Unit 4 Cycle 1 – Ratios and Proportions
Lesson 4.1.1 – Basic Unit Rates
Vocabulary
Rate
Unit Rate
Active Instruction
ta(1) Stella's family drove 184.3 miles to the beach for the family vacation. The trip took 4.8 hours. Eli's family
drove 212.5 miles to the mountains for their vacation. Their trip took 5.1 hours. Which family reached their
destination at a faster rate?
th(1) Shoppers can buy a five-ounce tube of Brand A toothpaste for $5.29 or a three-ounce tube of Brand B for
$4.37. Write a unit price that describes the cost per ounce for each type of toothpaste. Which is the better deal?
Team Mastery
(4) Oscar claims he is a faster painter than Paloma. Oscar painted a 111.6 square-foot wall in 2.2 hours, and
Paloma painted a 127.33 square-foot wall in 2.4 hours. Write unit rates that describe how fast Oscar and Paloma
paint per square foot. Is Oscar right?
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
Today we found unit rates and unit prices. A unit rate is a rate that compares a quantity to 1 unit of another
quantity, like miles driven per 1 hour or the cost of 1 pound of flour. We find unit rates and unit prices for
many reasons. They help us compare two different rates. They can also help us problem solve situations
involving rate. Here is an example!
Jeff types 225 words in 5 minutes and Betsy types 204 words in 4 minutes. What is the unit rate for each
person and who types faster?
Jeff:
; So Jeff types 45 words per minute
Betsy:
; So Betsy types 51 words per minute
When we find the unit rate, the denominator in each rate is the same. This makes the rates much easier
to compare. We can see that Betsy types at a faster rate than Jeff.
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Lesson 4.1.1 - Homework
Directions for questions 1–6: Write a unit rate that
describes the situation.
6)
Sun records the amount of food her pet
mouse, Spot, eats after different numbers of
hours in the table below
1)
Hikers take 3.5 hours to hike a mountain
trail 2.6 miles long. Write a unit rate that
describes their distance per hour.
2)
Ming is at the pet store and sees the
following prices on different sized bags of cat
food:
40 pounds $51.99
20 pounds $24.29
5 pounds $6.97
If Ming is interested in saving money, which bag
should he buy?
3)
Omar walked across the room, a distance
of 6.75 yards in 5.3 seconds. Write a unit rate to
describe how quickly Omar walked in 1 second.
Explain your thinking.
4)
Olga mows 3 lawns in 4.36 hours. Write a
unit rate to describe how many lawns she mowed
in an hour. At this rate, how many laws will she
mow in 15 hours?
5)
Shawn wants to enter a race. To qualify, a
runner has to be able to run 700 feet per minute.
In his tryout run, Shawn runs 2,100 feet in 5
minutes. Will he qualify for the race?
Grams
Hours
2.75
2.5
8.25
7.5
13.75
12.5
Write a unit rate to show how much Spot eats per
hour for each time recorded. Does Spot always
eat at the same rate
Mixed Review
7)
Write the following ratio in three ways.
The punch recipe calls for 4 cups of apple juice
and 3 cups of grape juice.
8)
(
9)
Evaluate the expression.
)
(
)
(
)2
Find the product
.
10)
Find the quotient.
A 2 pound bag of carrots is on sale for
$1.79. The store also carries a 5 pound bag of
carrots for $4.40. Write a unit price that describes
the cost per pound for each bag of carrots. Which
is the better buy? Explain your thinking.
11)
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Unit 4 Cycle 1 – Ratios and Proportions
Lesson 4.1.2 – Unit Rates and Fractions
Active Instruction
ta(1) Ray walks
mile in hour on the treadmill. At what unit rate is Ray walking?
th(1) Alex rode his bike
miles in
hour. Kenya rode her bike
miles in
hour. Write unit rates that
describe their speed. Who rode faster?
Team Mastery
(3) Mrs. Wu finds that bag of mulch will cover square yards of garden. Write a unit rate that describes the
number of bags of mulch to cover 1 square yard of garden.
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
Today you found unit rates that involved one or two fractions. You know that rates involve dividing one
quantity into another so you used the process for dividing fractions: Invert the divisor fraction and multiply.
Here is an example!
A car drove 15 miles on gallon of gas. Write a unit rate to describe the miles the car can go on 1 gallon of
gas.
To find the rate of miles per gallons, we set up our rate with miles in the numerator and gallons in the
denominator. Then we divide to find the unit rate.
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Lesson 4.1.2 - Homework
1)
Car A can go 18.9 miles on
gas, and car B can go 21 miles on
gallon of
gallon of
gas. Write unit rates that describe the miles per 1
gallon of gas for each car. Which car has the
lower miles per gallon?
2)
Wen can make
dozen sandwiches in
hour. Write a unit rate that describes how many
sandwiches she can make in an hour.
