Unit 3 Unit Rates And Percentages — Unit Plan

Title

Assessment

Lesson 1

Anchoring Units of Measurement

So Much in Common

Lin and Elena have discovered they have so much in common.

They each walk 500 units to school.

Who walks 500 feet? $\underline{\hspace{.5in}}$

Who walks 500 yards? $\underline{\hspace{.5in}}$

Explain your reasoning.

A grid of three buildings labeled "School", "Lin's house", and "Elena's house". The grid has three rows and five columns. In row 1, there is an image of a school labeled "School" in column 3. In row 2, there is an image of a house labeled "Lin's house" in column 1. In row 3, there is an image of a house labeled "Elena's house" in column 5.
They each have a fish tank holding 20 units of water.

Whose tank holds 20 gallons? $\underline{\hspace{.5in}}$

Whose tank holds 20 cups? $\underline{\hspace{.5in}}$

Explain your reasoning.
They each have a dog that weighs 24 units.

Whose dog weighs 24 pounds? $\underline{\hspace{.5in}}$

Whose dog weighs 24 kilograms? $\underline{\hspace{.5in}}$

Explain your reasoning.

Show Solution

Lin walks 500 feet, and Elena walks 500 yards. Sample reasoning:
- Yards are longer than feet, and Elena’s house is farther away than Lin’s.
- It looks like 500 foot-long rulers could reach from Lin’s house to the school, but it would take 500 yardsticks to reach from Elena’s house to the school.
Elena’s fish tank holds 20 gallons, and Lin’s fish bowl holds 20 cups. Sample reasoning:
- Gallons are bigger than cups, and Elena’s tank is larger.
- It looks like Elena’s fish tank could hold 20 large milk jugs of water while Lin’s fish bowl could only hold 20 school milk cartons of water.
Lin’s dog weighs 24 pounds, and Elena’s dog weighs 24 kilograms. Sample reasoning:
- Kilograms are heavier than pounds, and Elena’s dog is bigger.
- It looks like Lin’s dog would weigh as much as 24 boxes of crayons while Elena’s dog would weigh as much as 24 textbooks.

Lesson 2

Measuring with Different-Size Units

Which Measurement is Which?

A dog weighs 38 when measured in one unit and 84 when measured in a different unit. Which measurement is in pounds, and which is in kilograms? Explain your reasoning.

$\phantom{0}38 \, \underline{\hspace{1in}} \hspace{0.5in} \phantom{0}84 \, \underline{\hspace{1in}}$
A bird weighs 6 when measured in one unit and 170 when measured in a different unit. Which measurement is in ounces, and which is in grams?

$\phantom{00}6 \, \underline{\hspace{1in}} \hspace{0.5in} 170 \, \underline{\hspace{1in}}$
A kiddie pool holds 180 or 680 units of water, depending on which unit is used to measure. Which measurement is in gallons, and which is in liters?

$180 \, \underline{\hspace{1in}} \hspace{0.5in} 680 \, \underline{\hspace{1in}}$
A storage container that holds 29 or 1,024 units, depending on which unit is used to measure. Which measurement is in cubic feet, and which is in cubic meters?

$\phantom{0}29 \, \underline{\hspace{1in}} \hspace{0.4in} 1,024 \, \underline{\hspace{1in}}$

Show Solution

38 is in kilograms and 84 is in pounds. Sample reasoning: A kilogram is heavier than a pound, so we need fewer kilograms to measure the same weight.
6 is in ounces and 170 is in grams
180 is in gallons and 680 is in liters
29 is in cubic meters and 1,024 is in cubic feet

Lesson 3

Converting Units

Buckets

A large bucket holds 5 gallons of water, which is about the same as 19 liters of water.

A small bucket holds 2 gallons of water. About how many liters does it hold?

Show Solution

About $\frac{38}{5}$ liters (or 7.6 or equivalent). Sample reasoning:

gallons	liters
5	19
1	$\frac{19}{5}$
2	$\frac{38}{5}$

Section A Check

Section A Checkpoint

Problem 1

A volume of 4 liters is approximately 17 cups. Han needs to fill a tank with 10 liters of water. How many cups is that? Show your reasoning.

