099i - Problem Solving with 3D Position and Location

Lesson 99 — Problem Solving and Consolidation: 3D Position and Location

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

Learning Intentions

• To apply 3D coordinate reasoning to solve practical position and location problems.
• To consolidate the connection between informal 3D description, formal coordinates, and calculated distances/midpoints.

Success Criteria

I can:

1. Interpret a real-world position problem and represent it using 3D coordinates.
2. Calculate distances and midpoints between points in 3D to solve applied problems.
3. Reason about symmetry, alignment, and relative position using coordinates.
4. Justify my solution using both calculation and geometric reasoning.

Warmup

(6 minutes — “always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Justify with an example or counterexample.

1. Two points with the same -coordinate lie at the same height.
2. If a point’s -coordinate is , the point lies on the -axis.
3. The midpoint of two points always has three coordinates that are whole numbers, if the original points do.
4. The distance between two distinct points in 3D can be zero.

Answers: 1. Always — measures height, so equal means equal height, regardless of and . 2. Never — means the point lies in the -plane, not necessarily on the -axis (which additionally needs and ). 3. Sometimes — true only when both corresponding coordinates sum to an even number, e.g. midpoint of and is , but midpoint of and is . 4. Never — distinct points are, by definition, not in the same location, so the distance formula always gives a positive result for two different points.

Activities

Activity 1 — Applied Problems: Navigation and Construction (14 min)

Pairs. Every answer must show full working and a one-sentence justification in context.

Problem 1. A warehouse drone starts at position (metres: east, north, height) and needs to deliver a parcel to shelf position . Find the straight-line distance the drone would travel if it could fly directly, ignoring obstacles.

Problem 2. Two security cameras are mounted at and in a warehouse (metres). A motion sensor is to be installed at the exact midpoint between them, at the same height. Find the sensor’s coordinates.

Problem 3. A climbing wall has two handholds at and (metres: across, depth into wall, height). A climber claims the second handhold is directly above the first. Is this correct? Justify using the coordinates.

Problem 4. A rectangular prism-shaped storage crate has one corner at and the opposite corner at (metres). Find (a) the crate’s dimensions (b) the length of the internal diagonal from corner to corner (c) the coordinates of the exact centre of the crate.

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what would “directly above” mean in coordinates?	Same and same (depth), only (height) differs.
Compare the two points’ -coordinates.	Both are — matches.
Compare the two points’ -coordinates.	versus — these differ.
So is the claim correct?	No — the second handhold is also m further into the wall’s depth direction, not purely above.
Looking back — how would you describe the true relationship?	The second handhold is up and further along the depth direction, not vertically above.

Answers: 1. m. 2. Midpoint . 3. Not correct — -coordinates differ ( vs ), so the second handhold is not purely above the first; it is also offset in the depth direction. 4(a) m (b) m (c) .

Activity 2 — Guided Practice: Reasoning about Symmetry and Alignment (10 min)

I do: “A drone hovers at . A charging pad sits at , directly below.” Since and match exactly and only differs, the drone is directly above the pad, at a height of m. This is a vertical alignment check — matching and coordinates guarantees one point is directly above/below the other.

We do: Together check whether and are vertically aligned, and find the vertical distance between them if so.

You do: For each pair of points, state whether they are vertically aligned (directly above/below each other), and explain your reasoning:

(a) and

(b) and

(c) and

(d) and

Activity 3 — Inquiry Task: Locating the Missing Vertex (15 min)

Pairs. A rectangular prism-shaped shipping container is being modelled in a 3D coordinate system, with axes aligned to its edges. Six of its eight vertices are known:

Two vertices are missing. Use reasoning about the shape of a rectangular prism — not guesswork — to find their coordinates.

Then answer:

• What is the length of the container’s longest internal diagonal (corner to opposite corner)?
• A cable needs to run from the vertex to the centre of the top face. Find the coordinates of that centre point, and the length of the cable (assuming it runs in a straight line, not along the edges).
• Extend: if the container’s dimensions were doubled in every direction, what would happen to the length of the internal diagonal? Justify using your formula, not just intuition.

