SUBTRACTION  ADDITION  

DIVISION  

Turn  Around  Words  

Parenthesis  Words  

MULTIPLICATION  

EQUALS  

To translate an equation:

1. Read the written expression left to right.

2. Translate written words into mathematical operators

3. SPECIAL NOTE! If the phrase “less than” or “fewer than” or “subtracted

from” is used, you must invert your translation

TRANSLATING VERBAL PHRASES

ExpressionOperation Verbal Phrase

of 3 and The a number n

A n xum pb le ur s 10

Subtraction The of 7

and a n aumber

T lw ee sl sv e t han a number x

Multiplication 1. Five a n umber y

The of 2 and a number n

Division of a The number a and 6

A number y by 8

Order is important when writing subtractionand division expressions.

Addition 1.

2.

1.

2.

2.

1.

2.

Quick Check!

1. A number increased by 9 _____________________

2. The product of a number and 4 _____________________

3. The difference between 10 and a number _____________________

3-2 Translating Equations

4. 2p________________________________________________

5. 2 - p________________________________________________

1

To translate more complex expressions:

1. the phrase “the quantity of” signifies a grouping using parenthesis

2. commas in written expression signify a grouping using parenthesis

3. the words “sum”, “difference”, “product”, “quotient” signify a grouping usingparenthesis

Practice 1 Translate verbal phrases into expressions

Translate the verbal phrase into an expression.

Verbal Phrase Expression

a. 6 less than the quantity8 times a number x

b. 2 times the sum of 5and a number a

c. The difference of 17 andthe cube of a number n

The words “the quantity” tell you what to group when translating verbal phrases.

1. The sum of 3 and 9 times X

2. The quotient of 7 and K minus 5

______________________________

______________________________

3. 16 divided by the difference of A and B

4. The quantity of X plus Y, subtracted from 11

__________________________

__________________________

Quick Check!

5.

6.

2

WRITE EQUATIONS When writing equations, use variables to represent the unspecified numbers or measures referred to in the sentence or problem. Then write the verbal expressions as algebraic expressions. Some verbal expressions that suggest the equals sign are:

• is • is equal to • is as much as

• equals • is the same as • is identical to

Translate Sentences into Equations Translate each sentence into an equation.

a. Five times the number a is equal to three.is equal

Five times a to three

b. Nine times y subtracted from 95 equals 37. Rewrite the sentence so it is easier

to translate. 95 minus nine times y equals 37.

95 minus nine times y equals 37.�����

�����

Translate each sentence into an equation. Solve the equation. Show your check.

1. Two times a number t decreased by eight equals seventy.

2. Five times m plus 4 is the same as 70.

Quick Check! GUIDED PRACTICE KEY

3

Lesson 3.2 – Translating two-step equations

Translate each sentence into a two-step equation and then solve the two-step equation.

1) Four more than twice a number is 8.

__________________________________________

2) Three more than four times a number is 15.

__________________________________________

3) Twice a number increased by 5 is 7.

__________________________________________

4) Eleven is one less than four times a number.

__________________________________________

5) Six more than the quotient of a number and 2 is 10.

__________________________________________

6) The quotient of a number and 8, decreased by 5 is 6 .

__________________________________________

4

7) Seven minus six times a number is 19.

__________________________________________

8) Seven increased by twice a number is 1.

__________________________________________

9) The difference between 5 times a number and 3 is 12.

__________________________________________

10) Four less than the quotient of a number and 3 is 10 .

__________________________________________

11) One is 3 less than the quotient of a number and 6.

___________________________________________

12) The difference between twice a number and 11 is 23 .

_________________________________________

5

©2010 Texas Education Agency. All Rights Reserved 2010

Problem-Solving Boards

Problem A: The total cost of shipping a package to Hendersonville includes a fee of $______ plus $0.75 per ounce. Write an equation to represent c, the total cost of shipping a package that weighs ______ ounces. See

Fee: $6.80

Cost per ounce: ______

Number of ounces: w

Total cost: c

Plan

Do

c = 0.75(____) + _____

Reflect

My equation is reasonable because _________________________________________

_________________________________________

Problem B: Nicole saved _______% of the amount of earnings from her summer lifeguarding job. She also saved $_______ that she earned babysitting. If Nicole saved a total of $_______, write an equation to determine x, the amount of money Nicole earned lifeguarding. See

