student workbook with scaffolded practice unit 2bunit 2b rational and radical relationships l 1 o r...

Student Workbookwith Scaffolded Practice

Unit 2B

1

1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-7456-8 U2B

Copyright © 2014

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

EDUCATIONWALCH

This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Officers. All rights reserved.

2

Program pages

Workbook pages

Introduction 5

Unit 2B: Rational and Radical RelationshipsLesson 1: Operating with Rational Expressions

Lesson 2B.1.1: Structures of Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . U2B-5–U2B-22 7–16

Lesson 2B.1.2: Adding and Subtracting Rational Expressions . . . . . . . . . . . . U2B-23–U2B-44 17–26

Lesson 2B.1.3: Multiplying Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . U2B-45–U2B-60 27–36

Lesson 2B.1.4: Dividing Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . U2B-61–U2B-79 37–46

Lesson 2: Solving Rational and Radical EquationsLesson 2B.2.1: Solving Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . U2B-87–U2B-106 47–56

Lesson 2B.2.2: Solving Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . U2B-107–U2B-126 57–66

Lesson 2B.2.3: Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . U2B-127–U2B-155 67–76

Station ActivitiesSet 1: Rational Expressions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . U2B-180–U2B-185 77–82

Set 2: Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2B-192–U2B-197 83–88

Coordinate Planes 89–102

Table of Contents

CCSS IP Math III Teacher Resource© Walch Educationiii

3

The CCSS Mathematics III Student Workbook with Scaffolded Practice includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes:

• Warm-Ups

• Problem-Based Tasks

• Practice Problems

• Station Activity Worksheets

In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB and SRB. This supports:

• Taking notes during class

• Working problems for preview or additional practice

The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts. Sections for you to take notes are provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end of the full unit, should you need to graph.

The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder.

CCSS IP Math III Teacher Resource© Walch Educationv

Introduction

5

© Walch Education

UNIT 2B • RATIONAL AND RADICAL RELATIONSHIPSLesson 1: Operating with Rational Expressions

U2B-5CCSS IP Math III Teacher Resource

2B.1.1

Name: Date:

Warm-Up 2B.1.1

Ms. Appleton teaches high school math. This year, she has 24 students in her afternoon geometry class.

1. If 1

8 of Ms. Appleton’s students are absent one day, how many students are present?

2. To celebrate Sarah’s birthday, Ms. Appleton brought in a cake. If the girls in the class ate 1

2 of

the cake, and the boys ate another 1

3, how much of the cake remains for Ms. Appleton to share

with the other teachers?

3. It’s early in the school year, and the expression x4 – 4x2 describes the number of days the students have been in school. How can this expression be rewritten as an equivalent expression?

Lesson 2B.1.1: Structures of Rational Expressions

7



2B.1.1© Walch Education

Name: Date:

Scaffolded Practice 2B.1.1Example 1

Simplify the rational expression x2

8 by eliminating any common factors found in both the numerator

and denominator.

1. Rewrite the original rational expression as a product of two expressions.

2. Identify any factors common to the numerator and denominator.

3. Simplify the resulting expression.

continued

9


U2B-12CCSS IP Math III Teacher Resource 2B.1.1

© Walch Education

Name: Date:

Example 2

Simplify the rational expression x

2

8 by eliminating any common factors.

Example 3


x x

5

3 102

−− −

.

Example 4


x

3

3

−−

.

Example 5

Simplify the rational expression x y

xy

12

4

2 5

2 .

10

© Walch Education



2B.1.1

Name: Date:

Problem-Based Task 2B.1.1: Rewriting a Rational Expression

As part of their preparations for an upcoming math test, Faith, Jia, and Kayla work together to

simplify the rational expression t

t t

16

6

4

2

−− −

.

• Faith reasons that, because there is a factor of t 4 in the numerator but a factor of only t 2 in the denominator, there is no way to simplify this expression. She says it is already as simple as it can be. Is Faith’s analysis correct?

• After factoring the numerator, Jia says that the expression is equivalent to t t

t

( 2)( 4)

3

2− +−

.

Is Jia correct?

• Kayla tries a different method. After much factoring, she arrives at the expression t t t

t

8 4 2

3

2 3− + −−

. Is Kayla correct?

Is Faith’s analysis correct?

Is Jia correct?

Is Kayla correct?

11

© Walch Education



Name: Date:

For problems 1–6, simplify each rational expression. State any restrictions on x.

1. x

x

35

7

2. x

x

3 12

2 8

++

3. x x

x

72 +

4. x x x

x

18 6 9

3

4 2− +

5. x

x x

8 2

2 10 82

++ +

6. x x x

x x

2 13 7

7

3 2

2

+ −+

Use your knowledge of rational expressions to complete problems 7–10. State any restrictions on x.

7. Show that the expression x x x

x x

6 7

4 3

3 2

2

− −+ +

and the expression x x

x

7

3

2 −+

are equivalent.