3)
Jasmine paid $5.34 for
cheese and $1.47 for
pound of
Carla painted
pound of blueberries.
yd2 of the wall in
in
Mixed Review
7)
Find the sum. (
8)
Find the difference
Asia wants to divide
)
(
)
(
)
Gustav needs a new pair of jeans. The
local department store is having a 15% off
everything sale. If the jeans originally cost
$29.00, how much will Gustav save because of
the discount?
hour.
10)
of a cake among 4
people. How much cake does each person get?
Evaluate the expression.
(
(
What unit rate describes the rate at which Carla
is painting the wall?
5)
hours, what is its speed?
9)
What is the price per pound for each item? Which
is the most expensive? Explain your thinking
4)
Wolves have good endurance and have
been known to travel long distances—up to about
60 miles— in one night. If a wolf trots
miles
6)
11)
)(
)
)
Trina’s science club is building bluebird
houses to put in the local park. For the first
dozen bird houses, Trina finds they used
foot
of wooden trim. Trina needs to buy enough
wooden trim for one dozen bird houses. Write a
unit rate that describes how much wooden trim
Trina needs. Explain how you got your answer.
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Unit 4 Cycle 1 – Ratios and Proportions
Lesson 4.1.3 – Problem Solving with Unit Rates
Active Instruction
ta(1) Anthony needs to buy some notebooks and wants to get the best buy.
th(1) To qualify for the state swimming competition in the 100-meter freestyle race, swimmers must have an
average speed of 1.7 meters per second in the trial races. The results of the three trials for each swimmer are
shown below.
(a) What is the average speed for each swimmer?
Team Mastery
(3) Caitlin keeps a record of when she babysits and how much she earns. Use Caitlin's babysitting data to answer
the following questions.
(b) What unit rate describes what Caitlin charges for each hour of babysitting? Explain your thinking.
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
5|Page
Today we solved real-world problems that involved rates and unit rates. These rates included money
amounts, fractions, and decimals. We also used graphs, charts, and other data organizers to help us!
For example, we can learn a lot about rates for buying bird seed from this graph:
We can see on the graph that thistle seed costs $2.10 per pound, and sunflower seed costs $1.40 per
pound. We can also see that there is about a $2 difference in price between 3 pounds of thistle seed and 3
pounds of sunflower see. Additionally, we can predict that 8 pounds of sunflower seeds will cost about
$11.00. What else can you learn from the graph? What other rates could you write?
Lesson 4.1.3 - Homework
Lorenzo is a painter. He recently painted
a garage floor that was 10.5 feet by 12.8 feet. It
took him
hours to complete the job. He
charged his customer $66.25 for the job. His next
job is painting the four walls of a basement with
the following dimensions: two walls are 9 feet by
10.2 feet and two walls are 9 feet by 6.7 feet.
c) How much profit did the store make on
the baseball gloves?
1)
a) Write a unit rate to describe what area of
surface Lorenzo can paint each hour.
b) How long will it take him to paint the
basement walls? Explain your thinking.
c) What will he charge for the time it took to
paint the basement?
2)
A sporting goods store ordered 48 youth
baseball gloves from the distributer for $861.60.
The store sold half of the gloves for a total of
$544.08 and put the rest on sale at a discount of
10% off per glove.
a) At what unit price did the store sell the
first half of the gloves?
b) At what price did the store sell the second
half of the gloves?
Mixed Review
3)
Alex’s family produces 15 gallons of
recycling in 6 days. Write a unit rate that
describes the amount of recycling his family
produces per day.
4)
Cynthia wants to buy a new bed for her
dog. The one she wants originally cost $17.79,
but is now on sale for 33% off. What did she pay
after the discount?
5)
Write a numeric expression to answer the
question. At 10 A.M., the temperature was –8°F.
By 1 P.M., the temperature had risen 4 degrees.
By 3 P.M., the temperature had gone up another
2 degrees. At 6 P.M., the temperature had
dropped by 1 degree. At 10 P.M., the
temperature had dropped another 3 degrees.
What was the temperature at 10 P.M.?
6)
Find the sum.
(
)
6|Page
Unit 4 Cycle 2 – Ratios and Proportions
Lesson 4.2.1 – Defining Proportional Relationships
Vocabulary
Proportion
Proportional Relationship
Active Instruction
th Isa painted a figure with a square inside it. She wanted to paint a new version that was proportional to the
original, but larger. Which could be the larger version Isa painted; A or B? How do you know? What does
proportional mean?
ta(1) Andie bought 5 notebooks for $6.50 and Quincy bought 3 notebooks for $3.90. Did they pay proportional
amounts for the notebooks?
th(1) Chris has drawn two circles.
Is there a proportional relationship between the circumferences and diameters of the circles? Explain your thinking.
Team Mastery
(4) Terry works on the factory floor making widgets and earns $700 for a 40-hour work week. Jamie works in the
factory office and receives $628.13 for 37.5 hours per week. Does the company offer proportional salaries to its
employees? Explain your thinking.
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
7|Page
Today we learned to identify proportional relationships. Proportions are math statements that two ratios are
equivalent.
For example,
.
