In this activity, students use the relationships between the areas of geometric shapes to reason about groups of halves, thirds, and sixths, building their intuition for division of fractions. The focus is on the “How many groups?” interpretation of division.

Students start by using pattern blocks to represent multiplication of a whole number and a fraction. For example, if a hexagon represents 1 whole and 6 triangles make a hexagon, then each triangle represents . They can then use 6 triangles to represent .

Later, students use the blocks to reason in the opposite direction. They consider questions such as, “How many s are in 4?” and connect the answer to the unknown factor in equations such as . These kinds of questions serve as a stepping stone to more abstract questions, such as “What is 4 divided by ?” and equations such as .

As students make sense of the quantities in pattern-block arrangements and the equations that represent them, they practice reasoning quantitatively and abstractly (MP2).

This activity works best when each student has access to pattern blocks. If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use an applet to investigate fractional relationships with pattern blocks. The applet allows students to place and compare pattern blocks with or without a grid.