Upload
caitlin-summers
View
231
Download
4
Tags:
Embed Size (px)
Citation preview
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
• 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.
• 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.
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.
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?
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!
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.
Self Assessment: Try p. 273 #1-4 on your own. Then you may check with a partner.
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
• 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 multi-step problems in various contexts
• 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?
Self Assessment: Try p. 278 #1-5 on your own. Then you may check with a partner.
Standards & Vocabfor 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 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?
Self Assessment: Try p. 296 #1-4 on your own. Then you may check with a partner.
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.
Self Assessment: Try p. 301 #1-5 on your own. Then you may check with a partner.