Integrated Math Concepts - Module 5

Integrated Math Concepts

Module 5

Fractions

Second Edition

National PASS Center 2006


Solve Problems

Organize

Model

Compute

Communicate

Measure

Reason

Analyze

National PASS Center BOCES Geneseo Migrant Center 27 Lackawanna Avenue Mount Morris, NY 14510 (585) 658-7960 (585) 658-7969 (fax) www.migrant.net/pass

Authors: Justin Allen Diana Harke Editor: Sally Fox Desk Top Publishing: Sally Fox Developed for Project MATEMÁTICA ((Math Achievement Toward Excellence for Migrant Students And Professional Development for Teachers in Math Instruction Consortium Arrangement), a Migrant Education Program Consortium Incentive project, by the National PASS Center under the leadership of the National PASS Coordinating Committee with funding from Region 20 Education Service Center, San Antonio, Texas. Copyright © 2006 by the National PASS Center. All rights reserved. No part of this book may be reproduced in any form without written permission from the National PASS Center.


Module 5

Fractions

Second Edition

National PASS Center 2006

BOCES Geneseo Migrant Center 27 Lackawanna Avenue Mount Morris NY 14510


Solve Problems

Organize

Model

Compute

Communicate

Measure

Reason

Analyze

Acknowledgements The materials included in this Integrated Math Concepts course were gathered, in part, from the National PASS Center’s Algebra I and Geometry courses which were written by Diana Harke. Ms. Harke currently is an instructor of mathematics at the State University of New York at Geneseo where she also supervises student teachers. She is a former junior and senior high school math teacher with experience in the United States and Canada. Ms. Harke’s courses produced thus far for the National PASS Center (NPC) have been very well received across the country, increasing the percentage of PASS mathematics courses being utilized throughout the migrant education network and beyond. It should be noted that two of the recent National Migrant PASS Students of the Year, Benancio Galvin of Marana, Arizona (2004) and Yesenia Medina of San Juan, Texas, and Wild Rose, Wisconsin (2006), have moved ahead toward their dreams of completing their high school graduation requirements thanks to their success with Ms. Harke’s Algebra I course. To meet the needs of migrant students requiring a more condensed resource to strengthen their math skills, the original curriculum materials were adapted, edited, modified, and expanded by Mr. Justin Allen. Mr. Allen is a certified secondary level math teacher and is currently pursuing a graduate degree in secondary education at the State University of New York at Geneseo. He taught middle school math and Algebra in Canandaigua, New York, for three years and, most recently, high school math in Livonia, New York. Mr. Allen assisted in the editing of the PASS Algebra II course which was released early in 2006. Acknowledgement is offered also to Ms. Sally Fox, Coordinator, National PASS Center, for her commitment to the development of quality curriculum. As with all materials produced by the NPC, her involvement with Integrated Math Concepts at all levels has played a key role in the addition of this offering to the growing number of courses available to migrant students and others seeking to master the necessary skills to become productive members of society.

Robert Lynch, Director

Module 5 – Fractions

Table of Contents

Page

Introduction i

Objectives 1

Review 17

Practice Problems 18

Answers to “Try It” Problems 21

Answers to Practice Problems 24

Glossary of Terms 27

i

Integrated Math Concepts – Introduction

The PASS Concept PASS (Portable Assisted Study Sequence) is a study program created to help you earn

credit through semi-independent study with the help of a teacher/mentor. Your teacher/mentor

will meet with you on a regular basis to: answer your questions, review and discuss

assignments and progress, and administer tests. You can undertake courses at your own pace

and may begin a course in one location and complete it in another.

Strategy

Mathematics is not meant to be memorized; it is meant to be understood. This course

has been written with that goal in mind. Mathematics must not be read in the same way that a novel is read. In order to read a

mathematics text most effectively you must pay close attention to the structure of each

expression and to the order that operations are performed. You might think of mathematics as

you would a foreign language. Every symbol in a mathematical expression is meant to

communicate a message in that language; therefore, to understand the language you must

understand the symbolism. Always read with a pencil and scrap paper in hand. Make notes in the margins of your

book where you have questions and write “what if” variations to problems to discuss with your

teacher/mentor.

ii

Course Content Integrated Math Concepts is divided into ten modules. Each module teaches concepts

and strategies that are essential for establishing a firm foundation in each content area.

The following is a description of the ten modules in Integrated Math Concepts:

Module 1 Real Numbers

Learn to recognize and differentiate between natural numbers, whole numbers, integers,

rational numbers, irrational numbers, and real numbers.

Relate the number line to the collection of real numbers. Module 2 Sets

Recognize a well-defined set

Learn set notation and terminology

Study some subsets of real numbers – prime and composite numbers Module 3 Variables and Axioms

Learn

• why, when, and how to use a variable

• the definition of an axiom

• some specific axioms Module 4 Properties of Real Numbers

Learn the characteristics and uses of the following properties of real numbers:

• the commutative property

• the associative property

• the distributive property

• identity elements

• inverses

• the multiplication property of zero

• to understand why division by zero is not allowed

• to introduce the uniqueness and existence properties

iii

Module 5 Fractions

Become comfortable with fractions by

• understanding their make-up

• comparing their sizes

Prepare for operations with algebraic fractions

• by understanding the concepts behind the algorithms

• by determining if solutions are reasonable

Module 6 Decimals

Become comfortable with decimals and decimal operations

• by understanding the relative size of decimals

• by understanding why the algorithms or rules dealing with decimals work

• by testing answers for reasonableness

Module 7 Order of Operations

Understand why problems need to be performed in a certain order

Evaluate numerical expressions using order of operations

Evaluate variable expressions for specific values

Module 8 Equations

Translate algebraic expressions and equations, as well as consecutive integer questions

Solve:

• One-step equations

• Two-step equations

• Complex equations (combining like terms, use of the distributive property,

variables on both sides)