Show Solution

About 42.5 cups. Sample reasoning:

volume (liters)	volume (cups)
4	17
2	8.5
10	42.5

Problem 2

A distance of 8 kilometers is approximately 5 miles. A runner ran 3 miles. About how many kilometers did they run? Show your reasoning.

Show Solution

About 4.8 or

\frac{24}{5}

kilometers (or equivalent). Sample reasoning: 1 mile is

\frac{8}{5}

or 1.6 kilometers, so 3 miles is

3 \boldcdot (1.6) = 4.8

Lesson 4

Comparing Speeds and Prices

Sparkling Water

Bottles of sparkling water usually cost $1.69 each. This week, 4 bottles cost $5.

Are bottles of sparkling water cheaper or more expensive this week? How much cheaper or more expensive? Show your reasoning.

<p>An image of an ad for sparkling water.</p>

Show Solution

They are $0.44 cheaper per bottle. Sample reasoning: Since 4 bottles cost $5, each bottle costs $5 \div 4$ , or $1.25, this week. The difference is $0.44, because $1.69-1.25 = 0.44$ .

Lesson 5

Interpreting Rates

Gasoline by the Gallon

Two gallons of gasoline cost $6.

Complete the table with the missing volume of gasoline or missing price.
Explain the meaning of each of the numbers you found.

gasoline (gallons)	price (dollars)
2	6
	1
1

Show Solution

Completed table:

gasoline (gallons) price (dollars)

2 6

$\frac13$ 1

1 3
The price of $\frac13$ gallon of gasoline is $1. One gallon of gasoline costs $3.

Lesson 7

More Rate Comparisons

Rulers by the Pack

A store sells wooden rulers in packs of 10 and packs of 6.

A pack of 10 rulers costs $8.49 and a pack of 6 rulers costs $5.40.

Which is the better deal? Explain how you know.

Show Solution

The pack of 10 rulers is a better deal. Sample reasoning:

In a pack of 10, each ruler is about $0.85 because $8.49 \div 10 = 0.849$ . In a pack of 6, each ruler is $0.90 because $5.40 \div 6 = 0.90$ .
If we need 30 rulers, we can buy 3 packs of 10 or 5 packs of 6 rulers. Buying 3 packs of 10 would cost less than $26 because $3 \boldcdot (8.49) = 25.47$ . Buying 5 packs of 6 would cost $27 because $5 \boldcdot (5.40) = 27$ .

Lesson 8

Solving Rate Problems

Going Up?

The fastest elevators in the Burj Khalifa can travel 330 feet in just 10 seconds.

How far does the elevator travel in 11 seconds? Explain or show your reasoning.

Show Solution

363 feet. Sample reasoning:

If the elevator travels 330 feet in 10 seconds, then it is traveling 33 feet per second. Adding 33 feet per second to 330 feet in 10 seconds gives 363 feet in 11 seconds.

Section B Check

Section B Checkpoint

Problem 1

At a store, ribbon is sold by the yard. Diego pays $3 for 12 yards of ribbon. Select all the unit rates that describe this situation:

A.4 dollars per yard

B.0.40 dollar per yard

C.0.25 dollar per yard

D.4 yards per dollar

E.0.25 yard per dollar

Show Solution

C, D

Problem 2

A worker at a clothing store can fold 60 shirts in 4 minutes. At this rate:

How many shirts can the worker fold in 15 minutes? Show your reasoning.
How long would it take the worker to fold 5 shirts? Show your reasoning.