Percent saved from lifeguarding: ______

x: amount earned ________

Amount saved from babysitting: _________

Total amount of savings: $150

Plan

Do

______ = 75 + _____ (x)

Reflect

My equation is reasonable because _________________________________________

_________________________________________

Total Savings: $150 =

Amount saved from babysitting: $75

Lifeguard earnings: ______

20% of this

Total cost: c Fee: _____ Cost per ounce:

______

w ounces

=

3-3 A

5a

©2010 Texas Education Agency. All Rights Reserved 2010

Problem-Solving Boards (continued)

Problem C: A rectangle has a perimeter of _________ units. The length is _________ units more than the width, w. Write an equation to represent the perimeter of the rectangle in terms of its width.

See

Perimeter: 34 units

Length: w + 3

Width: _______

Plan

Do

34 = 2 (______) + 2 (_______)

Reflect

My equation is reasonable because _________________________________________

_________________________________________

Problem D: The cost of Ellie’s tuition and books for her night class is $525. She made an initial payment of $_____ and is paying the remaining balance with monthly payments of $______. If she has made ______ payments, write an equation to represent b, her remaining balance. See

Cost of tuition and books: $525

Initial payment: $_______

Monthly payment: $75

Number of payments made: _____

b: _______________________

Plan

Do

125 + _____m + b = 525

Solve for b:

b = ____________________________

Reflect

My equation for the remaining balance is reasonable because _________________________________________

_________________________________________

_______

_______

w w

Tuition and books: $525=

Initial

payment:

_____

Monthly

payment:

______

m payments

Balance:

b

5b

©2010 Texas Education Agency. All Rights Reserved 2010

Jasmine wants to purchase a new guitar. The guitar costs $800. She has saved $150 and plans on saving an additional $50 each week.

Write an equation to determine the number of weeks, w, that it will take for Jasmine to save enough money to purchase the guitar.

Processa) Highlight the information in the problem:

costs $800, saved $150, $50 each week,number of weeks, w.

b) Translate the important information as numbersand/or symbols:

c) Determine the number of weeks, w, it will take forJasmine to save enough money to purchase the guitar.

$800 is $150 and $50 # of weeks

Reginald wants to buy a shirt that is on sale for 15% off the regular price, p.

Write equations to find the amount of discount, d, that Reginald saved and s, the sale price of the shirt.

Processa) Highlight the information in the problem: shirt, is, 15% off, regular price, p, amountof discount, d, s, sale price.

b) Translate the important information asnumbers and/or symbols.

c) Write an equation to represent s, the saleprice of the shirt.

d) Use your equations to determine the sale priceof a shirt that was regularly $30.

amount of discount

is 15% (as a

decimal)

regular price

sale price

is regular price

minus amount of discount

(use part b)

Stephen is a pizza delivery driver. He earns $8 per hour, h, plus an additional $0.75 for each delivery, d.

Write an equation to represent Stephen’s earnings of $286 before taxes last week.

Process

a) Highlight the information in the problem:$8 per hour, h, plus, $0.75 for eachdelivery, d, $286.

b) Translate the important information as numbersand/or symbols:

c) If Stephen made 40 deliveries, determine howmany hours he worked.

$8 per hour

plus $0.75 for each delivery

total $286

3-3 A

d) Use your equations to find the regular price of ashirt if the sale price is $34..

6

©2010 Texas Education Agency. All Rights Reserved 2010

A rectangle has an area of 105 square units. The length, l, of the rectangle is 1 more than twice the width, w.

Write an equation to represent the area of the rectangle in terms of the width.

Processa) Highlight the information in the problem:area, 105, length, l, is, 1 more than twicethe width, w

b) Translate the important information asnumbers and/or symbols.

c) If the width is 7 units, determine the length of therectangle.

d) Write an equation to represent the area in terms ofthe width.

length is 1 more than

twice the width

area is length (use expression

from part b)

times width

Digital songs cost $1.50 each and digital videos cost $2.50 each.

Write an equation to represent the number of songs, s, and the number of videos, v, that Marcella could purchase with $20.

Processa) Highlight the important information in

the problem: songs, $1.50 each,videos, $2.50 each, number of songs,s, number of videos, v, $20.

b) Translate the important information as numbersand/or symbols.

c) Determine the number of songs that could bepurchased if 6 videos are purchased.

