Your group gets three sheets of stiff cardboard — three planes, if you squint — and permission to slice them through one another. In three-space, what are all the ways three planes can sit?
The task
Catalogue every configuration. Start with two sheets: parallel, or meeting in a line — is there anything else? Then bring in the third, and for each arrangement answer the only question that matters: what do all three share — a point, a line, a whole plane, or nothing at all? Draw each configuration, count them, and defend the claim that your catalogue is complete. Then the bridge to algebra: a plane is the solution set of one equation , so three sheets are three equations. Match each configuration in your catalogue to what solving the system would report — exactly one solution, infinitely many, or none — and find the configurations the count alone cannot tell apart.
Facilitation notes — for the teacher
Sheets with pre-cut slots beat imagination — hands find arrangements that sketches miss. The configuration groups reliably overlook is the triangular prism: three sheets meeting pairwise in three parallel lines, sharing nothing — leave straws or skewers out so groups can lay one along each intersection line and see the three lines refuse to meet. The fight worth having: “no common point” comes in flavours, and so does “infinitely many”. Fast groups: which solution counts map to more than one picture, and what extra question separates them? Groups who met the cross product can chase the elegant test: dot one normal against the cross product of the other two, and ask what a zero is whispering.
What mathematics tends to surface
The complete catalogue, argued rather than listed — coincident, parallel, pencils of planes sharing a line, the prism, and the single clean point. Then the correspondence that powers the rest of the unit: intersection is shared solutions, and the count — none, one, infinitely many — is geometry’s shadow in the algebra. Equations of Planes and Intersections of Lines and Planes make both directions official.
Where it leads
Solving three equations in three unknowns now comes with pictures attached, and every answer earns a geometric interpretation. In The Flight Path, one line meets one plane in one point — and that point is where your wheels touch.
The answer is not on this page
The catalogue, its count, and the matching all happen at the boards — cardboard first, algebra second.
Curriculum connection
C3.3
determine, through investigation using a variety of tools and strategies (e.g., modelling with cardboard sheets and drinking straws; sketching on isometric graph paper), different geometric configurations of combinations of up to three lines and/or planes in three-space (e.g., two skew lines, three parallel planes, two intersecting planes, an intersecting line and plane); organize the configurations based on whether they intersect and, if so, how they intersect (i.e., in a point, in a line, in a plane)
Link to original
C4.4
recognize a scalar equation for a plane in three-space to be an equation of the form whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes Sample problem: Determine the equation of a plane that intersects the planes , , and , , in a single point. Determine the equation of a plane that intersects and in more than one point.
Link to original