Show Solution

225 shirts. Sample reasoning:
$\frac{1}{3}$ minute (or equivalent). Sample reasoning:
- Dividing 60 by 12 gives 5. Dividing 4 by 12 gives $\frac{1}{3}$ .
- It takes $\frac{4}{60}$ or $\frac{1}{15}$ of a minute to fold 1 shirt, so to fold 5 shirts it would take $5 \boldcdot \frac{1}{15}$ , or $\frac{5}{15}$ , minute, which is $\frac{1}{3}$ minute.
- There are 240 seconds in 4 minutes, so it takes the worker $240 \div 60$ or 4 seconds to fold 1 shirt. To fold 5 shirts takes $5 \boldcdot 4$ , or 20, seconds.

Lesson 10

What Are Percentages?

Kiran & Mai’s Coins

Kiran and Mai each have some coins.

Kiran has 80 cents. What percent of a dollar does he have?
Mai has 140% of a dollar. How much money does she have?

Use the double number line diagram to show your reasoning.

Show Solution

80 percent of a dollar
140 cents or $1.40

Lesson 11

Representing Percentages with Double Number Line Diagrams

Recycling Goal

Noah set a goal of collecting 60 plastic bottles for recycling each week. He has reached 125% of his goal for this week. How many bottles has he collected?

Use the double number line diagram, if you find it helpful.

Show Solution

75 bottles. Sample reasoning:

Lesson 12

Representing Percentages in Different Ways

Small and Large

A small tank holds 36 liters of water. This is 75% of the water that a large tank holds.

How much does the large tank hold? Show your reasoning.

Show Solution

48 liters. Sample reasoning:

If 36 liters is 75% of the water in the large tank, then 12 liters is 25% and $4 \boldcdot 12$ or 48 liters is 100%.

Lesson 13

Benchmark Percentages

Around the Clock

Answer each question and explain or show your reasoning.

How long is 10% of 60 minutes?
How long is 75% of 60 minutes?
If 25% of a show is 11 minutes, how many minutes is the show?

Show Solution

6 minutes. Sample reasoning: It is $\frac{1}{10}$ of an hour.
45 minutes. Sample reasoning: It is $\frac{3}{4}$ of an hour.
44 minutes. Sample reasoning:
- If $\frac{1}{4}$ of the show is 11 minutes, the show must be $4 \boldcdot 11$ or 44 minutes.

Lesson 14

Solving Percentage Problems

Walking to School

It takes Jada 20 minutes to walk to school.

It takes Andre 80% as long to walk to school. How long does it take Andre to walk to school?
Jada’s walk to school takes 250% as long as Tyler’s walk. How long does it take Tyler to walk to school?

Show Solution

16 minutes. Sample reasoning:
- 10% of 20 minutes is 2 minutes. $8\boldcdot 2 = 16$ , so it takes 16 minutes for Andre to walk to school.
- Using a table:
time (minutes) percentage

20 100

2 10

16 80
8 minutes. Sample reasoning:
- If 20 minutes is 250% of Tyler’s walk, then 4 minutes is 50% of Tyler’s walk and 8 minutes is 100% of Tyler’s walk.

Lesson 16

Finding the Percentage

Library and Cafeteria

There are 50 people in a school.

The library has a capacity of 23 people. What percentage of the school population is that? Show your reasoning.
The cafeteria has a capacity of 65 people. What percentage of the school population is that? Show your reasoning.

Show Solution

46%. Sample reasoning: $\frac{23}{50} = \frac{46}{100} = 0.46$ and $(0.46) \boldcdot 100 = 46$ .
130%. Sample reasoning:
- $\frac{65}{50} = \frac{130}{100}$ or 1.3. Multiplying 1.3 by 100 gives 130.
- If 50 people is 100%, then 1 person is 2% of the school population and 65 people is $65 \boldcdot 2$ or 130% of the school population.

Section C Check

Section C Checkpoint

Problem 1

There are 50 states in the U.S.

46% of all the states have an ocean coastline. How many states is this? Show your reasoning.
There are 7 states with a lake shoreline. What percentage of all the states is this?

