their safety.
• “Resolvamos algunos problemas sobre estaturas y atracciones de un parque de diversiones” // “Let’s solve some problems about height and amusement park rides.”

### Activity

• 10 minutes: independent work time
• 5 minutes: partner discussion
• “Comparen su estrategia con la de su compañero” // “Compare your strategy with your partner’s strategy.”
• Monitor for students who use the relationship between addition and subtraction to represent the situations.

### Student Facing

Lin y sus compañeros de clase están de paseo en el parque de diversiones. Para poder montar en las atracciones del parque, los visitantes deben tener por lo menos cierta estatura. Usa la tabla para responder preguntas sobre la estatura de cuatro estudiantes.

atracción estatura requerida
1. Andre mide $$3\frac{3}{8}$$ pulgadas menos que la estatura requerida para montar en la montaña rusa. ¿Qué tan alto es Andre?
2. Lin mide $$\frac{18}{8}$$ pulgadas más que Andre. ¿Qué tan alta es Lin?
3. El año pasado, Elena medía $$1\frac{3}{4}$$ pulgadas menos que la estatura requerida para poder montar en los carros chocones. Desde entonces, ella ha crecido $$4\frac{1}{2}$$ pulgadas. ¿Qué tan alta era Elena el año pasado? ¿Qué tan alta es ahora?

4. Mai es lo suficientemente alta para montar en todas las atracciones este año. Mai medía $$51\frac{7}{8}$$ pulgadas el año pasado. ¿Al menos cuántas pulgadas creció Mai?

### Student Response

If students add or subtract quantities in each problem in ways that do not match the situation, consider asking:

• “¿Quién debe ser más alto: _____ o _____? ¿Cómo lo sabes?” // “Who should be taller _____ or _____? How do you know?”
• “¿Tu respuesta coincide con las estaturas de la situación? ¿Cómo lo sabes?” // “Does your answer match the heights in the situation? How do you know?”
• “¿Cómo puedes usar un diagrama como ayuda para representar las distintas estaturas que hay en este problema?” // “How could you use a diagram to help you represent the different heights in this problem?”

### Activity Synthesis

• Invite students to share their equations and explain their solution for Lin’s height.
• Display: $$50\frac{5}{8} + \frac{18}{8}$$
• “¿Cómo podrían pensar en una forma de encontrar el valor de la estatura de Lin?” // “How might you reason about how to find the value of Lin’s height?” (Think about adding $$\frac{3}{8}$$ to $$\frac{5}{8}$$ to make 51, then it'd be $$51\frac{15}{8}$$. We are adding more than 1 because $$\frac{18}{8}$$ is more than $$\frac{8}{8}$$.$$\frac{18}{8} = \frac{8}{8} + \frac{8}{8} + \frac{2}{8}$$ or $$2\frac{2}{8}$$, and $$50\frac{5}{8} + 2\frac{2}{8} = 52\frac{7}{8}$$)

($$50\frac{5}{8} + \frac{18}{8} = 50\frac{5}{8} + 2\frac{2}{8} = 52\frac{7}{8}$$)

• Select students to share both addition and subtraction equations for Mai’s current height and discuss how both equations ($$54 - 51\frac{7}{8} = 2\frac{1}{8}$$ and $$54 = 51\frac{7}{8} + 2\frac{1}{8}$$) could be used to represent this situation.

## Lesson Synthesis

### Lesson Synthesis

“Para resolver problemas hay que razonar. Hoy resolvimos problemas en los que había sumas y restas de fracciones y de números mixtos” // “Problem solving is about reasoning. Today we solved problems involving addition and subtraction of fractions and mixed numbers.”

“¿Qué usaron para darle sentido a los problemas? ¿Qué les ayudó a darle sentido a las estrategias que los demás compartieron?” // “What did you use to make sense of the problems? What helped you make sense of the strategies that others shared?” (The diagrams and representations helped me visualize the situation and make sense of the math in the problems. It helped to see others' equations and compare them to what I wrote.)

“¿Cómo les ayudó entender la equivalencia de fracciones a resolver los problemas?” // “How did understanding fraction equivalence help you solve the problems?” (It helped me write equations that made it easier to add or subtract. It helped me compare fractions and make sense of what the problems were asking.)