Lesson 4

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In this case, the given decimal pushes students to think in terms of increments of tenths (0.1) and to relate the fractional measurement to nearby whole numbers.

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Por qué puede que 1.8 metros sea un estimación muy baja?” // “Why might 1.8 meters be too low of an estimate?”
• “¿En qué parte de la imagen de la persona podríamos marcar una altura de 1 metro? ¿Y en la imagen del águila?” // “Where might a height of 1 meter be on the image of the person? On the image of the eagle?”
• Consider asking:
  • “¿Alguien hizo una estimación menor que 2? ¿Alguien hizo una estimación mayor que 3?” // “Is anyone’s estimate less than 2? Is anyone’s estimate greater than 3?”
  • “Teniendo en cuenta esta discusión, ¿alguien quiere ajustar sus estimaciones?” // “Based on this discussion does anyone want to revise their estimate?”

This activity prompts students to apply what they know about tenths and hundredths and decimal notation to arrange two sets of numbers in order, first from least to greatest, and then the other way around.

A number line is given here, but students are likely to start seeing its limits as a tool for comparing and ordering decimals. It takes time to plot each value on the number line, the scale of the number line accommodates only a small range of numbers (numbers like 1.25 and 12.05 would go beyond the line), and there are other ways to discern how two decimals compare—by reasoning about the name of the decimals in tenths and hundredths, and by relating to benchmarks such as whole numbers and 5 tenths (0.5, 1.5, 2.5, and so on).

• Groups of 2
• Display the six decimals in the first problem.
• “¿Cómo podemos describir estos decimales en términos de décimas y centésimas? Leamos cada uno en voz alta” // “How do we name these decimals in terms of tenths and hundredths? Let’s read each one aloud.”
• Display the six decimals in the second problem.
• “Con su compañero, tomen turnos para leer cada decimal. Descríbanlos en términos de décimas y centésimas” // “Take turns reading each decimal with your partner. Name them in terms of tenths and hundredths.”
• 1 minute: partner work time

• “Tómense unos minutos para completar la actividad en silencio. Luego, compartan sus respuestas con su compañero” // “Take a few quiet minutes to complete the activity. Then, share your responses with your partner.”
• 5 minutes: independent work time
• 2–3 minutes: partner work time
• Monitor for students who order the decimals by:
  • plotting the decimals on the number line
  • using and comparing the word names of the decimals
  • relating each decimal to benchmarks such as 0, 0.5, and 1
• Ask them to share their strategies during the synthesis, in the order as shown.

Students may arrange the numbers by looking only at the digits in the numbers, without attending to the relative sizes of each decimal. (For example, they may say that 0.45 is greater than 0.9 because 45 is greater than 9.) Consider asking them to name the numbers and think about them in terms of tenths and hundredths, or to express them in fraction notation.

• Select previously identified students to share their responses and reasoning.
• “Después de observar estas estrategias, ¿cuál o cuáles prefieren usar para ordenar decimales? ¿Por qué?” // “After seeing these strategies, which one(s) do you prefer to use for ordering decimals? Why?”

In this activity, students compare and order decimals in the context of running times. Unlike in preceding activities, in which most decimals they encountered were less than one or were in the low ones, here the numbers all have two-digit whole numbers, prompting students to be more attentive to the place value of the digits. The context of track and field may be unfamiliar, so time is built into the launch for orienting students and for supporting them in making sense of the problem.

When students look carefully at the meaning of each digit in the numbers and interpret them in terms of the running context they are reasoning abstractly and quantitatively and observing place value structure (MP2, MP7).

This activity uses MLR6 Three Reads. Advances: reading, listening, representing

• Groups of 2
• Display a picture of a standard 400-meter running track.
• “¿Cuánto tiempo creen que tardarían en correr una vuelta de aproximadamente 400 metros? Piénsenlo por un momento. Luego, compartan su estimación con su compañero” // “How long do you think it would take you to run a lap, about 400 meters? Think about it for a moment, and then share your estimate with your partner.”
• 1 minute: partner discussion
• Explain that in track and field, runners compete to run different distances: 100 meters, 200, 400, 800, and more. The United States, Jamaica, and the Bahamas have produced some of the fastest track runners in the world.

• Display only the opening paragraph, the eight running times, and the table, without revealing the questions.
• “Vamos a leer este problema 3 veces” // “We are going to read this problem 3 times.”
• 1st Read: “La tabla muestra a ocho de las mejores corredoras de la prueba de 400 metros para mujeres. Abajo se muestran sus mejores tiempos de carrera, que las ubican entre las 25 mejores del mundo en esta prueba” // “The table shows eight of the top runners in the Women’s 400-Meter event. Their best running times, listed here, put the runners in the world’s top 25 in this event.”
• “¿De qué se trata la historia?” // “What is this story about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• 2nd Read: Read the opening paragraph a second time.
• “Nombren las cantidades. ¿Qué cosas podemos contar o medir en esta situación?” // “Name the quantities. What can we count or measure in this situation?” (times in seconds, years)
• 30 seconds: quiet think time
• Share and record all quantities.
• Reveal the questions.
• 3rd Read: Read the entire problem aloud, including the questions.
• “¿Cómo podríamos identificar correctamente cuál es el tiempo de cada corredora?” // “How might we go about matching the times to the right runners?” (Arrange the times in order, from shortest to longest.)

• “Completen la actividad con su compañero” // “Work with your partner to complete the activity.”
• 6–8 minutes: partner work time

• Display the table from the activity.
• Invite students to share their ordered list and discuss how they went about arranging the numbers.
• Highlight explanations that are based on place-value reasoning or on understanding of tenths and hundredths.