chapter 5
Proportions and Similarity
During this chapter,
Students will…
• distinguish between situations that are proportional or
not proportional
• use proportions to solve problems
• apply proportionality to measurement in multiple
contexts, including scale drawings and constant speed
• solve problems involving similar figures
• determine how changes in dimensions affect the
perimeter, area, and volume of common geometric
figures
Standards & Vocab
for 5-1-B: Rates
• GLE 0706.2.3- Develop an
understanding and apply
proportionality
• GLE0706.2.4- Use ratios, rates, and
percents to solve single-and multistep problems in various contexts
• SPI 0706.2.7- Use ratios and
proportions to solve problems.
Main Idea:
Determine unit
rates.
• rate
• unit rate
Your goals!!!
• I will be able to show what I know about finding unit rates
by correctly solving at least 5 out of the 7 real world
problems shown in the Power Point.
• With a partner, I will be able to create a visual to explain
how to find the unit rate from a newspaper ad. I will
explain my answer in sentence form and with a
mathematical expression.
• I will be able to explain the difference between rate and unit
rate after this lesson.
• Tonight, I will get at least 75% of the homework problems
correct.
• Next week, I will get at least 80% of the rate problems
correct on our Friday Quiz!
Explore!
• Do you know where to find your pulse?
•
Your neck or your wrist
• For two minutes, count the number of beats.
• Then, write the ratio of beats to minutes as a
fraction.
A ratio that compares two
quantities with different
kinds of units is called a
RATE!
160 beats
2 minutes
When a rate is simplified
so that it has a
denominator of 1 unit, it is
called a UNIT RATE
80 beats
1 minute
5-1-b
Rate
rates
Unit Rate
Abbreviation
Name
number of miles
1 hour
miles per hour
mi/h or mph
average speed
number of miles
1 gallon
miles per gallon
mi/gal or mpg
gas mileage
number of dollars
1 pound
price per pound
dollars/lb
unit price
find a unit rate
Adrienne biked 24 miles in 4 hours. If
she biked at a constant speed, how
many miles did she ride in one hour?
24 mi
???
24 miles in 4 hours =
=
4 hours 1 hour
Adrienne biked 6 miles in 1 hour.
Find each unit rate. Round to
the nearest hundredth if
necessary.
1. $300 for 6 hours
2. 220 miles on 8 gallons
Answers:
1. $50 per hour
2. 27.5 miles per gallon
find a unit rate
Find the unit price if it cost $2
for eight juice boxes. Round to
the nearest cent if necessary.
Remember: $2 for 8 boxes
So...
$2
???
=
8 boxes 1 box
The unit price is $0.25 per juice box!
Practice!!!
Find the unit price. Round to the nearest hundredth if necessary.
1. Find the unit price if a 4-pack of mixed fruit sells for $2.12.
2. Julia read 52 pages in 2 hours. What is the average number of
pages she reads per hour?
3. Find the unit price per can if it costs $3 for 6 cans of soda.
Answers:
1. $0.53
2. 26 pages/hr
3. $0.50 per
compare with unit rates
Mrs. Smith is shopping
for Layla’s dog food.
The prices of 3 different
bags of dog food are
given in the table. Mrs.
Smith wants to save
some money so she
need to know which
size has the lowest price
per pound?
Bag Size (lb)
Price ($)
40
$49.99
20
$23.44
8
$9.88
40 lb bag - $1.249 per pound
20 lb bag - $1.172 per pound
8 lb bag- $1.235 per pound
So…the 20 pound bag is the best buy!
practice comparing
Ms. Holloway wants to buy
some peanut butter to donate
to the Second Harvest Food
Bank so that her homeroom
will win the food drive. If
Ms. Holloway wants to save
as much money as possible,
which brand should she buy?
Name
Event
Brand
Sale Price
Kroger Brand
12 ounces for $2.19
Peter Pan
18 ounces for $2.79
Jif
28 ounces for $4.69
Planters
40 ounces for $6.60
Peter Pan will be the best buy!
Time (s)
Finn
50-m Free style
40.8
Briley
100-m Butterfly
60.2
Ethan
200-m Medley
112.4
The results of a swim meet
are shown. Who swam
the fastest? Be sure to
show all of your work!
Self Assessment: Try p. 268 #1-6 on your own. Then you may check with a partner.
