Mixed numbers on the line

• CCSS-M · 4.NF.B.3
  Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

4.NF.B.3 treats a fraction a/b with a greater than 1 as a sum of unit fractions, which is what makes 7/3 land at two and a third: seven thirds is two whole ones plus one more third. The line turns that sum into a position.

This line runs to 3, so starting from 0 alone the first estimates will be rough. The card faces show the improper form with the mixed number as a hint, so children convert as they place. Drop early.

1. Start: 0 at 5.9%
2. Drop 1: 1 at 35.3%
3. Drop 2: 2 at 64.7%
4. Drop 3: 3 at 94.1%
5. Drop 4: 1 1/2 at 50.0%

When to dropAfter two cards, drop the wholes in turn, 1 then 2 then 3, so each whole appears and the mixed numbers snap between them. Drop 1 1/2 last for fine tuning near the start of the line.

Drop unlocks after 2 cards placed.

• 1/2

  at 20.6%
  Middle of the first whole, well to the left.
• 5/4

  at 42.6%
  One and a quarter. A quarter past 1.
• 7/3 (2 1/3)

  at 74.5%
  Two and a third. Just past 2, a third into the third whole.
• 9/4 (2 1/4)

  at 72.1%
  Two and a quarter. Just past 2, a little left of 7/3.
• 11/4 (2 3/4)

  at 86.8%
  Two and three quarters. Between 9/4 and 3, near the end.

• How many wholes are in 7/3? What is left over?
• 9/4 and 7/3 are close. Which is bigger, and why?
• Between which two whole numbers does 11/4 sit?

• Looks like
  7/3 placed in the first whole.
  Means
  Reading 7/3 as a fraction below 1 rather than converting to two and a third.
  Respond with
  Ask how many thirds make one whole, then how many wholes are inside seven thirds.
• Looks like
  9/4 and 11/4 placed the wrong way round.
  Means
  Not tracking the quarters past the whole.
  Respond with
  Ask how many quarters past 2 each one is, and count the quarters aloud from 2.

Ask the class to read each card as a mixed number, and check that the mixed number matches the two whole numbers the card sits between.

An improper fraction is a real number with a home on the line, and rewriting it as a mixed number tells you which two whole numbers it sits between.