Lesson 5

To the Number Line

Warm-up: Notice and Wonder: Two Number Lines (10 minutes)

Narrative

The purpose of this warm-up is to elicit the idea that number lines can be partitioned into intervals smaller than 1, which will be useful when students see number lines partitioned into fractions in a later activity. While students may notice and wonder many things, the idea that fractions can be represented on the number line is the important discussion point. Students do not need to identify the tick mark as showing \(\frac{1}{2}\) in the warm-up, as that will be the focus later in the lesson.

This prompt gives students opportunities to look for and make use of structure (MP7). The specific structure they might notice is that each number line is partitioned in half.

Launch

  • Groups of 2
  • Display the image.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time

Activity

  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Facing

What do you notice? What do you wonder?

Number lines.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “What do you know about the number each tick mark represents?” (On the first number line, it is 5 because it is halfway between 0 and 10. On the second number line, I think it is \(\frac{1}{2}\) because it is halfway between 0 and 1.)

Activity 1: Card Sort: Number Lines (10 minutes)

Narrative

The purpose of this activity is for students to further develop the idea that fractional amounts can be represented on a number line. Students sort a given set of cards showing number lines. They first sort in a way of their choice, which might include number of parts or length of the number line. Monitor for different ways groups choose to categorize the number lines, but especially for categories that distinguish between number lines with whole number partitions and fractional partitions.

When students identify common properties of the number lines for their sorts, such as the numbers listed on the tick marks or the total number of tick marks, they look for and make use of structure (MP7).

MLR2 Collect and Display. Circulate, listen for and collect the language students use as they sort the number lines. On a visible display, record words and phrases such as: parts less than one, smaller than one, whole numbers, partitions, partitioned into fractions, and equal parts. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading

Required Materials

Materials to Copy

  • Card Sort: Number Lines

Required Preparation

  • Create a set of cards from the blackline master for each group of 2. 

Launch

  • Groups of 2
  • Distribute one set of pre-cut cards to each group of students.

Activity

  • “Work with your partner to sort some number lines into categories that you choose. Make sure you have a name for each category.”
  • 3-5 minutes: partner work time
  • Select groups to share their categories and how they sorted their cards.
  • Choose as many different types of categories as time allows. Be sure to highlight categories created based on whether the tick marks represent whole numbers or fractions.
  • If not mentioned by students, ask, “Can we sort the number lines based on what the tick marks represent?”
  • “Let’s look at B and E. Both are partitioned into 4 parts. What do the unlabeled tick marks in E represent?” (1, 2, 3) “What do you think those in B represent?” (\(\frac{1}{4}\), \(\frac{2}{4}\), \(\frac{3}{4}\), or amounts less than 1)
  • “Take a minute to sort your cards by number lines where the tick marks only represent whole numbers and number lines where the tick marks represent fractions.”
  • 1-2 minutes: partner work time

Student Facing

Your teacher will give you a set of cards that show number lines. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.

ANumber line. Evenly spaced tick marks labeled 0 and 8 with one tick mark in between. First tick mark, 0. Last tick mark, 8.

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

DNumber line. 0 to 2 with another tick mark between each number. First tick mark, 0. Last tick mark, 2.
ENumber line. Evenly spaced tick marks. First one, 0. Last one, 4. Three unlabeled tick marks between 0 and 4.

FNumber line. Tick marks at 0 and 1 with 5 unlabeled tick marks in between.
GNumber line. 0 to 6. First tick mark 0, two unlabeled tick marks, last tick mark, 6.

HNumber line. Evenly spaced tick marks 0 to 2 with two tick marks between 0 and 1 and two tick marks between 1 and 2. First tick mark, 0. Last tick mark, 2.
INumber line. Evenly spaced tick marks labeled 0, 2, and 4, with 3 unlabeled tick marks between 0 and 2 and between 2 and 4.

Two students sorting cards.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students don’t identify number lines that would have fractions marked, consider asking:
  • “What other numbers can we find on this number line?”
  • “What could the marks in between the whole numbers be?”

Activity Synthesis

  • “How did you know if a number line had tick marks that represent fractions?” (If there is one or more tick marks between two back-to-back whole numbers like 0 and 1, or 1 and 2, then the tick marks between them represent fractions.)
  • Attend to the language that students use to describe their number lines, giving them opportunities to describe the number lines more precisely.
  • Highlight the use of phrases like “parts less than 1” or “partitioning one part into smaller parts less than 1.”
  • Consider displaying a number line with fractions that are less obvious, such as number line I. Ask students to help identify the fractions on that number line, and label 1 and 3 so that the tick marks between the whole numbers are clear.

Activity 2: Fold and Label the Number Line (25 minutes)

Narrative

The purpose of this activity is to transition students from thinking about fractional lengths on fraction strips to thinking about fractions as numbers on the number line. Students build on their experience of folding fraction strips to fold number lines into halves, thirds, fourths, sixths, and eighths and then label unit fractions.

