Unit 4 Proportional Relationships And Percentages — Unit Plan

Title

Assessment

Lesson 1

Lots of Flags

Colorado State Flag

The side lengths of the state flag of Colorado are in the ratio $2:3$ . If a flag is 12 feet long, what is its height?

Show Solution

8 feet

Lesson 2

Ratios and Rates with Fractions

Comparing Orange Juice Recipes

Clare mixes $2 \frac12$ cups of water with $\frac13$ cup of orange juice concentrate.
Han mixes $1 \frac23$ cups of water with $\frac14$ cup of orange juice concentrate.

Whose orange juice mixture tastes stronger? Explain or show your reasoning.

Show Solution

Han's mixture tastes stronger. Sample reasoning: Clare uses $7 \frac12$ cups of water per cup of orange juice concentrate, because $2\frac12 \div \frac13 = 7 \frac12$ . Han uses $6 \frac23$ cups of water per cup of orange juice concentrate, because $1\frac23 \div \frac14 = 6 \frac23$ . Han's mixture has less water for the same amount of orange juice concentrate.

Lesson 3

Revisiting Proportional Relationships

The Price of Wire

It costs $3.45 to buy

\frac34

foot of electrical wire. How much would it cost to purchase

7\frac12

feet of wire? Explain or show your reasoning.

Show Solution

$34.50. Sample reasoning:

It costs 10 times as much to buy 7.5 ft of wire as to buy $\frac34$ ft of wire because $\frac34 \boldcdot 10 = 7.5$ and $3.45 \boldcdot 10 = 34.50$ .
The wire costs $4.60 per foot because $3.45 \div 0.75 = 4.60$ . At this rate, it will cost $34.50 for 7.5 ft wire because $4.60 \boldcdot 7.5 = 34.50$ .

Section A Check

Section A Checkpoint

Problem 1

Clare makes glitter glue by mixing $\frac78$ pound of glitter with $\frac13$ gallon of glue. At this rate,

How much glitter would she mix with 1 gallon of glue?
How much glue would she mix with 3 pounds of glitter?

Show Solution

$\frac{21}{8}$ pounds (or equivalent)
$\frac87$ gallons (or equivalent)

Problem 2

Farm A harvested $x$ bales of cotton. Farm B harvested $\frac35$ more than that.

Select all the expressions that represent how much cotton Farm B harvested.

\frac35 x

\frac85 x

1 + \frac35

x + \frac35 x

0.6x

1.35x

Show Solution

B, D

Problem 3

What is the decimal representation of

\frac{5}{12}

? Show your reasoning.

Show Solution

0.41\overline{6}

, because

$\displaystyle \require{enclose} \begin{array}{rll} 0.4166 \\[-3pt] 12 \enclose{longdiv}{5\phantom{.1000}} \\[-3pt] \underline{-0}\phantom{.1000} \\[-3pt] 50\phantom{.100} \\[-3pt] \underline{-48}\phantom{.100} \\[-3pt] 20\phantom{.00} \\[-3pt] \underline{-12}\phantom{.00} \\[-3pt] 80\phantom{.0}\\[-3pt] \underline{-72}\phantom{.0} \\[-3pt] 80\phantom{.}\\[-3pt] \underline{-72}\phantom{.}\\[-3pt] 8\phantom{.} \end{array}$

Lesson 6

Increasing and Decreasing

Fish Population

The number of fish in a lake decreased by 25% between last year and this year. Last year there were 60 fish in the lake. What is the population this year? If you get stuck, consider drawing a diagram.

Show Solution

There are 45 fish in the lake this year. Sample reasoning:

The number of fish decreased by 15, because $0.25 \boldcdot 60 = 15$ . That means there are 45 fish left, because $60 - 15 = 45$ .
There are only 75% as many fish this year, because $100 - 25 = 75$ . We can multiply $0.75 \boldcdot 60 = 45$ .
Here is a tape diagram that shows there are 45 fish left:

Lesson 7

One Hundred Percent

More Laundry Soap

A company claims that their new box holds 20% more laundry soap. If the new box holds 54 ounces of soap, how much did the old box hold?

Explain or show your reasoning. If you get stuck, consider using the double number line.

<p>A double number line for “laundry soap in ounces” with 7 evenly spaced tick marks.</p>

Show Solution

45 ounces. Sample reasoning: After a 20% increase, the new value is 120% of the original.