Socratic scaffolding:

Prompt	Purpose
Understand: what property must all 8 vertices of a rectangular prism satisfy?	Each vertex’s , , values must each come from exactly one of two possible values (the prism’s two extents in each direction).
What are the two possible values in each direction, based on the known vertices?	, , .
Devise a plan: list all 8 possible combinations, and find which two are missing.	Systematically pair each with each and — combinations total.
Carry it out	The missing vertices are and .
Looking back	Do all 8 vertices now form a sensible box shape when checked against the known 6?

Top face centre ; cable length m.

Doubling extension: doubling every dimension gives diagonal — exactly double the original diagonal, since doubling each term inside the square root before summing is equivalent to factoring out from the sum, and .

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Find the distance between and . What do you notice, and why?
2. Two points are and . Are they vertically aligned? Find the vertical distance between them.
3. Find the midpoint of and .
4. Reasoning. A rectangular prism has 6 known vertices as in Activity 3’s structure, with , , . State the coordinates of all 8 vertices without being told any of them individually.

Answers: 1. — this reduces to the familiar -- Pythagorean triple since for both points. 2. Yes — matching and ; vertical distance . 3. . 4. , , , , , , , .

Common Misconceptions

Misconception	How to pre-empt it
Assuming two points are “aligned” if any one coordinate matches.	Require all relevant coordinates to match for a specific type of alignment (e.g. both and for vertical alignment).
Guessing missing prism vertices instead of reasoning from the two-values-per-axis structure.	Model the systematic “list all combinations” method explicitly.
Forgetting the -term when two points happen to share the same height, mistakenly treating it as a genuinely 2D problem without checking.	Always write out the full 3D distance formula first, then simplify once a zero term is confirmed.
Believing doubling a shape’s dimensions doubles its diagonal length “because everything doubles”.	Require the algebraic check (factoring from the sum) rather than accepting the claim on intuition alone, since this reasoning does not hold for all transformations (e.g. non-uniform scaling).
Confusing the midpoint formula (averages) with the distance formula (differences, squared, summed, rooted).	Keep both formulas displayed side by side during problem-solving activities.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A fly starts at corner of a room measuring and walks in a straight line to the opposite corner. Find the distance travelled, correct to decimal place.

Answer

E2 (Kangaroo style). Four points are the vertices of a square lying flat in the plane : , , , . Find the coordinates of the point directly above the square’s centre, at a height of .

Answer

Centre of the square (in ) , at height . Directly above at height means increasing only: .

E3 (Challenge). Three vertices of a rectangular prism are known: , , . A fourth known vertex is . Determine all 8 vertices, then find the total edge length (sum of all 12 edges) of the prism.

Answer

From the given points, , , and since has , . All 8 vertices: , , , , , , , .

A rectangular prism has edges of each of its distinct lengths (, , ):

Homework

1. Find the distance between and .
2. Find the midpoint of and .
3. Two points are and . Explain why they are vertically aligned and find the vertical distance between them.
4. A rectangular prism has known vertices including , , . List all vertices.
5. Reasoning. A student says “the distance formula in 3D is just Pythagoras’ theorem used twice.” Explain what they mean, and whether this is an accurate description.
6. Challenge. A ladder leans from the ground point to a point on a wall at … but a gust of wind shifts its base to while the top stays fixed at . Find the new length of the ladder, and state whether the ladder has become longer, shorter, or stayed the same length, given it is rigid. What does this tell you about the top point’s actual new height?

Answers: 1. . 2. . 3. Both share , so they lie on the same vertical line; vertical distance . 4. , , , , , , , . 5. Accurate in spirit: the 3D formula can be seen as applying Pythagoras once to combine and displacements into a “flat” diagonal, then applying it again to combine that flat diagonal with the displacement — two successive right-angled-triangle calculations chained together. 6. Since the ladder is rigid, its length cannot change — it must still be units from base to a fixed top point only if the base stays at ; since the base moved but the length must stay at (rigid ladder) and the top’s stayed at , the top’s height must actually have changed — solving gives , so the top has slipped down slightly, to about m, not stayed at m as first stated.