Representing objects in two dimensions: a week of ready-to-teach maths
Five lessons for Year 7 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 isometric dot paper, patterns of six squares to fold, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets at 100%: isometric dot paper (Day 2), two per student, and the patterns of six squares (Day 3), one per pair.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
A solid has three dimensions and a page has two, so every drawing of a solid gives something up. Front, side and top views show true shapes and sizes, but each shows only one direction and they must be read together. An isometric drawing looks solid and keeps lengths true along three directions, but it bends right angles into 60° and 120° and can hide cubes at the back. A net shows every face flat and true, ready to cut and fold, but not what the solid looks like. Choosing a representation means asking what the reader needs.
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.
How we draw views
The front view is seen from straight in front, with columns left to right. The right side view is seen from the right, so the front of the solid is on its left. The top view is seen from directly above, with the back of the solid at the top of the page. Each view is drawn on a square grid, one square for each cube face that shows. Interlocking cubes are needed on Days 1 and 4.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen and on the desk |
|---|---|---|---|
| 1 | Front, side and top views | Build small cube models, draw their front, right side and top views on square grids, and see that one view is not enough | Many views of one object; interlocking cubes |
| 2 | Isometric drawings | Draw cuboids and cube models on isometric dot paper, measure the angles a drawing bends, and draw a solid from its views | Isometric versus a true view; isometric dot paper |
| 3 | Nets | Predict which patterns of six squares fold into a cube, fold them to check, find opposite faces, and draw nets of prisms and pyramids | A net folds into the solid; patterns to cut and fold |
| 4 | From views to solids | Build solids that fit a set of views, test height maps against views, and find the fewest and most cubes that fit | Rebuild the solid from its views; interlocking cubes |
| 5 | Choosing a representation | Weigh up what views, isometric drawings and nets show and hide, and choose one to suit a purpose | Trade-offs between representations; self-check quiz as a class game |
How the week builds
Days 1 and 2 draw solids two ways, as flat views and as isometric drawings; Day 3 unfolds them into nets; Day 4 runs the views backwards to rebuild solids; and Day 5 weighs up what each representation shows and hides. It builds on cross-sections and right prisms (AC9M6SP01) and leads into the volume of right prisms in Year 8 (AC9M8M02).
Materials for the week (one trip)
- From the classroom: pencils, rulers, protractors, scissors, sticky tape, this pack printed at 100%.
- Interlocking cubes, about 12 for each pair (Days 1 and 4).
- Cut-out sheet 1, two per student: isometric dot paper (Day 2).
- Cut-out sheet 2, one per pair: six patterns of six squares (Day 3).
- A cereal box or a tissue box to unfold, if you have one (Day 3).
Words for the word wall
solid, face, edge, vertex, view, front view, side view, top view, plan, isometric, net, fold, cube, cuboid, prism, pyramid, height map, representation, advantage, disadvantage
Dear families
This week in maths, Year 7 is drawing solid objects on flat paper in different ways (views from the front, side and top, isometric drawings and nets) and deciding what each way shows well and what it hides.
Try this at home
- Unfold an empty box carefully and look at its net. How many faces does it have, and which faces are opposite?
- Stack a few blocks or books and sketch them from the front, from the side and from above.
- Look at flat-pack furniture instructions together. Which drawings show the parts, and which show how they fit?
- Ask: why might a floor plan be more useful than a photo when buying a couch?
Drawings this week
Fill one row a day with an object, how you drew it, and one thing the drawing does not show.
| Day | The object | How we drew it | What it does not show |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Representing objects in two dimensions teaching pack · seegongsik.com/au/y7/space/AC9M7SP01/pack
Front, side and top views
A view shows a solid from one direction as a flat picture. Students build small cube models, draw their front, right side and top views on square grids, and see that one view alone does not fix the solid.
We are learning to
- draw the front, right side and top views of a solid made of cubes,
- read a set of views and picture the solid,
- explain why one view is not enough.
Success criteria
- I can draw the three views of a cube model on a square grid.
- I can find two solids with the same front view and a view that tells them apart.
You need
Interlocking cubes (about 12 for each pair), the worksheet and a pencil.
Lesson flow (about 60 minutes)
| 10 min | Three directions Show the board (see the box). Ask: “Which views would change if we added one cube behind the front row?” |
| 35 min | Build and draw Pairs build each solid in Part A with cubes, turn it to look from each direction, and draw the views; then Part B. |
| 15 min | One view is not enough Part C as a class; early finishers do Part D. Ask: “How many views do you need before only one solid fits?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Drawing the solid in 3D instead of a flat view.