• Multi-step equations

Translate algebraic inequalities

Solve and graph solutions to one and two-step inequalities

iv

Module 9 Geometry

Describe points, lines, and planes

Sketch and label points, lines, and planes

Use problem solving to explore points, lines, and planes

Define line segments, rays, and angles

Recognize and examine types of angles

Explore problems using angle properties

Explore line relationships

Module 10 Properties of Polygons

Recognize and classify 2-dimensional shapes –

circles, triangles and quadrilaterals

Find 2-dimensional shapes in the environment

Explore the sum of the measures of the angles of triangles and quadrilaterals

Classify a polygon according to the number of its sides

Count diagonals in polygons

Find the measures of the interior and exterior angles in polygons

Course Organization Each module begins with a list of the objectives. This is a short list of what you will

learn. Definitions, theorems, and

mathematical properties appear as

strips of paper tacked to the page so that

they may be easily found. Examples are used to illustrate each new concept. These are

followed immediately by “Try It” problems to see if you

understand the concept. You are to write the answers to the “Try

It” problems right in your book and then check your answers with

the detailed solutions farther back in the module.

A set is a collection of objects.

v

Many lessons include the following types of inserts.

“Think Back” boxes – denoted with an arrow pointing backwards. These are

reminders of things that you have probably already learned.

“Problem solving tips” – denoted with a light bulb

“Calculator tips” – denoted with a small calculator

“Algorithms” – denoted with a fancy capital A. An algorithm is a rule (or step by

step process) used to solve a specific type of problem.

"Facts” – denoted by a small flashlight

At the end of each module you will be asked to highlight parts of the lesson as a way to

review the terminology and concepts that you just studied. You will also be asked to write

about something that you learned in your own words or list any questions to ask your

teacher/mentor about something that you did not understand. This last step is extremely

important. You should not continue on to the next activity or module until all your questions

have been answered and you are sure that you thoroughly understand the concept you just

finished. Finally, you will be asked to practice what you have learned. Athletes in every sport

must practice their skills to become better at their sport. The same is true of mathematicians.

In order to become a good mathematician, you must practice what you have learned so that it

becomes easier and easier to solve problems. You should keep a math journal or notebook

where you will do your practice problems. Detailed answers to the practice problems will be

found toward the end of the module just ahead of the glossary section.

vi

A glossary / index of the mathematical terms used in this course has been provided at

the end of each module. It contains definitions as a reference to help your understanding of

these specialized mathematical terms. Unlike other PASS courses, there is no separate Mentor Manual for this course as all of

the answers to practice problems are provided within each module. Should you require

additional support, do not hesitate to ask your mentor or teacher. That is why they are there.

Testing When you have completed all the exercises and practice problems in a module and you

and your teacher/mentor feel that you have a good grasp of the material, you will take a test

covering what you should have learned in that module.

Test taking tips 1) Make sure all of your questions have been answered and that you feel confident that

you understand the concepts on which you are to be tested.

2) Do not rush.

3) Be neat. Sometimes handwritten numbers or letters are misread.

4) Be organized. Do computations on a separate piece of paper or, if there is room on

your test sheet, in the space provided, so as to keep the flow of the problem clearly

in focus.

5) Check your answers to see

a) if you actually answered the question that was asked, and

b) that the answer is reasonable.

6) Be aware of the particular types of errors that you are prone to make. Arithmetic

mistakes are often repeated if you merely repeat the computations. Use your

calculator to prevent these types of errors and concentrate on

a) choosing the correct operations,

b) following the proper order of operations, and

c) applying valid mathematical techniques.

National PASS Center

Module 5 - Fractions

1

Fractions

Objectives

Become comfortable with fractions by

• understanding their make-up

• comparing their sizes

Prepare for operations with algebraic fractions

• by understanding the concepts behind the algorithms

• by determining if solutions are reasonable

Notice the word “ratio” is part of the word “rational”. If a and b are integers and b ≠ 0 , then

ab

is a rational number. Some examples of rational numbers are 1 4, ,3 7

− and 154

. Rational

numbers may also be written as decimals that either terminate, such as 0.35, or repeat, such as

0.33 or 0 .52 . In this lesson you will look at rational numbers in the first form or as

fractions.


Solve Problems

Organize

Model

Compute

Communicate

Measure

Reason

Analyze

A number that can be written as the ratio of two integers is called a rational number.

MATEMÁTICA August 2006


2

The denominator of a fraction indicates the number of equally sized pieces into which the

whole is divided. The numerator indicates the number of the equally sized pieces that are

under discussion.

Example 1

Compare the size of these fractions using number sense. 1 1 1, , ,2 3 4

and 112

.

Solution

A common denominator may be found and the

numerators compared to see which represents more of

the equal sized pieces. For example 25

35

< since 2 pieces

is less than 3 pieces and the pieces are the same size.

However, finding a common denominator is not always

necessary if number sense is used.

When a number is written as a fraction, the top number is called the numerator

and the bottom number is called the denominator.

34

numeratordenominator

3 pieces shaded

4 equal pieces

3

4

If two positive fractions have the

same denominator, theone with the larger numerator is larger.

5 49 9

>



3

If two positive fractions

have the same numerator, the one with

the smaller denominator is larger.

2 23 5

>

If the numerator remains unchanged, the larger the denominator the smaller the pieces.

Therefore 112

14

13

12

< < < .

Example 2

Compare the size of these fractions using number sense.

25

23

27

, ,

Solution

Thirds are larger than fifths are larger than sevenths. Given two pieces of each then

27

25

23

< <

Determine which fraction is larger in each case using number sense.

1. 111

or 19

2. 1319

or 1119



4

3. 617

or 615

Sometimes comparing two fractions with a third number such as 0 1 12

, , or is helpful.

A number is less than zero if it is negative and more than zero if it is positive.

2 03

− < and 1 027

>

A fraction is greater than one if the numerator is greater than the denominator and less than one

if the numerator is less than the denominator.