$1.50 number of songs

and $2.50 number of videos

is $20

The total cost, c, to rent a carpet cleaner at Company A is $12 plus an additional $3 per day.

Write an equation to represent the total cost to rent a carpet cleaner for d days.

Processa) Highlight the important information in theproblem: cost, c, is, $12 plus $3 per day,d.

b) Translate the highlighted areas in numbers and/orsymbols:

cost is $12 plus $3 per day

c) Determine the total cost if the carpet cleaner isrented for 4 days.

7

Daily Notetaking Guide L1

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Lesson Objective Write an equation to solve a problem1

Lesson 3-3B Write an Equation

Writing Expressions Define a variable and write an algebraic expression for the phrase “four times the length of a rope in inches, increased by eight inches.”

Let � length of rope in inches. d Define the variable.

� �

� 8

d Write an algebraic expression.

Evaluate the expression if the length of a rope is 9 inches.2

1

Quick Check.

1. Define a variable and write an algebraicexpression for “a man is two years younger thanthree times his son’s age.”

2. Evaluate the expression to find theman’s age if his son is 13.

Step 1 Explore the Problem To solve a verbal problem, first read the problem carefully and explore what the problem is about.

• Identify what information is given.• Identify what you are asked to find.

Step 2 Plan the Solution: One strategy is to write an equation.

Then use the variable to write expressions for the unknown numbers in the problem.

Step 3 Solve the Problem: Use the strategy you chose in Step 2 to solve the problem.

Step 4 Examine the Solution: Check your answer in the context of the original problem.

• Does your answer make sense? • Does it fit the information in the problem?

Choose a variable to represent one of the unspecific numbers in the problem. This is called defining a variable.

8

Daily Notetaking GuideL1

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1. Suppose that Mr. Reynold’s bill was $161.80 and he drove the van 350 mi.a. How would the equation change?

b. Solve the new equation to find how many days he rented the van.

Van Rental A moving van rents for $29.95 a day plus $.12 a mile. Mr. Reynolds’s bill was $137.80 and he drove the van 150 mi. For how many days did he have the van?

1. What is the goal of this problem? 2. How many miles did Mr. Reynolds drive?

3. What does the van cost without mileage? 4. What is the charge for each mile?

Make and Carry Out a Plan

2. Write an equation.

d �

Mr. Reynolds had the van for days.

Multiply 0.12 and 150.

Simplify.

137.80�29.95d �

Subtract______ from each side.

Simplify.

Divide each side by

3. Solve the equation.

29.95d �

� �

Words

Equation

cost perday

29.95

cost per mile

number of days

? ?

�

� 0.12�

Understand the Problem

1

? # of miles driven

.�

total cost

1. Identify the Variable: Let ______ = the number of days Mr. Reynolds had the van.

9

Shopping The Healy family wants to buy a TV that costs $200. They already have $80 saved toward the cost. How much will they have to save per month for the next six months in order to have the whole cost saved?

plus

Identify the Variable: Let ______ = the amount to save per month.

Words

Equation

is

4

times

1. What is the goal of this problem? 2. How much does the TV cost?

3. How much money have they saved? 4. How many months will they save money?

Understand the Problem

10

3. Basketball During the first half of a game you scored 8 points. In thesecond half you made only 3-point baskets. You finished the game with23 points. Write and solve an equation to find how many 3-point basketsyou made.

Identify the Variable: Let ______ = how many 3-point baskets you made.

4.

5.

11

1. Analyze the following problem.Misae has $1900 in the bank. She wishes to increase her account to a total of $3500 bydepositing $30 per week from her paycheck. Will she reach her savings goal in one year?a. How much money did Misae

have in her account at thebeginning?

b. How much money will Misae add toher account in 10 weeks? in 20 weeks?

c. Write an expression representing the amountadded to the account after w weeks have passed.

d. What is the answer to the question? Explain.

Quick Check!

2. OPEN ENDED Write a problem that can be answered by solving x � 16 � 30.

HISTORY Refer to the information at the right.

In the fourteenth century, the part of the Great Wall of China that was built during Qui Shi Huangdi’s time was repaired, and the wall was extended. When the wall was completed, it was 2500 miles long. How much of the wall was added during the 1300s?

