# Lesson 6

Problemas con grupos iguales de fracciones

## Warm-up: Verdadero o falso: Dos y tres factores (10 minutes)

### Narrative

The purpose of this True or False is to elicit strategies and understandings students have for finding products of a whole number and a fraction and identifying equivalent expressions. This work help students deepen their understanding of the properties of operations and will be helpful later when students solve problems with a whole number multiplied by a fraction.

In this activity, students have an opportunity to look for and make use of structure (MP7) as they consider how fractions are decomposed into various factors and multiplied in parts.

### 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 tu razonamiento.

• $$\frac{10}{12} = 5 \times \frac{2}{12}$$
• $$1 \times \frac{10}{12} = 5 \times \frac {2}{12}$$
• $$\frac{24}{4} = 6 \times 3 \times \frac{1}{4}$$
• $$12 \times 2 \times \frac{1}{4} = 8 \times 3 \times \frac{1}{4}$$

### Activity Synthesis

• “¿Qué estrategias usaron para decidir si las afirmaciones eran verdaderas o falsas?” // “What strategies did you use to determine if the statements were true or false?”

## Activity 1: Receta de pan de banano (15 minutes)

### Narrative

The purpose of this activity is for students to use their understanding of multiplication of a unit fraction by a whole number to solve problems. Students use what they know to find a product given the factors and find the factors when given the product. This reinforces the idea that any fraction $$\frac{a}{b}$$ is a multiple of $$\frac{1}{b}.$$

Students may interpret quantities greater than 1 as a combination of whole numbers and fractions (for example, $$\frac{4}{3}$$ cups as 1 whole cup and $$\frac{1}{3}$$ cup) or express them as mixed numbers (such as $$1\frac{1}{3}$$). Both are acceptable. If possible, ask students whether $$1\frac{1}{3}$$ and $$\frac{4}{3}$$ express the same amount, but it is not necessary to discuss the term mixed numbers at this point. (Students will be introduced to mixed numbers in upcoming lessons.)

### Launch

• Groups of 2
• “¿Alguna vez han usado una receta para preparar algo? ¿Qué hay en una receta?” // “Have you followed a recipe to make something before? What is in a recipe?” (A list of ingredients, amounts of each, and instructions for putting the ingredients together)
• “Si tienen una receta para 5 porciones o para 5 personas, pero necesitan preparar una mayor cantidad, ¿qué harían?” // “If a recipe is for 5 servings or 5 people, but you need more than that, what would you do?” (Adjust the amount of ingredients.)
• “En general, decimos que con las cantidades que aparecen en una receta preparamos ‘1 tanda’” // “We often refer to the amounts specified in a recipe as ’1 batch’.”
• “¿Qué significa preparar 2 tandas de una receta?” // “What might it mean to make 2 batches of a recipe?” (Make twice as much, or need twice as much ingredients)

### Activity

• “En silencio, trabajen unos minutos en la actividad. Luego, discutan con su compañero cómo pensaron” // “Take a few quiet minutes on work on the activity. Then, discuss your thinking with your partner.”
• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who discuss:
• that each quantity in Monday’s table will be multiplied by 2
• that each quantity in Tuesday's table need to be multiplied by 4 because the amount of butter tells us that the number of batches is 4

### Student Facing

En una panadería preparan pan de banano. Esta es la receta para preparar 1 tanda.

 Receta: 1 banano $$\frac{2}{3}$$ de taza de mantequilla $$\frac{3}{2}$$ cucharaditas de bicarbonato de sodio $$\frac{5}{8}$$ de taza de azúcar 2 huevos grandes $$\frac{5}{2}$$ tazas de harina común
1. El lunes prepararon 2 tandas de pan de banano en la panadería. Completa la tabla para mostrar la cantidad que se usó de cada ingrediente.

Pan de banano del lunes

bananos _______
mantequilla _______ taza(s)
azúcar _______ taza(s)
huevos _______
harina _______ taza(s)

2. El martes necesitaron $$\frac{8}{3}$$ tazas de mantequilla para hacer suficiente pan de banano para el día. ¿Cuántas tandas prepararon? Explica o muestra tu razonamiento.

