Lesson 9

All Kinds of Numbers on the Number Line

Warm-up: Which One Doesn’t Belong: Many Number Lines (10 minutes)

Narrative

This warm-up prompts students to compare four number lines. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the number lines in comparison to one another. During the synthesis, ask students to explain the meaning of any terminology they use, such as parts, partitions, mark, label, halves, fourths, or whole.

Launch

  • Groups of 2
  • Display the image.
  • “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
  • 1 minute: quiet think time

Activity

  • “Discuss your thinking with your partner.”
  • 2–3 minutes: partner discussion
  • Share and record responses.

Student Facing

Which one doesn’t belong?

ANumber line. Tick marks at zero and one half, a point plotted at one half.

BNumber line. Tick marks labeled zero and one. Point plotted at one.

CNumber line. 0 to 1 with 3 evenly spaced tick marks between. First tick mark, 0. Last tick mark, 1.

DNumber line. Evenly spaced tick marks labeled zero, one, and two. Point plotted at the one.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “How did you know that the number line in A stopped at 1?” (The location of \(\frac{1}{2}\) shows that the whole number line is split in half right there and I know that 2 halves is 1. I can only fit one more half next to the first one.)
  • Consider asking: “Let’s find at least one reason why each one doesn’t belong.”

Activity 1: Locate 1 Again (20 minutes)

Narrative

The purpose of this activity is for students to locate 1 on a number line given the location of a non-unit fraction less than 1 or greater than 1. In either case, it is likely students will reason about unit fractions to locate 1.

In the first problem, students may use the size of thirds to locate 1. In the second problem, they reinforce their knowledge that the denominator of a fraction tells us the number of equal parts in a whole and the size of a unit fraction, and that the numerator gives the number of those parts (MP6). Students typically use the denominator to partition a number line, but here they need to use the numerator.

This activity uses MLR1 Stronger and Clearer Each Time.
Advances: reading, writing

Action and Expression: Develop Expression and Communication. Synthesis. Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Conceptual Processing, Memory

Launch

  • Groups of 2

Activity

  • “Take a few minutes to locate 1 on these number lines. Then use any of the number lines to explain how you located 1.”
  • 5–7 minutes: independent work time

Student Facing

  1. Locate and label 1 on each number line.


    1. Number line. Two tick marks, zero and two thirds.

    2. Number line, two tick marks. One tick mark, 0. Second tick mark, five fourths.

    3. Number line, two tick marks, zero and eleven eighths.
  2. Use any of the number lines to explain how you located 1.

Fractions.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students say they aren’t sure how to get started, consider asking:
  • “What do we know about this fraction?”
  • “How could that help us find 1?”

Activity Synthesis

MLR1 Stronger and Clearer Each Time

  • “Share your written reasoning for one of the number lines with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
  • 3–5 minutes: structured partner discussion
  • Repeat with 2–3 different partners.
  • “Revise your initial draft based on the feedback you got from your partners.”
  • 2–3 minutes: independent work time
  • Invite students to share their revised explanations of how they located 1 on the number lines.

Activity 2: Locate $\frac{3}{4}$ [OPTIONAL] (15 minutes)

Narrative

The purpose of this activity is for students to use the location of a unit fraction to locate another fraction with a different denominator on the number line. Students can use their knowledge from the previous activity to place 1 on the number line and then use that to partition the interval from 0 to 1 to find other numbers. Because students have only located fractions with the same denominator on a single number line, they may want to use more than one number line in this activity. They may or may not label the points they find along the way to \(\frac{3}{4}\). Encourage them to use whatever strategy makes sense to them.

Monitor for students who use a single number line to show both thirds and fourths and those who use separate number lines. Select them to share during activity synthesis.

This activity is optional because it goes beyond the depth of understanding required to address grade 3 standards.

MLR8 Discussion Supports. Synthesis: As students share the similarities and differences between the strategies, use gestures to emphasize what is being described. For example, point to each fraction and show with your fingers the partitions such as thirds and fourths, that are being discussed.
Advances: Listening, Representing

Launch

  • Groups of 2
  • “Now we’re going to try something a little bit different. Let’s use the location of a unit fraction to find a fraction with a different denominator.”

Activity

  • “Take a few minutes to locate \(\frac{3}{4}\) on this number line. Use any strategy that makes sense to you.”
  • 3–5 minutes: independent work time
  • As students work, consider asking:
    • “How did you decide whether to use one or two number lines?”
    • “Did you locate any numbers before locating \(\frac{3}{4}\)?”
    • “Where is 1 on your number line(s)?”
  • “Share your strategy with your partner.”
  • 2–3 minutes: partner discussion

Student Facing

Number line. Tick marks labeled zero and one third.

Number line. One tick mark labeled zero.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Ask the two selected students to display their work side-by-side for all to see.
  • “What do these strategies have in common? How are these representations different?” (They both located 1 before locating \(\frac{3}{4}\). In the first strategy, the thirds are one number line and used to find 1. Then, 1 is on the second number line and used to find \(\frac{3}{4}\). In the second strategy, the thirds and fourths are marked on the same number line.)
  • Consider asking: “What questions do you have about these representations?”

Lesson Synthesis

Lesson Synthesis

Display fraction strips and a number line.

“Work with your partner to brainstorm all the things you’ve learned about fractions so far. Then, we’ll share and record our ideas.” (The numerator is the top part of a fraction and the denominator is the bottom part. Fractions can be represented with diagrams, fraction strips, and number lines. Number lines can be partitioned to show unit fractions and non-unit fractions, and fractions less than 1 and greater than 1. Non-unit fractions are built from unit fractions.)

Share and record ideas.

Cool-down: Where is 1 Now? (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Section Summary

Student Facing

In this section, we located and labeled fractions on the number line. We learned how to partition the number line from 0 to 1 to locate unit fractions.

Number line. Scale 0 to 1, evenly spaced tick marks. First tick mark, 0. Second tick, one sixth, point plotted. 4 unlabeled tick marks. Last tick mark, 1.

Number line. Scale 0 to 2. First tick mark, 0. Second tick, one sixth, point plotted. 4 unlabeled tick marks. Seventh tick mark, 1. Last tick mark, 2.

Then we used the location of unit fractions to locate other fractions.

Number line. Scale 0 to 2 by sixths, evenly spaced tick marks. First tick mark, 0. Last tick mark, 2. Points plotted at 2 sixths and 7 sixths.

We also learned that some fractions are at the same location as whole numbers on the number line. Here, we can see that \(\frac{6}{6}\) shares the same location as 1 and \(\frac{12}{6}\) shares the same location as 2.

At the end of the section, we used our understanding of unit fractions to locate 1 on the number line when we only knew the location of a fraction.