Chapter 5 Proportions and Similarity. During this chapter, Students will… distinguish between situations that are proportional or not proportionaldistinguish

chapter 5Proportions 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 & Vocabfor 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 multi-step problems in various contexts

• SPI 0706.2.7- Use ratios and proportions to solve problems.

• rate• unit rate

Main Idea:Determine unit rates.

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!When a rate is simplified so that it has a denominator

of 1 unit, it is called a UNIT RATE

5-1-b rates

Rate Unit Rate Abbreviation Name

miles per hour mi/h or mphaverage speed

miles per gallon mi/gal or mpg gas mileage

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?

Adrienne biked 6 miles in 1 hour.

Find each unit rate. Round to the nearest

hundredth if necessary.1.$300 for 6 hours2.220 miles on 8 gallons

Answers:1.$50 per hour2.27.5 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.

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. $0.532. 26

pages/hr3. $0.50 per

can

compare with unit rates

Bag Size (lb) Price ($)

40 $49.99

20 $23.44

8 $9.88

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?HINT: Find

out the unit

price for

each….the

price per

pound!

40 lb bag - $1.249 per pound20 lb bag - $1.172 per pound8 lb bag- $1.235 per pound

So…the 20 pound bag is the best buy!

practice comparingMs. 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?

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!

The results of a swim meet are shown. Who swam the fastest? Be

sure to show all of your work!

Name Event Time (s)

Finn 50-m Free style 40.8

Briley 100-m Butterfly 60.2

Ethan 200-m Medley 112.4

Self Assessment: Try p. 268 #1-6 on your own. Then you may check with a partner.

Standards & vocab for 5-1-C: Relationships



• 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.


• proportional

• non proportional

Main Idea:Identify proportional and non proportional relationships

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.

Cost ($) Pizzas

8 1

16 2

24 3

32 4For 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?

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.

Cost ($) Pizzas Ordered

10 1

17 2

24 3

31 4For each number of

pizzas, write the relationship of the

cost and number of pizzas as a ratio in

simplest form.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.


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.

Remember that when

you are dividing a

mixed number, you

can change it to an

improper fraction.

Don’t forget to

multiply by the

reciprocal!

On your own, simplify each of the ratios written above. Are they equal? Is this a proportional or non proportional relationship?


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!

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.


Week Money

1 $120

2 $160

3 $180

Cost Hours Worked

$45 1

$75 2

$105 3

Standards & Vocab for 5-1-D: Solving proportions

• SPI 0706.1.1 Use proportional reasoning to solve mixture/concentration problems




• equivalent ratios

• proportion• cross

products

Main Idea:Use proportions to solve problems.

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.

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.

a.) 3.6b.) 85c.) 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?

FYI:There are four

different blood types: A, B, AB, and O. People with Type O

are considered universal donors. Their blood can be transfused into people with any

blood type.

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?


Standards & Vocabfor Wildlife Sampling



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 capture-recapture 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 Recaptur

ed 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

• scale drawing

• scale model

• scale• scale

factor

Main Idea:Solve problems involving scale drawings.

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 modelA 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 FACTOR- A 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.

Tip: Scale factors can be used to calculate actual distances from the distances shown in a scale drawing or map. If, for example, a drawing has a scale factor of 1/96, then something that measures 1 inch in the drawing will actually measure 96 inches, or 8 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

• similar figures• corresponding

sides• corresponding

angles• indirect

measurement• Side-Side-Side

Similarity (SSS)• Angle-Angle

Similarity (AA)• Side-Angle-Side

Similarity (SAS)

Main Idea:Solve problems involving similar figures

5-3-A Similar figures

SIMILAR FIGURES-Figures that have the same shape but not necessarily the same size

similar figures

So… since corresponding sides are proportional, if you have to find a missing side length, write and solve a proportion.

Congruent or Similar?

• Congruent figures are the same SIZE AND SHAPE

• Similar figures are the SAME SIZE but not necessarily the same shape.

find missing measures

In the second

example, the triangles

are positioned

differently. You might

want to re-draw the

figures to help you set

up the proportions

correctly!

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?


Standards & Vocabfor 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.

• perimeter• area

Main Idea:Find the relationship between perimeters and areas of similar figures

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 perimeterTwo 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?

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?

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.

Reminders:1. What is the scale factor?2. Are you finding what happens to the PERIMETER or AREA?

3. Look @ your notes and think about if you multiply by the scale factor or (scale factor)2.


Documents

Chapter 5 Proportions and Similarity. During this chapter, Students will… distinguish between situations that are proportional or not proportionaldistinguish