8. What simplest rational expression represents the depth of a pond that is x x

x

3 8

5

2 − meters deep?

9. What simplest rational expression represents the diameter of a circle that has a radius of x

x

4 8

3 6

−−

centimeters?

10. What simplest rational expressions can be used to represent the length and width, in inches, of

a rectangle with sides that are x x

x

6

2

2 − −+

, x x

x

20

2 10

2 + −+

, x

x

6 96

48 12

2 −+

, and x2 6

2

−?

Practice 2B.1.1: Structures of Rational Expressions

13

Notes

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15

Notes

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16

© Walch Education



2B.1.2

Name: Date:

Warm-Up 2B.1.2

Every pizza at Percy’s Pizza Palace is cut into 12 slices, regardless of the pizza’s size. Use this information to answer the following questions.

1. If Caiden eats 1

3 of a pizza, and Nate eats another

1

4 of the same pizza, what fraction of the

pizza have they eaten together?

2. Mrs. Booker ordered a pizza at Percy’s. She ate 1

2 of it, and saved

1

4 of the pizza for lunch the

next day. What fraction remains?

3. The Shepard family—Jake, Jena, Jamie, and their mother—ordered 2 pizzas. Jake ate 2

3 of

1 pizza and Jena ate 1

2 of 1 pizza. If Jamie ate

3

4 of a pizza, what was left for Mrs. Shepard?

Lesson 2B.1.2: Adding and Subtracting Rational Expressions

17




Name: Date:



3

5

2+ .

1. Identify any invalid values of the expression.

2. Find a common denominator.

3. Rewrite each term of the expression using the new denominator.

4. Check to see if the result can be written in a simpler form.

continued

19



© Walch Education

Name: Date:

Example 2


5

13

++ .

Example 3

Simplify the rational expression x x x

5 3 1

42 + + .

Example 4


x x

3

2 1

8

3++

−.

Example 5


x

x x x

1

3

10 6

2 3

4

12−+

−− −

−+

.

20

© Walch Education



2B.1.2

Name: Date:

Problem-Based Task 2B.1.2: Who Is Right?

Three students in Mr. Kunal’s class were given the task of determining the result when the rational

expression x

3

7− is subtracted from

x

x x

4 9

10 212

−− +

. Their findings are below. Which students, if any,

are correct?

• Deion calculates the result as x

x x10 212 − +.

• Daniel thinks the answer is x

x x

18

10 212

−− +

.

• Dylan is pretty sure his response of x

x x x10 21

2

3 2− + is absolutely correct.

Which students, if any, are correct?

21

© Walch Education



Name: Date:

For problems 1–7, simplify each rational expression. Whenever possible, reduce the expression to its lowest terms. State any restrictions on x.

1. x x

3 7

5+

2. x

x

x

11

1−

−

3. x

x

3

8

42 +−

4. x

x

1

5

1++

5. x

xx

9

15

++

6. x

x

x

x x

5

1

3 1

22++

+− −

7. x

x x x

x

x

8 1

6 7 3

4

3 1

11

2 32

+− −

−+

+−


8. Sam walks 3

8 km to school. After school, he walks another

x 1

3

+ km to get to work. What

simplified rational expression describes the total length of both of his walks?

9. Emma’s strawberry farm has a total of 5x hectares of arable land. This past season, Emma

planted x

x

3

7

2 ++

hectares of strawberries. What simplified rational expression represents the

total amount of land, in hectares, that was not planted last season?

10. The width of a rectangle is x 2

5

+ cm. The rectangle’s length is

x x

x

3 2

3

2 + ++

cm. What expression

represents the perimeter of this rectangle?

Practice 2B.1.2: Adding and Subtracting Rational Expressions

23

Notes

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25

Notes

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26

© Walch Education



2B.1.3

Name: Date:

Warm-Up 2B.1.3

The area of a rectangle is the product of its length and width. Use this relationship to answer the following questions.

1. A rectangle has a length of 7

8 cm and a width of

2

3 cm. What expression represents the

rectangle’s area?

2. A square has a perimeter of 17 cm. What expression represents the square’s area?

3. Another rectangle has a width of x

2 m and a height of

3

4 m. What expression represents this

rectangle’s area?

Lesson 2B.1.3: Multiplying Rational Expressions

27



© Walch Education

Name: Date:



3

1•8

+.

1. Identify any invalid values of the expression.

2. Rewrite the second term as a fraction.

3. Multiply the expressions to form a single rational expression.

4. Check for any factors that might make it possible to further simplify the resulting expression.

continued

29




Name: Date:

Example 2


xx

3•5

−.

Example 3


xx

5•

12

+.

Example 4


x

x

x

1•

2 3

12

+ −−

.

Example 5

Simplify the rational expression x x

x

x

x x

4 5

1•

8

9 8

2

2

− −+

++ +

.