Standards & vocab for 51-C: Relationships
• GLE 0706.2.3- Develop an understanding
and apply proportionality
• GLE0706.2.4- Use ratios, rates, and
percents to solve single-and multi-step
problems in various contexts
• SPI 0706.1.3 Recognize whether
information given in a table, graph, or
formula suggests a directly proportional,
liner, inversely proportional, or other
nonlinear relationship.
• SPI 0706.2.7- Use ratios and proportions
to solve problems.
Main Idea:
Identify
proportional and
non proportional
relationships
• proportional
• non
proportional
5-1-c Proportional and
Non proportional Relationships
Mrs. Bybee and Ms. Holloway
are planning a year-end pizza
party . Little Italy Pizza offers
delivery and charges $8 for each
medium pizza.
For each number of pizzas,
we are going to write the
relationship of the cost and
number of pizzas as a ratio
in simplest form. What do
you notice?
Cost ($)
Pizzas
8
1
16
2
24
3
32
4
cost of order 8 16 24 32
= = =
=
pizzas ordered 1 2
3
4
The cost of an order is proportional to the
number of pizzas ordered.
Two quantities are proportional if they have a constant ratio. If the
relationship in which the ratio is not constant, the quantities are
nonproportional.
proportional or
nonproportional?
Papa John’s sells medium pizzas
for $7 each but charges a $3
delivery fee per order. Is the cost
of an order proportional to the
number of pizzas ordered?
Explain.
For each number of
pizzas, write the
relationship of the cost
and number of pizzas as
a ratio in simplest form.
Cost ($)
Pizzas
Ordered
10
1
17
2
24
3
31
4
cost of order
10
®
or 10
pizzas ordered
1
17
or 8.5
2
24
or 8
3
Since the ratios of the two quantities are NOT the same, the cost of an
order is NOT PROPORTIONAL to the number of pizzas ordered.
31
or 7.75
4
proportional or
nonproportional?
You can use the recipe shown to make a healthier
version of a popular beverage. Is the amount of
mix used proportional to the amount of sugar used?
Explain.
On your own, simplify each of the
ratios written above. Are they equal? Is
this a proportional or non proportional
relationship?
proportional or
nonproportional?
Look at the chart to the right. Is the
amount of sugar used proportional to the
amount of water used? Show all of your
work on your paper!
Week
Money
1
$120
2
$160
3
$180
At the beginning of the year, Isabel had $120 in the
bank. Each week, she deposits another $20. Is her
account balance proportional to the number of weeks
of deposits? This time, create your own chart and
then find the ratios!
A cleaning service charges $45 for the first hour and
$30 for each additional hour. Is this fee proportional to
the number of hours worked? Make a table of values
to explain your reasoning.
Cost
Hours
Worked
$45
1
$75
2
$105
3
Self Assessment: Try p. 273 #1-4 on your own. Then you may check with a partner.
Standards & Vocab for 5-1-D:
Solving proportions
• SPI 0706.1.1 Use proportional
reasoning to solve
mixture/concentration problems
Main Idea:
Use proportions
to solve problems.
• SPI 0706.2.7- Use ratios and
proportions to solve problems.
• GLE 0706.2.3- Develop an
understanding and apply
proportionality
• GLE0706.2.4- Use ratios, rates, and
percents to solve single-and multistep problems in various contexts
• equivalent
ratios
• proportion
• cross
products
5-1-d Solve proportions
Kohl’s advertised a sale as shown at the left.
1. Write a ratio in simplest form that compares the cost to
the number of bottles of nail polish.
2. Suppose Kate and some friends wanted to buy 6 bottles
of polish . Write a ratio comparing the cost to the number
of bottles of polish.
3. Is the cost proportional to the number of bottles of polish
purchased? Explain.
$5
$15
=
2 bottles of polish 6 bottles of polish
The ratios of the cost to the number of bottles of polish for two and six
bottles are both equal to 5/2.
They are equivalent
ratios because they have the same value!
proportions
There are two ways
to tell if two ratios
form a proportion.
Either you must:
1. Show that cross
products are equal
2. Show that they
simplify into
equivalent
fractions.
write and solve a
proportion
After 2 hours, the air
temperature had risen 7°F.
Write and solve a proportion to
find the amount of time it will
take at this rate for the
temperature to rise an
additional 13°F.
