Scale and scale factor

Understanding Size

Models and Scale

Enlargementscale factorReductionScaleProportionsScale Diagram/Model

Vocabulary

Model• Representation of something else • Usually too big or too small to analyze

easily

Enlargement

To make something bigger so that one can analyze/observe the details.

Reduction

To make an object small so that one can observe/analyze the details.

• In math, scale shows the relationship between two things as well.

• With maps, it is usually between a distance measured on the map and the actual distance on the ground.

scale map distance ground

distance map

I Can Solve Problems Using Scale Drawings!

• We know about scales at the supermarket. They measure weight.

• They show the relationship between how much you are buying and how much you have to pay.


• We also know about the scales we stand on. They measure our weight.

• They help to show the relationship between our health and Grandma’s potato salad last week!

• A scale drawing represents something that is too large or too small to be drawn at its actual size.

• Maps and blueprints are examples of scale drawings.

All scale drawings must have a scale written on them. Scales are usually expressed as ratios.Normally for maps and buildings the ratio:

Drawing length: Actual length

For maps the ratio is normally in the ratio:Map distance: Actual Distance

Example: 1cm : 100cm

The ratio 1cm:100cm means that for every 1cm on the scale drawing the length will be 100cm in real life

Example: 1:10000

The ratio 1:10000 means that the real distance is 10000 times the length of one unit on the map or drawing.

Understanding Scales

• Scale factor is the ratio of change

• The number you multiply by to relate the first shape to the second is the scale factor.

Scale factor = new measurement old measurement

- Scale factor more than 1 => shape gets bigger

(Enlargement)- Scale factor less than 1 => shape gets smaller (Reduction)- Congruent shapes are similar shapes with SF = 1

Old measurement x SF = new measurementSF

new

old

Scale Factor

• The scale can be written as a scale factor, which is the ratio of the length or size of the drawing or model to the length of the corresponding side or part on the actual object.

• Scale Factor needs to be the SAME UNITS!

This HO gauge model train is a scale model of a historic train. A scale model is a proportional model of a three-dimensional object. Its dimensions are related to the dimensions of the actual object by a ratio called the scale factor. The scale factor of an HO

gauge model train is . 1

87

This means that each dimension of the model is of the corresponding dimension of the actual train.

187

A scale is the ratio between two sets of measurements. Scales can use the same units or different units. The photograph shows a scale drawing of the model train.

A scale drawing is a proportional drawing of an object. Both scale drawings and scale models can be smaller or larger than the objects they represent.

If you have ever seen Jurassic Park, you saw how big the dinosaurs were compared to the people. Pretend that they made a large Human to watch over the animals. What would be the scale factor if a 64 inch person was made to be 160 feet?

The scale factor tells you how many times bigger than “normal” that person really is.You must make all units of measure the same….

64 inches

160 feet64 inches

160 x 12

64 inches1920 inches

= =

Now take the:

64 inches

1920 inches And simplify

1/30 inchesThis means that the person was created 30 times his normal size.

Scale Factor

Keep like

units in the same

fraction.

Inches = yards

Inches yards

Setting up Proportions

• There is more than one way to set up a proportion correctly!

• Cross Multiply!• Use common

sense!

Remember…

• Tom is drawing a blueprint for a rectangular shed he wants to build. The scale factor is 1 ft. to ¼ inch. If the dimensions of the blueprint are 1 ¼ in. by 2 inches, what are the actual dimensions of the shed going to be?

Write a Proportion Using Scale Factor

¾ inch to 1 foot

• If the length in inches is 2 ¼ inch, what would the actual length be in feet ?

Write a Proportion Using a Scale Factor

Scale Drawings

Scale Drawings

On Maps

Footprints of houses

Vehicle design

What are scale drawings?Scale drawings are everywhere!

Can you think of any more?

Scale 1 cm = 1 m

6cm

Length of units = 6 m5

Scale in everyday life: kitchen design

Scale 1 : 1 000 000

Scale in everyday life: maps

• The blueprint of the pool shows each square has a side length of ¼ inch.

• If the scale is written as ¼ in = 2 ft, what is actual width of the pool?– (To figure this out, what

else do you need to know?)

Scale 2 cm = 1 m

pool path

decking

7

Scale in everyday life: plans

Using A Scale Drawing

When objects are too small or too large to be drawn or constructed at actualsize, people use a scale drawing or a model.

The scale drawing of this tree is 1:500If the height of the tree on paper is 20 inches, what is the height of the tree in real life?

The scale is the relationship between the measurements of the drawing or model to the measurements of the object.

In real-life, the length of this van may measure 240 inches. However, the length of a copy or print paper that you could use to draw this van is a little bit less than 12 inches

• Map Scales (Legends) are used to find distances on a map.

• For example, if your map legend tells you that ½ of an inch represents 50 miles, how could you find the mileage for a 2 inch distance on the map?

Map Scales

Ratios and proportions can be used to find distances using a scale.

Example:

1 inch = 15 milesThe distance from Jacksonville to Smithtown on a map is 4 inches. How many miles are between these cities?

1 in.15 mi.

= 4 in n

1n =

60

n = 60

The distance between the

two cities is 60 miles.

Map Scales

• Suppose the distance between Coral Springs and Fort Lauderdale is about 4.1 centimeters on the map.

• What is the actual distance on the ground if the scale is 1 cm = 4.5 km?

scale map distance ground

distance map

• Use the scale as a fraction.

• Use cross-products to calculate.

km 18.45 Distance

4.1x 4.5 ?x 1 Distance

km ?

cm 4.1

kilometers 4.5

centimeter 1


• Width of the pool on the blueprint = 1.75 inches.

• How can you use cross products to figure out how wide the pool really is?


feet 14 pool of Width

14/4 ?x 1/4

3/4 1x 2 ?x 1/4

feet ?

inches 3/4 1

feet 2

inch 1/4

I Can Solve Problems Using Scale Drawings!(SOL 7.6)

• You can convert the units in a scale to simplify it.

• When you do that, you end up with a scale factor.

• It is a ratio written in its simplest form.

96 :1or 96

1 factor Scale

96

1

inches 24

inch 1/4x

4

4

inches 24

inch 1/4

feet 2

inch 1/4


• 1) Find the scale factor of the blueprint of a school bus parking lot if the scale is written as “1 inch = 8 feet”.

• 2) On a scale drawing of a new classroom, the scale is 1 centimeter = 2.5 meters. What is the scale factor?


• 1) Scale factor = 1/96. That means that each measurement on the blueprint is 1/96th of the actual measurement of the parking lot.

• 2) 1 centimeter / 2.5 meters: = 1 cm / (2.5 m x 100) cm

= 1 cm / 250 cm

= 1/250


• If you know the actual length of an object and you know the scale, you can build a scale model.

• Scale models are used to represent things that are too large or too small for an actual-size model.

• Examples are cars, planes, trains, rockets, computer chips, heart cells, bacteria.


• Designers are creating a larger model of a computer memory board to use in design work. The board measures 5 ¼ inches in length.

• If they use a scale of 20 inches = 1 inch, what is the length of the model?

inches 105 length Model

? 1 4

15 20

inches 1/4 5

inches ?

inch 1

inches 20


• Things to remember:–When solving proportions, give your answer in

the correct unit of measurement.–Scale factors do not have units.–Equivalent scales have the same scale factor.

• For example 1 inch = 8 feet and ¼ inch = 2 feet both equal 1/96 (or 1:96)

–Scale is the ratio between the drawing/model measurement to the actual measurement. • Not always the ratio of smaller to larger!

Education

Scale and scale factor