3D coordinates: a week of ready-to-teach maths

Five lessons for Year 8 Space. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus 3D grids drawn the same way as the interactive unit, twelve triple cards, steps for a free 3D graphing tool, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page, and book computers or tablets for Day 4.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet. The grids are on the worksheets.
3. Print the two cut-out sheets, one copy per pair: two blank 3D grids (any day) and the triple cards (Day 2).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

On a flat page, a pair (x, y) fixes a point. In the space around us we need a third number. Three axes meet at right angles at the origin (0, 0, 0): here x runs across, y runs back, and z runs up. The ordered triple (2, 3, 1) means 2 along x, then 3 along y, then 1 up z, always in that order. A triple with a zero sits on a flat plane through two of the axes, and one with two zeros sits on an axis. The same position can also be told in other ways: in words, as moves from a starting point, or from a view from the top and a view from the front.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

The drawings

The 3D grids in this pack, like the unit, draw the floor at a slant: x straight across, y back and up to the right, and z straight up. A slanted grid is a picture, not a measurement: lengths along y look shorter than they are. Plot by counting squares, never by measuring with a ruler.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 grows a pair into a triple; Day 2 sets up the axes and plots points; Day 3 describes a solid with its corners; Day 4 moves the same work into a 3D graphing tool; and Day 5 compares different ways of saying where something is. It builds on the Cartesian plane and transformations in Year 7, and on drawing solids from different views.

Materials for the week (one trip)

• From the classroom: pencils, rulers, coloured pencils, this pack printed.
• Cut-out sheet 1, one per pair: two blank 3D grids.
• Cut-out sheet 2, one per pair: 12 triple cards (Day 2), and scissors.
• A computer or tablet with a free 3D graphing tool (dynamic geometry software), one per pair (Day 4).
• Optional: a box and some interlocking cubes to show a corner and the three edges that meet there.

Words for the word wall

dimension, axis, axes, origin, ordered triple, coordinate, x-coordinate, y-coordinate, z-coordinate, plane, corner, vertex, edge, centre, top view, front view, position, location

Dear families

This week in maths, Year 8 is describing where things are in three dimensions, with three numbers: how far across, how far back and how far up. They also use a free 3D graphing tool on a computer.

Try this at home

• Pick a corner of a room as the start. Describe where a light switch is: how far along, how far out, how far up.
• Find a seat number on a ticket for a concert or a game: which number tells the row, the seat and the level?
• Ask: why does a drone need a height as well as a position on a map?
• Build a small tower of blocks and describe each block with three numbers.

My positions this week

Fill one row a day. Name something, give its position as a triple, and say what each number means.

Printed from the free seegongsik 3D coordinates teaching pack · seegongsik.com/au/y8/space/AC9M8SP03/pack

From two numbers to three

A pair of numbers fixes a point on a flat page; in space we need three. Students read points on a flat grid, then find books in a bookcase by slot, row and shelf, in words and as ordered triples.

We are learning to

• read and plot points as pairs on a flat grid,
• explain why a position in space needs three numbers,
• describe a position in words and as an ordered triple.

Success criteria

• I can explain why two numbers are not enough to find a book when each shelf has two rows.
• I can turn “slot 4, back row, shelf 3” into a triple and back again.

You need

The worksheet each; a bookcase or a box of compartments to point to, if you have one.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the bookcase.

Watch for these ideas

• Thinking a pair is always enough because the page is flat.
• Writing the three numbers in any order.
• Counting from 1 at the origin instead of from 0.

Answers

• Part A: A(1, 4), B(5, 2), C(3, 0), D(0, 3); P is 4 across and 1 up; C is on the x-axis and D on the y-axis.
• Part B: K slot 4, back row, shelf 3, (4, 2, 3); L slot 2, front row, shelf 1, (2, 1, 1); M slot 5, front row, shelf 4, (5, 1, 4). Book N is at (1, 2, 2).
• Part C: no: (4, 2, 3) is slot 4, back row, shelf 3, but (3, 2, 4) is slot 3, back row, shelf 4. The bookcase has 5 × 2 × 4 = 40 places.
• Part D: a plane needs a position east or west, a position north or south, and a height.

