Lesson 13

The purpose of this warm-up is to elicit ideas that students have about rulers and measurements of \(\frac{1}{2}\), \(\frac{1}{4}\), and \(\frac{1}{8}\), which will be useful when students measure objects and generate and analyze line plots in a later activity. While students may notice and wonder many things about the images, focus the discussion on how to name the fractional measures in each image and the progression of the different levels of precision.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿Por qué la medida del crayón cambia en cada nueva imagen?” // “Why does the crayon measurement change each time?” (Sample responses: The crayon is not changing. The measurements on the ruler are becoming more precise. We’re using smaller and smaller pieces to measure the crayon.)
• “¿Qué creen que representan las marcas de cada regla?” // “What do you think the tick marks on each ruler represent?” (They represent inches, halves, fourths, and eighths of an inch.)
• “¿Qué cosas medirían con la primera regla? ¿Qué cosas medirían con la última regla? ¿Por qué?” // “What are some things that you would measure with the first ruler? What about things you would measure with the last ruler? Why might that be?” (Sample response: I would use the first ruler if the measurement doesn’t need to be exact and use the last ruler if it needs to be pretty exact.)

In this activity, students first measure colored pencils to the nearest \(\frac{1}{4}\) inch, collect a set of data in a table, and then plot them on a line plot. Then, they measure the colored pencils again, but this time to the nearest \(\frac{1}{8}\) inch. They plot their data on a new number line and attend to a greater level of precision as they do so. Students then reflect on the difference in the measuring process and in the measurements on the same set of pencils. Students attend to precision when they measure the pencils to the appropriate fractional unit (MP6).

• Each student needs a used colored pencil.

• Groups of 5
• Give a used colored pencil to each student.
• Prepare a number line on a poster and display it for students to see. Label the tick marks with whole numbers.
• “Si quisiéramos hacer un diagrama de puntos para mostrar medidas al \(\frac{1}{4}\) de pulgada más cercano, ¿qué más sería útil hacer?” // “If we wanted to make a line plot and show measurements to the nearest \(\frac{1}{4}\) inch, what else might we do that would be helpful?” (Partition the space between two consecutive whole numbers into 4 equal parts.)
• Partition the number line in increments of \(\frac{1}{4}\). 
• “¿Qué tal si quisiéramos mostrar medidas que incluyeran \(\frac{1}{8}\) de pulgada?” // “What if we wanted to show measurements that include \(\frac{1}{8}\) inch?” (Partition the space between two whole numbers into 8 equal parts.)

• “En grupo, midan los lápices de colores al \(\frac{1}{4}\) de pulgada más cercano. Anoten sus medidas en la primera tabla y después grafíquenlas en el primer diagrama de puntos” // “Work with your group to measure colored pencils to the nearest \(\frac{1}{4}\) inch. Record your measurements in the first table and then plot them on the first line plot.”
• 5–7 minutes: group work time
• “Ahora vuelvan a medir los lápices, pero esta vez midan al \(\frac{1}{8}\) de pulgada más cercano. Anoten sus medidas en la segunda tabla y después grafiquen los datos nuevos en el segundo diagrama de puntos” // “Now measure the pencils again, but this time to the nearest \(\frac{1}{8}\) inch. Record your measurements in the second table and then plot the new data on the second line plot.”
• 5–7 minutes: group work time

Students may identify the nearest inch and half inch but need support identifying the quarter or eighth inch. Consider asking students to think about what the halfway points between \(\frac{1}{2}\)-inch increments on the ruler represent, and then asking them again about the halfway points between  \(\frac{1}{4}\)-inch increments.

• Allow students to record their two sets of data on two different class line plots. (If dot stickers are available, consider using them—one sticker for each data point.)
• “¿Cómo cambiaron sus datos y sus diagramas de puntos cuando midieron los lápices de colores al \(\frac{1}{8}\) de pulgada más cercano?” // “How did your data and line plots change when you measured colored pencils to the nearest \(\frac{1}{8}\) inch?” (Sample responses:
  • We got different numbers.
  • The marks or points on the line plots are distributed differently. The points for some of the same pencils show up as different lengths in the second line plot.)
• “¿Qué es retador cuando medimos al \(\frac{1}{8}\) de pulgada más cercano?” // “What is challenging about measuring to the nearest \(\frac{1}{8}\) inch?” (The tick marks are smaller and harder to see on the ruler.)
• “¿Por qué piensan que medimos al \(\frac{1}{8}\) de pulgada más cercano?” // “Why do you think we measure to the nearest \(\frac{1}{8}\) inch?” (We measure to be more accurate.)
• “Veamos otros datos de longitudes que tienen medidas en medios, cuartos y octavos de pulgada” // “Let’s look at some other length data with measurements in halves, fourths and eighths of an inch.”

