pa aioswa07 dng ch07 114-129 part 2 notes.pdf · practice 1 translate verbal phrases into...
TRANSCRIPT
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
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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 .
_________________________________________
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©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..
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©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.
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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.
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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
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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.
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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.
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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 ______ = ___________________________________________
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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 ______ = ___________________________________________
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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.
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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
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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.
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