30

© Walch Education



2B.1.3

Name: Date:

Problem-Based Task 2B.1.3: The Area of a Triangle

One leg of a right triangle is represented by the expression x

x

3

1+. The other leg is represented by the

expression x

x

12

−. What is the area of the triangle? The length of each leg is in centimeters.

What is the area of the triangle?

31

© Walch Education



2B.1.3

Name: Date:

For problems 1–5, simplify each rational expression. Whenever possible, reduce the expression to its lowest terms. If necessary, factor the terms in each expression before multiplying. State any restrictions on x.

1. 3

•5

6 92x

x

x +

2. 3

•2 7

1

x

x

x

x−++

3. 3

4•

12 2

2

x x

x

x

x

+−

+

4. 12

•8 4

5

x x

x

+

5. 2 6

6•

4 12

30 4 2

2

2

x

x

x x

x x

+−

− −+ −

Practice 2B.1.3: Multiplying Rational Expressions

continued

33

© Walch Education



Name: Date:

Use your knowledge of rational expressions and factoring to complete problems 6–10. State any restrictions on x.

6. For what values of x is x

x x

2 8

7 102

−+ +

an invalid expression?

7. Show that the rational expression 5 5

•3

1•

1

5

3 2

2

x

x

x x

x

x

x

+ +−

− is equivalent to the rational expression

x + 3.

8. What simplified rational expression represents the area of a rectangle with a width of x

2 inches

and a length of x

x

2 1

5

+−

inches?

9. The momentum of an object is the product of the object’s mass and its velocity. What

simplified expression describes the momentum (in g • cm/s) of a moving toy car if the rational

expression x x

x

4 4

9

2

2

+ +−

describes the toy car’s mass in grams, and the expression x x

x

5 6

2

2 + ++

describes the toy car’s velocity in centimeters per second (where x > 3)?

10. What simplified expression describes the volume of a rectangular box with a length of x x

x 5

2 +−

meters, a width of x

x

7

1+ meters, and height of

x x

x x

30

7

2

2

+ −+

meters?

34

Notes

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35

Notes

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36

© Walch Education



2B.1.4

Name: Date:

Warm-Up 2B.1.4

Yuri will be making a glass window hanging with a metal frame for his art class. He is deciding which shape to use: a hexagon, a triangle, or a square. His window hanging must have all sides the same length.

1. If Yuri decides to use a hexagon (a 6-sided figure), what is the length of one side if the hexagon’s perimeter is 25 cm?

2. If he decides on a triangular shape, what is the length of one side if the perimeter of the

triangle measures 8

3 m?

3. Yuri’s math teacher has a square frame with a perimeter of (8x + 12) cm. The teacher will give Yuri the frame if Yuri can tell him what expression represents the length of one side. What is the correct expression?

Lesson 2B.1.4: Dividing Rational Expressions

37




Name: Date:


Simplify the rational expression x x3

7 3

2

÷ .

1. Rewrite the division problem as a multiplication problem, using the reciprocal of the divisor.

2. Identify any invalid values of the rewritten expression.

3. Multiply the terms of the rewritten expression.

4. Check for any factors that might make it possible to further simplify the resulting expression, and then factor them out.

continued

39



© Walch Education

Name: Date:

Example 2


x x x1

5

12 2+÷

− +.

Example 3


x x

x

x

3

6

3

2

2

2

−− −

÷−+

.

Example 4


x x

x x

x x

9 14

7

2

4 4

2

2

2

2

+ ++

÷+ −−

.

Example 5

Use synthetic division to rewrite a x

b x

x x x

x

( )

( )

2 3

1

3 2

=+ − +

+ in the form

a x

b xq x

r x

b x

( )

( )( )

( )

( )= + .

40

© Walch Education



2B.1.4

Name: Date:

Problem-Based Task 2B.1.4: Fuel Economy

A car’s fuel economy, measured in miles per gallon, is the ratio of the distance the car can go on a

specified amount of fuel. Piper’s car can travel x x

x

3 7 4

10

2 + +−

miles using x

16 gallons of gas. What is

her car’s fuel economy?

What is her car’s fuel economy?

41

© Walch Education



Name: Date:

For problems 1–4, simplify each rational expression. Whenever possible, reduce the expression to its lowest terms. State any restrictions on x.

1. x

x

2

7

1÷

2. x x

3

5

5÷

3. x x8 3

5 9

+÷

4. x

x

x

x

2 4

1 2

−+

÷+


5. Show that the expression 2 15

2 8•

32 2

13 40

4

8

2

2

2

2 2

x x

x x

x

x x

x

x x

− −−

−− +

÷+−

is equivalent to the expression x + 3.

Practice 2B.1.4: Dividing Rational Expressions

continued

43

© Walch Education



2B.1.4

Name: Date:

6. Use synthetic division to rewrite the rational expression x x

x

2 7 15

5

2 − −−

in the form q xr x

b x( )

( )

( )+ .