In this activity, students create a line plot using measurements to the nearest \(\frac{1}{4}\) and \(\frac{1}{8}\) inch. This task prompts students to use their understanding of fraction equivalence to plot and partition the horizontal axis.

• Groups of 2
• “En la tabla hay una lista de varias longitudes diferentes” // “The table lists many different lengths.”
• “¿Qué observan sobre las longitudes de los lápices?” // “What do you notice about the pencil lengths?” (Sample responses:
  • Some repeat more than one time.
  • The numbers are mixed numbers.
  • There are no whole numbers.)
• 1 minute: quiet think time
• “Hay algunas longitudes que son más comunes o que ocurren más veces que las demás” // “There are some lengths that are more common or occur more often than others.”
• “Díganle a su compañero cuál es la longitud más común” // “Tell a partner the length that is most common.” (\(6\frac{3}{4}\))

• Groups of 2
• 5 minutes: independent work time
• Monitor for students who use equivalence to plot measurements to the nearest eighth inch.
• “Compartan sus diagramas de puntos con su compañero y hagan los ajustes que necesiten” // “Share your line plots with your partner and make revisions as needed.”
• 2 minutes: partner discussion

If students do not yet identify “most common” length as the length with the most data points on the line plot, consider asking: “¿Cuál medida apareció más veces que las demás?” // “Which measurement occurred more often than others?” and “¿Cuál medida de los lápices se graficó más veces?” // “Which pencil measurement was plotted most often?”

• “¿Cuántos lápices de colores midieron en la clase de Andre?” // “How many colored pencils were measured in Andre’s class?” (10, because there were ten data points and each represented a pencil that was measured.)
• “¿Cuál fue la medida más común en el primer conjunto de datos?, ¿en el segundo conjunto de datos?” // “What was the most common measurement in the first set of data? In the second set of data?” (In the first set of data: \(6\frac{3}{4}\) inches. In the second set of data: \(6\frac{6}{8}\).)
• “¿Cómo les ayudaron las equivalencias a graficar los datos de las medidas en octavos de pulgada?” // “How did you use equivalence to help as you plotted measurement data in eighths of an inch?” (I know that two eighths are equivalent to one fourth and this helped to find eighths on the line plots.)
• Use the line plot image to clearly label \(\frac{1}{2}\), \(\frac{1}{4}\) and \(\frac{1}{8}\) with help from students.

In this activity, students continue to analyze line plot data and use the data to answer questions. Each data set involves lengths measured to the nearest \(\frac{1}{4}\) and \(\frac{1}{8}\) inch. As students organize and analyze data, they revisit ideas about fraction equivalence and addition and subtraction of fractions. When students relate the data to the context it represents and carefully interpret the elements of a graph, they reason abstractly and quantitatively and attend to precision (MP2, MP6). 

• Groups of 2
• “Observen el diagrama de puntos. Piensen en un par de cosas que sepan que son verdaderas sobre los lápices de colores de Noah. Básense en los datos que ven” // “Take a look at the line plot. Think of a couple of things that you know to be true about Noah’s colored pencils based on the data you see.”
• 1 minute: quiet think time
• Invite 2–3 students to share their responses.

• “Trabajen individualmente durante unos minutos antes de compartir sus ideas con su compañero” // “Take a few minutes to work on your own before sharing ideas with your partner.”
• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
  • decompose the mixed numbers to find the difference between  the longest and shortest points of data
  • recognize and label eighths on the number line as the halfway points between consecutive fourths

• Select students to share strategies for finding the difference between the longest and shortest lengths? 
• Make connections between strategies, being sure to emphasize strategies that involve decomposing fractions in different ways. 
• Consider asking: “¿Qué tienen en común estas estrategias?” // “What are the similarities between these strategies?”