From two numbers to three

A. On a flat grid

B. A bookcase

In this model each book is a dot. x is the slot (1 to 5 from the left), y is the row (1 at the front, 2 at the back) and z is the shelf (1 to 4 from the bottom).

C. The order matters

Mia says (4, 2, 3) and (3, 2, 4) are the same book, because they use the same numbers. Is that right? How many book places does the bookcase have?

D. Stretch: in the sky

A town on a map needs two numbers. Why does a plane in the sky need three? Say what each number could be.

Axes and ordered triples

Three axes meet at the origin, and every triple is a set of moves from it. Students plot points on a 3D grid by stepping along x, then y, then up z, read points back, and sort triple cards by where each point lies.

We are learning to

• name the three axes and the origin (0, 0, 0),
• plot and read a triple by moving along x, then y, then up z,
• say where a point lies from its zeros: on an axis, on a flat plane, or out in space.

Success criteria

• I can plot (4, 2, 3) on a 3D grid and read it back.
• I can explain why (3, 0, 2) lies on a flat plane and (0, 0, 3) lies on an axis.

You need

Cut-out sheet 2 (the triple cards) and scissors, one set per pair; the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Going up z before along y, or swapping the numbers.
• Measuring along the slanted y-axis with a ruler instead of counting.
• Thinking a 0 means the point is not there.

Answers

• Part A: A (3, 0, 0), B (2, 3, 0), C (4, 2, 3), D (0, 1, 2); A is on the x-axis, B on the floor, D on the side wall (x = 0), and C off every plane.
• Part B: E (5, 1, 2), F (2, 4, 1), G (0, 3, 3).
• Part C: on an axis 1, 6, 11; on a flat plane 3, 4, 7, 10 (3 the floor (z = 0), 4 the side wall (x = 0), 7 the front wall (y = 0), 10 the floor (z = 0)); out in space 2, 5, 8, 9, 12.
• Part D: on the floor 3 × 3 = 9 points; in the whole block 3 × 3 × 3 = 27.

Axes and ordered triples

A. Plot the points

Plot A (3, 0, 0), B (2, 3, 0), C (4, 2, 3), D (0, 1, 2). For each, draw the path from the origin: along x, then along y, then up z.

B. Read the points

C. Where is it?

Sort the twelve triple cards. Record the card numbers. For each card on a flat plane, say which one: the floor (z = 0), the side wall (x = 0) or the front wall (y = 0).

D. Stretch: counting points

How many points with whole-number coordinates from 0 to 2 lie on the floor (z = 0)? How many in the whole block from (0, 0, 0) to (2, 2, 2)?

Corners of a box

A box set square to the axes is described completely by its corners. Students name the corners of a drawn box, list all eight from two opposite corners, find the centre and the number of unit cubes, and see which number changes along each edge.

We are learning to

• write the triples of the corners of a box,
• list all eight corners from two opposite ones,
• find a box’s centre and the number of unit cubes it holds.

Success criteria

• I can list the eight corners of the box from (0, 0, 0) to (5, 3, 2).
• I can say which coordinate changes as I move along an edge.

You need

The worksheet each; a box and some interlocking cubes to show, if you have them.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Listing only the corners you can see, and missing the hidden back corner.
• Taking a box’s far corner as its size when it does not start at the origin.
• Halving only one coordinate to find the centre.

Answers

• Part A: P (5, 0, 0), Q (0, 3, 2), R (5, 3, 2), S (0, 0, 2), T (5, 3, 0).
• Part B: (0, 0, 0), (5, 0, 0), (5, 3, 0), (0, 3, 0), (0, 0, 2), (5, 0, 2), (5, 3, 2), (0, 3, 2); centre (2.5, 1.5, 1); 5 × 3 × 2 = 30 unit cubes. Along each edge one coordinate changes and two stay the same.
• Part C: 3 by 4 by 3, so 36 cubes; corners (1, 2, 0), (4, 2, 0), (4, 6, 0), (1, 6, 0), (1, 2, 3), (4, 2, 3), (4, 6, 3), (1, 6, 3); centre (2.5, 4, 1.5).
• Part D: centre (1.5, 1.5, 1.5); (2, 1, 4) is not inside: its z is 4, above the top at z = 3.