The proportions we worked on included fractions, decimals, and whole numbers. Identifying proportions extends our
use of unit rates. There are different ways to determine if a pair of fractions or ratios are proportional. One way is to
compare the unit rates, or divide each ratio and write it as a decimal. If the decimal equivalents for the two ratios are
equal, then this is a proportional relationship.
For example:
Both 3 ÷ 4 and 75 ÷ 100 equal 0.75, so these two ratios are proportional.
Another way to test if two ratios or fractions are proportional is to cross multiply to find out if the ratios
are proportional.
For example:
Are
4 • 75
proportional? Let's cross multiply to find out
3 • 100
300 = 300, so these two ratios are proportional.
Lesson 4.2.1 - Homework
Directions for questions 1–6: Determine if the
relationships are proportional.
1)
Gretchen can run
miles in 12 minutes.
Tamara can run
miles in 16 minutes. Do
Gretchen and Tamara run at proportional
speeds?
2)
Are these ratios proportional?
Are these two triangles proportional to each
other?
Mixed Review
3)
Dana bought the last 3 red delicious
apples at the ABC Grocery for $2.07 and bought
9 more at the XYZ Grocery for $6.21. Are the
prices for these apples at the two stores
proportional? Explain your thinking.
4)
The school cafeteria sold 65 ham
sandwiches for $149.50 and 81 peanut butter
sandwiches for $170.10. Does the cafeteria sell
ham sandwiches and peanut butter sandwiches
at a proportional cost?
5)
7)
On a trip to Maine, the Montoya family
drove 712 miles and used 29 gallons of gas.
Write a unit rate that describes the number of
miles the family drove for each gallon of gas they
used.
8)
The price of potatoes has risen 4.4% from
the previous price of $0.48 per pound. What is
the new price per pound of potatoes?
9)
(
(
Are these ratios proportional?
10)
6)
Evaluate the expression.
)
)
Solve the equation.
Bia has drawn two isosceles triangles.
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Unit 4 Cycle 2 – Ratios and Proportions
Lesson 4.2.2 – Solving Proportions
Active Instruction
ta(1) The two rates below are proportional because the unit rates are equal when we divide the miles by minutes.
This means that walking 2 miles in 40 minutes is the same rate as walking 5 miles in 100 minutes.
th(1) Lan's recipe makes 12 dozen cookies, but she only wants to make 2 dozen cookies today. Use proportions to
adjust the recipe so that she makes only 2 dozen cookies. Explain your thinking.
Chocolate Cookies: Makes 12 dozen
ounces of chocolate cake mix
cups vegetable oil
6 eggs
Team Mastery
(3) The mass and volume of pure substances are proportional. This relationship is called density: Density = mass
divided by volume, or
3
Table salt (sodium chloride or NaCl) has a density of 2.16 g/cm . What is the mass of table salt if its volume is
3
38.6 cm ? Explain your thinking.
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
Today you learned to solve proportions for a missing value. Understanding how to set up and solve
proportions can help you with many everyday problems such as figuring out the price of a new number of
items, the exchange rate between U.S. currency and the currencies in other countries, changing the
ingredients in recipes, and many other situations.
Here is an example!
If Harold used 3.5 gallons of gas to drive 77 miles, he can find out how many gallons he will need to drive
100 miles by solving a proportion:
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Remember, always set up the two ratios in your proportion with the same pattern. For example, miles over
gallons = miles over gallons. This problem was solved by looking for a common denominator, then working
to isolate the missing variable to solve for x.
Lesson 4.2.2 - Homework
Directions for questions 1–6: Use proportions to
solve the missing value or values.
1)
What is the missing value in this
proportion?
6)
The band at Central School has 20
members. North School is much larger, and has
50 members in the band. The two bands have
proportional amounts of the same instrument.
Use proportions to figure out how many students
play each instrument at North School.
Central School Band:
2)
Baker Barb uses 162 cups of flour to bake
54 loaves of bread each day. Tomorrow, she
needs to bake an additional 8 loaves. How much
flour will she need tomorrow?
3)
What is the missing value in this
proportion? Explain your thinking.
6 violin players
8 clarinet players
Mixed Review
7)
Identify which of the following are
proportional to the ratio 75:375 and which are
not.
(a) 10:50
4)
Dalonte is making a salad to bring to a
barbeque. His recipe makes enough salad for 6
people, but there will be 32 people at the
barbeque. Use proportions to adjust the recipe so
that Dalonte has enough salad to feed 32 people.
Salad: Feeds 6 people.
6 cups salad greens
3 medium tomatoes, sliced
cup red onion, sliced
cup croutons
5)
Olivier bought a pair of shoes in France
for 23.99 Euro. If $1 U.S. equals 0.77 Euro, use a
proportion to find out how much the shoes would
cost in U.S. dollars. Round to the nearest
hundredth.
2 tuba players
4 flute players
(b) 25:125
(c)
(d) 150:400
8)
Write a numeric expression to answer the
question, then solve.