7. Use polynomial division to rewrite the rational expression x x

x

8 1

1

2 − −+

in the form q xr x

b x( )

( )

( )+ .

8. One way to find an average velocity is to take the quotient of distance traveled and the time

required to do so. Find the velocity (in kilometers per hour) of a car that covers a distance of

2x + 4 kilometers in a total time, in hours, described by the expression x

x

3 6

2

−−

(where x > 2).

9. Density is the ratio of an object’s mass to its volume. What simplified rational expression

represents the density (in grams per meter3) of an alloy ingot whose mass, in grams, is described

by the expression x x

x

6 15

4

2 + −+

, with a volume in cubic meters represented by the expression x x

x

2 11 12

5

2 − + (where x > 4)?

10. The area of a right triangle, in square meters, is represented by the rational expression x x

x

3 5

2

2 − +−

, where x > 2. If the length of one leg, in meters, is described by the rational

expression x

x

5

1

2 −−

, what simplified rational expression represents the other leg?

44

Notes

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45

Notes

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46

© Walch Education

UNIT 2B • RATIONAL AND RADICAL RELATIONSHIPSLesson 2: Solving Rational and Radical Equations


2B.2.1

Name: Date:

Warm-Up 2B.2.1

Jordan and his sister, Julia, have been assigned the chore of raking their lawn. Suppose it takes Jordan 90 minutes to rake the lawn by himself. If Julia rakes it by herself, it also takes her 90 minutes.

1. If Jordan and Julia work together, how long will it take them to rake the lawn? Assume that their rates are not affected by working together.

2. Suppose again that it takes Jordan 90 minutes to rake the lawn by himself. Jordan’s brother, Ian, however, can rake the lawn faster than Jordan, completing the job in just 45 minutes. If Jordan and Ian work together, how long will it take them to rake the lawn?

Lesson 2B.2.1: Solving Rational Equations

47



© Walch Education

Name: Date:


Solve the proportion =x

4

6

8 for x.

1. Determine the least common denominator.

2. Multiply each ratio by the common denominator.

3. Simplify the resulting equation.

4. Solve the resulting equation.

5. Verify the solution(s).

continued

49




Name: Date:

Example 2

Solve the rational equation − =t t

5 3

12

9

3 for t.

Example 3

Solve the rational equation −+

=y y

2 1

2

1

3 for y.

Example 4

Solve the rational equation −

++

=− −p

p

p p p

1

4 2

6

2 82 for p.

50

© Walch Education



2B.2.1

Name: Date:

Problem-Based Task 2B.2.1: Snow Removal

When Scot and his younger brother Levi work together, they can remove snow from a driveway in 40 minutes. But if Scot were working alone, he could do the same job in half the time it would take Levi to do it alone. How long would it take Levi to remove the snow from the driveway if he were working by himself? Assume the rate at which each brother works is not affected by working together.

How long would it take Levi to remove the snow from the driveway if he were working by himself?

51

© Walch Education



Name: Date:

Solve each rational equation for x.

1. +

+−

=x

x

3

3

5

45

2. + =+x

x

x

3

22

2

1

3. −−

=−−

xx

x

x

2

3

1

3

Use the given information and what you know about rational equations to solve problems 4–10.

4. Gideon can rake his lawn in 3 hours. His friend Katsuro can rake a lawn the same size in just 2 hours. Suppose they worked together to rake Gideon’s lawn. What rational equation represents the information presented in this problem?

5. Can x = –2 be a solution for +

−−

=−x

x

x x

3

2

6

4

1

22 ? Explain.

6. It takes Lana 8 hours to paint a particular size room, but her co-worker Nila can do the same job in 7 hours. If they paint the room together, how long will it take to finish painting the room?

7. The reciprocals of two consecutive, positive odd integers have a difference of 2

63. What are the

integers?

8. Kris has access to two different water sources for filling up his swimming pool for summer. One source, using a rigid pipe, supplies water 50% faster than the other source, which uses a flexible hose. Using both the pipe and the hose, Kris can fill the pool in 6 hours. How long would it take to fill the pool if Kris only used the pipe?

9. Reena rode her skateboard to her uncle’s house, 12 miles away. A wheel broke just as she arrived, so Reena borrowed her uncle’s bicycle to return home. Traveling by bicycle, she was able to go twice as fast, and returned home in 2 hours less time than it took to go to her uncle’s house. How long did it take Reena to get home on the bicycle?

10. A passenger train is 40 mph faster than a freight train. In the time it takes the slower freight train to travel 161 miles, the passenger train can travel 301 miles. What is the speed of each train?