- Mirroring the right side view. Seen from the right, the front of the solid is on the left.
- Missing a taller stack at the back: it shows above the front row in the front view.
Answers
- Part A (column heights, the right side from the front row): Solid 1, front 2, 1, 1, right side 1, 2, top 3 squares wide and 2 deep, without the front right; Solid 2, front 2, 3, right side 2, 1, 3, top all 2 squares wide and 3 deep; Solid 3, front 3, 2, right side 2, 3, top all 2 squares wide and 2 deep.
- Part B: answers vary with where the stacks of 2 stand. The right side view can be 1, 2 or 2, 1 or 2, 2.
- Part C: both front views are 2, 1; the right side views tell them apart (Solid X 1, 2, Solid Y 2, 1), while the top views match.
- Part D: 7 solids fit; the fewest cubes is 6 (stacks of 2 on a diagonal) and the most is 8.
Front, side and top views
Build each solid with cubes. Draw each view on its grid: the front from straight in front, the right side from the right, the top from above.
A. Draw the views
B. Build to the views
C. One view is not enough
D. Stretch
A solid stands on a 2 by 2 square, with a cube on every square. Its front view and its right side view are both 2 by 2 squares. What is the fewest number of cubes it could have? The most?
Isometric drawings
An isometric drawing shows a solid from a corner: vertical edges stay vertical, the other two directions slope at 30°, and lengths are true along all three. The cost is that right angles become 60° or 120°, and cubes at the back can hide.
We are learning to
- draw cuboids and cube models on isometric dot paper,
- keep lengths true along the three directions,
- explain what an isometric drawing bends or hides.
Success criteria
- I can draw a cuboid of a given size on isometric dot paper.
- I can explain why a right angle is drawn as 60° or 120°.
You need
Isometric dot paper (cut-out sheet 1, two per student), interlocking cubes, protractors and rulers.
Lesson flow (about 60 minutes)
| 10 min | From a corner Show the board (see the box). Ask: “The front of the box is a rectangle. Why is it drawn as a parallelogram?” |
| 35 min | Draw Students complete Parts A and B on the dot paper and the worksheet. |
| 15 min | Views to drawing Part C in pairs, building the solid first; early finishers do Part D. Ask: “What could this drawing hide that the views would show?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Sloping the vertical edges. Only the two horizontal directions slope.
- Drawing edges that are hidden behind the solid.
- Taking a 60° angle on the paper as the angle on the solid. On a box, every corner is 90°.
Answers
- Part A: drawings depend on the corner chosen; check that each has vertical edges and two sloping directions, with edges of 3, 2, 1; 2, 2, 3; 4, 1, 2 units (one unit is the gap between two dots).
- Part B: a = 120°, b = 60°, c = 60°, d = 120°; on the real box every one of them is 90°.
- Part C: one solid fits, back row 3, 1, 1; front row 1, 1, 0, so 7 cubes.
- Part D: drawings vary; the two corners show different faces, and one drawing may hide a cube the other shows.
Isometric drawings
Use the isometric dot paper on cut-out sheet 1, with the long side of the page upright.
A. Draw cuboids
- A cuboid 3 long, 2 deep and 1 high (in cube edges). Mark each length on your drawing.
- A cuboid 2 long, 2 deep and 3 high (in cube edges). Mark each length on your drawing.
- A cuboid 4 long, 1 deep and 2 high (in cube edges). Mark each length on your drawing.
B. Measure the angles
C. From views to a drawing
D. Stretch
Make an L-shape from 4 cubes: 3 in a row with one more on top of an end cube. Draw it on the dot paper from two different corners. What does each drawing show that the other does not?
Nets
A net is a flat pattern of faces that folds into a solid, each face once. Six squares can be joined edge to edge in 35 ways, and 11 of them fold into a cube. Students predict and test patterns, find opposite faces, and draw nets of a prism and a pyramid.
We are learning to
- decide whether a pattern of six squares is a net of a cube,
- work out which faces end up opposite each other,
- draw nets of prisms and pyramids.
Success criteria
- I can predict whether a pattern folds into a cube, and check by folding.
- I can draw a net of a triangular prism.
You need
Cut-out sheet 2 (one per pair), scissors, sticky tape, and a cardboard box to unfold if you have one.