A SCUBA diving instructor has the class start
their practice at 9.4 feet below the surface. He
directs the student divers to ascend 3.9 feet, then
descend 17.3 feet and wait for him to join them.
Then the students dive another 10.7 feet to take
pictures of clown fish. Next the students ascend
2.9 feet to take pictures of a sea star. At what
depth are the divers now?
9)
Mrs. Wang buys 3 boxes of the same
cereal for $8.37. Write a unit rate that describes
the price per box.
10)
A book store has a 10% off sale. If a book
costs $16.97, how much money will Li save by
buying the book on sale?
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Unit 4 Cycle 2 – Ratios and Proportions
Lesson 4.2.3 - Proportions in Tables and Graphs
Active Instruction
ta(1) How can you tell if the ratios in the table are
proportional? Does the table show equivalent ratios
of miles driven to hours?
ta(2) How can you tell if the data in the graphs are proportional?
th(1) Mrs. Cruz recorded how much tax she paid on some major purchases. She bought a used car for $13,000, a
new refrigerator for $899, and a new furnace for $3,200. She paid $650, $44.95, and $160 in sale taxes respectively.
a. Make a table and draw a graph on grid paper of the price and sales tax data. On the graph, let the x-axis
show the price of the item (in thousands of dollars) and the y-axis show the sales tax.
b. Are the sales tax-price ratios proportional or non-proportional? Explain how you know.
Team Mastery
(3) Are the ratios shown in the table proportional or
not proportional? Explain your thinking.
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
Today you identified proportional and non-proportional
relationships in tables and graphs. If the ratios in
tables have the same value, then the data are
proportional.
Here is an example!
11 | P a g e
The unit rate is the same for each set of values. The unit rate is 1 item sells for $16. Now we can predict the cost for
other numbers of the same item. For example, 10 items will cost $160. If the data were graphed, it would show a
straight line that would intersect with the origin, because 0 items would cost $0. The line tells us that there is a
constant increase in the price as the number of items bought increases.
Because there is a proportional increase, we can
predict the cost of other amounts of items. For
example, we can find 2 items on the x-axis and find
that the price will be about $30.
Lesson 4.2.3 - Homework
Directions for questions 1–4: Draw a table or graph if
needed, and then determine if the data is proportional
or not proportional.
4)
1) This graph shows a bike ride Dylan did. Was the
distance he traveled proportional to his time?
pounds. Patient C is 25 and weighs 123 pounds.
A doctor charts the age and weight of some of
his patients. Patient A is 18 years old and weighs 107
pounds. Patient B is
years old and weighs 120
a. Draw a graph on grid paper of this
information. Let the x-axis show the patients'
age and the y-axis show their weight.
b. Do the patients have proportional ages and
weights?
Mixed Review
5)
To prepare for the winter, a bear consumes
140,000 calories in a week. Write a unit rate that
describes how many calories a bear eats in a day.
6)
Solve.
7)
What is 13.5% of 76?
8)
Identify which of the following are proportional
to the ratio 8:10 and which are not.
2)
Are the data in this chart proportional or not
proportional? Explain how you know.
a. 12:15
b. 56:60
c. 36.8:46
d.
9)
a)
Mr. Werth bought gas for his car at three
different gas stations. At the first station, he paid
$37.45 for 10 gallons of gas. At the second station, he
pumped 8 gallons and paid $29.96. He bought 16
gallons for $59.92 at the third gas station.
The following data was gathered in an
experiment on how fast a free-falling object, such as a
ball, falls. Explain how to figure out if this data is
proportional.
Make a table of the prices and amounts of gas Mr.
Werth purchased at each station. Is the price per
gallon of gas proportional at the three stations?
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Unit 4 Cycle 2 – Ratios and Proportions
Lesson 4.2.4 - Problem Solving with Proportions 1
Active Instruction
ta(1) This graph shows a proportional relationship.
th(1) Newton's second law of motion says that the
acceleration (think about a car changing its speed) of
an object depends on the amount of force (measured
in newtons) and the mass of the object (measured in
kilograms - kg). Note: 1 newton of force is about what
you feel when you hold asmall apple in your hand.
Below is the data for force and acceleration
(measures in meters per second squared - m/s2) for
two different masses:
a) Create one graph with two line on grid paper to show the data for both the 2 kg mass and the 4 kg mass (use
different colors or make on line a dashed line).
b) What do both lines tell you about the relationship of force and acceleration?
c) What can you conclude about the difference between the 2 kg line and the 4 kg line. Explain your thinking.
Team Mastery
(3) An aquarium has to make repairs to a 4,200
gallon tank. It has set up pumps to pump out the
water. This graph shows drainage times.
c) Using the same set of pumps, how long would it
take to drain the 6,000 gallon Sting Ray Reef tank?
Describe two different ways of finding the answer.
Explain your thinking.
13 | P a g e
Lesson Quick Look
Team Name
Team Did Not Agree On
Questions…
Team Complete?