Show Solution

23 states. Sample reasoning:
- The ratio 23 to 50 is equivalent to 46 to 100.
- number of states percentage
  
  50 100
  
  1 2
  
  23 46
- $\frac{46}{100} \boldcdot 50 =23$
14% of all the states. Sample reasoning:
- The ratio 7 to 50 is equivalent to 14 to 100.
- If 50 states is 100% of all the states, then 1 state is 2% of all the states and 7 states is $7 \boldcdot 2$ or 14% of all the states.

Problem 2

Priya has a plant that measures 26 cm tall. This is 130% as tall as its height when Priya bought it. How tall was the plant when it was bought? Show your reasoning.

Show Solution

20 cm. Sample reasoning:

Using a double number line diagram:
Using a table:

height (cm) percentage

26 130

2 10

20 100

Lesson 17

Painting a Room

No cool-down

Unit 3 Assessment

End-of-Unit Assessment

Problem 1

There are 15 pieces of fruit in a bowl and 6 of them are apples. What percentage of the pieces of fruit in the bowl are apples?

0.06%

0.4%

40%

Show Solution

40%

Problem 2

Select all of the trips that would take 2 hours.

Drive 60 miles per hour between Buffalo and Seneca Falls, which are 120 miles apart.

Walk 3 miles per hour to school, which is 1.5 miles away.

Take a train going 80 miles per hour from Albany to New York City, which are 160 miles apart.

Show Solution

A, C

Problem 3

Lin’s family has completed 70% of a trip. They have traveled 35 miles. How far is the trip?

24.5 miles

50 miles

59.5 miles

200 miles

Show Solution

50 miles

Problem 4

Lin runs 5 laps around a track in 6 minutes.

How many minutes per lap is that?
How many laps per minute is that?
At that rate, how long does it take Lin to run 21 laps?

Show Solution

$\frac{6}{5}$ or 1.2 minutes per lap
$\frac{5}{6}$ laps
25.2 minutes ( $21 \boldcdot (1.2)= 25.2$ )

Problem 5

A car is traveling at a constant speed. The table shows how far it travels in some amounts of time.

What is the speed of the car?

time (hours)	distance (miles)
2	84
3	126

Show Solution

42 miles per hour

Problem 6

Which weighs more: a watermelon that weighs 7.5 kilograms or a baby that weighs 12 pounds? Explain your reasoning.

Note: 1 pound is about 0.45 kilograms.

Show Solution

The watermelon weighs more. Sample reasoning:

12 lbs is about 5.4 kg.
Without computing anything, it can be reasoned that $12 \boldcdot (0.45)$ is less than 6, so the baby must weigh less than the watermelon.

Minimal Tier 1 response:

Work is complete and correct.
Sample: A 12-pound baby weighs less than 6 kilograms, so the watermelon weighs more.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Correct unit conversion with incorrect interpretation; unit conversion multiplies kilograms by 0.45 or otherwise “goes the wrong way”; arithmetic errors in unit conversion.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not involve unit conversion, either through estimation (as in the minimal Tier 1 response) or explicitly; unit conversion is attempted but with an incorrect conversion factor.

Problem 7

On Saturday, Elena read 40% of a 225-page book. That day, Jada read 45% of a 200‑page book.

Who read more pages that day? Explain or show your reasoning.

Show Solution

They read the same number of pages (90). Sample reasoning:

Calculate $\frac{P}{100}$ times the number of pages for each reader.
Create tables, double number line diagrams, or tape diagrams to find the pages read for an intermediate percentage (5%, 10%, or 20% for Elena, and 1% or 5% for Jada), and then scale those numbers to 40% and 45%, respectively.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: $\frac{40}{100} \boldcdot 225=90$ , so Elena read 90 pages. $\frac{45}{100} \boldcdot 200 = 90$ , so Jada read 90 pages.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Work uses a valid representation or sequence of calculations but contains one error that propagates. A written-only response is reasoned correctly but contains arithmetic errors.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: A representation is used to keep track of the information but contains flawed reasoning. Calculations show incorrect values being associated with 100% or misinterpretation of the values to be found and compared.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: No visual representation or written explanation. Explanation does not involve reasoning about ratios and rates.