Whole Numbers and Fractions

10 min

Narrative

This Warm-up elicits observations about the different ways whole numbers can be expressed as fractions. Students have previously seen number lines on which 1, 2, and 3 were labeled, with fractions in halves, thirds, fourths, sixths, and eighths. They understand that a denominator of 2 corresponds to 2 equal parts in the length representing 1 whole. The number line marked with 11\frac{1}{1}, 21\frac{2}{1}, and 31\frac{3}{1} is shown, together with those marked with halves, thirds, and fourths, to highlight that a denominator of 1 means each whole has 1 part.

In the Activity Synthesis, students learn that fractions with 1 as a denominator can be used to represent whole numbers (21=2\frac{2}{1} = 2).

Launch

  • Groups of 2
  • Display the image.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Task

What do you notice? What do you wonder?

4 number lines. Evenly spaced tick marks. 0 to 3 ones by ones, 0 to 6 halves by halves, 0 to 9 thirds by thirds, 0 to 12 fourths by fourths.

Solution Steps (5)
  1. 1
    Observe 4 number lines with denominators 1, 2, 3, 4
    →Lines go from 0 to 3 ones, 0 to 6 halves, 0 to 9 thirds, 0 to 12 fourths
  2. 2
    Notice fractions on first line
    →Labels are 1/1, 2/1, 3/1 (whole numbers as fractions)
  3. 3
    Notice vertical alignment across lines
    →1/1 = 2/2 = 3/3 = 4/4 all at position 1
  4. 4
    Interpret denominator of 1
    →Denominator 1 means 1 part per whole (no partitioning)
  5. 5
    Connect fractions to whole numbers
    →2/1 = 2, 3/1 = 3 because 2 parts of size 1 = 2

Sample Response

Students may notice:

  • There are four number lines with a different number of tick marks on each.
  • The last three number lines are partitioned into halves, thirds, and fourths.
  • The first number line has labels with 1 for the denominator.
  • Some labels that line up vertically show a pattern (for instance, the first set shows 11,22,33,44\frac{1}{1}, \frac{2}{2}, \frac{3}{3}, \frac{4}{4}).
  • The last fraction on each number line has a numerator that is 3 times the denominator.

Students may wonder:

  • Are the fractions in the first number line 1, 2, and 3? Why are they written as fractions with 1 for the denominator?
  • How do we name the fractions with 1 for the denominator?
  • Why are there no whole-number labels?
Activity Synthesis (Teacher Notes)
  • “What could it mean to have a denominator of 1?” (The whole hasn’t been partitioned. The whole has been partitioned into 1 part.)
  • Have students label the locations of 11\frac{1}{1}, 21\frac{2}{1}, and 31\frac{3}{1} on the first number line, with 1, 2, and 3, respectively.
Number line.
  • “The length from 0 to 1 hasn’t been partitioned, so each part has a length of 1. This is what a denominator of 1 means. If we have 1 part of 1, the numerator is 1. If we have 2 parts of 1, the numerator is 2, and so on.”
  • “What other fractions on these number lines are equivalent to 1?” (22\frac{2}{2}, 33\frac{3}{3}, 44\frac{4}{4})
Standards
Addressing
  • 3.NF.3.c·Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
  • 3.NF.A.3.c·Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. <span>Examples: Express <span class="math">\(3\)</span> in the form <span class="math">\(3 = 3/1\)</span>; recognize that <span class="math">\(6/1 = 6\)</span>; locate <span class="math">\(4/4\)</span> and <span class="math">\(1\)</span> at the same point of a number line diagram.</span>

20 min

15 min