Lesson 1

Sumemos, restemos y multipliquemos fracciones

Warm-up: Conversación numérica: Fluidez y fracciones (10 minutes)

Narrative

This Number Talk encourages students to think flexibly about numbers to multiply. The understandings elicited here will be helpful throughout this unit as students build toward fluency with multiplying fractions and whole numbers.

Students use what they know about fractions and equivalent fractions to apply the properties of operations to find the products (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.

• $$5 \times \frac{10}{5}$$
• $$9 \times \frac{6}{3}$$
• $$8 \times \frac{11}{4}$$
• $$6 \times \frac{12}{10}$$

Activity Synthesis

• “¿En qué se diferencia la última expresión de las demás?” // “How is the last expression different from the others?” (Sample responses:
• The denominator in the fraction is not a factor of the whole number.
• The whole number is not a multiple of the denominator in the fraction.
• The value of the product is not a whole number.)
• “¿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 el problema de otra forma?” // “Did anyone approach the problem in a different way?”
• “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

Activity 1: ¡Hagamos turbantes africanos! (15 minutes)

Narrative

In this activity, students multiply fractions by whole numbers and compare fractions to solve problems (MP2). They make comparisons by reasoning about the denominator of fractions or about equivalence.

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.

Launch

• “Miren la foto de las dos mujeres que tienen turbantes africanos. ¿Qué observan? ¿Qué se preguntan?” // “Look at the picture of the two women with head wraps. What do you notice? What do you wonder?”
• Collect observations and questions from 1–2 students.
• “En muchas culturas africanas, las mujeres envuelven su cabello en telas coloridas como parte de su vestuario diario” // “In many African cultures, women wrap their hair with colorful fabric when they dress for the day.”
• “¿Han visto tradiciones similares a esta? ¿Cómo acostumbran ustedes vestirse a diario?” // “Have you seen a similar practice such as this one? What is your routine for dressing for the day?”
• Allow 1–2 students to share.
• “En esta actividad, vamos a pensar en la longitud de los turbantes africanos” // “We will be thinking about the length of head wraps in this activity.”

Activity

• 5 minutes: independent work time
• 5 minutes: partner work
• Monitor for diagrams and multiplication equations that represent each situation.

Student Facing

Jada y Lin vieron una foto de turbantes hechos de tela con estampados de cera africana y quieren hacer sus propios turbantes africanos.

1. Jada cose 5 pedazos de tela que miden cada uno $$\frac{2}{6}$$ de yarda. Escribe una ecuación que muestre la longitud total de tela que usó Jada.

2. Lin cose 3 pedazos de tela que miden cada uno $$\frac{2}{3}$$ de yarda. Escribe una ecuación que muestre la longitud total de tela que usó Lin.

3. ¿Quién usó más tela? Explica o muestra cómo razonaste.

Student Response

If students use repeated addition to find the product of a whole number and a fraction, consider asking:

• “¿Cómo encontraste la longitud de la tela?” // “How did you find the length of the fabric?”
• “¿De qué otra forma podrías haber encontrado la longitud?” // “What is another way you could have found the length?”
• “¿Cómo podrías encontrar la longitud de la tela con una multiplicación?” // “How could you find the length of the fabric using multiplication?”

Activity Synthesis

• Select 2–3 students to share their equations and reasoning.

Activity 2: Formemos 2 yardas de tela (10 minutes)

Narrative

The purpose of this activity is to practice adding and subtracting fractions. Students reason about different combinations of fractions to make 2. Students can find many combinations by looking for ways to add fractions that have the same denominator. Although not required by the standards for grade 4, the activity also invites students to use their understanding of equivalence to combine fractions with unlike denominators in preparation for grade 5. In the synthesis, emphasize the ways students used addition and subtraction and reasoned about equivalent fractions.

Representation: Develop Language and Symbols. Before showing the problem, activate or supply background knowledge. Tell students that it will be helpful to think flexibly about different ways to make the number 2. Offer the examples $$1\frac{1}{2} + \frac{1}{2}$$ and $$\frac{6}{3}$$. Invite students to write down at least three additional different ways to represent 2 and explain their reasoning to a partner.
Supports accessibility for: Conceptual Processing, Organization, Memory

• Groups of 2

Activity

• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
• use multiplication to show combining multiples of the same length
• use equations with addition and multiplication
• share how they thought of fractions that were equivalent to $$\frac{1}{2}$$, 1, or 2 when adding fractions

Student Facing

Las mamás de Jada y Lin le enseñaron a la clase de cuarto grado cómo combinar y usar pedazos de tela para formar turbantes africanos. Estas son las longitudes de los pedazos de tela.