Today we solved real-world problems that involved proportions. We used graphs, charts, and other data organizers
to help us. When events occur in proportional relationships, we can use that relationship to predict other events.
Understanding proportional events helps us understand everyday experiences such as the time it takes to travel a
certain distance when going different speeds. Here is an example!
The local deli sells fresh ham each day. They keep track of the amount of ham and cost for each customer. Here is
the data, including the unit rates.
We see that the prices for the different amounts of
ham are not proportional. The unit prices to buy 2.5
pounds, 5 pounds, and 9 pounds of ham are all $1.78
per pound, but the cost to buy 10, 11, or 16 pounds of
ham is $1.62 per pound.
If we graph the data we can see that the graph does
not make a straight line. This reminds us that all of
the data is not proportional
We can even think about reasons why the first part of
the data seem to be proportional and the second half
seems to be proportional, but all of the data is not
proportional. Maybe the deli gives a discounted price
to larger orders of ham!
Lesson 4.2.4 - Homework
1) Reggie noticed his kitchen faucet was dripping.
He put a 4-cup measuring cup under the faucet
and left it overnight. After 9 hours, the measuring
cup had 2.5 cups of water.
a. What is the unit rate for the leak in cups per
second?
b. At this rate, how much water will he lose in a
week? A Year?
c. If 16 cups are in one gallon, how many
gallons will Reggie lose in a year? Explain
your thinking.
2)
Last week, Mrs. Park bought 12 ounces of
strawberries for $2.28. This week she bought 28
ounces of strawberries for $5.32. Mrs. Park also
bought blueberries each week; the price of blueberries
was $4.60 for 20 ounces last week and $1.15 for 5
ounces this week
a. Create one graph with two lines on grid paper
to show the relationship between the amount
of berries purchased and the cost of the
berries (use different color for your lines, or
make on line dashed).
b. Are the prices for strawberries and blueberries
proportional for the different amounts
purchased?
c. What is the price per pound for the
strawberries? (16 oz = 1 lb)
3)
Tanya paid $26.25 for a 15-pound bag of rice.
a. What is the unit price?
b. How many ounces of rice could she buy for
$1.25? (16 oz = 1 lb)
c. Create a graph on grid paper to show the
relationship between the pounds of rice (x)
and the total cost (y).
Mixed Review
4)
Over 3 minutes, Lamar counted 219 heart
beats. What is his heart rate per minute
5) Find the difference.
6) Solve
(
)
.
14 | P a g e
Unit 4 Cycle 3 – Ratios and Proportions
Lesson 4.3.1 - Constant of Proportionality
Vocabulary
Constant of proportionality (k)
Active Instruction
ta(1) U.S. law requires all American flags be
proportional. Flag makers must follow a specific ratio
of length to width.
th(1) Juan is comparing prices of wooden molding
for a construction project. This graph shows two
moldings: simple and fancy.
(a) For each type of molding, find the constant of proportionality for the ratio of cost to amount
purchased. Explain your thinking.
Team Mastery
(4) What is the constant or proportionality? Explain
your thinking.
15 | P a g e
Lesson Quick Look
Team Name
Team Did Not Agree On
Questions…
Team Complete?
Write the vocabulary introduced in this lesson:
Today you learned that there is another name for unit rate – the constant of proportionality, sometimes called k.
There is a constant value for the ratio of two proportional amounts. If a gallon of paint costs $21.95, that price is
constant whether you buy one gallon for $21.95 or 10 gallons for $219.50. Each can has a constant price, so the
constant of proportionality, k, is $21.95 per gallon for this paint. Knowing the value of k in this situation helps us figure
out how much we would spend for different amounts of paint or how much paint we can buy for a certain amount of
money.
Here is an example!
If y = 420, and x = 3, then we can solve for k in the equation:
Remember, the constant of proportionality only
applies to ratios that are proportional, so you have to
determine that first. Some things do not change
proportionally. One example is buying something in a
larger package. Often, but not always, larger
packages have a lower unit rate than smaller
packages. It is best to check by doing the math.
Lesson 4.3.1 - Homework
Directions for questions 1–6: Find the constant of
proportionality for each.
3)
1)
What is the constant of proportionality
between x and y?
2)
Inga makes three sizes of pizza: small,
medium, and large. She made a table to show the
diameter of each pizza and the amount of pepperonis
she put on it.?
a. What is the constant of proportionality
between the area and the price of the
countertop? Explain your thinking.
b. What does the constant of proportionality
represent for this graph?
a.
b.
What is the constant of proportionality between the
size of the pizza and the number of pepperonis?
What does the constant of proportionality represent
for the relationship shown in the table?
16 | P a g e
4)
Farouk can do more sit-ups than anyone in his
class. He can do 54 sit-ups in 45 seconds and 108 situps in a minute and a half.
a. What is the constant of proportionality
between the number of sit-ups (x) Farouk can
do and the time (y) it takes him to do them?
b. What does the value for k in this problem
represent?
