In this activity, students examine number lines that have been partitioned into smaller and smaller parts. They see that this strategy can be used to generate many equivalent fractions and to verify if two fractions are equivalent.

Students encounter fractions with 5, 10, 15, and 20 for the denominator. Working with multiples of a number (in this case, 5) allows students to notice structure in how the partitioning of a part on a number line relates to equivalent fractions (MP7). (Students will not be assessed on fractions with denominator 15 or 20.)

If desired and logistically feasible, consider enacting Andre’s reasoning with one or more human number lines.

• Place a strip of masking tape or painter’s tape, at least 25 feet long, on the floor of the classroom or a hallway. 
• Ask a student to stand on each end of the tape. They represent 0 and 1.
• “How can we partition this line into fifths?” (Position 4 students on the tape, spaced apart equally between 0 and 1.)
• Ask each student on the line to say the number they represent. (0, , , , , 1) Give each student a sign with their fraction. (Consider distinguishing the sign for with a different color.)

  

• Invite 5 students to join the line, each person standing exactly in the middle of two others. 
• “What fraction do you represent now?” Ask every student to say the number they represent now. (0, , , , . . .  1) Give the student representing  another sign showing .
• Repeat to represent twentieths:
  • Ask 10 additional students to each join a space between two students representing tenths. (The students representing fifths and tenths should stay in place.)
  • Ask students to say aloud the number they represent now. (0, , , , . . .  1) 
  • Give the student representing another sign showing .
• “Can you explain why the student representing also ends up representing  and ?”
• If time allows, extend the activity to include fifteenths, with 2 additional students standing between each fifth.

• Invite a student to share their explanation of Andre’s strategy (first problem). Ask the class if they agree with the explanation or if they would change it or explain it differently.
• Ask another student to show how the number lines could help us see if two fractions are equivalent (second problem).
• Select previously identified students to share their reasoning for the last problem. Record their reasoning.
• Solicit some initial impressions on how the strategies are alike and different, but save further comparisons for future lessons.

In this activity, students continue to use the idea of partitioning a number line into smaller increments to reason about and generate equivalent fractions. Through repeated reasoning, students begin to see regularity in how the process of decomposing parts on a number line produces the numbers in the equivalent fractions (MP8). The task encourages students to think of the relationship between one denominator and another in terms of factors or multiples (even if they don’t use those terms which connect to work in a previous unit). 

Partitioning a number line into smaller parts becomes increasingly inconvenient when the denominator gets larger. As students begin to think about the relationship between tenths and hundredths, they see some practical limitations to using a number line to find equivalent fractions and are prompted to generalize the process of partitioning. (Students are not expected to draw a full number line with 100 parts.)

• “How did you find a fraction that is equivalent to one-tenth?”
• “Into how many equal parts should each one-tenth section on the line be split to get hundredths? How do you know?”

• Invite previously identified students to share their strategies for answering the first two problems.
• If not done by students in their explanations, consider asking students to revoice their reasoning in terms of factors and multiples.
• See Lesson Synthesis.