Crossing Dimensions: A Math Odyssey into Cross Products

By Angela Smart, Heritage College

$Crossing Dimensions: A Math Odyssey into Cross Products$

At a Glance

Discipline

Mathematics

Instructional Level

College & CEGEP

Course

Linear Algebra and Vector Geometry

Tasks in Workflow

Social Plane(s)

Group

Type of Tasks

Solving problems
Creating & designing
Reviewing & assessing peers
Discussing

Technical Details

Useful Technologies

Zometools

Class size

Small (20-49)

Time

Single class period (< 90 mins)

Inclusivity & Accessibility

Diversity of engagement
Variety of representations

Instructional Purpose

Exploration & inquiry

Overview

The aim of this activity is to allow students to scaffold a geometric visual understanding of vector concepts, in this case, the cross product. The visualization of abstract concepts benefits the students as it helps them grasp the concept by building it themselves in a simplified foundational way.

This activity employs the Active learning approach of inquiry based learning in a lab setting as well as problem based learning. Students are introduced to the geometrical concept of orthogonality and cross products and are then asked to scaffold it using their Zometools in groups of 3-5. They circulate around the classroom to compare their results with other groups. Throughout each of the building steps, there are open-ended questions interspersed. The instructor then links the geometric construction with the algebraic concept to recontextualize the activity.

This activity takes place over the course of one class in a collaborative lab environment.

Instructional Objectives

Students will be able to:

Use their Zometools to scaffold geometrical visual representations of the cross product of vectors in R3.
Use this visual representation, guided by the instructor’s orchestration, to link the geometric and algebraic worlds.

Workflow & Materials

Activity Workflow

View on CourseFlow

Interactive media

Zometool

Angela Smart

Document

Cross Product Instruction Manual

Angela Smart

Contributor's Notes

Benefits

Collaborative classroom setting: helps students scaffold difficult theoretical concepts into simplified geometric models.
Open-ended questions: the activity gives students the opportunity to form the theoretical connections themselves, thus inviting them to a deeper grasp of the concept.

Challenges

ZomeTools Breaking: The biggest challenge is that if the ZomeTools are not handled responsibly, the ends can break off. The company, ZomeTools, knows this and has a free replacement program. Despite this issue, they are still the best tool for vector geometry (I’ve tried many).
As with any active learning activity, classroom management is vital to ensuring the activities success. Make sure the students understand the expectations of the activity as well as the expectations of group work. This can be done at the beginning of this activity or during an earlier class with just a quick reminder before this activity begins.

Tips

Zometools: have enough for each group and demonstrate proper handling as they can break easily.
Classroom setup: use round tables or group desks together for students to work rather than individual desks.

Applied Strategies

Inquiry-Based Learning

Research, Problem solving and Lab-based methods

Published: 03/09/2024

Copyright: © 2025 Smart. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication on this website is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.