5)
Mixed Review
6)
If Calvin drives 82.5 miles in 1.5 hours, how
far will he go if he drives at the same rate for 2.5
hours
7)
Are the ratios shown in the table proportional
or not proportional?
8)
Manuel ate 18.75% of the 32 cookies his
mother made. How many cookies did he eat?
9)
Tanya owes her mother $15.78. She pays half
of it back to her mother. How much does she still
owe? Write a numeric expression for this situation and
then evaluate the expression?
What is the constant of proportionality between x and
y?
17 | P a g e
Unit 4 Cycle 3 – Ratios and Proportions
Lesson 4.3.2 - Represent a Proportion as an Equation
Vocabulary
Slope
Active Instruction
th(1) Below are the prices of gasoline in 2013 and 1974.
ta(1) This graph shows the cost of downloading music
from two different online companies.
a)
Write an equation for each year (2013 and 1974)
that expresses the relationship between the
number of gallons of gas (x) and total price (y).
b) Draw a graph on grid paper where x is the number
of gallons and y is the price. Show a line for each
year (use separate colors or a dashed line).
c) Explain what the slope means for each line.
Team Mastery
How can we use the equation
to describe these
two different proportional relationships?
(4) a. Write an equation for the relationship shown
in the graph.
b. What is the slope? Explain your thinking.
ta(2) This graph shows the cost of downloading music
from two different online companies. What's another way
to describe unit rates and constant proportionality?
company A:
company A:
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Lesson Quick Look
Team Name
Team Did Not Agree On
Questions…
Team Complete?
Write the vocabulary introduced in this lesson:
Today you learned how to describe proportional relationships with an equation. You can use data from tables,
graphs, and written descriptions to find the constant of proportionality (k) by dividing one of the y-values by its xvalue. Then you can write an equation of the form y = kx. This equation reveals how the y-value changes whenever
the x-value changes. It allows you to make predictions about future changes in the quantities in proportional
relationships.
We also learned another word that describes unit rate and constant of proportionality—slope. Slope refers to the
relative steepness of a line on a graph. In a proportional relationship, slope visually shows the rate of constant
change: a steep, straight line has a higher slope (a larger value of k) and a less steep, straight line has a lower slope
(a lower value of k). The rate of change is lower in a line with a low slope value as compared to a line with a larger
slope value.
Here is an example! The table below shows a
proportional relationship.
y divided by x for both pairs is 0.067, so that is the
constant of proportionality. We can write an equation
for this relationship: y = 0.067x. This means that for
every increase in x, y increases by 0.67. If we know
the values of x and k, we can find the values for y. The
slope in this relationship is 0.067. When we graph the
relationship, we can see that for everyone 1 that x
rises, that y rises by 0.067.
Lesson 4.3.2 - Homework
Directions for questions 1–6: Use the information to
answer each question.
1)
Mrs. Jung knits mittens for her grandchildren.
She needs
a) Write an equation for the relationship between
the yards of yard and amount of mittens
knitted.
b) What is the slope?
yards of yarn to knit 6 pairs of
mittens, and 1,970
yards to knit 7 pairs of mittens?
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2) Wood Needed for a Fence
5)
Students at Kipling School are selling
popcorn. A portion of each student’s sales goes to a
fund to buy more books for the school library.
Cameron sold $23 worth of popcorn and was able to
donate $18.40 to the library fund. Masha sold $51.90
and donated $41.52?
a) Write an equation for the relationship between
the amount each student raised and the amount
he or she gave to the library fund.
b) What is the slope?
6) Swimming Laps
a) Write an equation for the relationship between
the weight and cost of peaches.
b) What is the slope?
3)
The Smith, Johnson, and Wilson families have
been saving to go to the Ice Show. The table below
shows the number of tickets they bought and the
prices they paid.
a) Write an equation for the relationship between
the time and amount of laps swam.
b) What is the slope?
a) Write an equation for the relationship between
the number of tickets and the total price.
b) What is the slope?
Mixed Review
7) Mary bought 1.75 pounds of apples for $5.32.
What is the constant of proportionality in terms of
price per pound?
4)
An orchard charges a proportional amount for
pick-your-own peaches. The table below shows
weight and cost of each basket of peaches.
8)
Jun paid a 10% shipping charge for the
$45,683 he spent on detergent for his laundry
business. How much did he pay for shipping?
9)
Are the following ratios proportional? How can
you tell?
a) Write an equation for the relationship between
the weight and cost of peaches.
b) What is the slope? Explain your thinking.
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Unit 4 Cycle 3 – Ratios and Proportions
Lesson 4.3.3 - Interpret Points of a Proportional Relationship
Active Instruction
ta(1) Raemonn has a lawn mowing business. The
amount of money he charges for hours worked is
represented in the graph below.
a.
b.
What does the point (0,0) mean on the graph in this
situation?
Explain what (1,r) represents on a graph of a
proportional relationship.
a. What does the point (0,0) represent on the
graph in this situation?
b. What is the value of r in the point (1,r) for both
Veronika and Dnia? What does this point
represent for each of them? Explain your
thinking.
