Views of a solid

Classroom resources

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Students will be able to

• I can draw or read the top, front and side views of a pile of cubes.
• I can say what a view hides.
• I can say one advantage and one disadvantage of a net and of a 3D drawing.

Timing

Predict and Run 10 min · Investigate 10 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Build a small pile from linking cubes to hold up.

The one idea: a view shows only what you can see from one side, so the front and side views show the tallest pile in each column and hide the rest. The common mistake is adding the piles (Noor's mistake).

The board's rows run from the back (top of the grid) to the front. The front view looks from the front; the side view looks from the right. The top view puts the back row at the top.

The view orientation is the hinge for this node: students who mix up front and side cannot compare representations.

Running the lesson

1. Predict: ask what could be hiding behind the front pile.
2. Run: ask students to say which view they think will change before they press.
3. Investigate: ask Noor's question again with a column of 1, 3, 1.
4. Modify: ask for the fewest cubes first, then the most.
5. Make: ask which drawing they would give a builder.

The hard part (on purpose)

Knowing which side is which. Turn a real pile of cubes round on the desk beside the board; the side view is seen from the right.

When students get stuck

Answers to the checks

1. one view: You see only the front view of a pile of cubes. Can you tell how many cubes there are?
   No. A front view shows the tallest pile in each column; cubes behind it, or lower down, are hidden.
2. three views: Change the piles so the top view has 4 squares and the front view has 5. Which piles did you change?
   
   The top view counts the occupied cells; the front view counts, for each column, only the tallest pile.
3. Hidden cubes: A column has piles of 3, 1 and 2 cubes, one behind another. How many squares does the front view show for that column?
   3
   The tallest pile is 3, so the front view shows 3 squares for that column.
4. Find the mistake: Noor works out the front view of a column of piles. Click the line where her reasoning goes wrong.
   line 3
   Line 3: a front view shows only the tallest pile in each column, not all of them added up.
5. Which drawing: Which drawing tells you exactly how many cubes there are?
   A top view with the height written on each pile
   With a height on every pile, adding the numbers gives the total.
6. Six cubes: Build a shape with 6 cubes whose front view and side view each have 3 squares.
   
   Six piles of one cube spread over the 3 by 3 base, with every row and every column used, give 3 squares in the front view and 3 in the side view.
7. The fewest: Build the fewest cubes that give a top view of 4 squares.
   
   Each square of the top view needs at least one cube, so the fewest is 4.
8. The most: Build the most cubes you can with a top view of 4 squares and no pile taller than 3.
   
   4 piles of 3 cubes make 12.
9. From a top view: The top view of a solid has these heights. Back row: 2, 1, 0. Middle row: 0, 3, 0. Front row: 0, 0, 1. Build it. How many cubes?
   
   2 + 1 + 3 + 1 = 7 cubes. The front view shows the tallest pile in each column: 2, 3 and 1, so 6 squares.
10. Nets or drawings: What is one advantage of a net over a 3D drawing?
    It shows every face at its true size, so you can find the surface area
    A net lays every face flat, true size; a 3D drawing shows what the solid looks like but squashes the faces.
11. Side view: Look at the stack on the board. How many squares does its side view show?
    3
    From the right, the rows give tallest piles 2, 0 and 1: 3 squares.
12. Cube count: A top view with heights is 3 1 on the top row and 0 2 on the bottom row. How many cubes are there?
    6
    3 + 1 + 0 + 2 = 6 cubes.
13. The top view: Which view shows which cells of the base have cubes?
    The top view
    Looking down, the top view shows a square for every occupied cell.
14. Review: nets: Review: which of these six-square nets folds into a cube?
    A cross: one square above and one below a row of four
    Only the cross-shaped net covers all six faces once.
15. two designs: Build a solid whose top, front and side views each have 5 squares.
    
    One answer, rows from back to front: 0 0 1, then 0 0 2, then 1 2 1. Five occupied cells; the tallest pile in each column adds to 5, and in each row to 5.

Solutions

One possible solution for each task. Other correct programs are fine.

build to a description › Six cubes

Open task: any program that does what the step asks.

build to a description › The fewest

Open task: any program that does what the step asks.

build to a description › The most

Open task: any program that does what the step asks.

build to a description › From a top view

Open task: any program that does what the step asks.

build to a description › Nets or drawings

Open task: any program that does what the step asks.

two designs

Open task: any program that does what the step asks.

Stretch and support

Stretch: Two different piles can have the same top, front and side views. Can you find a pair?

Support: Use a real pile of linking cubes and look at it from each side before using the board.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Views of a solid".

The ideas:
- Isometric and plan/elevation drawings: I can Students draw and interpret isometric drawings and plan/elevation views (front, back, side, top) of 3D objects.
- Advantages and disadvantages of different representations: I can Students reason about when each type of 2D representation is most useful and what information each preserves or loses.

What understanding looks like:
- Correctly draws at least two views of a 3D object.
- Reconstructs a 3D object from its views.
- Does not confuse view orientation (e.g. top vs front).
- States at least one advantage and one disadvantage for two different representations.
- Selects an appropriate representation for a given purpose and justifies the choice.
- Recognises that no single representation shows all information about a 3D object.

Mistakes students often make here:
- A student adds all the piles to find a view.
- A student confuses front and side views.
- A student thinks the isometric drawing shows every cube.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Nets of prisms and other 3D objects), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Views of a solid".

Exit ticket

A front view shows 3 squares. Does the pile have 3 cubes? Why or why not?

Not necessarily. The front view shows only the tallest pile in each column; there could be more cubes behind.

What each step is for

• one view: Committing to whether the front view gives the number of cubes shows whether they know a view hides cubes.
• three views: Changing cubes and watching the three views shows what each view reveals and what it hides.
• what a view shows: A view shows only the tallest pile in each column (Noor adds the piles); the top view with numbers shows the most.
• build to a description: Building to a description uses the views to reason about hidden cubes: the fewest, the most, and the trade-offs of each drawing.
• views: Reading views and counting cubes, mixed; adding the piles is the planted slip; the cube net review keeps nets fresh.
• two designs: Building a solid whose three views match, then judging nets against drawings, brings the lesson's representations together.

Where it leads

This completes AC9M7SP01.

Where it leads: building solids from their views, and surface area and volume of prisms (Measurement); drawing in perspective in Year 8.