3 14

< and 4 13

>

A positive fraction is equal to 1

2 if the numerator is exactly one-half the denominator. If the numerator is more than half the denominator the fraction is greater than 1

2 . If the numerator is less than half the denominator the fraction is less than 1

2 .

3 1 4 1 5 1 , ,

8 2 8 2 8 2< = >



5

Example 3

Decide whether each fraction is less than, greater than, or equal to 12

.

37

Solution

Half of 7 is 3.5.

Since 3 < 3.5, 37

12

<

59

Solution

Half of 9 is 4.5. Since 5 > 4.5, 59

12

>

Decide whether each fraction is less than, greater than, or equal to 12 .

Justify your answer using number sense.

4. 1632

5. 815

6. 1022



6

Example 4

Use number sense to tell which fraction is larger in each pair.

615

47

or

Solution

615

12

47

12

< >and . Therefore 47

is larger.

1113

98

or

Solution

11 113

< and 9 18

> . Therefore 98

is larger.

145

97

or −

Solution

1 045

> and 9 07

− < . Therefore 145

is larger.

615 1

247



7

7. 920

919

75

35

1217

510

, , , , ,−

a. Which number is less than zero?

b. Which number is greater than one?

c. Which of the remaining numbers are 1 1 1,

2 2 2,=> < ?

d. Which is larger, 9 920 19

or ? 1

2

0 1

e. Order the fractions from smallest to largest.

Sometimes it is not possible to use any of these number sense techniques to compare the size

of fractions. For example, it would be difficult to compare the sizes of 1115

712

and with the

previous techniques. Both of these fractions are more than 12 and less than 1. They do not

have a common denominator nor do they have a common numerator. Although the pieces of

the 12ths are larger, there are more of the 15ths.



8

If the denominators of

two fractions are

relatively prime the lcd

is the product of the two

denominators.

Example 5

Compare the sizes of 1115

and 712

by finding a common denominator.

Solution Method 1:

Find the lowest common denominator (lcd). This is a good technique to use when the

fractions must be added or otherwise manipulated.

Since 15 3 5 and 12 2 2 3= ⋅ = ⋅ ⋅ , the lcm (lowest common multiple) is 2 2 3 5⋅ ⋅ ⋅ or 60.

1115

44

4460

712

55

3560

⋅ = ⋅ =and

Since 35 < 44, 712

1115

< .

Method 2:

Use the product of the denominators as a common

denominator. This is a handy technique if the “only”

goal is to figure out which fraction is larger.

1115

1212

132180

⋅ = and 712

1515

105180

⋅ =

Since 105 < 132 then 712

1115

< .

Notice that in this example the lcd, 60, and the common

denominator found by multiplying the two denominators

together, 180, differ by a factor of 3 (the factor they have in

common).

Think Back

Integers are relatively

prime if they have no

common factors. Since 15

and 12 have a common

factor of 3 they are not

relatively prime.



9

8. Determine which fraction, 38

512

or , is smaller using both of the

techniques described in Example 5.

Let’s take a look at performing some operations using fractions.

Here is an algorithm for adding and subtracting fractions.

To add fractions:

1. Write each fraction using a common denominator.

2. Add the numerators.

3. Keep the common denominator.

4. Reduce the fraction if necessary.

To subtract fractions:

1. Write each fraction using a common denominator.

2. Subtract the numerators.

3. Keep the common denominator.


Algorithm

A rule (or step by step process) used to help solve

a specific type of problem. Algorithm



10

Why is it necessary to find a common denominator when adding and subtracting fractions?

Example 6

Find the sum of 1 26 6

+ and then the difference between 7 212 12

− .

Solution For both problems, each of the fractions contains a common denominator. All that

needs to be done is to follow the algorithm by working with the numerators and then

reducing, if possible.

1 2 36 6 6

+ = since 3 is a factor of 6, the fraction can be reduced to 12

.

7 2 512 12 12

− = , there are no common factors, thus 512

is the answer.

What happens if you perform an operation and the numerator is larger than the denominator?

Example 7

Find the sum of 2 23 3

+

Solution Recall that when the numerator is greater than the denominator the result is larger then

one; therefore, you should have an improper fraction which results in a mixed number.

2 2 4 is an improper fraction3 3 3

+ = ← . Convert 43

to a mixed number by asking how

many times 3 goes into 4. It goes in once, with a remainder of 1 over the common

denominator, 3 and looks like this: 4 113 3

= .



11

Why is it necessary to find a common denominator when adding and subtracting fractions?

Example 8

Find the sum. 45

215

+

Solution The sum will be a fraction with a denominator that describes the number of pieces of equal size. However 5ths and 15ths do not have pieces of the same size. A common size or common denominator must be found. Since the sum should be in lowest terms (reduced), it is best to choose the least common multiple of 5 and 15, which is 15, as the common denominator.

45

215

45

33

215

+ = ⋅ +

= +1215

215

=1415

Had the product of the denominators been chosen as the common denominator the following results: 45

215

45

1515

215

55

+ = ⋅ + ⋅

= +6075

1075

=7075

=1415

The final result is the same but the arithmetic is more complicated.



12

Think Back

An improper fraction is a fraction in which the numerator (top #) is

larger than the denominator (bottom #). Improper fractions are greater

than 1 and can be turned into mixed numbers.

Example 9

Which one of the following is the best answer for this sum? 3 28 9

+

a. The sum is less than 1.

b. The sum is greater than 1.

c. It is impossible to tell without actually adding.

Solution

Since the fractions are both less than 12

the sum is less than 1.

The best answer is: a.

Example 10

Use a TI – 30XIIS calculator to find the following sum. 2 512

31720

+

Solution Press the following keys:

b b b b2 A 5 A 12 3 A 17 A 20c c c c+ =

The screen then shows 6 4 15∪ / which means the sum is 6 415

. The TI – 30XIIS is a

great calculator. It performs the operation intended and even reduces the fraction, if

possible. To find the sum without the calculator would require several different steps

to arrive at the same answer.

