Cubes, Volume, and Repeated Addition

“With well-designed activities, appropriate tools, and teachers’ support, students can make and explore conjectures about geometry and can learn to reason carefully about geometric ideas from the earliest years of schooling. Geometry is more than definitions; it is about describing relationships and reasoning. The notion of building understanding in geometry across the grades, from informal to more formal thinking, is consistent with the thinking of theorists and researchers.”

From the National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, 2000

This unit’s activities focus on developing spatial visualization, communication skills, and strategies for finding volume. Students find volume by counting the number of cubes that compose an object. They discover that different shapes can have the same volume. Concrete explorations with volume lay the foundation for later study.

This unit also provides students a context to explore partitioning larger numbers to add. Students find the volume of buildings, towers, and cubic animals by grouping, skip counting, and using repeated addition strategies. These repeated addition strategies not only lay the foundation for multiplication but also for mental math strategies for addition. Students who are proficient with partitioning numbers can more easily break numbers into manageable parts to facilitate mental arithmetic.

In the first lesson, students discover that a specific volume (8 cubic units) can result in a variety of shapes. Students use connecting cubes to make buildings that have a volume of 8 cubic units. Students also copy their classmates’ cube model buildings. They organize the buildings by height, which is measured by the number of floors in each model. Students guide the teacher in making a group data table to display the information.

In the second lesson, students build cube models of classroom objects. They evaluate how closely each cube model reflects the size of the actual object. Students make decisions about whether the model is smaller than, larger than, or very close to the size of the real object.

In the third lesson, students work collaboratively to construct buildings based on three-
dimensional drawings. They determine the volume of these buildings using counting strategies, such as grouping the cubes and skip counting.

In the fourth lesson, students build on their strategies for determining volume by grouping and skip counting. They write number sentences to show repeated addition. They read A World of Cubic Animals in the Adventure Book story. In this story, Manuel dreams that the animals in his life have changed into animals made of cubes. Students use the drawings in the book to re-create the cubic animals and compare the volumes.