Solve each proportion below.
x 9
a.) =
4 10
2 5
b.)
=
34 y
7 n
c) =
3 2.1
a.) 3.6
b.) 85
c.) 4.9
solve using proportions
During a blood drive, the ratio of Type O
donors to non-Type O donors was 37:43.
About how many Type O donors would
you expect in a group of 300 donors?
Solve using
proportions
Janie can decorate 8 T-shirts in 3 hours. Write
and solve a proportion to find the time it will
take her to decorate 20 T-shirts at this rate.
A recipe serves 10 people and calls
for 3 cups of flour. If you want to
make the recipe for 15 people, how
many cups of flour should you use?
Recycling 2,000 pounds of paper saves
about 17 trees. Write and solve a
proportion to determine how many
trees you would save by recycling
5,000 pounds of paper.
Write and use an
equation
Beth bought 8 gallons of gasoline for $31.12. Write an equation relating the
cost to the number of gallons of gasoline. How much would Beth pay for 11
gallons at this same rate?
So…How much
would Beth pay
for 11 gallons at
this same rate?
Olivia typed 2 pages in 15 minutes. Write an equation relating the number of
minutes m to the number of pages p typed. If she continues typing at this rate,
how many minutes will it take her to type 1- pages? to type 25 pages?
Self Assessment: Try p. 278 #1-5 on your own. Then you may check with a partner.
Standards & Vocab
for Wildlife Sampling
• SPI 0706.2.7- Use ratios and
proportions to solve problems.
• GLE 0706.2.3- Develop an
understanding and apply
proportionality
Main Idea:
Use proportions
to estimate
populations.
5-1-d Extend:
Wildlife Sampling
Naturalists can estimate the population in a wildlife preserve by using the capturerecapture technique. You will model this technique using dried beans in a bowl to
represent bears in a forest.
1. Fill a small bowl with dried beans Scoop out some of the beans. These represent the
original captured bears. Count and record the number of beans. Mark each bean with an
X on both sides. Then return these beans to the bowl and mix well.
2. Scoop another cup of beans from the bowl and count them. This is the sample for Trial A.
Count the beans with the X’s. These are the recaptured bears. Record both numbers in a
table.
3. Use the proportion below to estimate the total number of beans in the bowl. This
represents the total population P. Record P in the table.
4. Return all of the beans to the bowl.
5. Repeat steps 2-4 nine times!
Trial
Sample
Recaptured
P
A
B
…..
TOTAL
Standards & Vocab for 5-2-b:
scale drawings
• SPI 0706.1.4- Use scales to read
maps
• .GLE 0706.2.3- Develop an
understanding and apply
proportionality
• SPI 0706.2.7- Use ratios and
proportions to solve problems
Main Idea:
Solve problems
involving scale
drawings.
• scale
drawing
• scale model
• scale
• scale factor
5-2-b scale drawings
Scale drawings and scale models are used to represent objects that
are too large or too small to be drawn or built at actual size. The scale gives
the ratio that compares the measurement of the drawing or model to the
measurements of the real object. The measurements on a drawing or model
are proportional to the measurements on the actual object.
What is the actual distance between Hagerstown and
Annapolis?
1. You would first need to use a centimeter ruler to
find the distance on the map between the two
cities.
2. Then, you would write and solve a proportion
using the scale.
use a map scale
On the map of Arkansas shown,
find the actual distance between
Clarksville and Little Rock. Use a
proportion to solve.
(The ruler measures 4cm.)
Refer the the map of South
Carolina. What is the actual
distance between Columbia and
Charleston. Use a proportion to
solve. (The ruler measures 3.8 cm.)
use a scale model
A graphic artist is
creating an
advertisement for a
new cell phone. If she
uses a scale of
5 inches = 1 inch,
what is the length of
the cell phone on the
advertisement?
TRY THIS ONE!
A scooter is 3 ½ feet long. Find the length of a scale
model of the scooter if the scale is 1 inch = ¾ feet.
4 2/3 inches.
scale factor
SCALE FACTORA scale written as a ratio
without units in simplest
form
Find the scale factor of a
model sailboat if the
scale is 1 inch = 6 feet.
Find the scale factor of a
model car if the scale is
1 inch = 2 feet.
Find the scale factor of a
blueprint if the scale is ½
inch = 3 feet.