 Receta: 1 banano $$\frac{2}{3}$$ de taza de mantequilla $$\frac{3}{2}$$ cucharaditas de bicarbonato de sodio $$\frac{5}{8}$$ de taza de azúcar 2 huevos grandes $$\frac{5}{2}$$ tazas de harina común
3. Teniendo en cuenta el número de tandas que prepararon el martes, completa la cantidad de cada ingrediente en la tabla.

Pan de banano del martes

bananos _______
mantequilla $$\frac{8}{3}$$ tazas
azúcar _______ taza(s)
huevos _______
harina _______ taza(s)

### Student Response

If students are unsure how to complete the last table, check if they recognize that the given $$\frac{8}{3}$$ cups represent the amount of butter for 4 batches. If so, ask: “¿Cuántos bananos se necesitan para preparar 4 tandas?” // “How many bananas are needed in 4 batches?” and “¿Cuántos huevos se necesitan?” // “How many eggs are needed?” “¿Cómo te puede ayudar esto a encontrar los ingredientes que se necesitan para preparar 1 tanda, 2 tandas y así sucesivamente?” // “How might you use this to help to determine the ingredients needed for 1 batch, 2 batches and so on?”

### Activity Synthesis

• Display the table for Monday and ask students to share responses. Record their responses for all to see.
• “¿Cómo cambia el numerador en las cantidades de todos los ingredientes?” // “How is the numerator changing in all of the ingredients?” (It is multiplied by 2 in each problem.)
• “¿Por qué el denominador es diferente en las cantidades de todos los ingredientes?” // “Why is the denominator different in all of them?” (A different unit and unit fraction was used to measure each ingredient.)
• Ask students to share the expressions for the ingredients in the table for Tuesday. Record each expression and its value as an equation:
• $$4 = 4 \times 1$$
• $$\frac{8}{3} = 4 \times \frac{2}{3}$$
• $$\frac{12}{2} = 4 \times \frac{3}{2}$$
• $$\frac{20}{8} = 4 \times \frac{5}{8}$$
• $$8 = 4 \times 2$$
• $$\frac{20}{2} = 4 \times \frac{5}{2}$$
• “¿Por qué hay dos cantidades de ingredientes que no están en forma de fracción?” // “Why are two of the ingredients not in fraction form?” (They have whole-number units.)

## Activity 2: ¿Cuánta leche se usó? (20 minutes)

### Narrative

In this activity, students are presented with descriptions of situations and equivalent multiplication expressions. They match each description to an expression that could represent the situation and see that more than one expression can be used, depending on how they interpret the situation. Likewise, students find that one expression can be used to represent different descriptions (MP2).

Students discuss their matching decisions, analyze how the expressions are related, and consider revising the matches they made if appropriate. When students discuss and justify their decisions they are creating viable arguments and critiquing one another’s reasoning (MP3).

MLR7 Compare and Connect. Synthesis: Lead a discussion comparing, contrasting, and connecting the different representations. Ask, “¿Cómo se ve la situación en la representación?” // “How does the situation show up in the representation?”, “¿Qué tienen en común todas estas representaciones?” // “What do each of these representations have in common?”, and “¿En qué son diferentes?” // “How were they different?”
Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for the group work in this activity. Record responses on a display and keep visible during the activity.
Supports accessibility for: Social-Emotional Functioning

### Required Materials

Materials to Gather

### Required Preparation

• Write the 5 expressions from the activity on separate posters and post them around the room:

$$4 \times (2\times \frac{1}{10})$$

$$4 \times \frac{2}{10}$$

$$8 \times \frac{1}{10}$$

$$2 \times (4 \times \frac{1}{10})$$

$$2 \times \frac{4}{10}$$

### Launch

• Groups of 2
• Read the first problem aloud to students.
• “Compartan con su compañero la expresión que escogieron” // “Share the expression you selected with a partner.”
• Students may select any of the expressions because each is equivalent to $$\frac{8}{10}$$. If this happens, ask, “¿Alguna expresión parece representar la situación mejor que las demás?” // “Does one expression seem to represent what is happening in the situation better than others?” ($$8 \times \frac{1}{10}$$)

### Activity

• 5 minutes: independent work time
• Ask students to stand with the poster showing the expression that they believe represents how much milk was used on Tuesday, and to discuss with others there why they chose this expression.
• Ask students to partner with a student from a different poster to explain why they made a different choice.
• “¿Alguien quiere reconsiderar lo que pensó sobre la expresión que escogió?” // “Does anyone wish to revise their thinking about the expression they selected?”
• “Expliquen por qué la nueva expresión que escogieron es mejor que la que tenían antes” // “Can you explain why you think that a different expression is a better choice now?”
• Repeat this process for each problem.