Practice 2B.2.1: Solving Rational Equations

53

Notes

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55

Notes

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56

© Walch Education



2B.2.2

Name: Date:

Warm-Up 2B.2.2

A large canvas leaning against a wall forms a right triangle. The horizontal distance from the base of the canvas to the wall represents one leg of the triangle. The vertical distance from the floor to the point of contact between the canvas and the wall represents the other leg. The canvas itself represents the hypotenuse. The diagram that follows illustrates the triangle created by the canvas, floor, and wall. Use the diagram and the Pythagorean Theorem, a2 + b2 = c2, to solve the problems that follow.

a

b c

1. If the base of the canvas is 1.5 ft from the wall, and the top of the canvas touches the wall 3.6 ft from the floor, how long is the canvas?

2. Suppose the base of this same canvas were pulled away from the wall until the base was 2.34 ft from the wall. How high along the wall would the canvas now reach?

3. If a 6-foot-long canvas were set against the wall so that the angle formed by the canvas and the floor was 45º, how far from the wall would the base of the canvas be?

Lesson 2B.2.2: Solving Radical Equations

57



© Walch Education

Name: Date:


Solve the radical equation − =x 5 2 for x.

1. Isolate the radical expression.

2. Raise both sides of the equation to a power to eliminate the radical.

3. Solve the resulting equation.

4. Verify the solution(s).

continued

59




Name: Date:

Example 2

Solve the radical equation − =x 5 2 for x.

Example 3

Solve the radical equation − =x x2 for x.

Example 4

Solve the radical equation − =x 7 3 2 for x.

Example 5

Solve the radical equation + = +x x5 3 5 for x.

Example 6

Solve the radical equation + + =x3 6 10 73 for x.

60

© Walch Education



2B.2.2

Name: Date:

Problem-Based Task 2B.2.2: A Towering Cone

The height, h, of a right circular cone is the perpendicular distance between the base and the peak. The slant height of such a cone, c, is the distance along the outer edge from the peak to the perimeter of the circular base, as shown in the diagram below. The algebraic relationship between the cone’s

height, its slant height, and the radius of its base is described by the equation = +c r h2 2 , where c is the slant height, r is the radius of the base, and h is the height of the cone from base to tip.

hc

r

A cone-shaped top is to be added atop a tower that has a diameter of 9 m. What is the height of the cone if its slant height is 7.5 m?

What is the height of the cone if its slant height

is 7.5 m?

61

© Walch Education



2B.2.2

Name: Date:

For problems 1–3, solve each radical equation for x. Round answers to the nearest thousandth.

1. − =x 3 6

2. + =x x6

3. + + =x4 1 5 13

Use the given information and what you know about radical equations to solve problems 4–10. Round answers to the nearest thousandth.

4. A rectangular window is twice as wide as it is long. The area enclosed by the window is numerically equivalent to the length of the shortest side of the window. Find the area of the window.

5. A right triangle has a hypotenuse measuring 17 inches long. If one leg measures 8 inches, what is the length of the other leg?

6. When an object is dropped from a great height, gravity causes the object to speed up as it falls.

When the object strikes the surface, the velocity at which the object was traveling at the moment of

impact (ignoring air resistance) can be determined using the formula =v gh2 , where v represents

the object’s velocity (in feet per second), g is the object’s acceleration due to gravity (in feet per

second per second), and h is the height of the object (in feet). On Earth, an object’s acceleration due

to gravity is 32 ft/s2. From what height must a stone be dropped to reach a velocity of 128 ft/s at the

moment it hits the ground?

7. The time it takes an object to fall from a great height depends on the object’s acceleration due

to gravity, as well as the height from which the object is dropped. The formula =h gt1

22 can

be used to describe this relationship, with t representing time (in seconds), h representing the

height of the object (in feet), and g representing the object’s acceleration due to gravity, which is

32 ft/s2 on Earth. If a cliff diver leaps into the water from a height of 164 feet, how long will it

take him to reach the water? Assume that air resistance is negligible and does not have an effect.

Practice 2B.2.2: Solving Radical Equations

continued

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8. Accident investigators can estimate the speed of a vehicle involved in a crash by examining the skid marks left by the tires on the pavement. The algebraic relationship is shown by the formula =s fd30 , with s representing the speed in miles per hour and d representing the skid distance

in feet. The value f is determined by the friction of the pavement: wet concrete is much more slippery than hot asphalt. A typical value for f, assuming a dry roadway, is 0.7. After a minor car accident on a dry day, Charlie insisted to his parents that he wasn’t exceeding the 35 mph speed limit. An accident investigator found that the skid marks left by Charlie’s tires were 75 feet long. Does the investigator’s skid analysis indicate Charlie was telling the truth?

9. In an electrical circuit, the current is related to the power and resistance according to the rule

=IP

R, where I represents the current (measured in amps), P represents power (measured in

watts), and R represents resistance (measured in ohms). What is the electrical power drawn by a

device that has a resistance of 15 ohms with a current of 4 amps?

10. For a sphere with radius r, the volume, V, can be found using the formula π=V r4

33 . Find the

radius of a sphere with a volume of 288π.