+

521217320

2526051360

76 19 4 45 5 5 1 660 15 15 15

→ → + =

The Texas Instruments (TI) TI – 30XIIS will find sums, differences, products, and quotients of fractions and mixed numbers and give the results in fractional form.



13

Example 11

Find the difference between 1 28 22 3

− .

Solution

Find the common denominators. When you subtract the numerators, notice that the bottom is larger than the top. You can’t take away from something you don’t have, so you have to borrow from the whole number. The short cut is to take the denominator and the numerator and add them together to get the new numerator for the top fraction, then proceed with the operation. Remember, the calculator will assist you.

9. Solve and check with a calculator. 916

1124

−

10. Is this sum >1 or <1? 2 43 5

+

The following is an algorithm for multiplying fractions.

Why is it unnecessary to find a common denominator when multiplying and dividing

fractions?

To multiply fractions:

1. Multiply their numerators.

2. Multiply their denominators.


a c acb d bd

⋅ = Algorithm

1 38 82 62 42 23 6

→

− → −

3 98 76 64 42 26 6

556

→

− → −



14

Example 12

Find this product. 12

34

×

Solution

The whole is now divided into 8 equal portions. Three of the portions are double shaded.

The product is 38

.

Since the whole was divided into 4 equal parts vertically and 2 equal parts horizontally, the whole is now divided into 8 equal parts and the product is written in terms of 8ths. The equal parts can thus be found by multiplying the denominators without finding a common denominator.

When multiplying fractions you may reduce before multiplying.

Example 13

Find this product: 38

23

×

Solution

If the steps are performed as given the following product is found.

38

23

624

× =

=14

If the reducing is done first, the result is as follows. 11

4 13 2 18 3 4

× =

The results are the same.

34

The double shaded portion is 12

of 34

Problem Solving Tip

When multiplying

fractions you may cancel

either vertically or across

the multiplication sign.



15

The Division Algorithm

Dividing by a number is equivalent to

multiplying by the reciprocal of the number.

a c a db d b c

÷ = ×

Algorithm

Problem Solving Tip

A good way to remember the algorithm is:

“Copy, Change, Flip”.

Copy the 1st fraction.

Change the division to multiplication.

Flip the numerator and denominator in the

2nd fraction.

Example 14

Show that these are equivalent. 6 23

6 32

÷ = ×

Solution

It is sometimes helpful to write a problem in a different form.

6 23

÷ may be written 6 23

If a number is multiplied by 1, its value remains unchanged. Choose the 1 so that the

denominator becomes 1.

2 6 6 233

÷ = 6 123

= ⋅

3 6 22 33 2

= ⋅

362

2 33 2

⋅=

⋅ =

⋅6 32

1 = ⋅6 3

2

Luckily, we do not have to do all this work every time we divide fractions. We can use the

following algorithm:



16

Example 15

Perform the division.

4 45 9

÷

Solution

4 4 4 9 9 4 or 15 9 5 4 5 5

÷ = × =

Again, remember that the calculator will assist you in checking your work.

11. Perform the divisions; check with your calculator.

a. 2 35

÷ =

b. 4 47 5

÷ =

c. 354

÷ − =

d. 5 36 7

÷ =

e. 388

÷ =



17

f. 3 85 9

− ÷ − =

Review 1. Highlight the following words and their definitions.

a. rational number

b. numerator

c. denominator

d. relatively prime

2. Highlight the Fact boxes.

3. Highlight

a. the addition and subtraction algorithm

b. the multiplication algorithm

c. the division algorithm

d. the Think Back box that gives the definition of an improper fraction.

4. Write one thing you learned or one question you have for your mentor from this module.



18

Practice Problems Fractions

Directions: Write your answers in your math journal.

Label this exercise Fractions: Set A, Set B, Set C, Set D, Set E, and Set F.

Set A

1. Which fractions are greater than 1?

a. 35

b. 67

c. 43

d. 193

e. −32

2. Which fractions are greater than 0?

a. 35

b. −811

c. 29

d. −1

100000

3. Which fractions are greater than 12 ?

a. 35

b. 67

c. 110

d. 49

4. In which fraction are the sizes of the pieces greater?

a. 35

or 47

b. 29

or 18

c. 710

or 111

5. Find the least common denominator of the following:

a. 56

and 79

b. 712

and 920

c. 1125

and 45

d. 421

and 14



19

Set B

1. Which fraction is larger? Justify your answer.

a. 35

or 37

b. 29

or 19

c. 43

or 27

d. 25

or −1

10

e. 310

or 311

f. 712

or 812

2. Compare the following fractions using either the lcd or the product of denominators.

Which is larger?

a. 49

or 712

b. 720

or 925

c. 47

or 35

Set C

1. Put the following fractions in order from least to greatest:

711

, −23

, 65

, 611

, −15

, 27

, 25

2. Explain the difference between the lowest common denominator and common

denominator.

Set D

1. Perform the following calculations manually. Then check by using a calculator.

a. 16

518

+ b. 512

12

− c. 712

34

+ d. 57

821

−

e. 34

23

× f. 12

56

× − g. 79

23

÷ h. 35

512

÷



20

Set E

1. Use your calculator to find the simplest form of the following.

a. 9 34

5 12

− b. 7 13

3 12

+ c. 7 12

115

÷ d. 3 25

6 12

×

Set F

1. Which is larger, 1 72 18

or ? Is the difference 1 72 18

− positive or negative?

2. Which of the following is the best estimate for the sum 23

35

+ ?

a. It is less than 1.

b. It is greater than 1.

c. It is impossible to tell without actually adding.

3. Is

1 2 78

more or less than 1? Justify your answer without actually dividing.



21

1. If a whole is cut into 11 pieces, the pieces are smaller than if the whole is cut into 9 pieces.

Since one piece of each is given, 19

is larger.

2. The pieces are the same size. Therefore 13 of these pieces are more than 11 of them

or 1119

1319

< .

