Understanding Rates and Unit Rate
ACTIVITY 19
Zooming!
Lesson 19-1 Understanding Rates and Unit Rates
Learning Targets:
Understand the concept of a unit rate a associated with the ratio a : b
b
with b ≠ 0.
Use rate language in the context of a ratio relationship.
Give examples of rates as the comparison by division of two quantities
having different attributes.
My Notes
•
•
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SUGGESTED LEARNING STRATEGIES: Marking the Text,
Interactive Word Wall, Graphic Organizer, Self Revision/Peer Revision
For the last 28 years, students have participated in the Science Olympiad.
The 2012 Science Olympiad drew 6,200 teams from 50 states. In this
activity, you will use ratios and rates to describe some Science
Olympiad events.
One Science Olympiad event requires teams to build a mousetrap car that
both is fast and can go certain distances.
Several teams of students have decided to compete in the Mousetrap Car
event. They will use mousetraps to act as the motor of their car. Seven of
the students want to use wooden mousetraps, and 9 of the students want
to use plastic mousetraps.
© 2014 College Board. All rights reserved.
1. Write a ratio in fraction form that shows the relationship of the
number of team members who want wooden traps to the number of
team members who want plastic traps.
When wooden traps are compared to plastic traps, you compare different
types of traps (wooden and plastic), but they have the same unit (traps).
This is a ratio because the units are the same.
2. The coaches know that the students will need extra traps. These are
needed so that the students can practice. Write a ratio equivalent to
the one you wrote in Item 1 that shows the relationship of wooden
traps to plastic traps, assuming each member will need 8 traps.
Activity 19 • Understanding Rates and Unit Rate
233
Lesson 19-1
Understanding Rates and Unit Rates
ACTIVITY 19
continued
My Notes
3. Use this ratio to determine:
a. How many of each type of trap to buy.
b. The total number of traps needed. Show your work.
Another way to figure out the total number of traps needed is to write a
ratio comparing traps to people.
4. Write the average number of traps per 1 person as a ratio in
fraction form.
A rate is a comparison of two
different units, such as miles per
hour, or two different things
measured with the same unit,
such as cups of concentrate per
cups of water.
You have just written is a special type of ratio known as a rate. This rate
shows a relationship between quantities measured with different units
(traps and people).
When the rate is per 1 unit, such as traps per 1 person, it is called a
unit rate. Unit rates are easy to spot because they are often written with
the word per or with a slash (/) (for example, traps per team member or
traps/team member).
Rates are called unit rates when
they are expressed in terms of
1 unit.
5. Name at least 2 other rates expressed with the word per.
Examples of unit rates are 60 miles
per hour and 12 words per second.
6. Describe a situation that uses a unit rate.
Check Your Understanding
7. A factory can produce small wheels for the mousetrap cars at a rate
of 18,000 wheels in 3 hours. What is the unit rate per hour?
8. Use the unit rate you found in Item 4 to find the total number
of traps needed for the Mousetrap Car event. Fill in the values
you know.
Unit Rate
Traps/1 Person
Rate for Total
Total Traps/Total People
traps
traps
=
person 16 people
9. How does this compare to your answer in Item 3b? Explain.
234
Unit 4 • Ratios
© 2014 College Board. All rights reserved.
MATH TERMS
Lesson 19-1
Understanding Rates and Unit Rates
ACTIVITY 19
continued
10. Use the Venn diagram below to compare and contrast ratios, rates,
and unit rates. Give an example of each in the diagram.
My Notes
MATH TIP
© 2014 College Board. All rights reserved.
Remember that the regions of the
Venn diagram that are outside the
common regions should be
information that is unique to
that topic.
LESSON 19-1 PRACTICE
$40
11. Find the missing value.
=
8 mousetraps 1 mousetrap
12. The science teacher bought 20 mousetraps for $59.99. What was the
unit cost for each mousetrap?
48 mousetraps x mousetraps
13. Solve:
.
=
6 people
1 person
14. Students spent an average of $5.50 to buy materials for the Science
Olympiad. If they each built 3 mousetrap cars, what was their unit
cost per car?
15. Give an example of a rate that is not a unit rate. Explain your choice.
16. Make sense of problems. Do rates always have to be expressed as
a quotient? Explain how you know.
17. A recipe has a ratio of 2 cups of flour to 3 cups of sugar. How much
flour is there for each cup of sugar?
18. A punch recipe has a ratio of 3 pints of sparkling water to 5 pints of
fruit juice. How much sparkling water is there for each pint of fruit
juice?
Activity 19 • Understanding Rates and Unit Rate
235
Lesson 19-2
Calculating Unit Rates
ACTIVITY 19
continued
My Notes
Learning Targets:
Solve unit rate problems.
Convert units within a measurement system, including the use of
proportions and unit rates.
•
•
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Interactive Word Wall, Visualization, Identify a Subtask, Create a Plan
When a problem involves working
with money, the unit rate is called
the unit price. The unit price tells
you the cost of one item, in this
case the price of 1 bottle.
nose cone
compressed air
Another Science Olympiad event is Bottle Rockets. To compete in this
event, a team must have a large supply of plastic bottles. The coaches and
students decide to take advantage of specials on bottled drinks at two
local stores. They will drink the contents of the bottles at their practices
and meetings and use the bottles themselves to make the rockets.
