098i - Using a 3D Coordinate System with Digital Tools

Lesson 98 — Exploring 3D Position with Dynamic Geometry Software

Strand: Space | Descriptor: AC9M8SP03 | Duration: 45 minutes

Equipment: laptops/tablets with GeoGebra 3D Calculator (app or geogebra.org/3d) for every student or pair.

Learning Intentions

• To use dynamic geometry software to plot points and simple solids in a three-dimensional coordinate system.
• To interpret and manipulate a 3D view (rotating, zooming) to verify coordinate positions.

Success Criteria

I can:

1. Enter 3D coordinates into GeoGebra 3D to plot points accurately.
2. Rotate and navigate the 3D view to check a point’s position from multiple angles.
3. Use digital tools to construct a simple solid (e.g. a cuboid) from vertex coordinates.
4. Use the software to verify a distance or midpoint calculation between two 3D points.

Warmup

(5 minutes — predict then check, pairs with devices)

Before opening GeoGebra, predict on paper: for the point , which axis does it sit closest to, and does it touch the “floor” (-plane)?

Open GeoGebra 3D Calculator. In the input bar, type A=(3,0,4) and press enter.

1. Was your prediction correct? Rotate the view (click-drag) to check the point from above and from the side.
2. Type B=(0,0,0) to show the origin. Does the software draw the axes exactly as we sketched by hand last lesson?

Answers: 1. The point lies in the -plane (since ), not touching the floor (since ) — rotating confirms it floats above the -axis. 2. Yes — GeoGebra’s default view shows (red, towards viewer), (green, across) and (blue, up), matching the convention introduced in Lesson 97.

Activities

Activity 1 — Explicit Instruction: Navigating and Plotting in GeoGebra 3D (10 min)

I do: Demonstrate on the shared screen, narrating each step:

1. Open a new GeoGebra 3D Calculator file.
2. Use the input bar to plot a point: type P=(2,3,1) and press enter. Note the point appears in both the 3D graphics view and the Algebra panel on the left, listing its exact coordinates.
3. Rotate the view by left-click-dragging in the graphics view. Zoom using the scroll wheel or pinch gesture.
4. Toggle the axes and grid using the view settings (the icon in the top-right of the graphics view) to make points easier to see against the plane.

Key digital-tools skill: the Algebra panel is the “single source of truth” — always check a point’s listed coordinates there, not just its visual position, since rotation can make points look misleading from certain angles.

We do: Together plot Q=(0,4,2) and R=(4,0,0). Rotate to view the scene directly from “above” (looking straight down the -axis) — discuss what you see. (Points appear to collapse onto a flat -view, since height is no longer visible from directly above — reinforcing why 3D coordinates need genuine 3D viewing, not just a flat picture.)

You do: In your own GeoGebra file, plot the four points from last lesson’s homework Q2: , , , plus the origin. Rotate to check each against your isometric-paper answers from Lesson 97. Screenshot or note any mismatches to discuss with your partner.

Activity 2 — Guided Practice: Building a Cuboid from Coordinates (12 min)

I do: Demonstrate constructing a simple rectangular prism (cuboid) with one corner at the origin, dimensions .

1. Plot the base rectangle corners: A=(0,0,0), B=(4,0,0), C=(4,3,0), D=(0,3,0).
2. Use the Extrude to Prism tool (or type Prism(A,B,C,D,2) in the input bar) to extrude the base upward by units, generating the full cuboid.
3. Rotate to inspect all vertices. Click on the top face’s vertices to read their coordinates in the Algebra panel — confirm they match the prediction , , , .

We do: Together predict the coordinates of all vertices of a cuboid with base corners E=(0,0,0), F=(5,0,0), G=(5,5,0), H=(0,5,0), extruded to height , before constructing it. Then build it in GeoGebra and check.

You do: Construct your own cuboid in GeoGebra with different dimensions of your choice (minimum ). Record all vertex coordinates in your workbook before checking them against the Algebra panel.

Activity 3 — Inquiry Task: Verifying Distance and Midpoint with Software (14 min)

Pairs, devices. Recall (or predict) how the 2D distance and midpoint formulas might extend to 3D.

