Where do the fourths go?

• CCSS-M · 3.NF.A.2
  Understand a fraction as a number on the number line; represent fractions on a number line diagram.
• CCSS-M · 3.NF.A.3.c
  Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers, e.g., recognize that 4/4 and 1 are located at the same point of a number line diagram.

3.NF.A.2 defines a fraction as a point on the line, which is exactly what placing these cards does. The 4/4 card carries 3.NF.A.3.c directly: the standard names recognising that 4/4 and 1 sit at the same point, and this task stages that as a small shock rather than a told fact.

Only 0 shows to start, so the first fourths are estimates of an unseen whole. That is deliberate. When 1 drops, the whole is fixed and the recalibration is the richest moment of the task.

1. Start: 0 at 8.3%
2. Drop 1: 1 at 91.7%
3. Drop 2: 1/2 at 50.0%

When to dropAfter two cards, drop 1 and watch the class rework everything now that the whole is fixed. Drop 1/2 last: it is the sharpener that pins the fourths either side of the middle. Hold it until 1/4 and 3/4 are placed, so its arrival exposes any that were squashed.

Drop unlocks after 2 cards placed.

• 1/4

  at 29.2%
  Halfway between 0 and 1/2. If they have 1/2, this is a halving move.
• 3/4

  at 70.8%
  Halfway between 1/2 and 1. Symmetrical to 1/4 about the midpoint.
• 1/3

  at 36.1%
  Left of 1/2 and right of 1/4. Does not land on a fourths mark, which is the point.
• 2/3

  at 63.9%
  Right of 1/2, and the mirror of 1/3 about the middle.
• 4/4

  at 91.7%
  Sits exactly on 1. This is the whole task in one card: 4/4 and 1 are the same number.

• How did you decide where that goes, with only 0 to go on?
• Now that 1 is here, what has to move?
• Where does 4/4 go? Is anyone surprised?

• Looks like
  1/3 placed to the right of 1/2.
  Means
  Reading the denominator as magnitude, so a bigger bottom number feels like a bigger fraction.
  Respond with
  Ask how many thirds make a whole, and whether one of three equal parts can be more than one of two equal parts.
• Looks like
  4/4 placed short of 1.
  Means
  Treating 4/4 as almost a whole rather than exactly a whole.
  Respond with
  Ask how many quarters are in one whole, and where the fourth quarter finishes.

Ask which card was hardest and why, before you show the score. The 1/3 and the 4/4 usually spark the best talk.

A fraction is a number with a position, not a shaded part of a shape. Fourths sit at even steps, and 4/4 is just another name for 1.