Lesson flow (about 60 minutes)
| 10 min | Unfold Show the board (see the box), then unfold a box along its edges. Ask: “Could we cut the box open another way and still get a net?” |
| 30 min | Predict and fold Pairs record a prediction for each pattern in Part A, then cut out and fold the patterns to check. |
| 20 min | Opposite faces, other nets Parts B and C; early finishers do Part D. Ask: “In a row of three squares, which two end up opposite?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Thinking any six joined squares fold into a cube.
- Missing the giveaways: five squares in a row, or a 2 by 2 block, never fold into a cube.
- Taking two squares side by side as opposite faces. Squares that share an edge always meet at an edge of the cube.
Answers
- Part A: A, B, D fold into a cube; C, E, F do not (two squares land on the same face, so one face is left open).
- Part B: 3 is opposite 1; 5 is opposite 2; 6 is opposite 4.
- Part C: a triangular prism has 2 triangles and 3 rectangles: 5 faces, 9 edges and 6 vertices; a square-based pyramid has 1 square and 4 triangles: 5 faces, 8 edges and 5 vertices.
- Part D (one way): 1, 5, 4, 2, 3, 6 for the top square, the row of four from the left, then the bottom square; any numbering works if opposite faces add to 7.
Nets
A. Predict, then fold
Use the six patterns on cut-out sheet 2.
| Pattern | I predict: net or not? | After folding | Why |
|---|---|---|---|
| A | |||
| B | |||
| C | |||
| D | |||
| E | |||
| F |
B. Opposite faces
C. Other solids
- Sketch a net of a triangular prism. How many faces, edges and vertices does the solid have? ______________________
- Sketch a net of a square-based pyramid. How many faces, edges and vertices does the solid have? ______________________
D. Stretch
Number the squares of pattern A from 1 to 6 so that, like a die, the numbers on opposite faces add to 7. Fold it to check.
From views to solids
Views can be read backwards: build a solid that fits the front, right side and top views, and record it as a height map, the top view with each stack’s height written in. Sometimes one solid fits; sometimes several do.
We are learning to
- build a solid that fits a set of views, and record it as a height map,
- check a height map against every view,
- find the fewest and the most cubes that fit a set of views.
Success criteria
- I can build and record a solid that fits three views.
- I can explain why some sets of views fit more than one solid.
You need
Interlocking cubes (about 12 for each pair) and the worksheet.
Lesson flow (about 60 minutes)
| 10 min | Test every view Show the board (see the box). Ask: “Guess C fits all three views. Is it the only solid that does?” |
| 30 min | Build to the views Pairs build each solid in Part A and write its height map, then check the maps in Part B. |
| 20 min | Fewest and most Part C as a class, collecting every solid the pairs find; early finishers do Part D. Ask: “Which squares must hold a tall stack, and which can hold just one cube?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Checking one view and stopping. A solid must fit every view.
- Reading the right side view mirrored: its left column is the front row.
- Expecting the top view to show heights. It shows only where stacks stand.
Answers
- Part A (height maps, rows from the back): (1) back 2, 1; front 1, 0; (2) back 1, 2; front 1, 3; (3) back 2, 3, 1; front 1, 1, 0.
- Part B: (a) fits; (b) fits; (c) breaks the right side view; (d) fits; (e) breaks the front and right side views.
- Part C: 5 solids fit; the fewest cubes is 7 and the most is 9.
- Part D: the fewest is 15 (stacks of 3 on a diagonal, single cubes elsewhere) and the most is 27.
From views to solids
The right side view is seen from the right, so its left column is the front row. Write each height map with the back row at the top.
A. Build it
B. Does it fit?
C. Fewest and most
How many different solids fit the views in Part B? Write each height map. What is the fewest number of cubes? The most?
D. Stretch
A solid stands on a 3 by 3 square, with a cube on every square. Its front view and its right side view are both 3 by 3 squares. What is the fewest number of cubes it could have? The most?
Choosing a representation
Each representation tells the truth about some things and hides others. Students weigh up views, isometric drawings and nets, choose one to suit a purpose, and justify the choice; then the quiz.
We are learning to
- describe what views, isometric drawings and nets each show and hide,
- choose a representation to suit a purpose, and justify the choice.
Success criteria
- I can give an advantage and a disadvantage of each representation.
- I can choose a representation for a purpose and explain why.
You need
The worksheet, interlocking cubes, and the unit’s self-check quiz on the board.
Lesson flow (about 60 minutes)
| 10 min | Pick the purpose On the unit, show “Trade-offs between representations”, which opens on “measure it”. Press “build it”, then “show how it looks”. Ask: “Is there a purpose where no representation wins outright?” |
| 30 min | Problems Students work through problems 1 to 6; 7 and 8 are the stretch. |
| 20 min | Class quiz game Run the self-check quiz as a class game (see the box), then wrap up. |
Two shorter sessions instead? Split after problem 3; start the second with problem 4.
