The purpose of this activity is to engage students in finding many ways to form one (whole) with fractional units. In doing so they will use simple equivalence and learn the recognise non-unit fractions as iteration of unit fractions.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- The student uses trial and error approaches to find other ways to make one.
Students who operate procedurally will be interested in finding other ways to make one rather than attending to ways in which the answers can be found systematically. They may not record their answers using expressions or equations.Click on the image to enlarge it. Click again to close.
An improvement is to name iterations of unit fractions as non-unit fractions, like this:
Click on the image to enlarge it. Click again to close.
The conceptual approach (show more)
- The student uses a systematic strategy to find all the possible ways to make one and uses expressions or equations to record the ways.
Students who work conceptually will find ways to systematically account for all possible ways to make one. They will use equivalence, particularly applying quarters and sixths in halves, and sixths in thirds. It is likely they will work with halves first then move to thirds then quarters. Alternatively, they may work with sixths first because they are the smallest pieces. Conceptually focused students tend to use symbols to express the names for one rather than draw diagrams. They notice when one possibility is the same as another.Click on the image to enlarge it. Click again to close.
Some students may consider order relevant in establishing unique names for one. That assumption will increase the number of possible ways they find.Click on the image to enlarge it. Click again to close.