Eighths and their friends

• 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 is served by placing each eighth as a point. The 4/8 card is 3.NF.A.3.b in the flesh: when it lands on the 1/2 drop, the class sees 4/8 = 1/2 as a fact about position, not a rule to memorise.

This works best after the fourths task. Only 0 shows at first, so eighths are hard to estimate until the whole and the quarter drops arrive.

1. Start: 0 at 8.3%
2. Drop 1: 1 at 91.7%
3. Drop 2: 1/2 at 50.0%
4. Drop 3: 1/4 at 29.2%

When to dropAfter two cards, drop 1 to fix the whole, then 1/2 when the class reaches 4/8 so the equivalence lands visually. Drop 1/4 last if 1/8 and 3/8 are still hard to separate; it gives them the quarter to halve.

Drop unlocks after 2 cards placed.

• 1/8

  at 18.7%
  Half of a quarter. Sits between 0 and 1/4.
• 3/8

  at 39.6%
  Between 1/4 and 1/2, closer to a half.
• 5/8

  at 60.4%
  Just past 1/2. The mirror of 3/8 about the middle.
• 7/8

  at 81.2%
  Between 3/4 and 1, near the end.
• 4/8

  at 50.0%
  Lands exactly on 1/2. The equivalence card: 4/8 and 1/2 are the same point.

• 1/8 is half of which fraction? Where does that put it?
• Where will 4/8 land? Why there?
• How is 3/8 related to 5/8?

• Looks like
  4/8 placed away from 1/2.
  Means
  Not yet seeing that different names can share a point.
  Respond with
  Ask how many eighths are in a half, then drop 1/2 and let the position settle it.

Ask the class to name every card that could have a second name. 4/8 is the obvious one; some will spot that 1/2 could be written as 2/4 too.

Eighths are half-steps between the fourths, and 4/8 is another name for 1/2.