3. The pieces are smaller if a whole is cut into 17 pieces than if it is cut into 15 pieces.

The same number of pieces of each is represented. Therefore 617

615

< . The larger fraction

is 615

.

4. Since 16 is half of 32, 1632

12

= .

5. Half of 15 is 7.5 and 8 is greater than 7.5. Therefore 815

12

> .

6. Half of 22 is 11 and 10 is less than 11. Therefore 1022

12

< .

7. a. 35

− is the only fraction less than zero. Therefore it is also the smallest.

b. 75

is greater than one since its numerator is greater than its denominator.

c. 510

12

= . 1217

12

920

919

12

> <and both and are .

d. 919

920

and have a common numerator but the size of pieces are larger in 19ths. Thus

920

919

< . Therefore the fractions are ordered this way.

e. − < < < < <35

920

919

510

1217

75

.



22

8. Method 1

Since 8 2 2 2= ⋅ ⋅ and 12 2 2 3= ⋅ ⋅ , the lcd is 2 2 2 3 24or⋅ ⋅ ⋅ .

Four or 2 2⋅ is a common factor and is used only once as a factor in the lcd.

The extra 2 is from the third 2 in 8.

3 3 98 3 24

⋅ = and 5 2 1012 2 24

⋅ = . Therefore 3 58 12

< .

Method 2

Use the product of the denominators as a common denominator.

3 12 368 12 96

⋅ = and 5 8 4012 8 96

⋅ = . Therefore 3 58 12

< .

*Notice 24 4 96⋅ = . The two denominators in the two methods differ by a factor of 4.

This is the factor they have in common.

9. The lcd is 48.

9 11 9 3 11 216 24 16 3 24 2

27 2248 48548

− = ⋅ − ⋅

= −

=

The results are the same with the calculator.

10. The sum is >1 since both fractions are > 12

.



23

11. a. 2 2 1 235 5 3 15

÷ = × = .

b. 1

1

4 4 4 5 57 5 7 4 7

÷ = × = .

c. 3 5 4 20 24 1 3 3 3

5 6÷ − = × − = − = − .

d. 5 3 5 7 35 176 7 6 3 18 18

1÷ = × = = .

e. 3 8 8 64 18 218 1 3 3 3

÷ = × = =

f. 3 8 3 9 275 9 5 8 40

− ÷ − = − × − =



24

Answers to Practice Problems Set A 1.

a. 35

< 1 b. 67

< 1 c. 43

> 1

d. 193

> 1 e. −32

< 1

2.

a. 35

> 0 b. −811

< 0 c. 29

> 0 d. −1

100000 < 0

3.

a. 35

> ½ b. 67

> ½ c. 110

< ½ d. 49

< ½

4.

a. 35

b. 18

c. 710

5. a. 18 b. 60 c. 25 d. 84

Set B 1.

a. 35

b. 29

c. 43

d. 25

e. 310

f. 812

2.

a. 49

< 712

, because 49

1636

= and 712

2136

=

b. 720

< 925

, because 720

35100

= and 925

36100

=

c. 47

< 35

, because 47

2035

= and 35

2135

=



25

Set C

1. −23

, −15

, 27

, 25

, 611

, 711

, and 65

2. The lowest common denominator of two or more fractions is the lowest common multiple

of the denominators. A common denominator is a multiple of the denominators, or simply

the product of the denominators.

Set D 1.

a. 818

49

= b. −1

12 c. 16

121 4

1211

3= =

d. 721

13

= e. 612

12

= f. −5

12

g. 2118

76

116

= = h. 3625

11125

=

Set E 1.

a. 4 14

b. 10 56

c. 6 14

d. 22 110

Set F

1. 1 , positive2

2. b, greater than 1 3. Less than 1. The denominator is larger than the numerator.



26

NOTES or questions for your teacher / mentor

End of Fractions



27

Glossary of Terms

Acute angle – an angle whose measure is between 0 o and 90 o . (Modules: 9, 10)

Acute triangle – a triangle with three acute angles. (Module 10)

Addition Operation – term + term = sum. (Modules: 5 – 10)

Additive Inverse (or opposite of a number, x) – the unique number -x, which when added to x

yields zero. ( ) 0x x+ − = . (Modules: 4, 8)

Adjacent angles – two angles with the same vertex and a common side between them. Angles

1 and 2 are adjacent angles. (Modules: 9, 10)

2

1

Algebraic Expression – a mathematical combination of constants and variables connected by

arithmetic operations such as addition, subtraction, multiplication, and division.

(Module 8)

Algorithm – a rule (or step by step process) used to help solve a specific type of problem.

(Modules: 5 – 10)

Alternate exterior angles – when a line intersects two parallel lines, eight angles are formed;

two angles that are outside (exterior) the parallel lines and on opposite sides (alternate)

of the intersecting line are called alternate exterior angles. (Module 9)


Integrated Math Concepts 28

Alternate interior angles – when a line intersects two parallel lines, eight angles are formed;

two angles that are between (interior) the parallel lines and on opposite sides (alternate)

of the intersecting line are called alternate interior angles. (Module 9)

Altitude – the perpendicular from a vertex to the opposite side (extended if necessary) of a

geometric figure. (Module 10)

altit

ude

altit

ude

Angle – the union of two rays with a common endpoint; angles are measured in a counter-

clockwise direction; the angle’s rays are labeled as initial and terminal sides with the

terminal side counter-clockwise from the initial side. (Modules: 9, 10)

initial side

terminal side

Apothem – the apothem of a regular polygon is the radius of an inscribed circle. (Module 10)

apothem

Arc – any part of a circle that can be drawn without lifting the pencil. (Module 10)



29

Area – the measurement in square units of a bounded region. (Module 3)

Associative Property of Addition – this property of real numbers may be written using

variables in the following way: ( ) ( )a b c a b c+ + = + + . Terms to be combined may be

grouped in any manner. (Module 4)

