Tag Archives: partitioning

Let students break things

One of the most important things you can do when you introduce fractions is to let students break things.   Give them lots and lots of experiences where they have to break up things evenly and share things fairly.  For example:

  • Have them fairly share 3 cookies with 2 friends.  Have them fairly share 5 granola bars with 4 friends.  Have them fairly share 1 granola bar with 3 friends.
  • Have them fairly share large pizzas and small pizzas.
  • Have them fairly break up groups of items.
  • Have them evenly break up spaces between numbers on the number lines.

Have them break whatever else you can think of.  Change the numbers of items and the number of people sharing them.  Change the things to be shared.    Just let them keep breaking things.    

While they are breaking things, build on their concept of fairness to develop a notion of what equal shares mean.  Ask them questions like:

  • Does equal mean the same number of pieces?  Is it fair if we both get three pieces but mine are bigger than yours?  Why or why not?
  • Do equal shares have to be the same shape? If we both have the same size brownie and I cut my brownie into 2 pieces by making a diagonal line and you cut yours into 2 pieces making a horizontal line, did we both make fair shares?  Why or why not?

And then let them break some more things.

So what do we do after they break all these things?

Let them put things back together.  Give them a broken piece and tell them it is the piece one person got after a candy bar was shared between 5 people.  Ask them to figure out how big the candy bar was.  Do this with the pizzas and the granola bars and the groups of items.

After enough breaking things and putting them back together, talk to them about how we name the size of the pieces we made.  If we break a cookie or a granola bar or a group of objects into five pieces, we made fifths.   Ask them questions like:  If I break something into fifths and you break something into fifths, can my fifths be bigger than yours? Show me. Why are they both called fifths?

Researchers have talked a lot about the importance of these activities in developing a strong concept of fractions.   They use the terms partitioning and iterating for the two activities.   Partitioning is the act of breaking a unit into pieces.  In the case of fractions we talk about equi-partitioning or breaking a unit into equal pieces.  The unit can be the pizza or the candy bar or the space between two numbers on a number line or a group of objects.    Iterating is what we do when we put the pieces back together (or later on when we put pieces together to form other fractions).

I’ll talk more about why these activities are important (and what to do next) another time, but if you want to know now, check out the research below.

Lamon, S. J. (1996). The development of unitizing: Its role in children’s partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170-193.

Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for research in mathematics education, 21(1) 16-32.

Olive, J. (2002). Bridging the Gap: Using Interactive Computer Tools To Build Fraction Schemes. Teaching Children Mathematics, 8(6), 356-61.

Siebert, D., & Gaskin, N. (2006). Creating, Naming, and Justifying Fractions. Teaching Children Mathematics, 12(8), 394-400.