--- title: "Coordinates in 3D" subtitle: "Angles and shapes · Solids · Lesson 21" --- # Coordinates in 3D **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Two drones are both above the same spot in a park. How can a controller tell where each one is, so they don't crash? Two numbers are not enough. :::teacher-note **In one sentence:** Students plot and read points (x, y, z) on 3D axes, and use 3D systems for drones, buildings, planes and 3D printers (AC9M8SP03, nodes 1 and 2). **Students should be able to say:** I can plot and read a point (x, y, z) in 3D. I can say what each coordinate means in a real system, such as a drone's position. I can describe positions with latitude, longitude and altitude. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict The $z$-axis points up. A point is $(x, y, z)$: - $x$: how far along the x-axis - $y$: how far along the y-axis - $z$: how far up The **origin** is $(0, 0, 0)$. Points with $z = 0$ are on the floor (the x-y plane). The order is always $x$, then $y$, then $z$. Real systems use the same idea with other names: | System | Three numbers | |---|---| | a drone | metres east, metres north, metres up | | a building | metres across, metres back, metres up (about 3 m a floor) | | a plane | latitude, longitude, altitude | Each real system has its own fixed order. Latitude (how far north or south) is always written before longitude (how far east or west), then altitude. :::keep-together **A point is at (2, 5, 3). How far up is it?** ☐ 2 ☐ 3 ☐ 5 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 3. The third number, z, is the height. The point is 2 along x and 5 along y. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Reading points :::keep-together ### Reading a point Here is the routine. 1. $\text{5 along } x$ — The first number. 2. $\text{0 along } y$ — The second number: no distance is 0. 3. $\text{2 up}$ — The third number, z. 4. $(5, 0, 2)$ — Write them in order, in brackets. **P is 4 along the x-axis, 1 along the y-axis and 6 up. What are its coordinates?** ☐ (4, 6, 1) ☐ (6, 4, 1) ☐ (1, 4, 6) ☐ (4, 1, 6) ::: :::teacher-note **Answer:** (4, 1, 6) Always x, then y, then z: (4, 1, 6). ::: :::keep-together ### Find the mistake **Kim records a drone's position. Click the line where she goes wrong.** ``` 1 A drone is 5 m east (x), 2 m north (y) and 3 m up (z) from its base. 2 Kim writes its position as (3, 5, 2), with the height first because it matters most. 3 Anyone reading (3, 5, 2) would put the drone 3 m east, 5 m north and 2 m up. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: the order is always (x, y, z), so the drone is at (5, 2, 3). Anyone reading (3, 5, 2) puts it 3 m east and 2 m up. ::: :::keep-together ### Latitude, longitude and altitude **A plane's position is given as latitude 34° S, longitude 151° E and altitude 10 000 m. Which number plays the part of z?** ☐ The longitude ☐ The altitude ☐ The latitude ☐ None of them ::: :::teacher-note **Answer:** The altitude Latitude and longitude place it over the ground; altitude says how high, like z. ::: :::typical ## Boxes, axes and floors **A box of 24.** **On the z-axis.** **Floors.** **z = 0.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## 3D positions :::keep-together ### A plane climbs **A plane is at latitude 33.9° S, longitude 151.2° E and altitude 9000 m. It climbs straight up 1000 m. Where is it now?** ☐ 33.9° S, 152.2° E, 9000 m ☐ 33.9° S, 151.2° E, 10 000 m ☐ 34.9° S, 151.2° E, 9000 m ☐ 33.9° S, 151.2° E, 1000 m ::: :::teacher-note **Answer:** 33.9° S, 151.2° E, 10 000 m Only the altitude changes: 9000 + 1000 = 10 000 m. Latitude and longitude still place it over the same spot (near Sydney). ::: :::keep-together ### Flying up **A drone at (6, 4, 10) flies straight up 5 m. What is its new z-coordinate?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 15 Up changes only z: 10 + 5 = 15. x and y stay 6 and 4. ::: :::keep-together ### A 3D printer **A 3D printer's nozzle is at (20, 35, 2) mm. It prints the next layer 0.2 mm higher. Where is it?** ☐ (20.2, 35, 2) ☐ (20.2, 35.2, 2.2) ☐ (20, 35, 2.2) ☐ (20, 35.2, 2) ::: :::teacher-note **Answer:** (20, 35, 2.2) A new layer is higher, so only z changes: 2 + 0.2 = 2.2 mm. ::: :::keep-together ### Review: views **Review: a pile of cubes is seen from above, from the front and from the side. Which view shows how tall the pile is?** ☐ All three equally ☐ The top view ☐ The front view (or the side view) ☐ None of them ::: :::teacher-note **Answer:** The front view (or the side view) From above you see the floor (x and y); from the front or side you see the heights (z). ::: ## Make: a corner of a box Then write the coordinates of all eight corners of the box, and check two of them on the board. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A drone is 8 m east, 3 m north and 20 m up from its base, with x east, y north and z up. Write its position. Then it flies 5 m north: where is it now?** {{lines-3}} ::: :::teacher-note **Answer:** (8, 3, 20), then (8, 8, 20). :::