Skip to content
Spark & Whetstonethe grown-up commons

← Fractions for Grown-ups

Fractions for Grown-ups

Three short lessons on fractions the way your child meets them now: a string that sings in fractions, a number line where 1/2 and 2/4 land on the same spot, and dividing by a half.

Spark & Whetstone · sparkwhetstone.org/learn/fractions-for-grown-ups · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. A string that sings

An old tradition credits Pythagoras with noticing that musical intervals are fractions of a string. Stop a vibrating string halfway along, and the half sings an octave higher. Stop it at two-thirds of its length, and you hear a fifth; at three-quarters, a fourth.

Fractions began as lengths you could hear. That is still a good way in: a fraction is a part of a whole, measured.

Question. Which length of string sings an octave above the whole string?

  1. Half the string.
  2. Two-thirds of the string.
  3. Twice the string.

Say it back. In your own words: how is a fraction like a length of string?

________________________________________________

________________________________________________

Lesson 2. Same spot on the line

Children now place fractions on a number line, not only as slices of pizza. Between 0 and 1, cut the line into 2 equal parts and 1/2 is the first mark. Cut it into 4 and 2/4 is the second mark.

They land on the same spot. That is what equal fractions mean: different names for one point on the line.

Question. On a number line from 0 to 1, which fraction lands on the same spot as 3/6?

  1. 1/3
  2. 1/2
  3. 3/4

Say it back. Say it back: what does it mean for two fractions to be equal?

________________________________________________

________________________________________________

Lesson 3. Dividing by a half

Dividing by a fraction puzzles many grown-ups because the answer gets bigger. Read 3 ÷ ½ as a question: how many halves fit in 3?

Draw three bars and cut each in half. There are six halves. So 3 ÷ ½ = 6. The rule "flip and multiply" gives the same answer, but the drawing shows why.

Question. How many quarters fit in 2? That is, what is 2 ÷ ¼?

  1. ½
  2. 8
  3. 6

Say it back. In your own words: why can dividing by a fraction give a bigger answer?

________________________________________________

________________________________________________

At the table: Fold a fraction strip

Players: a grown-up and a child, 7 or older. You'll need: four strips of paper the same length and a pencil.

  1. Leave one strip whole. Fold one in half, one in quarters and one in eighths; label the folds.
  2. Line them up and find every fraction that matches ½.
  3. Ask the child: how many eighths make a quarter?
  4. Use the strips to show how many quarters fit in two whole strips.

Answers

  1. Half the string. — Half the length sings an octave higher.
  2. 1/2 — Three of six equal parts is halfway, the same spot as 1/2.
  3. 8 — Each bar holds 4 quarters, so two bars hold 8: 2 ÷ ¼ = 8.