Corners of a box

A. Name the corners

B. All eight corners

List all eight corners of the box: the four on the base, then the four on top.

The centre of the box = ______   Unit cubes that fill the box = ______

C. A box anywhere

A box has opposite corners at (1, 2, 0) and (4, 6, 3). How long, wide and high is it? How many unit cubes fill it? List its eight corners and find its centre.

D. Stretch: inside or not?

A cube has opposite corners at (0, 0, 0) and (3, 3, 3). Where is its centre? Is the point (2, 1, 4) inside the cube? How can you tell from the numbers alone?

A 3D graphing tool

Dynamic geometry software draws points in space from their triples and lets us turn the view. Students plot points, build a cube and a staircase from their corners, and turn the view to check positions a flat drawing cannot settle.

We are learning to

• enter points as ordered triples in a 3D graphing tool,
• join points to build a solid from its corners,
• turn the view to check where a point really is.

Success criteria

• I can build the cube with corners from (0, 0, 0) to (2, 2, 2).
• I can explain why two points can look the same from one side but be in different places.

You need

A computer or tablet with a free 3D graphing tool per pair; the worksheet each. Try the steps in your tool first: most accept a point typed as A = (2, 3, 1). With no devices, do Parts B and C on the 3D grids from cut-out sheet 1.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the staircase.

Watch for these ideas

• Typing the numbers in the wrong order, or without the brackets and commas.
• Trusting one fixed view: two points can line up from one side.
• Missing the hidden corners of the cube.

Answers

• Part A: A and B line up from the front (only y differs); B and C line up from the top (only z differs).
• Part B: (0, 0, 0), (2, 0, 0), (2, 2, 0), (0, 2, 0), (0, 0, 2), (2, 0, 2), (2, 2, 2), (0, 2, 2); 12 edges, three meeting at each corner.
• Part C: the back copy is (0, 2, 0), (1, 2, 0), (1, 2, 1), (2, 2, 1), (2, 2, 2), (3, 2, 2), (3, 2, 3); join each point to its copy.
• Part D: the top is (2, 2, 3): halfway along x and y puts it over the centre of the base, and 3 up gives the height.

A 3D graphing tool

Open the 3D graphing tool. Type each point as a name and a triple, such as A = (2, 3, 1). Drag the view to turn it, and use the tool’s segment or line command to join points.

A. Points that line up

Plot A (2, 1, 2), B (2, 4, 2), C (2, 4, 0). Turn the view until two of them sit exactly in line. Which two line up from the front? Which two from the top? Why?

B. A cube

1. Plot the eight corners of the cube with one corner at (0, 0, 0) and the opposite corner at (2, 2, 2).
2. Join the corners to make the twelve edges. Turn the view to check none is missing.

C. A staircase

1. Plot and join, in order: (0, 0, 0), (1, 0, 0), (1, 0, 1), (2, 0, 1), (2, 0, 2), (3, 0, 2), (3, 0, 3). This is the side of a staircase on the front wall (y = 0).
2. Make a copy of every point with y = 2, join the copies in order, then join each point to its copy.

Write the triples of the copies:

D. Stretch: a pyramid

Build a pyramid on the square base (0, 0, 0), (4, 0, 0), (4, 4, 0) and (0, 4, 0), with its top 3 units above the centre of the base. What is the triple of the top? Why?

Different ways to say where

The same position can be told in more than one way. Students describe one point as a triple, as moves and as views, find a point from a top view and a front view, locate seats, and play the self-check quiz as a class game.

We are learning to

• describe a position as a triple, as moves from a start, and from views,
• find a triple from a top view and a front view,
• use triples for real places, such as seats.

Success criteria

• I can find (2, 4, 3) from a top view of (2, 4) and a front view of (2, 3).
• I can say what each number in a seat triple means.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the seats and the quiz.

