Bigger or smaller than a half?

• CCSS-M · 4.NF.A.2
  Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.A.2 names comparing to a benchmark fraction such as 1/2 as a valid strategy. This task is that strategy made physical: each card is judged against the 1/2 drop, and the line records the comparison as position.

Only 0 shows to start. See whether the class reaches for a half in their heads before you give it to them; the aim is for them to want the benchmark, then get it.

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: it is the referee. Reveal it once several cards are down, then check every card is on the correct side.

Drop unlocks after 2 cards placed.

• 2/3

  at 63.9%
  More than a half. Two of three parts passes the midpoint.
• 3/8

  at 39.6%
  Less than a half. Three of eight parts falls short of the middle.
• 5/6

  at 77.8%
  Well more than a half, close to 1.
• 2/5

  at 41.7%
  Just less than a half. Two of five parts is a little under the middle.
• 1/4

  at 29.2%
  Clearly less than a half, halfway between 0 and 1/2.

• Is this fraction more or less than a half? How do you know before the mark is here?
• 2/5 and 3/8 are both under a half. Which is closer to it?
• Which card is the closest to 1/2 from above?

• Looks like
  2/5 placed above 1/2.
  Means
  Seeing the 5 and treating fifths as large, so 2/5 feels big.
  Respond with
  Ask what half of five fifths would be, and whether 2/5 reaches two and a half fifths.
• Looks like
  3/8 and 2/3 placed at similar distances from the middle.
  Means
  Comparing numerators or denominators in isolation rather than the whole fraction.
  Respond with
  Ask how many eighths make a half, and how many thirds, then place each relative to that.

Sort the cards into two piles, under a half and over a half, and ask the class to justify each using the benchmark.

You can compare any two fractions by asking which side of 1/2 each one sits, without finding a common denominator.