Associative Property of Multiplication – this property of real numbers may be written in the

following way: ( ) ( )a b c a b c⋅ ⋅ = ⋅ ⋅ . Terms to be multiplied may be grouped in any

manner. (Module 4)

Axiom – a statement that is accepted as true, without proof. (Module 3)

Base – the numbers being used as a factor in an exponential expression. In the exponential

expression 2 5 , 2 is the base. (Module 7)

Base angles of an isosceles triangle – the angles opposite the equal sides of an isosceles

triangle are the base angles, which are also equal. (Module 10)

Base of an isosceles triangle – the congruent sides of an isosceles triangle are called the legs,

while the third side of the isosceles triangle is called the base. (Module 10)

Binary operation – an operation such as addition, subtraction, multiplication, or division that

changes two values into a single value. (Modules: 5 – 10)

Bisector – a line that divides a figure into two equal parts. (Module 10)

Centi – a prefix for a unit of measurement that denotes one one-hundredth 1100( ) of the unit.



Central angle – an angle whose vertex is the center of a circle and whose sides are radii of the

circle. (Module 10)

Chord – a line segment with endpoints on a circle. (Module 10)

Circle – the set of all points in a plane at a given distance (the radius) from a given point (the

center). (Module 10)

radius

center

Circumference – the distance around the edge of a circle. (Modules: 9, 10)

Closed dot – means the number is part of the solution set, thus it is shaded. (Module 8)

Coefficient – the numerical part of a term. (Module 8)

Combine like terms – means to group together terms that are the same (numbers with numbers

/ variables with variables) and are on the same side of the equal sign. (Module 8)

Complementary angles – two angles whose sum is 90 o . (Modules: 9, 10)

Common factor – identical part of each term in an algebraic expression; in the expression ab +

ac, the variable a is the common factor. (Module 8)

Commutative Property of Addition – terms to be combined may be arranged in any order; this

property of real numbers may be written using variables in the following way:

a b b a+ = + . (Module 4)



31

Commutative Property of Multiplication – terms to be multiplied may be arranged in any

order; this property of real numbers may be written using variables in the following

way: a b b a⋅ = ⋅ . (Module 4)

Comparison Axiom – if the first of three quantities is greater than the second and the second is

greater than the third, then the first is greater than the third; if a > b and b > c,

then a > c. (Module 3)

Composite number – a natural number greater than one that has at least one positive factor

other than 1 and itself. (Module 2)

Consecutive even integers – even integers that follow one another such as 2, 4, 6, etc.

(Module 8)

Consecutive integers – integers that follow each other on the number line such as 7, 8, 9, etc.

(Module 8)

Consecutive odd integers – odd integers that follow one another such as 5, 7, 9, etc.

(Module 8)

Constant – any symbol that has a fixed value such as 2 or π. (Modules: 3, 7, 8)

Coplanar – coplanar points are points in the same plane. (Module 9)

Corresponding angles – if a line intersects two parallel lines, eight angles are formed; two

non-adjacent angles that are on the same side of the intersecting line but one between

the parallel lines and one outside the parallel lines are called corresponding angles.

(Module 9)



Counting numbers (or natural numbers) – the set of numbers {1, 2, 3, 4, 5, …}. (Module 1)

Decagon – a ten-sided polygon. (Module 10)

Denominator – the bottom part of a fraction. (Modules: 5, 6, 7, 8)

Diagonal – a line segment with endpoints on two non-consecutive vertices of a polygon.

(Module 10)

Diameter – a line segment that passes through the center of a circle and whose endpoints are

points on the circle. (Module 10)

Difference – the answer to a subtraction problem. (Modules: 5, 6)

Distributive Property of Multiplication over Addition – a property of real numbers used to

write equivalent expressions in the following way: ( )a b c a b a c+ = ⋅ + ⋅ .

(Modules: 4, 8)

Distributive Property of Multiplication over Subtraction – a property of real numbers used to

write equivalent expressions in the following way: ( )a b c a b a c− = ⋅ − ⋅ .

(Modules: 4, 8)

Dividend – the number being divided in a quotient; in c

b a or a cb= , a is the dividend.

(Modules: 5, 6, 7, 8)

Division operation – Quotient

Dividend Quotient or Divisor DividendDivisor

= . (Modules: 5, 6, 7, 8)



33

Elements (of a set) – the objects that belong to a set. (Module 2)

Empty set – a set that has no elements in it. (Module 2)

Equal Quantities Axiom – quantities which are equal to the same quantity or to equal

quantities, are equal to each other. (Module 3)

Equation – a mathematical statement that two quantities are equal to one another. (Module 8)

Equiangular polygon – a polygon with all angles equal. (Module 10)

Equiangular triangle – a triangle with all angles equal. (Module 10)

Equilateral polygon – a polygon with all sides equal. (Module 10)

Equilateral triangle – a triangle with all sides equal. (Module 10)

Existence Property – a property that guarantees a solution to a problem. (Module 4)

Existential quantifier – ∀ is the existential quantifier; it is read “for all,” “for every,” or “for

each.” (Modules: 1, 2)

Exponent – tells how many times a number called the base is used as a factor; in 32 8= , three

(3) is the exponent. (Module 7)



Exterior angle – is an angle formed by one side of a polygon and an adjacent side extended.

(Modules: 9, 10)

A

B C

ED Factor – one of the numbers multiplied together in a product; if a b c⋅ = , then a and b are

factors of c. (Modules: 5, 6)

Fundamental Theorem of Arithmetic – every composite number may be written uniquely

(disregarding order) as a product of primes. (Module 2)

Geometry – the branch of mathematics that investigates relations, properties, and

measurements of solids, surfaces, lines, and angles. (Modules: 9, 10)

Gram (g) – a basic unit of mass in the metric system; 1 gram≈ .035 ounces.

