scale and scale factor
DESCRIPTION
Looking at models and scaleTRANSCRIPT
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Understanding Size
Models and Scale
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Enlargementscale factorReductionScaleProportionsScale Diagram/Model
Vocabulary
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Model• Representation of something else • Usually too big or too small to analyze
easily
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Enlargement
To make something bigger so that one can analyze/observe the details.
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Reduction
To make an object small so that one can observe/analyze the details.
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• 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
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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.
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I Can Solve Problems Using Scale Drawings!
• 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!
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• 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.
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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
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• Scale factor is the ratio of change
• The number you multiply by to relate the first shape to the second is the scale factor.
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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
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• 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!
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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
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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.
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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?
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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
= =
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Now take the:
64 inches
1920 inches And simplify
1/30 inchesThis means that the person was created 30 times his normal size.
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Scale Factor
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Keep like
units in the same
fraction.
Inches = yards
Inches yards
Setting up Proportions
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• There is more than one way to set up a proportion correctly!
• Cross Multiply!• Use common
sense!
Remember…
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• 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
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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
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Scale Drawings
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Scale Drawings
On Maps
Footprints of houses
Vehicle design
What are scale drawings?Scale drawings are everywhere!
Can you think of any more?
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Scale 1 cm = 1 m
6cm
Length of units = 6 m5
Scale in everyday life: kitchen design
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Scale 1 : 1 000 000
Scale in everyday life: maps
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• 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?)
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Scale 2 cm = 1 m
pool path
decking
7
Scale in everyday life: plans
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Using A Scale Drawing
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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?
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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
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• 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
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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
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• 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
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• 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
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I Can Solve Problems Using Scale Drawings!
• 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?
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I Can Solve Problems Using Scale Drawings!
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
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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
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I Can Solve Problems Using Scale Drawings!
• 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?
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I Can Solve Problems Using Scale Drawings!
• 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
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I Can Solve Problems Using Scale Drawings!
• 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.
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I Can Solve Problems Using Scale Drawings!
• 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
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I Can Solve Problems Using Scale Drawings!
• 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!