Lesson 8

Percent Increase and Decrease with Equations

Tyler's Savings Bond

Tyler's mom purchased a savings bond for Tyler. The value of the savings bond increases by 4% each year. One year after it was purchased, the value of the savings bond is $156.

Find the value of the bond when Tyler's mom purchased it. Explain your reasoning.

Show Solution

The bond was originally worth $150. Sample reasoning: To represent the situation, use the equation $1.04x=156$ , where $x$ represents the value of the savings bond when Tyler's mom purchased it. The solution is $x = 156\div1.04 = 150$ .

Lesson 9

Part of a Percent

Percentages of 750

A school has 750 students.

If the number increases by 4%, how many students will there be? Explain or show your reasoning.
If the number increases by 0.4%, how many students will there be? Explain or show your reasoning.

Show Solution

780 students. Sample reasoning: $750 \boldcdot 1.04 = 780$
753 students. Sample reasoning: $750 \boldcdot 1.004 = 753$

Section B Check

Section B Checkpoint

Problem 1

Last week it took a canoe team 56 minutes to paddle across the bay. This week it took them 49 minutes. By what percentage did their time decrease?

Show Solution

12.5%

Problem 2

This year, the canoe club has 32 members. This is a 28% increase from the number of members last year.

Draw a diagram that could help you find the number of members that were in the canoe club last year. (You don’t have to actually solve the problem, but the diagram should represent how the number of members from this year and last year relate to each other.)
Write an equation that could help you find the number of members that were in the canoe club last year. (You don’t have to actually solve the problem.)

Show Solution

Sample responses:
$1.28x = 32$ (or equivalent)

Lesson 10

Tax and Tip

A Restaurant in a Different City

At a dinner, the meal costs $22, and a sales tax of $1.87 is added to the bill.

How much would the sales tax be on a $66 meal?
What is the tax rate for meals in this city?

Show Solution

$5.61 ( $22\boldcdot 3 = 66$ and $1.87 \boldcdot 3 = 5.61$ )
8.5% ( $1.87 \div 22 = 0.085$ )

Lesson 12

Solving Multi-step Percentage Problems

Shoes on Sale

A pair of shoes normally costs $85. They are on sale for 20% off. A sales tax of 6% is added to the sale price.

How much will the shoes cost after the discount and the tax?

Show Solution

$72.08, because $0.8 \boldcdot 85 = 68$ and $1.06 \boldcdot 68 = 72.08$

Lesson 13

Measurement Error

Off by a Little Bit?

Clare estimates that her brother is 4 feet tall. When they get measured at the doctor’s office, her brother’s height is 4 feet, 2 inches.

Should Clare’s or the doctor’s measurement be considered the actual height? Explain your reasoning.
What is the error expressed in inches?
What was the error, expressed as a percentage of the actual height?

Show Solution

The doctor's measurement is more precise, while Clare's is only an estimate.
2 inches
4% (4 feet 2 inches is equivalent to 50 inches, and $2 \div 50 = 0.04$ .)

Lesson 14

Percent Error

Yarn Weight

A ball of yarn is supposed to weigh 3.5 ounces. Priya measures it and finds that it weighs 3.3 ounces. What is the percent error?

Show Solution

About 5.7%. Sample reasoning: $3.5 - 3.3 = 0.2$ and $0.2 \div 3.5 = 0.0\overline{571428}$ .

Section C Check

Section C Checkpoint

Problem 1

Han’s father uses a credit card to buy a suitcase that costs $143. If he does not pay it off by the end of the month, the credit card company will add 2% interest to his bill.

How much will the suitcase cost him with interest? Explain or show your reasoning.

Show Solution

$145.86. Sample reasoning:

2% of 143 is 2.86, and $143 + 2.86 = 145.86$
$143 \boldcdot 1.02 = 145.86$

Problem 2

A package is supposed to contain 17.6 ounces. A worker weighs it and finds that it only contains 16.5 ounces. What is the percent error?

6.25\%

6.\overline{6}\%

93.75\%

106.\overline{6}\%

Show Solution

6.25\%

Lesson 15

Changes on the Earth

The Great Salt Lake

This table shows information about the Great Salt Lake, in Utah.

date	depth at deepest point (feet)
June 1986	37
November 2022	14
June 2023	19

Find a percentage of increase or decrease that describes the situation. Write a sentence that clearly describes what the percent increase or percent decrease represents.

