Lesson 6

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying a fraction by a whole number. These understandings help students develop fluency and will be helpful later in the lesson when students explore the relationship between multiplication and division.

• 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.

• Display equation: \(\frac{15}{2} = 7 \frac{1}{2}\)
• “¿Cómo sabemos que esto es cierto?” // “How do we know this is true?” (They are both equal to \(15 \div 2\) or 7 is \(\frac{14}{2}\) so \(7 \frac{1}{2}\) is \(\frac{15}{2}\)).

The purpose of this activity is for students to apply what they learned in earlier lessons to represent a division situation. During the synthesis, students connect what they know about division to multiplication when they see that the situation of 2 people equally sharing the distance in a 3 mile race can also be described as each person running one-half of the three mile race. Students connect the language they used to describe the situation to multiplication and division expressions. This relationship between multiplication and division is the focus of the next several lessons.

This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, Writing.

• Display the image from the student workbook.
• “Cuéntenle a su compañero una historia sobre esta imagen” // “Tell a story about this image to your partner.”
• “En esta imagen se ven dos niños corriendo en una carrera de relevos. Están en el mismo equipo y ambos deben correr la misma distancia. El niño que lleva el testigo está terminando su turno. Cuando le pase el testigo a su compañera de equipo, ella tomará su turno” // “This image shows two children who are running in a relay race. They are on the same team. They each have to run the same distance. The boy holding the baton is finishing his turn. When he hands the baton to his teammate, his teammate will take her turn.”

• Groups of 2
• 2 minutes: independent think time
• 5–8 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for and collect the language students use to describe how to represent the situation and how far each person ran.
• Listen for:
  • 3 miles divided into 2 parts
  • half of 3 miles
  • one and one half miles
  • three halves 
• Record students’ words and phrases on a visual display and update it throughout the lesson. 

• “Estas son las palabras y las frases que ustedes usaron para describir la situación. ¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “These are the words and phrases that you used to describe the situation. Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Display: \(3 \div 2\), \(1\frac{1}{2}\), \(\frac{3}{2}\), and \(3\times \frac{1}{2}\)
• “¿Cómo representa la situación cada una de estas expresiones?” // “How does each of these expressions represent the situation?” (3 miles was divided into 2 equal parts, each person ran \(1\frac{1}{2}\) miles or \(\frac{3}{2}\) miles which can be thought of as \(3\times \frac{1}{2}\).
• Consider giving students time to record their answers in their journal.
• Display: “la mitad de 3 millas” // “one half of 3 miles”
• “Esta es otra forma de describir cuánto corrió cada persona” // “This is another way to describe how far each person ran.”
• Display: \(\frac{1}{2}\times3\)
• “También podemos usar esta expresión de multiplicación para representar la mitad de 3” // “We can also use this multiplication expression to represent one-half of 3.”
• “En la siguiente actividad, van a describir cómo distintos diagramas representan estas expresiones” // “In the next activity, you will describe how different diagrams represent each of these expressions.”

The purpose of this activity is for students to build on the previous activity to interpret diagrams as representing different expressions. Both diagrams in the activity represent solutions to the running situation in the previous activity. In this activity, the diagrams are interpreted without the running context. As students use the diagrams to interpret expressions, they begin to see relationships between the expressions \(\frac{3}{2}\), \(1\frac{1}{2}\), \(3 \times \frac{1}{2}\), \(\frac{1}{2}\times3\), and \(3 \div 2\). In this activity, students make sense of different ways to interpret a given diagram and relate the operations of division and multiplication (MP7).

• Groups of 2

• 5 minutes: independent work time
• 5–8 minutes: partner discussion
• Monitor for students who choose different diagrams to represent the same expression. 

If students do not explain where they see each expression in one of the diagrams, refer to the expressions and ask: “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “How are these expressions the same? How are they different?” Then, refer to one of the numbers in one of the expressions and ask students to describe how the diagrams represent the number.

• “¿En qué se parecen los diagramas?” // “What is the same about the diagrams?” (They both show 3 rectangles that are divided into 2 equal pieces.)
• “¿En qué son diferentes los diagramas?” // “What is different about the diagrams?” (The shaded pieces are next to each other in one of the diagrams.)
• Ask previously selected students to share their explanations.
• Display expression: \(3 \div 2\)
• “¿Cuál diagrama escogieron para la expresión \(3 \div 2\)? ¿Por qué?” // “Which diagram did you choose for the expression \(3 \div 2\)? Why?” (I chose diagram A because I see the 3 miles and the 2 equal parts. I chose diagram B because I can see the 2 equal parts that make 3 miles and can see that they make \(1 \frac{1}{2}\) miles.)
• Display expressions: \(3 \times \frac{1}{2}\), \(3 \div 2\), \(\frac{1}{2} \times 3\)
• “¿Cómo sabemos que todas las expresiones son iguales?” // “How do we know all the expressions are equal?” (Because the same diagram represents all of them or because they all equal \(\frac{3}{2}\) or \(1\frac{1}{2}\).)