Transcript Introduction to Scale Factor

What is a scale drawing?
http://www.basic-mathematics.com/scale-drawings.html
Scale Drawing
• A drawing that shows a real object with
accurate sizes except they have all been
reduced or enlarged by a certain amount
(called the scale).
The scale is shown as the length in the
drawing, then a colon (":"), then the
matching length on the real thing.
Example: this drawing has a scale of "1:10", so
anything drawn with the size of "1" would have a
size of "10" in the real world, so a measurement of
150mm on the drawing would be 1500mm on the real
horse.
Investigating Scale
Click on the Butterfly Below…
• The scale indicates how many units of length
of the actual object are represented by each
unit of length in the drawing.
Scale
• A scale of 1:1 implies that the drawing of the
grasshopper is the same as the actual object.
• The scale 1:2 implies that the drawing is
smaller (half the size) than the actual object
(in other words, the dimensions are multiplied
by a scale factor of 0.5).
• The scale 2:1 suggests that the drawing is
larger than the actual grasshopper -- twice as
long and twice as high (we say the dimensions
are multiplied by a scale factor of 2).
• If no units are listed in the scale, then you can
assume that the drawing and the object are
measured using the same units. For example,
the scale 1:2 might represent 1 cm:2 cm or 1
in.:2 in.
Teacher’s Domain videos
Island of the Little (48 s)
•
http://www.teachersdomain.org/resource/vtl07.math.number.rat.islandlitt/
Island of the Giants (3min 33s)
•
http://www.teachersdomain.org/resource/vtl07.math.number.rat.landgiants/
Vocabulary
• Scale drawing/scale model: is used
to represent an object that is too
large or too small to be drawn or built
at actual sizes
• Scale: gives the relationship between
the measurements on the drawing or
model and the measurements of the
real object
• Scale factor: the ratio of a length on a
scale drawing or model to the
corresponding length on the real
object
Ratios
• Rates are often written with a slash
rather than the word per:
– such as mi/h for miles per hour
– $2/dozen for $2 per dozen
– a car traveling 30 miles per hour
– making a long-distance telephone call that
costs 20¢ per minute
– skating at an ice rink that costs $10 for 2
hours
Proportions
A statement that shows two ratios are
equivalent.
• A proportion is often used when one
ratio is known and only part of a second
ratio is known, such as:
Example
“The ratio of girls to boys in a class is 6:8
and there are 12 boys in the class.” A
proportion can be set up and solved to
find how many girls there are in the
class.
• Set up the proportion.
Solving Proportions
_6_ = _X_
8
12
cross multiply:
6 x 12 = 8 x X
72 = 8x
isolate the variable:
72 = 8x
8 8
X = 9 girls
By using proportions, you can find
lengths needed to make a scale drawing
or can find the actual lengths of an
object based on a given scale drawing.
map scales
• If 1 cm on a map represents a distance
of 250 km, what is the approximate
distance of a length represented by 2.7
cm? We can set up a proportion to show:
1 cm = 2.7cm
250 km
X
• Solving the equation for x, we get
x = 250 • 2.7 = 675 km.
3 ways scale can be expressed
1. 1 cm = 1 km
2. ______ = 1 km
3. 1: 50,000
Practice Problem
• A student has a map on which
the scale is 2 cm = 5 km. Having
measured the distance between
two points on the map to be
7.5 cm how do you calculate the
real world distance from this
measurement?
Scale Factor
Example: Suppose a scale model has a scale of
2 inches = 16 inches. The scale factor is
2 or 1
16
8
The lengths and widths of objects of a scale
drawing or model are proportional to the
lengths and widths of the actual object.
Your Turn, Again!
In an illustration of a honey bee, the length of the
bee is 4.8 cm. The actual size of the honeybee is
1.2 cm. What is the scale of the drawing?
4.8 cm = 1cm
1.2 cm x cm
4.8x = 1.2
x = .25
The scale of the drawing is 1 cm = .25cm
Example 1: Find Actual Measurements
A set of landscape plans shows a flower bed that is
6.5 inches wide. The scale on the plans is 1 inch =
4 feet.
What is the width of the actual flower bed?
Let x represent the actual width of the flower bed.
Write and solve a proportion.
Plan width----> 1 inch = 6.5 inches<---plan width
Actual width--> 4 feet x feet <-----actual width
1x = 46.5 cross products
x= 26 The actual flower bed width is
26 feet.
From the last example, what is the
scale factor?
To find the scale factor, write the ratio of 1 inch
to 4 feet in simplest form.
1inch = 1 inch
Convert 4 feet
4 feet 48 inches to inches
The scale factor is 1 . That is , each
48
measurement on the plan is 1 the actual
measurement.
48
Example 2: Determine the Scale
In a scale model of a roller coaster, the highest hill
has a height of 6 inches. If the actual height of
the hill is 210 feet, what is the scale of the model?
Model height---> 6 inches = 1 inch <--model height
Actual height--->210 feet x feet <--actual height
6x = 210
6x = 210 x= 35
6
6 So, the scale is 1” =
35 feet
Your Turn!
On a set of architectural drawings for an office building,
the scale is 1/2” = 3 feet. Find the actual length of
each room.
.5” = 2”
Lobby: 2 inches
3ft x ft
.5x = 6
x = 12
The actual length
of the lobby is 12 ft
Cafeteria: 8.25 inches
.5” = 8,25”
3ft x ft
.5x = 24.75
x = 49.5
The actual length of the
cafeteria is 49.5 feet
Practice Problem
•
A mural of a dog was painted on a wall. The
enlarged dog was 45 ft. tall. If the average
height for this breed of dog is 3 ft., what is the
scale factor of this enlargement? Can you
express this scale in more than one way?
• The scale factor is 45:3. This can be simplified
to 15:1 or expressed in other ways, such as
7.5:0.5
Make a scale map of your desk
Place three or four objects on their desk.
Orient the objects parallel to the edges of the desk.
Use a 1:10 scale, with 1 cm on the map representing 10 cm of the desk
top.
To help students appreciate what the scale is doing and how the
numbers are used in calculating, the teacher may give students a
10 cm × 25 cm rectangle of paper to be one of the objects on the
desk. This gives students one object for which it is easy to work out
what the scaled version is; they may be able to generalize this to
their other objects with more awkward dimensions. A second map
using a different scale could then be produced, perhaps 2 cm = 5 cm
(which is 1:2.5).
Graph paper may help students in drawing their maps.