The purpose of this activity is for students to make sense of partitioning number lines that extend beyond 1. Clare’s and Diego's number lines show two common misconceptions that students often make while partitioning number lines into fractions. Clare partitions the entire number line into fourths, and Diego places four tick marks to show fourths. Students analyze these misconceptions (MP3) before they locate and label unit fractions on number lines of various lengths in the next activity.

• Ask students to share why Andre’s partitioning makes sense to them.
• Consider asking:
  • “Did anyone think of Andre’s reasoning in a different way?”
  • “How do we know that Andre’s number line is partitioned into fourths?”
• Ask students to explain why Clare’s and Diego’s partitionings do not show intervals of one-fourth.
• “We learned that when you partition a number line into fractions, you have to pay attention to where 0 and 1 are, and make sure to partition that distance into the right number of equal-length parts.”

The purpose of this activity is for students to partition the interval from 0 to 1 into equal parts to locate and label unit fractions. Students see number lines that vary in length, from 1 unit to 4 units, which provides an opportunity for them to practice accurately partitioning the unit on the number line, rather than the entire number line (MP6). Some number lines show numbers greater than 1, which gives students the opportunity to think about fractions greater than 1 even though they are not explicitly addressed in this lesson.

• “Tell me how you partitioned your number line.”
• “What did we learn in the last activity about partitioning number lines?”

• “Were there any number lines that you and your partner were not sure how to partition or about which you disagreed? How did you resolve your confusion or your disagreement?” (We weren’t sure how to partition the number line that goes up to 3 when we were locating . We talked together about partitioning just the space from 0 to 1 into half.)
• Consider asking: “What was different about partitioning, locating, and labeling fractions on number lines with numbers greater than 1 than on number lines that just go up to 1?” (We have to be careful to partition just the whole, not the whole number line.)
• Display the same unit fraction on number lines of length 1 and length 2, such as:

• “What do you notice?” (The top number line just shows 0 to 1. The bottom number line shows 0 to 2. The number 1 is located in the same place on both number lines. The number is located in the same place on both number lines.)
• “The location of any number on the number line doesn’t change just because we extend the number line. The number is located between 0 and 1 whether the number line goes up to 1 or to another number.”