Heptagon – a seven-sided polygon. (Module 10)

Hexagon – a six-sided polygon. (Module 10)

Hypotenuse – the side opposite the right angle in a right triangle. (Module 10)

Identity – an equation that is true for all values of the variable; every real number is a root of

an identity. (Module 4)

Identity Element for Addition – zero is the additive identity element because 0 may be added

to any number and the number keeps its identity; 0 0a a a+ = + = for any real number

a. (Module 4)



35

Identity Element for Multiplication – one (1) is the multiplicative identity element because

any number may be multiplied by 1 and the number keeps its identity; 1 1a a a⋅ = ⋅ = for

any real number a. (Module 4)

Improper fraction – a fraction in which the numerator (top #) is larger than the denominator

(bottom #). Improper fractions are greater than 1 and can be turned into mixed

numbers. (Module 5)

Inequality – a mathematical sentence that compares two unequal expressions.

(Modules: 2, 3, 8)

Inscribed angle – an angle whose vertex lies on a circle and whose sides are chords of the

circle. (Module 10)

Integers – the natural numbers, zero, and the additive inverses of the natural numbers;

{…-3, -2, -1, 0, 1, 2, 3…}. (Modules: 1 – 10)

Interior angle – an angle that lies inside a polygon and is formed by two adjacent sides of the

polygon. (Module 10)

Intersect – to cross; two lines in the same plane intersect if and only if they have exactly one

point in common. (Module 9)

Irrational number – a real number that cannot be written as the quotient of two integers; an

irrational number, written as a decimal, does not terminate and does not repeat.

(Module 1)



Isosceles trapezoid – a trapezoid whose non-parallel sides (or legs) are congruent.

(Module 10)

leg leg

Isosceles triangle – a triangle with two sides equal. (Module 10)

Kilo – a prefix for measurement that denotes one thousand (1000) units.

Kite – a quadrilateral with two pairs of adjacent sides congruent and no opposite sides

congruent. (Module 10)

Least Common Multiple (LCM) – the least common multiple of two or more positive values is

the smallest positive value that is a multiple of each. (Modules: 5, 6)

Legs of an isosceles triangle – the congruent sides of an isosceles triangle are called its legs.

(Module 10)

Like terms – terms which have identical variable factors. (Module 8)

Line – one of the undefined terms; consists of a set of points extending without end in opposite

directions. (Modules: 9, 10)



37

Line segment – a subset of a line that contains two points of the line and all points between

those two points. (Modules: 9, 10)

Liter (L) – a basic unit of volume in the metric system; 1 liter ≈ 1.06 liquid quarts.

Lowest common denominator (lcd)(of two or more fractions) – the least common multiple of

the denominators of the fractions. (Modules: 5, 6)

Major arc – an arc of a circle that is greater than a semicircle. (Module 10)

Meter (m) – a basic unit of length in the metric system; 1 meter ≈ 39.37 inches.

Milli – a prefix for a unit of measurement that denotes one one-thousandth 11000( ) of the unit.

Minor arc – an arc of a circle that is less than a semicircle. (Module 10)

Minuend – the number from which something is subtracted; in 5 3 2− = , five (5) is the

minuend. (Modules: 5 – 8)

Multiplicative inverse (or reciprocal of a real number x) – the unique number, 1x

, which,

when multiplied by x, yields 1. 1 1xx⋅ = if 0x ≠ . (Modules: 4, 8)

Multiplication operation – factor x factor = product. (Modules: 5 – 8)

Multiplicative property of zero – for any real number a , 0 0 0a a⋅ = ⋅ = . (Modules: 4 – 8)

Natural numbers (or counting numbers) – the set of numbers {1, 2, 3, 4, 5, …}. (Module 1)



Negative integers – the opposite of the natural numbers. (Modules: 1 – 8)

Nonagon – a nine-sided polygon. (Module 10)

Numerator – the top part of a fraction. (Module 5)

Obtuse angle – an angle that measures between 90 o and 180 o . (Modules: 9, 10)

Obtuse triangle – a triangle with one obtuse angle. (Module 10)

Octagon – an eight-sided polygon. (Module 10)

Open dot – means the number is not part of the solution set, thus it is not shaded. (Module 8)

Parallel lines – lines in the same plane that do not intersect; the two lines are everywhere

equidistant. (Modules: 9, 10)

Parallelogram – a quadrilateral whose opposite sides are parallel. (Module 10)

Pentagon – a five-sided polygon. (Module 10)

Percent – Percent means per 100 or divided by 100. The symbol for percent is %.

(Module 6)

Perfect square – a number whose square root is a natural number. (Module 1)

Perimeter – the sum of the lengths of the sides of a figure or the distance around the figure.

(Modules: 8, 10)



39

Perpendicular lines – two lines that form a right angle. (Modules: 9, 10)

Plane – one of the undefined terms; a set of points that form a flat surface extending without

end in all directions. (Modules: 9, 10)

Plane geometry – the branch of mathematics that deals with figures that lie in a plane or flat

surface. (Module 10)

Point – one of the undefined terms; a location with no width, length, or depth.

(Modules: 9, 10)

Polygon – a closed figure bounded by line segments. (Module 9)

Positive integers – the collection of numbers known as natural numbers. (Modules: 1 – 10)

Prime numbers – the natural numbers greater than one (1) that have exactly two factors, one

(1) and themselves. (Module 2)

Product – the result when two or more numbers are multiplied. (Modules: 3 – 10)

Quadrilateral – a polygon with four sides. (Module 10)

Quotient – the number resulting from the division of one number by another. (Modules: 1, 5)

Radical – the symbol that tells you a root is to be taken; denoted by . (Module 1)

Radicand – the number inside the radical sign whose root is being found; in 7x , 7x is the radicand. (Module 1)



Radius (radii) – a line segment with endpoints on the center of the circle and a point on the

circle. (Module 10)

Ratio – proportional relation between two quantities or objects in terms of a common unit.

(Module 5)

Rational numbers – the collection of numbers that can be expressed as the quotient of two

integers; when written as a decimal it will terminate or repeat. (Modules: 1, 5)

Ray – a subset of a line that consists of a point and all points on the line to one side of the

point. (Modules: 9, 10)

Real numbers – the combined collection of the rational numbers and the irrational numbers.