### Student Facing

En la panadería, además de pan de banano, también venden malteadas frescas. Cada malteada contiene $$\frac{1}{10}$$ de litro de leche.

Estas son cinco descripciones de las malteadas que se venden durante una semana y cinco expresiones que representan los litros de leche que se usan.

Empareja cada descripción con una expresión que la represente.

1. El lunes, en la panadería vendieron 8 malteadas. ¿Cuánta leche se usó?
2. El martes, dos clientes compraron 4 malteadas cada uno. ¿Cuánta leche se usó?
3. El miércoles, cuatro clientes compraron 2 malteadas cada uno. ¿Cuánta leche se usó?
4. El jueves, dos clientes compraron una malteada cada uno. Ese día, cada uno de ellos hizo el mismo pedido otras tres veces, para sus amigos. ¿Cuánta leche se usó?
5. El sábado, cuatro amigos compraron una malteada cada uno, para el desayuno. Después de la cena, volvieron y compraron lo mismo. ¿Cuánta leche se usó?

$$4 \times (2\times \frac{2}{10})$$

$$4 \times \frac{2}{10}$$

$$8 \times \frac{1}{10}$$

$$2 \times (4 \times \frac{1}{10})$$

$$2 \times \frac{4}{10}$$

### Student Response

If students do not see that each factor in the expressions can be interpreted in different ways, or that different expressions could represent the same quantity, invite them to use diagrams or record their thinking using diagrams to illustrate various groupings of the same quantity. Consider asking: “¿Dónde ves _____ grupos de _____?” // “Where do you see _____ (whole number) groups of _____ (fraction)?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Hoy asociamos expresiones con situaciones. Aprendimos que muchas expresiones pueden representar una misma situación” // “Today, we matched expressions to situations. We learned that several expressions can represent the same situation.”

Invite 1–2 students who chose different expressions for the same problem (one of the last two problems in the milkshake activity) to share. Record their ideas for all to see.

“¿Alguien puede explicar la forma en la que cada expresión le corresponde al problema?” // “Who can explain how each expression matches the problem?” (On Thursday, there were 4 separate orders of 1 serving each, or $$4 \times \frac{1}{10}$$, that were made by 2 people, or $$2 \times (4 \times \frac{1}{10})$$. This is also the same as $$2 \times \frac{4}{10}$$.)

“¿Observaron algo sobre las respuestas a los problemas?” // “Did you notice something about the answers to the problems?” (They are all the same. They are all $$\frac{8}{10}$$.)

“¿Por qué creen que todas son la misma?” // “Why do you think they are all the same?” (They all involve 8 groups of $$\frac{1}{10}$$.)

## Student Section Summary

### Student Facing

En esta sección, aprendimos a multiplicar un número entero por una fracción pensando en grupos de igual tamaño, como lo hicimos cuando multiplicamos dos números enteros.

Por ejemplo, podemos pensar en $$6 \times 4$$ como 6 grupos de 4. Un diagrama como este nos ayuda a mostrar que el producto es 24:

De la misma manera, podemos pensar en $$6 \times \frac{1}{4}$$ como 6 grupos de $$\frac{1}{4}$$. Los diagramas nos pueden ayudar a entender que el producto es $$\frac{6}{4}$$:

Después de estudiar patrones, vimos que cuando multiplicamos un número entero por una fracción, el número entero se multiplica únicamente por el numerador de la fracción y se deja el mismo denominador. Por ejemplo:

$$6 \times \frac{1}{2} = \frac{6}{2}$$

$$2 \times \frac{4}{5} = \frac{8}{5}$$

También aprendimos que:

• Todas las fracciones se pueden escribir como un producto de un número entero y una fracción unitaria. Por ejemplo, $$\frac{5}{4}$$ se puede escribir como $$5 \times \frac{1}{4}$$.
• Podemos escribir diferentes expresiones de multiplicación para representar la misma fracción. Por ejemplo, $$\frac{8}{3}$$ se puede escribir así:

$$8 \times \frac{1}{3}$$

$$4 \times 2 \times \frac{1}{3}$$

$$4 \times \frac{2}{3}$$

$$2 \times \frac{4}{3}$$