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Notes

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Notes

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2B.2.3

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Warm-Up 2B.2.3

Marco wants to sell his car, and plans to post signs around town. To make the most of the limited amount of poster board he has, Marco decides that each sign will be a right triangle with a hypotenuse of 13 inches. The diagram below shows the dimensions of Marco’s design. Use the diagram and the given information to solve the problems that follow.

13

x

y

1. Use the Pythagorean Theorem to write an equation relating the length of each leg of the triangular sign to the length of the hypotenuse. Solve this equation for y.

2. The perimeter of the triangular sign can be determined by finding the sum of the lengths of the sides of the triangle. If Marco’s sign has a perimeter of 30 inches, what is the equation, solved for y, that represents the perimeter of the triangle?

3. Use the equations written in problems 1 and 2 to complete the table below. Round values to the nearest hundredth.

x 1 2 3 4 5 6 7 8 9 10 11 12

Value of y based on equation from problem 1

Value of y based on equation from problem 2

4. At what values of x are the values of y for each equation equal? In the context of the problem, what do these x-values represent?

Lesson 2B.2.3: Solving Systems of Equations

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Use a graph to solve the system == −

f x x

g x x

( )

( ) 4 0.5. Verify that the identified coordinate pairs are

solutions.

1. Create a graph of the two functions, f(x) and g(x).

2. Find the coordinates of any apparent intersections.

3. Verify that the identified coordinate pair is a solution to the original system of equations.

continued

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Example 2

Create a table to approximate the real solution(s), if any, to the system = +

= +

f xx

g x x

( ) 11

( ) 1.

Example 3

A system contains the equations = + −y x2 3 and 4x – 4y = 19. Solve the system by creating a table of values on a graphing calculator.

Example 4

Solve the system of equations

=

= +−

f x x

g xx

( ) 0.5

( ) 12

2

by graphing.

Example 5

Solve the system of equations =

−= − + −

f xx

g x x x x

( )1

2( 1)

( ) 3 12 12 42 3

by graphing.

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2B.2.3

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Problem-Based Task 2B.2.3: Measuring a Masterpiece

A new sculpture was just unveiled at a local park. Part of the sculpture is a right triangle that has a hypotenuse of 29 meters. One leg is 1 meter longer than the other. What is the area of this triangular portion of the sculpture?

What is the area of this triangular portion of the sculpture?

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Practice 2B.2.3: Solving Systems of Equations

For problems 1–4, solve each of the following systems by graphing. If necessary, rewrite equations in the form “y =”. Round to the nearest hundredth.

1.

= −

= ++

f x x

g xx

x

x

( ) 3 5

( )1 4

6

2. 3y – x = 17 and = +y x 5

3. = −

=−

−−

f xx

g xx

x x

( ) 54

( )6

5

3

4.

=

=+

+

f x x

g xx

x

( )

( )2

2 3

Use the given information and what you have learned about systems of equations to solve problems 5 and 6.

5. The table below shows values for two functions, f(x) and g(x), at given values of x.

x –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8

f(x) 136 105 78 55 36 21 10 3 0 1 6 15 28 45 66 91

g(x) 330 206 118 60 26 10 6 8 10 6 –10 –44 –102 –190 –314 –480

Based on the information in the table, what can you conclude about solutions to the system of equations defined by f(x) and g(x)?

6. A system of two linear equations has no real solution. What can you conclude about the two lines?

continued

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2B.2.3

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For problems 7–10, determine the solution(s), if any, to each of the described systems. Round to the nearest hundredth.

7. Debbie says that the point (–1, 3) is the only solution to the system of equations =−

yx

x

2 1 and

= + −y x1 1 . Explain why her solution is (or is not) correct.

8. Solve the system defined by the equations =+

yx 6

3 and = −

+− −

yx

x x2

4 4

2 32 .

9. A right triangle has a hypotenuse of 37 cm. The length of one leg is 3 times the length of the other. What is the area of the triangle?

10. The period of a pendulum is given by the lowercase Greek letter tau, τ, and depends on the

length L of the pendulum, and the pendulum’s gravitational acceleration, g, according to the

formula τ π=L

g2 . For what length(s) in meters is the period of an Earth-bound pendulum

numerically equal to 4 times the pendulum’s length in meters? The gravitational acceleration for

an object on Earth is about 9.8 m/s2.

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Notes

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Notes

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UNIT 2B • RATIONAL AND RADICAL RELATIONSHIPSStation Activities Set 1: Rational Expressions and Equations

CCSS IP Math III Teacher ResourceU2B-180

© Walch Education

Name: Date:

Station 1At this station, you will find three sets of cards:

• 5 “Original Expression” cards with rational expressions, numbered 1–5

• 10 “Simplified Expression” cards, lettered A–J

• 10 “Restrictions on the Variable(s)” cards, lettered AA–JJ

You will work with your group to simplify the five original expressions. First, have one group member select an “Original Expression” card and turn it face up so everyone can see it. Then, work together to simplify the expression. Determine the matching “Simplified Expression” card and the correct “Restrictions on the Variable(s)” card. Record the letters and expressions for the matching cards in the table below. Take turns selecting “Original Expression” cards until all the “Original Expression” cards have been selected. Note that there are more cards to match than you will need, and two of the restriction cards are identical.