$$\frac{2}{6}$$ de yarda
$$\frac{2}{6}$$ de yarda
$$\frac{2}{6}$$ de yarda

$$\frac{11}{10}$$ yardas

$$1\frac{2}{5}$$ yardas
$$\frac{9}{10}$$ de yarda
$$\frac{2}{6}$$ de yarda

$$\frac{6}{12}$$ de yarda

$$\frac{3}{6}$$ de yarda
$$\frac{2}{6}$$ de yarda
$$\frac{2}{6}$$ de yarda

$$\frac{12}{12}$$ yarda

$$\frac{2}{6}$$ de yarda
$$\frac{3}{5}$$ de yarda

$$\frac{2}{6}$$ de yarda

Encuentra combinaciones de tela que tengan una longitud de 2 yardas. Encuentra tantas combinaciones como puedas. Cada pedazo de tela se puede usar solamente una vez. Escribe una ecuación para cada combinación.

Student Response

If students find fewer than 4 combinations, consider asking:

• “Ya has encontrado algunas combinaciones. ¿Cómo puedes usar una de ellas como ayuda para pensar en otra combinación?” // “How might you use a combination you have already found to help you think of another?”
• “¿Qué sabes sobre las fracciones que te quedan?” // “What do you know about the fractions you have left?”
• “¿Cuánto necesitas sumarle a una fracción para formar 1 yarda?, ¿y para formar 2 yardas?” // “How much would you need to add to one fraction to make 1 yard? How much to make 2 yards?”

Activity Synthesis

• Invite 2–3 previously identified students to share their equations and their reasoning.
• “¿Cómo supieron cuándo su fracción era equivalente a 2?” // “How did you know when your fraction was equivalent to 2?” (I looked for ways to make 1 first. When the numerator is twice as big as the denominator, the fraction is equivalent to 2.).
• “¿Cómo podemos usar la multiplicación para representar la combinación  $$\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}$$?” // “Why can we use multiplication to represent the combination $$\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}+\frac{2}{6}$$?” (There are 6 groups of $$\frac{2}{6}$$ or $$6 \times \frac{2}{6}$$. Both expressions are equal to $$\frac{12}{6}$$ or 2.)

Activity 3: Sigue las reglas (10 minutes)

Narrative

The purpose of this activity is to practice adding and subtracting fractions. Students reason about different combinations of fractions, including fractions greater than 1, and the relationship between addition and subtraction. Students also reason about equivalent decimal fractions to add and subtract fractions with unlike denominators (MP7).

Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who use benchmarks such as $$\frac{1}{2}$$ and whole numbers to reason about how to add and subtract fractions when working with the decimal fractions to share in the synthesis.

Student Facing

1. Estas son cuatro fracciones:

$$\frac{15}{12}$$

$$\frac{7}{12}$$

$$\frac{21}{12}$$

$$\frac{18}{12}$$

1. ¿Cuál es la suma de todas las fracciones?
2. Escoge dos fracciones que tengan una diferencia menor que $$\frac{1}{3}$$. Muestra o explica tu razonamiento.
3. Escoge dos fracciones que tengan una suma mayor que 3. Muestra o explica tu razonamiento.
2. Estas son cuatro fracciones nuevas:

$$\frac{5}{12}$$

$$\frac{8}{12}$$

$$\frac{3}{12}$$

$$\frac{2}{12}$$

Úsalas de tal manera que al sumarlas o restarlas su valor sea 1. Sigue estas reglas:

• Usa sumas, restas o ambas.
• Usa las cuatro fracciones.
• Usa cada fracción solamente una vez.
3. Intenta obtener 1 nuevamente. Usa las mismas reglas y las siguientes fracciones.

$$\frac{15}{10}$$

$$\frac{13}{100}$$

$$\frac{53}{100}$$

$$\frac{9}{10}$$

Activity Synthesis

• Invite previously identified students to share how they used the decimal fractions to make 1.
• “¿Cómo les ayudó a estos estudiantes la equivalencia de fracciones a obtener un total de 1 a partir de estas fracciones?” // “How did these students use equivalence to help make a total of 1 using these fractions?” (They thought about ways to make all the fractions have the same denominator. They thought about how the fractions compared to 1 to decide how to add or subtract.)

Lesson Synthesis

Lesson Synthesis

“Hoy sumamos, restamos y multiplicamos fracciones para resolver problemas” // “Today we added, subtracted, and multiplied fractions to solve problems.”

“¿Por qué es importante entender la equivalencia de fracciones cuando se hacen operaciones con fracciones?” // “Why is it important to understand fraction equivalence while operating with fractions?” (Sometimes we will need to compare products, sums, and differences to whole numbers. If we understand when a fraction is equivalent to a whole number, we can determine which ones are greater or less than that number. We can also use benchmarks such as $$\frac{1}{2}$$ and $$\frac{1}{3}$$ to help us reason about our responses.)