Ratios and Scale Factor

Multiple Choice

$$\Delta PQR\ is\ similar\ to\ \Delta STU$$  What side corresponds to side UT?

• PQ

• RQ

• QR

• QP

Multiple Choice

Scale Factor is a ratio comparing:

• the new value to the original value

• the original value to the new value

Scale drawings and models

To solve scale problems, use a proportion!

Set up a Scale proportion

• write the word ratio of units

• make the first ratio the scale

• put the numbers from the problem in the appropriate place in the proportion

Multiple Choice

A scale drawing has a scale of 1cm:2ft. What is the actual width of a pool if it is 6cm on the drawing?

• 3ft

• 6ft

• 12ft

To find the scale

• write the word ratio

• put 1 over x for the scale ratio

• put the numbers from the word problem in the second ratio

• solve the proportion algebraically

Multiple Choice

A model ship is 30cm long and the actual ship is 150m long. What is the scale?

• 1cm:5m

• 1cm:3m

• 1cm:50m

• 1cm:150m

To find the scale factor

• use conversions to write the scale with matching units (use a proportion)

• the scale factor is the ratio written without units

Multiple Choice

The scale in an advertisement is 1in:6ft. What is the scale factor?

• 1:72

• 1:6

• 1:36

• 1:12

Defining Scale Copies

A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.

For example, the two rectangles on the left are scale copies because every length when going from X to Y is multiplied by 3.

Non-Scaled Copy

Let's talk as a group about why the picture of the bird on the right is not a scaled copy of the original on the right.

Defining Corresponding Parts

The word corresponding parts refers to parts that match between two scaled figures. You can identify corresponding angles, corresponding sides, corresponding points, or corresponding line sedments.

Corresponding Distances

Look at the photo on the right and discuss as a group what the corresponding distances/lengths are in the photos. Why is it important to know what the corresponding distances/lengths are?

Defining Scale Factor

Scale Factor is the number which all sides in the original figure are multiplied to get to the scale copy.

What is our scale factor in this figure we looked at on the last slide if your original is the smaller triangle?

Multiple Choice

For example, if the original side length is 5, and the corresponding side length is 25, the scale factor is...

• 5

• 1/5

• 20

Defining Reciprocal

Dividing by a number does not give you the scale factor. However,the reciprocal of the number you divided by does. For example, if your shape is dividing the corresponding sides by 12, the reciprocal of 12 is 1/12, Meaning your scale factor is 1/12.

Fill in the Blank

Let us look at these two triangles again.

We found that going from the smaller triangle to the larger triangle gave us a scale factor of 4. However, if we switched it up and went from the larger triangle to the smaller triangle our scale factor would be the reciprocal of 3.

The reciprocal of 4 is ____.

Question: Here is an unlabeled polygon, along with its scaled copies Polygons A–D. For each copy, determine the scale factor. Explain how you know.

Scale and Scale Drawings

The Scale is the ratio of the length in a drawing (or model) to the length on the real thing.

A Scale Drawing is a drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale). 

Open Ended

This is a map of Malta, a scale drawing of Malta. Identify the scale.

Big Four Bridge

• This is an image of the Big Four Bridge in downtown Louisville aka the Walking Bridge

• The total length for the walking bridge and its ramps is 770 m.

• At its tallest point it is 167 m tall.

Check

• What is the scale we are using?

• What operation do we need to do to calculate the sides of the scale drawing?

• Apply it to our problem..

• 770/10 = 77 cm

• 167/10 = 16.7

Pool Problem: HW

The question:

Here is a scale drawing of a swimming pool where 1 cm represents 1 m.

How long and how wide is the actual swimming pool?

First Step

• First we need to measure the length and width of the scale drawing.

• What is the length of the scale drawing?

• What is the width of the scale drawing?

• Length = 10 cm Width = 5 cm

2nd & 3rd Steps

• Now we need to look at our scale and change the mesurements to the size of the real life pool.

• What is our scale?

• What operation do we use?

• The measurements are the same because the scale is 1 cm to 1 m, we just change the answer to m.

• Answer: Length = 10 m Width = 5 m

Pool Problem Pt. 2

The question: Make a scale drawing of the swimming pool where 1 cm represents 2 m.