Words that describe the equation:

Variable Let ____ � the additional length.

1000 mi

Amountadded

2500 mi

Source: NationalGeographic World

3.

The _______________________ plus the _______________________ equals _____________.

Equation:

Answer: The Great Wall of China was extended ____________________ miles in the 1300s.

12

3.2 A – Translate “story” problems Practice

For 1-6, define your variable. Write an equation to model the situation. Solve the equation.

1)

2)

Ella swims four times a week at her club’s pool. She swims the same amount on Monday, Wednesday,and Friday, and 15 laps on Saturday. She swims a total of 51 laps each week. How many laps does sheswim on Monday?

3)

Identify the Variable: Let ______ = ___________________________________________

At the market, Meyer buys a bunch of bananas for $0.35 per pound and a frozen pizza for

$4.99. The total for his purchase was $6.04. How many pounds of bananas did Meyer buy?

Identify the Variable: Let ______ = ___________________________________________

Laura is making a patio in her back yard using paving stones. She buys 44 paving stones and a

flower pot worth $7 for a total of $73. How much did each of the paving stones cost?

Identify the Variable: Let ______ = ___________________________________________

13

4)

5)

6)

A taxi service charges you $1.50 plus $0.60 per minute for a trip to the airport. If the total

charge was$13.50, how many minutes did the ride to the airport take?

Identify the Variable: Let ______ = ___________________________________________

Shannon gets $12 for every lawn she cuts. On day she received $5 in tips. If she earned total $101, how many lawns did she mow?Identify the Variable: Let ______ = ___________________________________________

Josh needs to raise $116 to go on a class trip. He has saved $95. To raise the rest of the money, Josh is

selling pens. If he earns $0.75 for each pen, how many pens must he sell?Identify the Variable: Let ______ = ___________________________________________

14

Guided Problem Solving

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Name Class Date

3-3 • Guided Problem Solving

Food You are helping to prepare food for a large family gathering. You can slice 2 zucchinis per minute. You need 30 sliced zucchinis. How long will it take you to finish, if you have already sliced12 zucchinis?

Understand

1. Circle the information you will need to solve.

2. What are you being asked to do?

3. What will your variable represent?

Plan and Carry Out

4. How many sliced zucchinis do you need?

5. How many sliced zucchinis do youalready have?

6. Write and simplify an expression for thenumber of zucchinis you still need to slice.

7. To calculate the number of minutes it will taketo slice the remaining zucchinis, what numberwill you divide your answer to Step 7 by?

8. Write an equation to solve the problem.

9. How long will it take you to finish slicingthe remaining zucchinis?

Check

10. Multiply your answer to Step 9 by your answer to Step 7. Doesyour answer match your result from Step 6?

Solve Another Problem

11. Jordan skates 6 mi/h. Today she has already skated 8 miles. Hergoal is to skate a total of 20 miles. How much longer does shehave to skate to reach her goal?

GPS

15

1. A _________________ is a rule showing relations among quantities. Example F ma

2. A _______________ (letter) is used to represent a quantity whose value may change or vary.

* In the formula F ma , the variables are _________________.

3-4A – Using Formulas

Vocabulary

Area of Rectangles:The base and the height are two important measurements of a rectangle. Sometimes, these measurements are called the length and the width.

height

The area of a rectangle is the product of its base and its height.

Explore: a) A rectangle with a base of 3 cm and a height of 2 cm has an

area of ____________________ = _________square centimeters (cm2)

b) A rectangle with a base of 6 units and a height of 4 units has an

area of ____________________ = ______________ square inches (in2)in

in

Relate: To find the area of a rectangle, a formula is used. Write a formula to find the area of any rectangle.

Let: _____ represents the base _____ represents the height _____ represents the area

base

1. Use the formula to find the width of the rectangle shown.? in.

9 in.

Area = 36 sq in.

16

height

baseThe perimeter of a rectangle is the distance around its outer edge.

Explore: a) A rectangle with a base of 3 cm and a height of 2 cm has a perimeter of

__________________________________________ = _______cm

Perimeter of Rectangles:

b) A rectangle with a base of 6 units and a height of 4 units has a perimeter of

_____________________________________________ = _________in

in

in

Relate: To find the perimeter of a rectangle, a formula can be used. Write a formula to find the perimeter of any rectangle.