Team Mastery
(4) Marta and Inez both volunteered to make 5 dozen
cookies for the school bake sale. Their efforts are
show in the graph below.
th(1) Veronika and Dnia raced against each other in
the 400 meter spring at a track meet. The following
graph represents their race results.
b) What is the value of r in point (1,r) for both
Marta and Inez? What does this point
represent for each of them? Explain your
thinking.
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Lesson Quick Look
Team Name
Team Did Not Agree On
Questions…
Team Complete?
Write the vocabulary introduced in this lesson:
Today you built on your knowledge of graphing
proportional relationships by using points on a line to
describe what is happening in the situation
represented by the graph. You specifically talked
about the points (0, 0) and (1, r), where r is the unit
rate.
Here’s an example:
On this graph, the point (0, 0) represents that
Raemonn has earned $0 for 0 hours of work. He
doesn’t make any money if he doesn’t do any work.
The point (1, r) on this graph is (1, 10). What this point
tells us is Raemonn earns $10 for every 1 hour of
work he does. That is the unit rate. So, we know that
when x = 1 on a graph of a proportional relationship,
we know that y represents the unit rate for that line.
Lesson 4.3.3 - Homework
Directions for questions 1–5: Use the graph to answer
each question.
1)
Konnor needs to buy balloons for a school
dance and has been comparing the prices at two party
supply stores, as shown in the graph below?
a) What does the point (0, 0) represent on the
graph in this situation?
b) What is the value of r in point (1, r) for Party
Down and Celebration Time? What does this point
represent for each of them? Explain your thinking.
2)
Johnny tracked the rainfall over two days in
Baltimore and made the following graph.
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a) What does the point (0, 0) represent on the
a) What does the point (0, 0) represent on the
graph in this situation?
b) What is the value of r in point (1, r) for Monday
and Tuesday? What does this point represent for
each of them?
3)
Lysette is comparing the speed of sail boats
and motor boats for an upcoming ocean excursion.
graph in this situation?
b) What is the value of r in point (1, r) for the
meteor shower? What does this point represent?
5)
Eduardo is taking part in the Great Backyard
Bird Count and has made a graph of the birds he saw
over a 15 minute period one day.
a) What does the point (0, 0) represent on the
graph in this situation?
b) What is the value of r in point (1, r) for the
sailboat and motor boat? What does this point
represent for each of them?
4)
Ki watched the Perseid meteor shower one
night and made a graph to chart how many meteors
she counted.
a) What does the point (0, 0) represent on the
graph in this situation?
b) What is the value of r in point (1, r) for
cardinals, finches, and crows? What does this
point represent for each of them?
Mixed Review
(
6) Find the difference.
7)
Find the product.
( )
(
)
)
8)
Monifa makes $11.25 for 2 hours of
babysitting. How much will she make for 4.5 hours of
babysitting?
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Unit 4 Cycle 3 – Ratios and Proportions
Lesson 4.3.4 - Problem Solving with Proportions 2
Vocabulary
Independent variable
Dependent variable
Active Instruction
ta(1) Apples are on sale: $2.50 for 2 pounds.
a. Which equation best describes this proportional relationship?
b. How do you decide which quantity is the y-value and which is the x-value?
th(1) People and objects have different weights on different planets and moons in space. Weight is the
effect of the pull of gravity on the mass of an object. Different planets and moons have different gravities,
due in part to their size. A smaller planet has a lower gravity than a large planet. A person weighing 100
pounds on Earth would weigh 38 pounds on Mars, 234 pounds on Jupiter, and 17 pounds on the moon.
a. Which value would be the independent variable and which would be the dependent variable? Write
equations for each of these relationships. Explain your thinking.
Team Mastery
(2) The Khan family is planning a camping trip
and wants to compare the cost of renting
campsites. They have gathered the following
data:
b) How much would each campground
charge for a 5-night stay? Explain your
thinking.
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Lesson Quick Look
Team Name
Team Did Not Agree On
Questions…
Team Complete?
Write the vocabulary introduced in this lesson:
Today you learned about independent and dependent variables to help you decide what will be the x- and y-values in
a proportional relationship. Changes in the independent variable are not affected by changes in the dependent
variable, but changes in the dependent variable are affected by changes in the independent variable.
Here is an example!
Bananas cost $7.45 for 5 pounds. The number of
pounds of bananas is the independent variable
because it is not dependent on the price, however the
price customers pay IS dependent on the number of
pounds that they buy.
Now you can find the constant of proportionality:
So the independent variable, x, is the number of
pounds; the dependent variable, y, is the total price.
Once you know the constant of proportionality, k, you can do many things. You can write an equation for the
relationship, graph the relationship, or find other values of x and y that share the proportional relationship.
