Thirds and sixths

• 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.b
  Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

3.NF.A.2 through the placement, and 3.NF.A.3.b through the 3/6 card landing on the 1/2 drop. The standard names 4/6 = 2/3 as an example of equivalence, and this line sets up that neighbourhood.

Thirds are harder to place than fourths because they do not sit on the halving points. Starting from 0 alone makes the early estimates rougher, and the disagreement that follows is good.

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 to fix the whole. Hold 1/2 as the final drop until 1/3 and 2/3 are placed, then drop it and check that 1/3 sits left of it, 2/3 sits right of it, and 3/6 lands right on it.

Drop unlocks after 2 cards placed.

• 1/3

  at 36.1%
  A third of the way along, left of the middle.
• 2/3

  at 63.9%
  Two thirds along, right of the middle and the mirror of 1/3.
• 1/6

  at 22.2%
  Half of 1/3. Sits between 0 and 1/3.
• 5/6

  at 77.8%
  Between 2/3 and 1, near the end. The mirror of 1/6.
• 3/6

  at 50.0%
  Lands on 1/2. Sixths and halves meet here: 3/6 = 1/2.

• Why is 1/3 not on the same spot as 1/2?
• 1/6 is half of which card? Where does that put it?
• Where does 3/6 go? Does it share a spot with anything?

• Looks like
  1/3 placed on the midpoint.
  Means
  Confusing the fraction with the middle of the line, or with 1/2.
  Respond with
  Ask how many thirds fill the whole, then whether one of three parts reaches the middle.

Ask which was harder to place, thirds or fourths, and why the thirds feel more awkward.

Thirds and sixths share a line, sixths are half-steps of thirds, and 3/6 is 1/2 by another name.