Students begin by considering how the fraction \(\frac{1}{2}\) can be labeled on the number line. They learn that each part of the number line has a length of one half, but the endpoint of the first one-half part is the location of the number \(\frac{1}{2}\) on the number line. This distinction is important for understanding fractions as numbers that can be represented as points on the number line and for using the number line precisely (MP6).

When folding the number lines, students also need to attend to the fact that it is the interval between 0 and 1 that needs to be partitioned, rather than the length of the entire strip of paper that contains each number line.

Representation: Develop Language and Symbols. Synthesis: Make connections between representations visible. Highlight the similarities and differences in the strategies students used to fold their number lines.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Required Materials

Materials to Gather

Materials to Copy

  • Fold and Label Number Lines

Required Preparation

  • Each student needs at least 5 number lines from 0 to 1. Each copy of the blackline master contains a few extra number lines, in case students fold incorrectly at first.
  • Create a number line folded into fourths and a fraction strip that shows fourths to display in the synthesis.

Launch

  • Groups of 2
  • “We’ve been thinking about where fractions are located on the number line. Let’s take some time to think about how to label fractions on the number line.”
  • “Take a minute to think about Andre and Clare’s number lines.”
  • 1 minute: quiet think time
  • “Talk to your partner about how each student’s labeling could make sense.”
  • 2–3 minutes: partner discussion
  • Share responses.
  • Display a number line with both the distance to \(\frac{1}{2}\) and the number \(\frac{1}{2}\) marked in a color, such as:
Number line.
  • “Andre was thinking about the parts that had length \(\frac{1}{2}\), so he labeled the parts from 0 to \(\frac{1}{2}\) and \(\frac{1}{2}\) to 1 with \(\frac{1}{2}\).”
  • “To locate and label the number \(\frac{1}{2}\), we find the endpoint of the first one-half part from 0 and label it.”
  • Give each student a copy of the blackline master and scissors.

Activity

  • “Cut the number lines apart.”
  • “Then, fold one into halves, one into thirds, one into fourths, one into sixths, and one into eighths.”
  • “As you fold, share your folding strategies with your partner.”
  • “Draw tick marks along your folding lines and label the unit fraction on each number line.”
  • 3–5 minutes: partner work time
  • Monitor for students who need support lining up the 0 and the 1 as they fold. Consider suggesting that they cut off the ends of the number line at 0 and 1, or marking 0 and 1 on both sides of each paper strip to make them easier to see while folding.

Student Facing

  1. Andre and Clare are talking about how to label fractions on the number line.

    Andre says \(\frac{1}{2}\) can be labeled like this:

    Number line. Tick marks at 0 and 1 with an unlabeled tick mark in between. Two labels of one half below number line with no corresponding tick marks.

    Clare says \(\frac{1}{2}\) can be labeled like this:

    Number line. Evenly spaced tick marks labeled zero, one half, and one. Point plotted at one half.

    How could each student’s labeling make sense?

  2. Your teacher will give you a set of number lines. Cut your number lines apart so that you can fold each one.

    As you fold, discuss your strategies with your partner.

    1. Fold one of the number lines into halves. Draw tick marks to show the halves. Label the number \(\frac{1}{2}\).
    2. Fold one of the number lines into thirds. Draw tick marks to show the thirds. Label the number \(\frac{1}{3}\).
    3. Fold one of the number lines into fourths. Draw tick marks to show the fourths. Label the number \(\frac{1}{4}\).
    4. Fold one of the number lines into sixths. Draw tick marks to show the sixths. Label the number \(\frac{1}{6}\).
    5. Fold one of the number lines into eighths. Draw tick marks to show the eighths. Label the number \(\frac{1}{8}\).

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Display a number line folded into fourths and a fraction strip of fourths.
  • “How was partitioning these number lines similar to partitioning our fraction strips?” (The number lines were folded just like the strips were, but instead of a rectangle it’s just a line. We labeled the location of the unit fractions at the end of the first part on each number line instead of the space.)

Lesson Synthesis

Lesson Synthesis

“Today we used what we know about fractions to think about where fractions are located on the number line.”

“What did you learn about locating and labeling fractions on the number line today?” (Fractions are between the whole numbers on a number line. We can fold number lines just like fraction strips to partition a whole into smaller parts. We can see the fraction as a distance and as a number located on the number line.)

Display a number line from 0 to 1 partitioned into thirds with the distance to \(\frac{1}{3}\) marked, such as:

Number line.

“How could we use this length to locate and label the number \(\frac{1}{3}\) on this number line?” (We could label the end of the first part with \(\frac{1}{3}\) to show it’s a third of the distance to 1 from 0 on the number line.)

“Locate and label \(\frac{1}{3}\) on the number line.”

Cool-down: Reflection (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.