E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Making Models from Views at an Appropriate Scale

E1.2 A1About 50 minutes

Learning goals

I can read the top, front and side views (or a perspective view) of an object and find its real dimensions.
I can choose an appropriate scale so a model fits the space and materials I have.
I can calculate the model's dimensions and build it.

Words to know

front view
What you see looking straight at the front of an object. It shows the width and the height.

side view
What you see looking straight at one side. It shows the depth and the height.

top view
What you see looking straight down. It shows the width and the depth. Also called a plan view.

perspective view
A drawing that shows three faces of an object at once, so it looks 3D (for example, on isometric dot paper).

scale
A ratio comparing a length on a model or drawing with the real length, like 1 : 50 (1 cm on the model stands for 50 cm in real life).

scale model
A 3D copy of an object in which every length is multiplied by the same scale factor.

Remember from earlier grades

In Grade 6 you built cube structures from their top, front and side views. In Grade 7 you drew views and perspective drawings to scale and built scale models of simple objects. In Grade 8 the focus is on choosing an appropriate scale: one that makes a model big enough to build and see the details, but small enough to fit the space and materials you have.

Let's start with a cabin

A family is planning a small cabin by a lake. Before they build it, they want a model on a display board that is 20 cm by 15 cm. The plans show three views. How small should the model be? A scale that is too large won't fit on the board; a scale that is too small makes a tiny model that is hard to build.

Views of the cabin, drawn to scale (the same scale in all three views). The front view shows the 6 m width and the 4 m height to the peak of the roof, with walls 2.5 m high. The side view shows the 4 m depth; the lower part is the wall and the upper part is the sloping roof. In the top view the line down the middle is the roof's ridge.

Which view shows what

Front view: width and height.
Side view: depth and height.
Top view: width and depth.

Each real dimension appears in two views, so you can check one view against another.

Choosing an appropriate scale

Find the largest real dimensions of the object (change them to centimetres).
Find the space you have for the model (the board, the shelf, the box).
Divide: real length ÷ space length. This tells you the smallest scale number that fits. For example, 600 cm ÷ 20 cm = 30, so you need a scale of at least 1 : 30.
Choose a convenient scale a bit beyond that, like 1 : 40 or 1 : 50, so the numbers are easy and there's a little room to spare.
Divide every real length by the scale number to get the model lengths.

Worked example

Example 1: Choose a scale for the cabin

The cabin is 6 m wide, 4 m deep and 4 m tall. The model must fit on a 20 cm by 15 cm board. Choose a scale and find the model's dimensions.

Real dimensions in centimetres: 600 cm wide, 400 cm deep, 400 cm tall (walls 250 cm).
Width: 600 ÷ 20 = 30. Depth: 400 ÷ 15 = 26.7. The scale must be at least 1 : 30 to fit.
1 : 30 would fill the board edge to edge (20 cm wide), with nothing to spare. 1 : 50 is easy to calculate and leaves room.
At 1 : 50: width 600 ÷ 50 = 12 cm; depth 400 ÷ 50 = 8 cm; height to the peak 400 ÷ 50 = 8 cm; walls 250 ÷ 50 = 5 cm.

A scale of 1 : 50 gives a model 12 cm wide, 8 cm deep and 8 cm tall, with 5 cm walls. It fits on the board with space around it.

Worked example

Example 2: From a model back to real life

A model of a picnic shelter is built at 1 : 25. The model's roof is 16 cm long. How long is the real roof?

At 1 : 25, every model length is multiplied by 25 to get the real length.
16 × 25 = 400 cm = 4 m.

The real roof is 4 m long.

Worked example

Example 3: Build from a top view with heights

A top view of a cube structure is drawn as a grid, and each square shows how many cubes are stacked there:

back row: 3, 1, 1
front row: 2, 1, 0

How many cubes are needed? What does the front view look like?

Add the stacks: 3 + 1 + 1 + 2 + 1 + 0 = 8 cubes.
The front view shows the tallest stack in each column, from left to right: the left column has stacks of 3 and 2, so it is 3 tall; the middle column is 1 tall; the right column is 1 tall.

8 cubes. The front view has columns of heights 3, 1, 1 from left to right.

Example 3. The top view shows how many cubes are stacked in each spot (8 in all). Looking from the front, you see the tallest stack in each column, so the front view has heights 3, 1 and 1.

Think like a mathematician

Use the same scale for every part of a model, or the parts won't fit together.
Measure lengths that aren't labelled (like a sloping roof) on a scale drawing of the view, rather than guessing.
A model's area shrinks by the scale factor squared. At 1 : 50, the model's floor has 1/2 500 of the real floor area. (You'll use this in the next lesson.)

