Mathematical goals

This lesson unit is intended to help you assess how well students are able to visualize two-dimensional cross-sections of representations of three-dimensional objects. In particular, the lesson will help you identify and help students who have difficulties recognizing and drawing two-dimensional cross-sections at different points along a plane of a representation of a three-dimensional object.

Introduction       

The lesson unit is structured in the following way:

• Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You review their solutions and create questions for students to consider, to help improve their work.
• During the lesson, students work in groups on a collaborative activity. They match representations of three-dimensional objects with two-dimensional cross-sections. Students then share their work with another group.
• In a whole-class discussion, students compare and evaluate the methods they have seen and used.
• In a follow-up lesson, students review their initial solutions, and then use what they have learned to either revise the same introductory assessment task or complete a different task.

Materials required

• Each student will need a copy of the two assessment tasks, Vessels of Water, Vessels of Water (revisited), a mini-whiteboard, a pen, and an eraser.
• Each small group of students will need the cut-up cards Flowing Water and Shape of the Surface of the Water (1) and (2), a large sheet of paper for making a poster, and a glue stick.
• There are some projector resources to support whole-class discussions.

For the introduction a real example, such as soda bottles filled to different levels with colored water, may help students visualize the two-dimensional cross-sections.

In the collaborative activity students may also benefit from seeing real examples of the three-dimensional objects such as cones, hemispheres, cuboids, triangular prisms, and rectangular pyramids.

Time needed

20 minutes before the lesson, a 70-minute lesson, and 20 minutes in a follow-up lesson or as homework. Timings are approximate and will depend on the needs of the class.