Lesson 1

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). 

• 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
• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿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.)
• Consider asking:
  • “¿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?”

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.

• “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.”

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

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?”

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

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.

• Groups of 2

• 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

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?”

• 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.)

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).

• 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.

• 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.)