What changed in the problem?

Work it out...

• We have a new scale, 1 cm is to 2 m.

• What operation is occuring?

• Apply it to the pool measurements:

• 10/2 = 5 cm is the length

• 5/2 = 2.5 cm is the width

• Now draw your scale drawing.

Answer

Your scale drawing should look like this.

​​The Unit Rate as the Scale Factor

What is SCALE FACTOR?

• ​A scale factor in math is the ratio between corresponding measurements of an object and a representation of that object.

• ​

• The scale factor can be calculated from the ratio of any length in the scale drawing to its corresponding length in the actual picture. The scale factor corresponds to the unit rate and the constant of proportionality. Scaling by factors greater than 1 enlarges the segment, and scaling by factors less than 1 reduces the segment.

​

​TYPES OF SCALE DRAWINGS

• If the scale drawing is LARGER than the original, then it is an enlargement.

• ​

• If the scale drawing is SMALLER than the original, then it is a reduction.

​Steps to check for proportionality for scale drawing and original object or picture

​

1. ​Create a table and record the lengths of the scale drawing on the table.

2. ​Record the corresponding lengths of the actual/original object or picture on the table.

3. ​Check for constant of proportionality by dividing scale drawing lengths by the original lengths.

Multiple Choice

What is the scale factor?

• 3

• 1/3

• 5

• 10

Multiple Choice

What is the scale factor from the large triangle to the small triangle?

• 1/2

• 2/3

• 3/4

• 2

Multiple Choice

Are the following similar? Why or why not?

• Yes, the sides are 2 times larger than the original

• No, one pair of corresponding sides are 2 times larger, but the other is the same length

• Yes, because the corresponding sides are the same length

• No, the numbers are even

Using Scale Factors:

Multiple Choice

You try to find the new dimensions of the box. Click the picture to zoom.

• 2.8 in. by 6.3 in

• 30 in. by 66 in.

• 35 in. by 77 in.

• 3.5 in. by 7.7 in.

Finding Scale factors

Multiple Choice

You try now. What is the scale factor of the rectangles in the image

• 3/2

• 2/3

• 1

• 3/4

Multiple Choice

Enter your answer from the last slide

• A

• B

• C

• D

Multiple Choice

Enter your answer from the last slide

• A

• B

• C

• D

Multiple Choice

Enter your answer from the last slide

• A

• B

• C

• D

Scale drawing

A scale drawing is the same shape but a different size as the original object

A drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale factor).

Scale

a ratio between two measurements, usually between a measurement on a scaled drawing and the actual measurement.

Often seen on maps and floorplans. Scale will always be a ratio between two numbers and include units.

(note: if the units are the same on both sides, they are sometimes left off)

Scale Factor

The scale factor is the number you multiply each measurement to scale up (enlarge) or scale down (reduce) a shape.

If you are scaling up a shape the scale factor must be more than 1

If you are scaling down a shape the scale factor must be less than 1 (usually a fraction)

Unit Rate

A Unit Rate is a special kind of ratio that must always have a 1 in it. It tells you have how much of something per 1 unit of something else.

Sometimes a scale is a unit rate, but only if the scale as a 1 on one side of it.

If 100 cars pass by in 2 hours. The unit rate is 50 cars per hour. 50 : 1

Units

A unit is any measurement that there is 1 of.

So 1 meter is a unit.

And 1 second is also a unit.

Units can measure length, width, height, depth, weight, and many other things

Some units are:

inches, feet, yards, miles, centimeters, meters, Kilometers, pounds, gallons, and even lightyears

How to solve?

• Second number

• divide by

• first number

Let's try 1:5

Multiple Choice

What is the scale factor of 3:1

• 0.333

• 3

Multiple Choice

6:1

• 6

• 0.1666

Multiple Choice

1:5

• 5

• 0.2

Multiple Choice

8:1

• 8

• 0.125

What is the value of the missing side length?

Multiple Choice

What is the value of the missing side length?

• 11

• 33

• 1/33

• 1/11

What is the value of m?

Multiple Choice

What is the value of the missing side length?

• 10

• 6

• 9

• 12