Show Solution

Sample responses:

From 1986 to 2022, the depth of the Great Salt Lake decreased by 62.2%.
From 1986 to 2023, the depth of the Great Salt Lake decreased by 48.6%.
From 2022 to 2023, the depth of the Great Salt Lake increased by 35.7%; however, this increase was equal to only 13.5% of its previous depth from 1986.

Lesson 16

Posing Percentage Problems

No cool-down

Unit 4 Assessment

End-of-Unit Assessment

Problem 1

Priya has a recipe for banana bread. She uses $7 \frac 1 2$ cups of flour to make 3 loaves of banana bread.

Andre will follow the same recipe. He will make $b$ loaves of banana bread using $f$ cups of flour.

Which of these equations represents the relationship between $b$ and $f$ ?

$b = \frac 2 9 f$

$b = \frac 2 5 f$

$b = \frac 5 2 f$

$b = \frac 9 2 f$

Show Solution

$b = \frac 2 5 f$

Problem 2

Diego measured the length of a pen to be 22 cm. The actual length of the pen is 23 cm.

Which of these is closest to the percent error for Diego’s measurement?

4.3%

4.5%

95.7%

104.5%

Show Solution

4.3%

Problem 3

A car is 180 inches long. A truck is 75% longer than the car.

How long is the truck?

135 inches

240 inches

255 inches

315 inches

Show Solution

315 inches

Problem 4

A circular running track is $\frac 1 4$ mile long. Elena runs on this track, completing each lap in $\frac 1 {20}$ of an hour.

What is Elena’s running speed? Include the unit of measure.

Show Solution

5 miles per hour. (We can divide the number of miles by the number of hours to find the number of miles per hour: $\frac 1 4 \div \frac 1 {20}$ .)

Problem 5

Today, everything at a store is on sale. The store offers a 20% discount off the regular price.

The regular price of a T-shirt is $18. What is the discounted price?
If the regular price of an item is $x$ dollars, what is the discounted price in dollars?
The discounted price of a hat is $18. What is the regular price?

Show Solution

$14.40 (80% of $18)
$0.8x$ or $(x - 0.2x)$ or $\frac 4 5 x$ (or equivalent)
$22.50 ( $0.8x = 18$ , so $x = 22.5$ .)

Problem 6

Lin’s father is paying for a $20 meal. He has a 15%-off coupon for the meal. After the discount, a 7% sales tax is applied.

What does Lin’s father pay for the meal? Explain or show your reasoning.

Show Solution

$18.19. Sample reasoning: The coupon takes away 15% of $20, which is $3. The cost after the coupon is $17. The added tax is $1.19 (7% of $17). The total is $18.19, the sum of $17 and $1.19.

Minimal Tier 1 response:

Work is complete and correct.
Sample: $20 - 3 = 17$ , and $17 \boldcdot (1.07) = 18.19$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Calculating the tax to be $1.40, using 20 instead of 17; answering $1.19, using the amount of the tax as the final answer; answering $18.19 without showing or explaining work; errors in computation but not in concept.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Ignoring the 15% discount or the 7% tax altogether; adding 15% to the total; using 15% of $20 as the amount before the tax; errors because of major misunderstandings about operations, such as adding $7 as the tax instead of 7%.

Problem 7

Tyler’s brother works in a shoe store.

Tyler’s brother earns a commission that is 2.5% of the amount he sells. Last week, he sold $900 worth of shoes. How much was his commission?
The store buys a pair of shoes for $50 and sells it for $80. What percentage is the markup?
Tyler’s brother earns $12 per hour. The store offers him a raise—a 5% increase. After the raise, how much will Tyler’s brother make per hour?

Show Solution

$22.50. The commission was $22.50, because $900 \boldcdot (0.025) = 22.5$ .
60%. The markup is $30, and $30 is 60% of $50.
$12.60. Because 5% of $12 is $0.60, Tyler’s brother now makes $12 + 0.60$ dollars per hour.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

$900 \boldcdot (0.025) = 22.5$
60%, a $30 increase.
$12.60. The raise is $0.60.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: One or two calculation errors in executing the parts of the problem; stating 60 instead of 60% for the markup without a very clear description that the 60 refers to 60%; answering 0.60, the value of the raise, instead of the total after the raise.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: One major conceptual error, perhaps due to a misinterpretation of the meaning of one of the terms in the problem; omission of one part of the problem; repeated calculation errors in execution.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more parts of the problem are omitted or with major conceptual errors.