Past the whole

• CCSS-M · 3.NF.A.2
  Understand a fraction as a number on the number line; represent fractions on a number line diagram.

3.NF.A.2 defines fractions as points on the line without capping them at 1. This task extends the fourths past the whole so children see that 5/4 is simply the fifth quarter, one place past 1.

This line runs to 2, so the first estimates from 0 alone will be rough. Drop early: the wholes are what make 5/4 and 7/4 placeable.

1. Start: 0 at 6.5%
2. Drop 1: 1 at 50.0%
3. Drop 2: 2 at 93.5%
4. Drop 3: 3/2 at 71.7%

When to dropAfter two cards, drop 1 to fix the first whole, then 2 for the far end, so the class can see there are two wholes to fill. Drop 3/2 last, the midpoint of the second half, to check 7/4 sits right of it and 5/4 sits left.

Drop unlocks after 2 cards placed.

• 1/4

  at 17.4%
  First quarter of the first whole.
• 1/2

  at 28.3%
  Middle of the first whole, well left of the line's centre.
• 5/4

  at 60.9%
  One and a quarter. Just past 1, a quarter into the second whole.
• 7/4

  at 82.6%
  One and three quarters. Between 3/2 and 2.
• 3/4

  at 39.1%
  Three quarters of the first whole, left of 1.

• 5/4 means five quarters. Where is the fifth quarter?
• How is the gap from 1 to 5/4 related to the gap from 0 to 1/4?
• Which is bigger, 7/4 or 3/2? How can you tell from the line?

• Looks like
  5/4 placed inside the first whole, left of 1.
  Means
  Assuming a fraction must be less than 1.
  Respond with
  Ask how many quarters are in one whole, and where the fifth quarter has to go.

Ask what 5/4 would be as a mixed number, and whether the line makes the one and a quarter obvious.

Fractions do not stop at 1. 5/4 and 7/4 live in the second whole, and quarters keep their even spacing all the way along.