Original expression

Simplified expression Restrictions on the variable(s)

1. x

x x

4

16 82 +

2. x y

x y( )

2 2

2

−−

3. x y

x y3 3

4 4−−

4. x

x

1

1

4

2

−−

5. y y

y y

3 12 12

3 12 36

2

2

− ++ −

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CCSS IP Math III Teacher Resource© Walch EducationU2B-181

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Station 2At this station, you will find a six-sided number cube. The table below contains six “numerator” expressions and six “denominator” expressions. The expressions correspond to each number on the number cube. You will use the number cube and table to create three rational addition problems that you will then simplify.

To create each addition problem, roll the number cube four times. Use the first two rolls to choose the two numerators and use the next two rolls to choose the two denominators for the expressions to be added. Record each addition problem on the next page. Make sure each group member writes the same problem. Each group member will simplify each problem individually, and then compare answers. Discuss the answers until all group members agree on the result and understand any errors; then, record your results.

Example

If you roll a 3 and a 6 to choose the numerators, and then roll a 1 and a 2 to choose the

denominators, the resulting addition problem would be x

x

x

x1

2

2 2

2

++

+.

Numerators Denominators

Number rolled Numerator Number rolled Denominator

1 1 1 x + 1

2 2 2 2x + 2

3 x 3 x2 + x

4 2x 4 x2 + 2x + 1

5 x2 5 4x2 + 8x + 4

6 2x2 6 x2 + 3x + 2

continued

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© Walch Education

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Roll the number cube four times and record each addition problem. Once all group members have recorded these problems, work individually to simplify the expressions. Then, compare your results.

1.

2.

3.

Rewrite the problems you created in 1–3, using the operation of subtraction instead of addition. Work on your own to simplify the new problems and then compare your answers with the group. Discuss the answers until all group members agree on the result and understand any errors; then, record your results.

4.

5.

6.

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Station 3 The following are eight simplified expressions.

1. x

1

1+ 2. 1 3. x + 9 4.

x

x

4

3

−+

5. x

2

36.

x

x

2

2+ 7.

x

x

7

2

+−

8. x

x

4

9

+

Select one expression and then write a problem involving multiplying or dividing rational expressions that will result in the chosen simplified expression. Each group member should choose a different expression and write a different problem.

Once everyone has written a problem, swap papers so each person in the group has a new paper. Simplify the problem your group member wrote, and show your work. Write your initials beside the problem number so it’s clear which group member simplified that problem.

Then, choose a new expression from the list above and write a new problem on your group member’s paper. Once everyone is ready, swap papers again. Check your group member’s work for the previous problem, and then simplify the next problem. Repeat this process until all problems 1–4 have been written, simplified, and checked.

Example

Suppose you chose option 6: x

x

2

2+. A problem that results in this simplified

expression is x x

x x

x

x

( 7)

( 2)( 4)

2( 4)

7

−+ +

•+−

. To make it trickier for your group member,

multiply the factors and write the problem as x x

x x

x

x

7

6 8

2 8

7

2

2

−+ +

•+−

.

continued

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Based on the simplified expression you chose, write a problem that involves multiplying or dividing rational expressions in the appropriate space below. Swap papers with your group members, and then simplify the problem your group member wrote. Show your work and write your initials next to the number of the problem you simplified.

1. Problem:

2. Problem:

3. Problem:

4. Problem:

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Station 4At this station, you will find a set of cards, each labeled with a different equation. Divide your group into two teams, shuffle the cards, and then follow the guidelines and rules to play this equation game.

How to Play

Place the cards face down in a pile. Each team takes turns choosing a card and solving the equation on the card. Record the problems and the number of solutions in the table provided. Your team gains 1 point for each solution an equation has; that is, an equation with 1 solution is worth 1 point and an equation with 2 solutions is worth 2 points. An equation with no solutions brings your total score back down to 0. Continue to add or subtract points from round to round based on the number of solutions. Once all the cards have been chosen and solved, the team with the score closest to 0 is the winner.

Rules

• Players may challenge their opponents by checking the other team’s work.

• There is no penalty for challenging an answer that is found to be correct. If an answer is challenged and found to be incorrect, the team with the incorrect answer must have their score revised based on the correct answer.

• Extraneous solutions do not count toward your score.

• Remember: If the equation has no solutions, all previous points are canceled, and your score is set to 0.