(Module 1)

Reciprocal (or multiplicative inverse of a real number x) – the unique number which, when

multiplied by x, yields 1; 1 1xx⋅ = if 0x ≠ . (Module 4)

Rectangle – a parallelogram with one right angle. (Modules: 3, 8, 10)

Reflex angle – an angle greater than a straight angle and less than two straight angles.

(Module 9)

Regular polygon – a polygon whose sides and angles are all equal. (Module 10)

Relatively prime – a pair of numbers with no common factor other than 1. (Module 5)



41

Repeating decimal – a decimal with an infinite number of digits to the right of the decimal

point created by a repeating set pattern of digits. (Modules: 1, 6)

Rhombus (rhombi) – a parallelogram having two adjacent sides equal. (Module 10)

Right angle – an angle whose sides are perpendicular; having a measure of 90 degrees.

(Modules: 9, 10)

Right triangle – a triangle with one right angle. (Module 10)

Scalene triangle – a triangle with no two sides of equal measure. (Module 10)

Secant – a straight line intersecting a circle in exactly two points. (Module 10)

Sector of a circle – the figure bounded by two radii and an included arc of the circle.

(Module 10)

Sector

Semicircle – an arc equal to half of a circle is called a semicircle. (Module 10)

Set – a collection of objects. (Module 2)

Sides of a polygon – the line segments forming a polygon are called the sides of the polygon.

(Module 10)



Similar figures – figures with the same shape but not necessarily the same size. (Module 10)

Similar polygons – polygons whose corresponding angles are congruent and whose

corresponding sides are proportional; the symbol ~ is used to indicate that figures are

similar. (Module 10)

Solution – a value that makes the two sides of an equation equal. (Modules: 5 – 10)

Solution set – the set of all roots of the equation. (Module 8)

Square – a rectangle having two adjacent sides equal. (Modules: 8, 10)

Square root – one of the two equal factors of a number. (Module 1)

Straight angle – an angle measuring 180 o . (Modules: 9, 10)

Subset – B is a subset of A, written B ⊆ A, if and only if every element of B is an element of A.

(Module 2)

Substitution Axiom – a quantity may be substituted for its equal in any expression. (Modules: 3, 4, 7 – 10)

Subtraction operation – Minuend

Subtrahend

Difference

− or Minuend – Subtrahend = Difference.

(Modules: 5 – 10)

Subtrahend – the number being subtracted in a subtraction problem; in 5 – 2 = 3, 2 is the

subtrahend. (Modules: 5, 6)



43

Sum – the result when two numbers are added. (Modules: 5 – 10)

Supplementary angles – two angles whose sum is 180 o . (Modules: 9, 10)

Term – a single number, a single variable, or a product of a number and one or more variables.

(Modules: 1 – 10)

Terminating decimal – a decimal with a finite (or countable) number of digits to the right of

the decimal point. (Module 6)

Transversal – a straight line that intersects two or more straight lines. (Module 9)

transversa l

Trapezoid – a quadrilateral with exactly one pair of parallel sides. (Module 10)

Triangle – a polygon with three sides. (Modules: 8, 10)

Trichotomy Property – for all real numbers, a and b, exactly one of the following is true;

a b= , a b< , or a b> . (Module 3)

Uniqueness Property – a property that guarantees that when two people work the same

problem they should get the same result. (Module 4)

Universal quantifier – ∃ is the universal quantifier. It is read, there exists or for some.

(Modules: 1, 2)

Variable – a letter or symbol used to represent a number or a group of numbers.

(Modules: 3, 7, 8)



Vertex – the turning point of a parabola; the common endpoint of the two intersecting rays of

an angle. (Module 10)

Vertex angle of an isosceles triangle – the angle formed by the equal sides of the triangle.

(Module 10)

Vertex of a polygon – a point where two sides of a polygon meet. (Module 10)

Vertical angles – two non-adjacent angles formed by two straight intersecting lines.

(Module 9)

Whole numbers – the collection of natural numbers including zero; {0, 1, 2, 3…}. (Modules: 1 – 10)

FORMULAS AND DISCOVERIES The Triangle Inequality:

The sum of two sides of a triangle must be greater than the third side. In ∆ABC

AB BC ACAB AC BCAC BC AB

+ >+ >+ >



45

Name Sketch Perimeter Area/ Surface Area Volume

Triangle

P a b c= + + 12A bh=

Does not have

volume

Square

P = 4s A = 2s Does not

have volume

Rectangle

P = 2l +2w A = lw Does not have

volume

Circle

C 2 rπ= 2A rπ= Does not

have volume

P 2 2a b= + A bh= Does not

have volume

1 1 2 2P s b s b= + + +

( )11 22A b b h= + Does not

have volume

P = r + s + t + u + v +

A = 12 ap

where p is the perimeter

Does not have

volume

The distance around a base

S. A. = area of bases ( 1 2B B+ ) + area of all lateral faces

V Bh= or

12V aph=

s

ss

sD C

BA

A B

CD

l

l

w w

A

B

C

a

b

ch

r

A B

CD

h

1b

2b

1s 2s

1B

2Blateral face

Parallelogram

Trapezoid

Regular Polygon

Prism

a

r s

t

u v

A

B C

D

h

b a



The distance around the base

S.A. = area of the base + area of all the lateral faces.

13V Bh=

or 16V aph=

C 2 rπ= S.A. = 22 2r rhπ π+

2V r hπ=

C 2 rπ= S.A. = 2 2r rlπ π+

213V r hπ=

Sphere

C 2 rπ= S.A. = 24 rπ 343V rπ=

End of Glossary

h

base

lateralface

h

r

r

h l

r

Pyramid

Cylinder

Cone

Integrated Math Concepts - Module 5

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Transcript of Integrated Math Concepts - Module 5