Watch for these ideas

• Expecting a top view to show the height.
• Reading a seat triple in the wrong order.
• Thinking two descriptions name different points because they look different.

Answers

• Part A: 3 across, 2 back, 4 up; from the top (3, 2); from the front (3, 4).
• Part B: (2, 4, 3); (5, 1, 0), a point on the floor, since the front view shows no height.
• Part C: 3 seats along; (12, 5, 3); (12, 7, 2) is 2 rows further back.
• Part D: (30, 40, 50); along the ground 50 m (30² + 40² = 50²); in a straight line about 70.7 m (50² + 50² = 5000).

Different ways to say where

A. One point, four ways

The point (3, 2, 4) is described four ways. Fill the gaps.

1. As a triple: (3, 2, 4)
2. As moves from the origin: ______ across, ______ back, ______ up
3. Looking down from the top, it sits over the floor point ( ______ , ______ )
4. Looking from the front, it lines up with the point ( ______ , ______ ) on the front wall, across and up

B. From two views

1. From the top a point looks like (2, 4), across and back. From the front it looks like (2, 3), across and up. What is its triple?
2. From the top a point looks like (5, 1); from the front, (5, 0). What is its triple, and where is the point?

C. Seats

Tickets give a seat as (seat, row, tier), with rows numbered from the front. My seat is (12, 5, 2) and my friend’s is (15, 5, 2). How many seats along is my friend? Which seat is directly above mine, on the next tier up? Is (12, 7, 2) in front of me or behind?

D. Stretch: a drone

A drone leaves the origin, flies 30 m east and 40 m north, then climbs 50 m. Write its position as a triple (east, north, up). How far is it from the start along the ground? In a straight line?

Two blank 3D grids

For any day, one sheet per pair

x runs across, y runs back at a slant, and z runs straight up. To plot a point, count along x, then along y, then up z, and mark it. Draw the path as dashed lines so others can check it.

Twelve triple cards

For Day 2, one set per pair

Cut out the cards. Is each point on an axis, on a flat plane through two axes, or out in space? The zeros will tell you.

Teacher note: on an axis 1, 6, 11; on a flat plane 3, 4, 7, 10; out in space 2, 5, 8, 9, 12. No row or column of cards gives a pile away.

What we know: 3D coordinates

Work on your own. Use the axes as in class: x across, y back, z up.

1. What is the triple for the origin? Name the three axes and the way each one runs. ______________________
2. Describe how to plot (4, 1, 3), starting at the origin. ______________________
3. Which of (0, 6, 0), (2, 0, 5), (3, 3, 3) lies on an axis? Which on a flat plane? Which out in space? ______________________
4. A box has one corner at (0, 0, 0) and the opposite corner at (6, 2, 3). List its four top corners. ______________________
5. Find the centre of that box, and how many unit cubes fill it. ______________________
6. From the top a point looks like (1, 5); from the front, (1, 2). What is its triple? ______________________
7. Leo typed B = (3, 2, 4) into a graphing tool, but wanted the point 3 across, 4 back and 2 up. What should Leo have typed? ______________________
8. A seat is (8, 3, 1) as (seat, row, tier). Which seat is directly above it, on the next tier? ______________________

Answers and marking guide

Answers with working

1. (0, 0, 0); x runs across, y runs back, z runs up, all at right angles.
2. 4 along x, then 1 along y, then 3 up z.
3. (0, 6, 0) on an axis; (2, 0, 5) on a flat plane, the front wall (y = 0); (3, 3, 3) out in space.
4. (0, 0, 3), (6, 0, 3), (6, 2, 3), (0, 2, 3).
5. Centre (3, 1, 1.5); 6 × 2 × 3 = 36 unit cubes.
6. (1, 5, 2): the top view gives x and y, the front view gives x and z.
7. (3, 4, 2): the second number is the distance back and the third is the height.
8. (8, 3, 2): the same seat and row, one tier up.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions and keeps the three numbers in their agreed order.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: from pairs to triples, axes and triples, corners of a box, the graphing tool, and ways to say where.