Team 1 Team 2

ProblemNumber

of solutions

Score ProblemNumber

of solutions

Score

Final score Final score

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UNIT 2B • RATIONAL AND RADICAL RELATIONSHIPSStation Activities Set 2: Solving Systems of Equations


© Walch Education

Name: Date:

Station 1Divide your group into two teams. Each team will solve the following systems of equations by graphing, and then estimate the points of intersection. When all four systems have been solved, check the other team’s work. The team whose estimated points of intersection are closest to the actual solution for each system scores a point. The team with the most points at the end wins.

Graph each system of equations that follows, by hand or using a graphing calculator. If using a graphing calculator, sketch your graph on the coordinate plane provided.

1.

y x

yx

2

1

2

= +

=+

–10 –9 –8 –7 –6 –5 –4 –3 –2 –10 1 2 3 4 5 6 7 8 9 10

–10–9–8–7–6–5–4–3–2–1

12345678910

2. y x

y x

1= −=

–10 –9 –8 –7 –6 –5 –4 –3 –2 –10 1 2 3 4 5 6 7 8 9 10

–10–9–8–7–6–5–4–3–2–1

12345678910

continued

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3. y

xy x x

1

15 43

=−

= − +

–10 –9 –8 –7 –6 –5 –4 –3 –2 –10 1 2 3 4 5 6 7 8 9 10

–10–9–8–7–6–5–4–3–2–1

12345678910

4. y

x

x

y x

2

12=−

=

–10 –9 –8 –7 –6 –5 –4 –3 –2 –10 1 2 3 4 5 6 7 8 9 10

–10–9–8–7–6–5–4–3–2–1

12345678910

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Station 2All of the systems of equations on the next page have a solution with an x-value between –3 and 3. Use tables to approximate this solution to the nearest hundredth. You may look at the graph on a graphing calculator to make your first guess. Note: The systems of equations may have other solutions as well.

Example

Given the system f x

x

x

g x x

( )1

4

( )1

41

2=+−

=− −

, graphing these two functions on a graphing

calculator shows that the point of intersection is between 1 and 2 on the x-axis.

Calculate the y-values of each function for x-values between 1 and 2. Determine when the y-values are closest together in value.

x f(x) g(x)1.4

1.5

1.6

–1.18

–1.43

–1.81

–1.35

–1.38

–1.40

Of these three x-values, the y-values for x = 1.5 are the closest together. To get a more accurate approximation, calculate the y-values for numbers close to 1.5.

x f(x) g(x)1.48

1.49

1.50

1.51

1.52

–1.37

–1.40

–1.43

–1.46

–1.49

–1.37

–1.37

–1.38

–1.38

–1.38

When x equals 1.48, both f(x) and g(x) are approximately equal to –1.37. The approximate solution to the system is (1.48, –1.37).

continued

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Fill in the table of values for each system of equations, then use the methods previously demonstrated to find an approximate solution that falls between –3 and 3 on the x-axis.

1. f x

xg x x x

( )1

1( ) 6 122

=−

= − +

3.

f xx x

xg x x

( )3 2

( ) 1 1

2

3=− +

= + −

x f(x) g(x)

x f(x) g(x)

2. f x x x

g x x

( ) 10

( ) 4 2 6

= − +=− + +

4.

f x x x

g xx

x

( )

( )1

2

2

= +

=+

x f(x) g(x)

x f(x) g(x)

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Station 3Below are six systems of equations. Divide your group into two teams. Team 1 will solve the first three systems algebraically. Then they will solve the next three systems by graphing on the graphing calculator and finding the point(s) of intersection. Team 2, on the other hand, will solve the first three systems by graphing on the graphing calculator and finding the point(s) of intersection. Then, they will solve the remaining three systems algebraically. Teams will then compare answers, discuss differences, and resolve any errors. Be sure to check for extraneous solutions when solving algebraically. Round the point(s) of intersection to the nearest tenth.

1. y x

y x

1

2 1

2= += −

2. y

x

xy x

3

42 3

=+−

= −

3. y x

y x

7

12

2= −= −

4. y

x

yx

x

1

2

12

=

=−+

5. y x

y x

1

5

2

2

= +

=− +

6. y x

y x

3

3

= −= −

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Station 4Work with your group to solve the following systems of equations by graphing them on a calculator and finding the point(s) of intersection. As you graph each system, be aware of the shape of each type of function. Make sure that the viewing window on the graphing calculator is an appropriate size that will allow you to see all points of intersection. For each system, round the point(s) of intersection to the nearest tenth.

1. y x

yx

x x

1

1

4 3

2

2

= +

=−

− +

2. y

x x

xy x x

2 1

2 3

2

2

3 2

=− +

= + +

3. y x x

y x

5 24

8

3 2= + +

= −

4. y

x

x x

y x

3

21

22

2=++ −

= +

5. y x

y x

3 5

6 15

= − += + −

6. How can you be sure that you have found all the solutions to a system of equations when you are solving with a graphing calculator?

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student workbook with scaffolded practice unit 2bunit 2b rational and radical relationships l 1 o r...

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