Answers: 1/72; 1/24; 1/72
construct a scale model
Zara is making a model of a Ferris wheel
that is 60 feet tall. The model is 15 inches
tall. Zara is also making a model of the
sky needle ride that is 100 feet tall using the
same scale. How tall is the model?
Try This One!
Julianne is constructing a
scale model of her family
room to decide how to
redecorate it. The room is
14 feet long by 18 feet wide.
If she wants the model to
be 8 inches long, about how
wide will it be?
Self Assessment: Try p. 287 #1-8 on your own. Then you may check with a partner.
Standards & Vocab for 5-3-a:
similar figures
• GLE 0706.4.1 Understand the
application of proportionality with
similar triangles.
• SPI 0706.4.1 Solve contextual
problems involving similar triangles
Main Idea:
Solve problems
involving similar
figures
•
•
•
•
•
•
•
similar figures
corresponding sides
corresponding angles
indirect measurement
Side-Side-Side
Similarity (SSS)
Angle-Angle
Similarity (AA)
Side-Angle-Side
Similarity (SAS)
5-3-A
SIMILAR FIGURESFigures that have the same
shape but not necessarily the
same size
Similar figures
similar figures
Congruent or
Similar?
So… since corresponding sides are proportional, if you
have to find a missing side length, write and solve a
proportion.
• Congruent figures
are the same
SIZE AND
SHAPE
• Similar figures are
the SAME SIZE
but not
necessarily the
same shape.
find missing measures
indirect measurement
Old Faithful in Yellowstone National Park
shoots water 60 feet into the air and casts a
shadow of 42 feet. What is the height of a
nearby tree that casts a shadow of 63 feet
long? Assume the triangles are similar.
Daley wants to resize a 4-inch-wide by
5-inch-long photograph so that it will
fit in a space that is 2 inches wide.
What is the new length?
2.5 in
indirect measurement
At a certain time of day, a cabbage palm tree that is 71 feet high casts a
shadow that is 42.6 feet long. At the same time, a nearby flagpole casts a
shadow that is 15 feet long. How tall is the flagpole?
Self Assessment: Try p. 296 #1-4 on your own. Then you may check with a partner.
Standards & Vocab
for 5-3-B: Perimeter & Area
of Similar Figures
• GLE 0706.4.3- Understand and use
scale factor to describe the
relationships between length, area,
and volume.
Main Idea:
Find the
relationship
between
perimeters and
areas of similar
figures
• perimeter
• area
5-2-b perimeter and
area of similar figures
Suppose you double each dimension of the rectangle at the
right. The new rectangle is similar to the original rectangle
with a scale factor of 2.
1. What is the perimeter of the original rectangle?
2. What is the perimeter of the new rectangle?
3. How is the perimeter of the new rectangle related to the perimeter
of the original rectangle and the scale factor?
In SIMILAR FIGURES, the perimeters are related by the scale factor!
What about the area? Use the example rectangle above to think about what happens
to the area?
So…the area of the
new rectangle is
equal to the area of
the original rectangle
times the square of
the scale factor!
Perimeter and area
of similar figures
determine perimeter
Two rectangles are similar.
One has a length of 6 inches
and a perimeter of
24 inches. The other has a
length of 7 inches. What is
the perimeter of this
rectangle?
• First, think: What is the
scale factor.
• Next, multiply the
perimeter by the scale
factor.
Triangle LMN is similar to triangle
PQR. If the perimeter of ΔLMN is
64 meters, what is the perimeter of
ΔPQR?
48m
determine area
The Eddingtons have a 5-foot by 8-foot porch on the
front of their house. They are building a similar porch
on the back with double the dimensions. Find the area
of the back porch.
Think: What is the scale factor?
What is the original area?
How is the area affected by the scale factor?
practice!
Two rectangles are similar. One has a
length of 10 inches and a perimeter of
36 inches. The other rectangle has a
length of 7.5 inches. What is the
perimeter of this rectangle?
Maria is painting a mural on her
bedroom wall. The image she is
reproducing is 1/20 of her wall and
has an area of 36 square inches. Find
the area of the mural.
The Coopers bought a 6-foot by 9-foot
rectangular rug. They would like to
buy a similar rug with double the
dimensions. What will be the area of
a new rug?
Self Assessment: Try p. 301 #1-5 on your own. Then you may check with a partner.