Lesson 2

The purpose of this Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. The warm-up also draws students' attention to a length between a full inch and one-half of an inch, preparing students to work with such lengths later.

• Groups of 2
• Display the image.
• “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• Consider asking:
  • “Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____?”
  • “Based on this discussion does anyone want to revise their estimate?”

The purpose of this activity is for students to partition the inches on a ruler into fourth or quarter inches, using what they know about fractions on a number line. Students then use the ruler to measure objects in the classroom to the nearest quarter inch. In the synthesis, students practice reading the measurements as fractions greater than 1 and mixed numbers. They also discuss how measuring in fourths of an inch is different than measuring in half inches.

Students will use the rulers they partition in the next activity.

• Each group of 2 will need a ruler that didn’t get partitioned in the previous lesson.

• Groups of 2
• Display the image.
• “Could we say that the paper clip is \(1\frac{1}{2}\) inches? Explain your reasoning.” (No, because it’s right in between the 1 inch mark and the \(1\frac{1}{2}\) inch mark. No, because it looks like it’s \(1\frac{1}{4}\) inches, because if we partition a half into 2 equal parts, it’s a quarter inch.)
• 1 minute: partner discussion
• Share responses.
• “How can we get a measurement closer to the length of the paper clip?“ (We can partition the half inches into smaller parts. We can measure in fourths.)
• Make sure each group has a ruler that didn't get partitioned from the previous lesson.

• “You have a ruler that was not partitioned in an earlier lesson.”
• “Work with your partner to partition that ruler to show fourths of an inch. Then, you will each choose two objects to measure to the nearest quarter inch.”
• “When we partition an inch into fourths, we can call the parts ‘fourths of an inch’ or ‘quarters of an inch’ or ‘quarter inches.’ They mean the same thing.”
• 5–7 minutes: partner work time

• “How did you partition the inches into fourths?”
• “How could partitioning into half inches first help partition into quarter inches?”

• Invite students to share an object and the length in quarter inches.
• For each measurement, ask the class to repeat the measurement as a fraction greater than 1 and a mixed number.
• “What’s different about measuring lengths in quarter inches and measuring in half inches?” (With quarter inches, it was more likely for the length of the object to match or be close to a mark on the ruler than to fall between two marks. There were also more marks to use to measure.)

The purpose of this activity is for students to practice measuring objects in the classroom using their partitioned rulers and find objects of certain fractional lengths. Because the specified lengths are in halves and fourths of an inch, students may use both rulers. As they realize that all lengths could be measured with the rulers showing fourths of an inch, they may opt to use only one ruler.

When students decide to measure in halves or fourths of an inch based on the length of an object, they attend to precision (MP6).

• Each group of 2 will need the rulers from previous activities: one that was partitioned into half inches and another partitioned into quarter inches.

• Groups of 2
• “You’ll now use the rulers you partitioned to find objects that have certain lengths in inches.”
• “Take a minute to read the instructions and look at the lengths in the table.”
• 1 minute: quiet think time
• Invite 1–2 students to explain the activity in their own words. Answer any questions students might have.

• “Work with your partner to complete the activity. Be sure to practice saying each measurement with your partner.”
• “If you find an object with the length described in the last four rows of the table, be sure to write down the measurement.”
• 10–12 minutes: partner work time

• Invite 3–4 students to share an object and the length of the object. Record each measurement and ask the class to say it together.
• “How did you decide which ruler to use—the one partitioned in half inches or the one partitioned into fourth or quarter inches?” (For lengths that are whole numbers or show \(\frac{1}{2}\) inch, we used the ruler partitioned into halves. For lengths that show \(\frac{1}{4}\), \( \frac{2}{4}\), or \( \frac{3}{4}\), we used the ruler partitioned into fourths. We used only the ruler partitioned into fourths because we can see whole numbers, halves, and fourths in it.)
• Consider asking: “Was there a particular measurement that was easy to find because a lot of objects have that length? Were there any that were hard to find?”