For two points and :

1. Choose two points, e.g. and . By hand, calculate the midpoint and the distance between them.
2. Verify using GeoGebra: type Midpoint(M,N) and Distance(M,N) in the input bar (after plotting M and N). Do the software’s answers match your hand calculation?
3. Choose two new points of your own, repeat the hand calculation and software check.
4. Investigate: what happens to the distance formula’s result if both points share the same -coordinate? Test it, then explain why using your understanding of the -plane.

Socratic scaffolding for the investigation in Q4:

Prompt	Purpose
Understand: what does it mean for two points to share a -coordinate?	Both points sit at the same height — the segment joining them is horizontal.
What happens to in the formula when ?	It becomes , dropping out of the sum entirely.
So what does the 3D distance formula reduce to?	The ordinary 2D distance formula, .
Does this match your geometric intuition?	Yes — a horizontal segment’s length only depends on horizontal displacement, exactly like a flat-plane distance.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. In GeoGebra 3D, what panel would you check to confirm a point’s exact coordinates, rather than relying on the rotated view alone?
2. Calculate by hand the midpoint of and .
3. Calculate by hand the distance between and . What do you notice about this answer, and why?
4. Reasoning. Explain why rotating the 3D view before reading a point’s coordinates is good digital-tools practice, rather than trusting a single static screenshot.

Answers: 1. The Algebra panel (left-hand list of exact coordinates). 2. Midpoint . 3. — this matches the familiar -- Pythagorean triple, since both points share and the problem reduces to a 2D distance calculation. 4. A single static view can make points appear closer, further, or aligned than they really are due to perspective and overlap; rotating to multiple angles (or checking the Algebra panel) confirms the true position rather than a misleading visual impression.

Common Misconceptions

Misconception	How to pre-empt it
Trusting the visual position of a point from one fixed viewing angle.	Insist on rotating every construction at least once, and cross-checking the Algebra panel.
Believing GeoGebra’s 3D distance/midpoint tools use different rules to the hand-calculated formulas.	Always verify by hand first, then confirm with software — showing they agree builds trust in both.
Forgetting that extruding a 2D base by height adds only to the -coordinate, not to or .	Predict all vertex coordinates before constructing, then check — mismatches reveal this error directly.
Assuming a “flat” top-down rotated view still shows full 3D information.	Use the warmup’s rotate-from-above task to show that height collapses out of a top-down view.
Typing coordinates in the wrong order, e.g. .	Model the consistent order every time; check the Algebra panel labels to confirm entry order was correct.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Two points have the same - and -coordinates but different -coordinates. Describe the direction of the segment joining them, and simplify the distance formula for this case.

Answer

The segment runs purely in the -direction. Since and , the distance formula reduces to — simply the difference in -coordinates.

E2 (AMC Junior style). A cuboid has vertices at and as opposite corners (a “space diagonal”). Find the length of this diagonal.

Answer

units.

E3 (Challenge). In GeoGebra, a student constructs a cuboid and finds the midpoint of one space diagonal is , with one vertex known to be the origin. Find the coordinates of the opposite vertex, and hence the cuboid’s dimensions.

Answer

Since the midpoint of and the opposite vertex is , doubling gives the opposite vertex . The cuboid’s dimensions are .

Homework

(Requires access to GeoGebra 3D, or paper/pencil equivalents using the formulas if unavailable.)

1. In GeoGebra, plot the points , , , and extrude to height to build a cuboid. Record all vertex coordinates.
2. Calculate by hand the midpoint of and , then verify with software if available.
3. Calculate by hand the distance between and .
4. Explain in your own words why the 3D distance formula has three squared terms instead of the two used in the 2D formula.
5. Reasoning. A student rotates a GeoGebra 3D scene and claims two points “look like” they’re in the same position. Explain what they should do to confirm or disprove this, and why relying on the visual alone is risky.
6. Challenge. A cuboid has one vertex at and the opposite vertex at . Find (a) the midpoint of the space diagonal (b) the cuboid’s dimensions (c) the length of the space diagonal.

Answers: 1. Vertices: , , , , , , , . 2. Midpoint . 3. . 4. 3D space has an extra independent direction (height/) beyond the 2D plane’s two, so an extra squared term accounts for displacement in that third direction, by the same Pythagorean reasoning extended one dimension further. 5. Check the exact coordinates in the Algebra panel, or rotate to a different angle — apparent overlap from one viewpoint does not guarantee the points are actually at the same location in 3D. 6(a) midpoint (b) dimensions , , , so (c) diagonal .