Watch for these ideas
- Thinking one representation is best for everything.
- Saying an isometric drawing shows true angles because the lengths are true.
- Forgetting that three views leave the back, left and bottom faces unseen.
Answers with working
- Front, side and top views: no, yes, no; Isometric drawing: yes, no, no; Net: no, yes, yes.
- front, side and top views with measurements, because they show true sizes and right angles.
- an isometric drawing (or a photo), because it looks like the solid at a glance.
- a net, because it shows every face flat, ready to cut out and fold.
- no: when it folds, two squares land on the same face and one face is left open.
- the right side view: Solid P 1, 2, Solid Q 2, 1, while both front views are 2, 1, 1.
- six rectangles in three matching pairs, 3 by 2, 3 by 1 and 2 by 1, with a total area of 2 × (6 + 3 + 2) = 22 square units.
- answers vary: turning it shows every side and hidden parts, but there are no exact measurements, and it needs a screen.
Choosing a representation
Give a reason with every choice.
Problems (7 and 8 are the stretch)
- Write yes or no in each box.
Representation Looks like the solid Shows true angles Shows every face Front, side and top views Isometric drawing Net - A builder needs the exact height and width of a wall unit. Which representation suits, and why?
- A toy shop wants a picture of a model on the lid of its box. Which representation suits, and why?
- A class will make gift boxes from card. Which representation suits, and why?
- Is this a net of a cube? Explain.
- These solids have the same front view and the same top view. Which view tells them apart? Draw it for both.
- Stretch. Sketch a net of a cuboid 3 by 2 by 1 on squared paper. What is the total area of the net?
- Stretch. An app shows a solid that you can turn with a finger. Give one advantage and one disadvantage compared with printed views.
Isometric dot paper
Day 2. Print at 100% and keep the long side of the page upright, so the vertical edges of a solid run up the page. The dots are 1 cm apart along all three directions of an isometric drawing.
Patterns of six squares
Day 3. Predict first: will the pattern fold into a cube? Then cut it out along the solid lines and fold along the dashed lines to check.
What we know: Representing objects in two dimensions
Work on your own. The right side view is seen from the right, so its left column is the front row.
Questions 1 to 3: draw the front, right side and top views of this solid.
- PQWhich pattern is a net of a cube? Why does the other not fold into one?
- A net has one hexagon and six triangles. Name the solid. How many edges and vertices does it have? ____________
- A solid stands on a 2 by 2 square, with a cube on every square. Its front view is 3, 3 and its right side view is 3, 3 (column heights). What is the fewest number of cubes it could have? The most? ____________
- Give one thing an isometric drawing does not show truly. ______________________
- A real estate brochure needs to show buyers what a new house looks like. Which representation suits, and what does it give up?
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- Front view: column heights 2, 3, 1, left to right.
- Right side view: 2, 3, from the front row to the back row.
- Top view: all 3 squares wide and 2 deep.
- P folds into a cube. Q does not: the two squares hanging below its ends fold onto the same face, leaving one face open.
- A hexagonal pyramid: 12 edges and 7 vertices (7 faces).
- Fewest 8 (stacks of 3 on a diagonal and single cubes on the other two squares), most 12 (every stack 3).
- Right angles are drawn as 60° or 120°; cubes at the back can be hidden.
- An isometric drawing or a 3D picture, so buyers see the house at a glance; it gives up true angles and exact measurements, which a floor plan shows.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Views (Q1 to Q3) | draws the solid in 3D, or mirrors the side | draws all three views on the grid | spots a stack that a view cannot show |
| Nets (Q4, Q5) | counts squares without folding | decides by folding in the mind, and names the solid | explains which faces would overlap |
| Views backwards (Q6) | finds one solid only | finds the fewest and the most cubes | explains which squares must hold tall stacks |
| Choosing (Q7, Q8) | names a representation without a reason | gives what it shows and what it hides | weighs two representations for one purpose |
Eight questions, four ideas. A student at standard draws the three views of a cube model, decides whether a pattern folds into a cube, reads views backwards, and chooses a representation with what it shows and what it hides.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Views | Isometric | Nets | Backwards | Choosing |
|---|---|---|---|---|---|
The five columns are the five days: views, isometric drawings, nets, views read backwards, and choosing a representation.