Lesson 4.3.4 - Homework
1)
Derrick drove 76.8 miles and used 2.4 gallons
of gas. Ling drove 82.6 miles and used 2.8 gallons of
gas.
a) Write an equation for each person. Are
Derrick’s and Ling’s situations proportional to
each other? How do you know?
b) How far will each drive on a full tank
(12 gallons)?
c) Who has the better gas mileage, Derrick or
Ling? If you graphed each relationship, whose
graph would be the steepest, Derick’s or Ling’s?
2)
Water makes up a large part of living things. A
55 kilogram person has 36.85 kilograms of water in
his or her body.
a) Write an equation for this relationship. What is
the slope?
b) What is the dependent variable in this
situation?
c) How much water would a 150 kg person
contain?
3)
There is a proportional relationship between
the mass of water and the mass of oxygen, a
component of water (H20); see the table below:
a) Write an equation for the relationship in the
table above.
b) What is the constant of proportionality for this
relationship?
c) Make a data table for the mass (g) of
hydrogen in water.
d) Write an equation for the relationship between
the mass of hydrogen and the mass of water.
Mixed Review
4)
At Famous Chocolates, you can buy 2.5
pounds of chocolate butter creams for $15.25. What is
the constant of proportionality and what does it mean?
5)
Five is what percent of 25.
6)
Solve.
( ) ( ) ( )
7)
Solve.
( )
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Unit 4 Cycle 3 – Ratios and Proportions
Lesson 4.3.5 - Think Like a Mathematician: Find the Patterns and Structure 1
Vocabulary
Pattern
Active Instruction
ta(1) Sweetness Bakery has an order to make a
cupcake display for the grand opening of the new
modern art museum downtown.
th(1) Nessa is building a house of cards with
two decks of playing cards. The first layer has 23
cards arranged the following way.
If the display continues this way, how many
cupcakes in all does Sweetness Bakery need to
bake to fill all six layers?
The second layer has 20 cards, and the third
layer has 17 cards. If her house continues like
this, how many layers of cards does Nessa
have? Explain your thinking.
Team Mastery
(4) A fishing crew went salmon fishing near the end of the annual salmon run on the river. The first
morning they caught 96 salmon, the second morning they caught 48 salmon, and the third morning they
caught 24 salmon. If the fishing continues this way, how many salmon will they catch on the sixth
morning? Explain your thinking.
Lesson Quick Look
Team Name
Team Complete?
Team Did Not Agree On
Questions…
Write the vocabulary introduced in this lesson:
Today we learned to use the Find a Pattern Strategy to solve a word problem. For example:
Ameen is trying to shorten the time it takes to do his chores. In week one, he finished them in 194 minutes. In week
two, he finished them in 190 minutes. In week three, he finished them in 186 minutes. If he keeps going like this, by
how many fewer minutes will it take him to do chores in week seven than in week one?
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If the problem looks like it has a pattern, create a table for the given information. At least 3 pieces of information are
needed to determine a pattern. In this problem, the pattern is that Ameen shortens his time each week by 4 minutes.
Next, fill in the table to find how long it takes Ameen to do his chores in week seven, 170 minutes. Finally, solve. It
took him 194 minutes to do his chores on week one. The difference in time between these two weeks is 194 – 170 =
24 minutes. So, Ameen takes 24 fewer minutes to complete his chores in week seven than week one
Lesson 4.3.5 - Homework
Directions for questions 1–6: Solve.
1)
In Baddington, it snowed 97.2 inches in
December, 32.4 inches in January, and 10.8 inches in
February. If it continues snowing like this, how many
inches total will the town get between the months of
December and March? Explain your thinking.
2)
Marisol does 7 sit-ups on day one, 14 sit-ups
on day two, and 28 sit-ups on day three. If she keeps
doing sit-ups like this, how many will she do on day
seven?
3)
The height of Mrs. Gallagher’s tree was 5.63
inches the first year. It was 9.5 inches tall the second
year, and 13.37 inches tall the third year. If it
continues to grow like this, how much taller will it be in
the seventh year than in the first year?
4)
A breakfast shop served 256 waffles on
Sunday, 243 waffles on Monday, and 230 waffles on
Tuesday. If they continue to serve waffles like this, on
what day will they serve 165 waffles?
5)
Darija’s garden grew 19 cucumbers the first
summer, 38 cucumbers the second summer and 76
cucumbers the third summer. If her garden continues
to produce like this, how many cucumbers will grow
the sixth summer?
6)
Truman hiked 43 minutes on day one, 54
minutes on day two, and 1 hour and 5 minutes on day
three. If he continues like this, how many minutes in
all will he hike for seven days?
Mixed Review
7)
Gordi typed 197 words in 3 minutes. What is
his unit rate for typing??
8)
What is the missing value in the proportion?
9)
You have a graph with the points (25.5, 48)
and (74, 139.12) and a line drawn between them.
What is the constant of proportionality for this graph?
10)
Izem feeds his two cats a proportional amount
of food based on their weight. Mittens weighs 7
pounds and is fed 3.5 scoops of cat food each day.
Tiger weighs 18.7 pounds and is fed 9.35 scoops of
cat food. Write an equation for the relationship
described in the problem.
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