Hands-on activity

Build the cabin model

You need: thin cardboard (a cereal box), a ruler, a pencil, scissors (a grown-up should help), tape or glue.

Use the scale 1 : 50 from Example 1. On the cardboard, draw the front and back walls as the pentagon in the front view: a 12 cm bottom edge, 5 cm sides, and two sloping edges that meet at the peak, 8 cm above the bottom edge, in the middle.
Draw the two side walls: rectangles 8 cm wide and 5 cm tall.
Measure the sloping edge of your pentagon (it should be about 6.7 cm). Draw two roof panels that are that long and 8 cm deep.
Cut out the pieces and tape them together. Check: does your model look like each of the three views?
Pick another object at home (a bed, a doghouse, a garden shed) and choose a scale so its model would fit in a shoebox.

What to look for: every model length is the real length ÷ 50, and the finished model matches all three views. For step 5, look for a sensible scale (the largest real length ÷ the shoebox length, rounded up to an easy number).

Just for fun

Many model trains are built at 1 : 87 (HO scale). At that scale, a person 1.7 m tall is only about 2 cm tall: a tiny passenger who never complains when the train is late.

Your turn

Which view shows an object's width and height?

A real wall is 6 m long. How long is it in a 1 : 50 model?

A real wall is 2.5 m tall. How tall is it in a 1 : 50 model?

A desk is 1.2 m long. How long is it in a 1 : 20 model?

A canoe is 4.8 m long. You want a model to fit on a 30 cm shelf. Which of these scales fit: 1 : 10, 1 : 20, 1 : 25? Which gives the biggest model that fits?

A top view with heights shows: back row 2, 2, 1 and front row 1, 0, 1. How many cubes are needed?

In the cabin views, which view shows the roof as a triangle on top of a rectangle?

More practice: warm-up

At a scale of 1 : 100, what does 1 cm on a model stand for?

A building is 8 m wide. How wide is a 1 : 100 model?

A 1 : 20 model is 15 cm long. How long is the real object?

Name the three views usually used to describe an object.

What does the top view of a can standing upright look like?

More practice: on your own

A school gym is 30 m long, 20 m wide and 8 m tall. A model must fit on a table 60 cm by 40 cm. Find the smallest scale that fits, then suggest a better one and give the model's dimensions.

A picnic table top is 1.8 m long. How long is it in a 1 : 12 model?

The views of a storage chest at 1 : 25 show: front view 9 cm wide and 6 cm high, side view 4 cm wide and 6 cm high. What are the real dimensions?

A model car is built at 1 : 24. The model is 18.5 cm long. How long is the real car?

A bookcase is 2 m tall, 90 cm wide and 30 cm deep. Choose a scale so a model fits in a shoebox that is 30 cm long, 18 cm wide and 10 cm tall (lying down, the bookcase's height can go along the box's length). Give the model's dimensions.

Challenge

The CN Tower in Toronto is about 553 m tall. A display case is 1 m tall inside. Choose a convenient scale and find the model's height.

Is a scale of 1 cm : 2 m the same as 1 : 200? Explain.

The cabin model at 1 : 50 has a floor that is 12 cm by 8 cm, an area of 96 cm². What is the real floor area? Why isn't it 96 × 50?

Check your understanding

Quiz

Which view shows an object's width and depth?

At a scale of 1 : 50, how long is the model of a 4 m wall?

A 1 : 25 model is 6 cm tall. How tall is the real object?

A boat is 9 m long. Which scale gives a model that fits on a 30 cm shelf?

What is the top view of a can standing upright?

A perspective view of an object shows:

For parents and teachers

Big idea: The Ontario curriculum (E1.2) asks Grade 8 students to make objects and models using appropriate scales, given their top, front and side views or their perspective views. The new skill is choosing a scale that suits the space and materials, not just using a scale someone gives them.

Common difficulties:

Forgetting to change metres to centimetres before dividing by the scale number.
Using different scales for different parts, so the pieces don't fit.
Mixing up which view shows which dimensions. Front: width and height. Side: depth and height. Top: width and depth.

How to help: Build the cabin model together, and let your child choose a scale for a model of their own room or a piece of furniture. Toy cars and train sets are labelled with scales (like 1 : 24 or 1 : 87): ask what the real length would be. Safety: help with scissors when cutting cardboard. This lesson also builds A1 skills: planning, making choices and justifying them.