Fraction Notation

Activities to try

1. 1

   Build the fraction, build the name

   Why it works: Fraction notation names one number. The bottom number tells how many equal parts one whole is cut into; the top counts how many we have. Pushing the numerator past the denominator shows 5/4 is a real number greater than one (equal to 1 1/4), and overlaying 2/4 on 1/2 shows different symbols can name the same quantity.

   1. Cut a bar into M equal parts (for example set it to 4) and shade N of them (for example 3), then write the symbol that matches: the parts taken on top, the equal parts in one whole on the bottom.
   2. As you shade, say the fraction as one spoken quantity ('three-quarters'), not 'three, four'.
   3. Label each number to its job: the numerator is 'the 3 parts we took' and the denominator is 'the 4 equal parts in one whole'.
   4. Now push the numerator past the denominator (shade 5 parts when the whole holds 4): the shading overflows into a second whole. Record that this is called an improper fraction (5/4) and write its mixed twin beside it (1 1/4).
   5. Sort every fraction you build into four trays: unit (numerator 1 and denominator more than 1, like 1/4), proper-but-not-unit (numerator more than 1 but less than the denominator, like 3/4), improper (numerator as big as or bigger than the denominator, like 5/4 or 4/4; 1/1 also goes here because it equals one whole), and mixed (like 1 1/4). A unit fraction like 1/4 is a special kind of proper fraction, so sort each card into its most specific tray.
2. 2

   Label the fraction's jobs

   Why it works: Each part of the written symbol has a fixed job tied to the model, and the symbol, spoken words and diagram are three views of the same single quantity.

   1. Look at 3/4 beside its shaded bar. Point to the numerator and say what it does ('the parts we took'); point to the denominator and say what it does ('the equal parts in one whole').
   2. Point to the whole bar and explain it is the one whole that was cut up.
   3. Answer the prompt 'What if the top number were bigger than the bottom?' and predict what the picture and the name would become.
   4. Link the three forms both directions: write the symbol, say the words, shade the picture, and go back again.
3. 3

   Classify and match for fluency

   Why it works: Secure classification and word-symbol matching show the vocabulary has attached to structure, not just to top/bottom positions.

   1. Sort a set of fraction cards into unit, proper, improper and mixed, giving one example of each; remember a unit fraction (numerator 1, denominator more than 1, like 1/4) is a special kind of proper fraction, so place it in the unit tray and keep the proper tray for proper fractions whose numerator is greater than 1 (a card such as 1/1 equals one whole and goes in the improper tray).
   2. Match spoken words to written symbols (for example 'three-quarters' to 3/4).
   3. Fill in the missing number: a shaded picture shows ?/? parts; write both the numerator and the denominator.
   4. Explain in your own words why 2/4 and 1/2 are equivalent.