Unit 4 Plan - Grade 7

Lesson 1

Lots of Flags

—

Colorado State Flag (1 problem)

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?

<p>The Colorado State flag has three equal horizontal stripes, blue, white, blue. In the center, a red C surrounds a yellow circle.</p>

Show Solution

8 feet

Lesson 2

Ratios and Rates with Fractions

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Comparing Orange Juice Recipes (1 problem)

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

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The Price of Wire (1 problem)

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 $4.60 \boldcdot 7.5 = 34.50$ .

Lesson 4

More than That, Less than That

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Swimming and Skating (1 problem)

Tyler swam for $x$ minutes, and Han swam for $\frac34$ less than that. Write an equation to represent the relationship between the amount of time that Tyler spent swimming ( $x$ ) and the amount of time that Han spent swimming ( $y$ ).
Mai skated $x$ miles, and Clare skated $\frac35$ farther than that. Write an equation to represent the relationship between the distance that Mai skated ( $x$ ) and the distance that Clare skated ( $y$ ).

Show Solution

Accept all equivalent forms of each response.

$y=\frac14x$ . (Han swam $\frac{3}{4}x$ minutes less than Tyler swam. Han swam $\frac{1}{4}x$ minutes because $x-\frac{3}{4}x=\frac{1}{4}x$ .)
$y=\frac85x$ . (Clare skated $\frac{3}{5}x$ miles farther than the number of miles Mai skated. Clare skated $\frac{8}{5}x$ miles because $x+\frac{3}{5}x=\frac{8}{5}x$ . )

Lesson 5

Say It with Decimals

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Reading More (1 problem)

Kiran read for $x$ minutes, and Andre read for $\frac58$ more than that. Write an equation that relates the number of minutes Kiran read with $y$ , the number of minutes that Andre read. Use decimals in your equation.

Show Solution

$y=1.625x$ (or equivalent). Andre read $\frac{5}{8}x=0.625x$ more minutes than Kiran read. $x + 0.625x =1.625x$ , so $y = 1.625x$ .

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.

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

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Fish Population (1 problem)

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

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More Laundry Soap (1 problem)

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

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Tyler's Savings Bond (1 problem)

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

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Percentages of 750 (1 problem)

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

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A Restaurant in a Different City (1 problem)

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 11

Percentage Contexts

—

The Cost of a Bike (1 problem)

The bike store marks up the wholesale cost of all of the bikes they sell by 30%.

Andre wants to buy a bike that has a price tag of $125. What was the wholesale cost of this bike?
If the bike is discounted by 20%, how much will Andre pay (before tax)?

Show Solution

$96.15 ( $125 \div 1.3 = 96.15$ )
$100 ( $125 \boldcdot 0.8 = 100$ )

Lesson 12

Solving Multi-step Percentage Problems

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Shoes on Sale (1 problem)

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

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Off by a Little Bit? (1 problem)

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 (1 problem)

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?

A.

6.25\%

✓

B.

6.\overline{6}\%

C.

93.75\%

D.

106.\overline{6}\%

Show Solution

A

Lesson 15

Changes on the Earth

—

The Great Salt Lake (1 problem)

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 Proportional Relationships And Percentages — Unit Plan