Kroker’s Market:
Slann’s Superstore:
2 bottles for $2.98
3 bottles for $4.35
$1.59 each
$1.59 each
1. From the advertisements above, predict which store has the less
expensive bottled drinks.
2. How can finding the unit rate for the drinks help you to determine
which store to order the bottled drinks from?
plastic soda bottle
water
fins
nozzle
expelled water
3. Use the price chart for Kroker’s Market.
a. Determine the unit price per bottle if you buy the drinks using
Kroker’s 2-bottle deal.
b. How much do the students save by using the 2-bottle deal instead
of buying 2 bottled drinks at the regular price?
4. Use the price chart for Slann’s Superstore.
a. Determine the unit price per bottle if you buy the drinks using
Slann’s 3-bottle deal.
b. How much do the students save by using the 3-bottle deal instead
of buying 3 bottled drinks at the regular price?
236
Unit 4 • Ratios
© 2014 College Board. All rights reserved.
MATH TERMS
Lesson 19-2
Calculating Unit Rates
ACTIVITY 19
continued
5. Reason quantitatively. To decide where they will get the better
deal, the students cannot simply compare unit rates. Since they need
a specific number of bottled drinks, the better deal may depend on
how many bottled drinks they are buying.
a. Determine how much it would cost to buy 7 bottles from Kroker’s
Market. (Hint: The students can use the deal for every 2 bottled
drinks they buy, but the seventh bottle will be at regular price.)
Show your work.
My Notes
b. Determine how much it would cost to buy 7 bottled drinks from
Slann’s Superstore. Show your work.
c. Where should the students buy their drinks if they want to buy
7 bottles? Explain.
The students now have all of the bottles that they need. They have just a
few more supplies to purchase.
© 2014 College Board. All rights reserved.
One needed supply is 1 -inch PVC pipe to build bottle launchers for
2
practice and competition. They do not need a specific amount of pipe,
because they will use the extra pipe in the future. They want to find the
best deals on this pipe by the foot.
6. The table shows rates for the cost of 1 -inch PVC pipe at three
2
different wholesalers.
Big S Supplies
Build It Again, Sam
Building Stuff
$1.45/2 feet
$3.98/5 feet
$1.77/2 feet
READING MATH
Symbols are sometimes used to
represent units in a measurement.
For example, ˝ is used for inches,
i.e., 9˝ = 9 inches. Similarly,
´ is used for feet, i.e., 8´ = 8 feet.
$28.77/50 feet
a. Find the unit rate for each of the prices at each of the suppliers
above. Show all of your work.
Big S Supplies
Build It Again, Sam
Building Stuff
Activity 19 • Understanding Rates and Unit Rate
237
Lesson 19-2
Calculating Unit Rates
ACTIVITY 19
continued
My Notes
b. Where should the PVC pipe be purchased? Explain why.
c. Explain why the numbers in the table make it easier to use unit
rates to compare prices than using equivalent ratios.
MATH TERMS
Proportions are two ratios that are
equal to each other.
3 = 9 is a proportion because the
5 15
two ratios are equal.
Now, look at just the two pipe prices at Big S Supplies. When trying to
decide which PVC pipe to buy at Big S Supplies, a proportion can also
be used.
In this case, let c represent the unknown cost of the pipe for 50 feet.
$1.45 = c
2 feet 50 feet
To determine a rule that can be used to solve for c, think about what you
already know about solving equations.
7. Write the steps you would use in solving this proportion.
8. Construct viable arguments. How can you use this proportion to
determine which is the less expensive PVC pipe at Big S Supplies?
Explain your reasoning.
9. The length of a car measures 20 feet. What is the length of a model
of the car if the scale factor is 1 inch:2.5 feet?
238
Unit 4 • Ratios
© 2014 College Board. All rights reserved.
$1.45 = c
2 feet 50 feet
Lesson 19-2
Calculating Unit Rates
ACTIVITY 19
continued
My Notes
Check Your Understanding
10. The table shows rates for the cost of buying toy rocket packages.
The packages cannot be broken up. What is the unit rate for each of
the prices shown?
ABC Toys
Z Science Supply
K Museum Store
$49.95/5 rockets
$29.99/2 rockets
$34.95/3 rockets
11. Where should the teacher buy the toy rockets? Explain why.
12. Suppose the teacher wanted to buy exactly 6 toy rockets.
Where should she buy them? Explain.
13. Explain why unit rates may be used to compare prices.
LESSON 19-2 PRACTICE
14. Gordon read 18 pages of a book about rockets in 40 minutes. What
was the unit rate per minute? Per hour?
15. Renaldo earned $45 organizing the science section of the library. If
he worked for 6 hours, what was his hourly rate of pay?
© 2014 College Board. All rights reserved.
16. Make sense of problems. The price of jet fuel in North America
during the last week of 2012 was recorded as $3,062 for 1,000 gallons.
What was the unit price of the jet fuel?
Activity 19 • Understanding Rates and Unit Rate
239