Lesson 8

The purpose of this Number Talk is to for students to demonstrate strategies and understandings they have for multiplying fractions. These understandings help students develop fluency and will be helpful later in this lesson when students solve problems involving fraction multiplication.

• 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.

• “¿Usaron la misma estrategia para encontrar el producto de la primera y de la última expresión? ¿Por qué sí o por qué no?” // “Did you use the same strategy to find the product of the first and last expressions? Why or why not?” (For the first one, it was easy to think about what \(\frac{1}{3}\) of \(\frac{3}{5}\) was. For the last one, I needed to multiply the numerators and then the denominators because it wasn’t easy for me to picture in my head.)

The purpose of this activity is for students to calculate areas in context. Students are not offered an area diagram but rather an image of a flag. Students may label the image of the flag with measurements or make their own area diagram. The lengths are not always presented in fraction form so students may rewrite them as fractions before calculating areas or they may use the distributive property of multiplication and multiply the whole number and fractional parts separately before adding them. Students reason abstractly and quantitatively when they interpret the given information about the flags and make calculations to solve problems (MP2).

• Display image of flags from student workbook.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (There are lots of different flags, some of the colors are the same, there are a lot of stripes, these flags are from 1968, have any of the flags changed since then?)
• “Estas son banderas de diferentes países. Vamos a resolver algunos problemas acerca de banderas” // “These are flags from different countries. We are going to solve some problems about flags.”
• Groups of 2

• 2 minutes: quiet think time
• 8–10 minutes: partner work time
• Monitor for students who:
  • notice that the blue stripe is twice as wide as the red stripe and use the area of the red stripe to find the area of the blue stripe
  • find the area of the full flag by converting \(7 \frac{1}{2}\) to a fraction
  • use the distributive property

• Invite previously selected students to share their solutions.
• Display equation: \(5 \times 7\frac{1}{2} = (5\times 7) + (5 \times \frac{1}{2})\)
• “¿De qué manera la ecuación representa el área de la bandera?” // “How does the equation represent the area of the flag?” (I can divide the flag into two pieces, one of them 7 inches long, and the other \(\frac{1}{2}\) inch long, and then add them to get the area of the flag.)
• Invite students to share responses for the area of the red stripe.
• “¿Cómo puedo usar el área de la franja roja para encontrar el área de la franja azul?” // “How can I use the area of the red stripe to find the area of the blue stripe?” (The blue stripe is twice as wide so its area is twice as much.)

The goal of this activity is to examine calculations with measurements of a flag and try to figure out what question the calculations answer. The answers include units and this can serve as a guide to students. Since the first calculation has an answer in inches, the question it answers must ask for a length. Since the second calculation has an answer in square inches, the question it answers must ask for an area. This is an important step in solving the problems as students can then look at the diagram and the measurements and decide what the question could be.

One important part of the modeling cycle (MP4) is interpreting information. That information may be presented in words or graphs or with mathematical symbols. In this case, students interpret equations in light of given numerical relationships and diagrams.

• Display image of the flag of Colombia from student workbook.
• “Esta es la bandera de Colombia. Esta representa la independencia de España, el 20 de julio de 1810” // “This is the flag of Colombia. It represents independence from Spain on July 20, 1810.”
• “¿Qué observan acerca del tamaño de las franjas?” // “What do you notice about the size of each stripe?” (The yellow stripe is about twice the size of each of the blue and red stripes.)
• Display the flag and information about the flag from the activity.
• “Un estudiante estaba respondiendo una pregunta sobre esta bandera y escribió \(\frac{1}{2} \times 3 \frac{1}{2} = 1 \frac{3}{4}\)” // “A student was answering a question about this flag and wrote \(\frac{1}{2} \times 3 \frac{1}{2} = 1 \frac{3}{4}\).”
• “¿Qué pregunta piensan que el estudiante estaba respondiendo?” // “What question do you think the student is answering?” (What is the width of the yellow rectangle?)
• “Van a resolver más problemas como este, en los que les dan la respuesta y ustedes tienen que escribir la pregunta” // “You are going to solve more problems like this one where you are given the answer and you have to write the question.”

• 4–5 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who use the units of the answers to help guide them to finding an appropriate question.

• Invite students to share their responses for the first question.
• “¿Cómo supieron que la pregunta era sobre la longitud?” // “How did you know the question was about length?” (The answer is \(\frac{7}{8}\) inch, which is the measurement of length. So the question had to be about length.)
• “¿Cómo decidieron cuál longitud?” // “How did you decide which length?” (The calculation took a quarter of the flag width. Since the red and blue stripes are each \(\frac{1}{4}\) of the width, the calculation could be for their width.)
• Invite students to share their responses for the second question.
• “¿Cómo supieron que la pregunta era sobre el área?” // “How did you know the question was about area?” (The answer is a measurement in square inches so that’s the area of something.)