Let: _____ represents the base _____ represents the height _____ represents the perimeter

In other words, the perimeter is the sum of the lengths of all of the sides of a figure.

2 Use the formula to find the base of the rectangle shown.

6 cm

Perimeter = 28 cm

? cm

17

Apply:

Area of a Square:

Apply: Calculate the area of a square that has side length s for each value of s.a. s =

b. s =

c. s =

d. Write a formula inwhich Athe area of a square and s is themeasure of the side of a square.

Perimeter of a Square:

Apply: Calculate the perimeter of a square that has side length s for each value of s.

a. s =

b. s =

c. s =

d. Write a formula in which Pthe perimeter of a square and sis the measure of the side of asquare.

Quick Check! The perimeter or area and the measure of one side of each rectangle is given. Find the length of the missing side.

3.

8 in.

Perimeter = 20 in.

? in. 4.

3 ft

Area = 24 ft2

? ft 5.

6 mm

Area = 30 mm2

? mm

18

Practice:

1. Whatisthemeasureofthesidesofasquarethathasanareaof49squarefeet?

Find the length of the missing side. Show your work.

3.

3 mm

Area = 12 sq mm

? mm 4.

4 yd

Perimeter = 22 yd

? yd

6.

19

Distance Equation

BASEBALL For Exercises 43–45, use the following information.In baseball, if all other factors are the same, the speed of a four-seam fastball is faster than a two-seam fastball. The distance from the pitcher’s mound to home plate is 60.5 feet.

43. How long does it take a two-seam fastball to go from thepitcher’s mound to home plate? Round to hundredth.

44. How long does it take a four-seam fastball to go from thepitcher’s mound to home plate? Round to hundredth.

45. How much longer does it take for a two-seam fastball to reach home plate thana four-seam fastball?

Two-Seam Fastball126 ft/s

Source: Baseball and Mathematics

Four-SeamFastball132 ft/s

20

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Skills Practice: Using Formulas

Chapter 3 50 Glencoe Pre-Algebra

NAME ______________________________________________ DATE ____________ PERIOD _____

3-4

Find the perimeter and area of each rectangle. Show all steps: write formula, substitute, solve.

3. 4. 5.

6.

a square that is 25 centimeters on each side

2 mm

6 mm5 ft

4 ft

45 m

75 m

84 mi

126 mi

For a rectangle: The formula P � 2(� � w) relates perimeter P, length �, and width w.

The formula A � �w relates area A, length �, and width w.

7. 8.

9. a rectangle that is 92 meters longand 18 meters wide

10.

22 yd

22 yd

48 mm

48 mm

21

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Find the missing dimension in each rectangle. Show all work: Formula, substitute, solve.

11.

14.

15. The area of a rectangle is 1260 in2.Its length is 36 inches. Find the width.

12 cm

Area �

276 cm2

25 ft

Area �

1125 ft2

13 m

Perimeter � 68 m

9. 10.

12.

13.

17. The perimeter of a rectangle is 100 centimeters. Its width is 9 centimeters. Find its length.

16. The area of a rectangle is 319 km2.Its width is 11 kilometers. Find its length.

Perimeter � 2402 mi

625 mi

Area � 216 ft2

8 ft

Area � 210 m2 10 m

For a rectangle: The formula P � 2(� � w) relates perimeter P, length �, and width w.

The formula A � �w relates area A, length �, and width w.

22

1. AIR TRAVEL:A plane is traveling 9 mi per min.How much time is needed to travel 216 miles?

2. JOGGING: What is the rate, in feet per second,of a girl who jogs 315 feet in 45 seconds?

Solve: Show all steps: write formula, substitute, solve.

3. AIR TRAVEL: What is the rate, in miles per hour,of a plane that travels 1680 miles in 3 hours?

4. TRAVEL: A train is traveling at 54 miles per hour.How long will it take to go 378 miles?

5. SWIMMING: What is the rate, in feet per second,of a swimmer who crosses a 164-foot-long poolin 41 seconds?

6. BALLOONING: A balloon is caught in a wind travelingat 25 feet per second. If the wind is constant,how long will it take the balloon to travel 1000 feet?

The formula d = rt relates distance d, rate r, and time t, traveled.

23