Lesson 8

The purpose of this Estimation Exploration is for students to use their experience with the number line, decimals, and fractions to estimate the value of a number located on the number line. Students may answer with a fraction but are likely to write a decimal since they have been working with decimals for the last several lessons. In the synthesis, students reflect on how having tick marks for each tenth would help improve their estimate.

• 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.

• “¿Tener una marca para cada décima en la recta numérica les ayudaría a mejorar su estimación? ¿Por qué sí o por qué no?” // “Would having tick marks for each tenth on the number line help improve your estimate? Why or why not?” (Yes, because I could see if it is one of the tenths and if not, where it is relative to those numbers.)

The purpose of this activity is for students to use place value understanding to accurately label number lines and then estimate the value of a labeled point. When they label the tick marks students will use their knowledge that a tenth is a tenth of one and a hundredth is a tenth of a tenth. When they estimate the value of the labeled point, students will also use their understanding that there are ten thousandths in each hundredth. This gives students an opportunity to make sense of each quantity and place it accurately on the number line (MP2).

The activity begins with a group discussion about how Jada labeled a number on the number line. This prepares students for the work of the activity by:

• highlighting how a decimal with a digit in the thousandths place is located between two decimals to the hundredths on the number line
• highlighting that the digits in a number of thousandths give information about which hundredths the decimal is between

As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

• Groups of 2
• Display the first image from workbook.
• “Jada ubica 15.53 en la recta numérica. ¿Creen que Jada ubicó el número con precisión? Expliquen su razonamiento” // “Jada locates 15.53 on the number line. Do you think Jada accurately located the number? Explain your reasoning.” (She placed 15.53 between 15.5 and 15.6 which is correct. She placed 15.53 closer to 15.6 than to 15.5 and this is not correct.)
• 1 minute: independent think time
• 1 minute: partner discussion time
• Make sure students identify that 15.53 should be closer to 15.5 than to 15.6.

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
  • accurately label the tick marks using their understanding of place value and the values of the end tick marks
  • use their understanding of place value to estimate each number

• Ask previously selected students to share.
• Display first image.
• “¿Cómo supieron qué números escribir debajo de las marcas?” // “How did you know how to label the tick marks?” (Since the ends of the number line are whole numbers, the tick marks are tenths. It’s the second and third tick marks so that meant they are 3.2 and 3.3.)
• “¿Cómo estimaron el valor del número?” // “How did you estimate the value of the number?” (It is halfway between 3.2 and 3.3 so that’s 3.25.)
• Display the last number line.
• “¿Cómo estimaron este número?” // “How did you estimate this number?” (The tick marks are 1.71 and 1.72 and it is really close to 1.72. So I put 1.719 which is just 1 thousandth from 1.72.)

The purpose of this activity is for students to round a decimal number to the nearest whole number, tenth, and hundredth. Number lines are provided for the first set of problems but not for the second set. Monitor for students who

• use the given number lines or draw number lines for the second set of problems
• use place value reasoning

• Groups of 2

• 5 minutes: independent work
• 2 minutes: partner discussion
• Monitor for who students:
  • who are using the number lines to determine proximity
  • who are using the digits in the number (place value reasoning)

• Ask previously selected students to share. Have the student who used the number line share first.
• “¿Cómo les ayudó la recta numérica a redondear los números?” // “How did the number line help you round numbers?” 
• “Cuando redondearon los números, ¿cómo les ayudó pensar en la posición y en el valor de los dígitos?” // “How did thinking about the place and value of the digit help you round numbers?”

The purpose of this activity is for students to practice rounding numbers to the nearest whole, tenth, or hundredth. Students may choose to use a number line or any other strategy that makes sense to them.

• Groups of 2

• 5 minutes: independent work
• 2 minutes: partner discussion
• Monitor for students who:
  • use a number line to round
  • reason using the digits and place value to round

• Ask previously selected to students to share.