integrated math concepts - module 5
TRANSCRIPT
Integrated Math Concepts
Module 5
Fractions
Second Edition
National PASS Center 2006
Integrated Math Concepts
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.
Integrated Math Concepts
Module 5
Fractions
Second Edition
National PASS Center 2006
BOCES Geneseo Migrant Center 27 Lackawanna Avenue Mount Morris NY 14510
Integrated Math Concepts
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.
Integrated Math Concepts
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
Integrated Math Concepts
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
>
National PASS Center
Module 5 - Fractions
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
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Integrated Math Concepts
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< = >
National PASS Center
Module 5 - Fractions
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
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Integrated Math Concepts
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
National PASS Center
Module 5 - Fractions
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.
MATEMÁTICA August 2006
Integrated Math Concepts
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.
National PASS Center
Module 5 - Fractions
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.
4. Reduce the fraction if necessary.
Algorithm
A rule (or step by step process) used to help solve
a specific type of problem. Algorithm
MATEMÁTICA August 2006
Integrated Math Concepts
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
= .
National PASS Center
Module 5 - Fractions
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.
MATEMÁTICA August 2006
Integrated Math Concepts
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.
National PASS Center
Module 5 - Fractions
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.
3. Reduce the fraction if necessary.
a c acb d bd
⋅ = Algorithm
1 38 82 62 42 23 6
→
− → −
3 98 76 64 42 26 6
556
→
− → −
MATEMÁTICA August 2006
Integrated Math Concepts
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.
National PASS Center
Module 5 - Fractions
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:
MATEMÁTICA August 2006
Integrated Math Concepts
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
÷ =
National PASS Center
Module 5 - Fractions
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.
MATEMÁTICA August 2006
Integrated Math Concepts
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
National PASS Center
Module 5 - Fractions
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
÷
MATEMÁTICA August 2006
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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.
National PASS Center
Module 5 - Fractions
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
.
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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
.
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Module 5 - Fractions
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
− ÷ − = − × − =
MATEMÁTICA August 2006
Integrated Math Concepts
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
=
National PASS Center
Module 5 - Fractions
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.
MATEMÁTICA August 2006
Integrated Math Concepts
26
NOTES or questions for your teacher / mentor
End of Fractions
National PASS Center
Module 5 - 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)
MATEMÁTICA August 2006
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)
National PASS Center
Module 5 - Fractions
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.
MATEMÁTICA August 2006
Integrated Math Concepts 30
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 32
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 34
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 36
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 38
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 40
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 42
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)
National PASS Center
Module 5 - Fractions
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)
MATEMÁTICA August 2006
Integrated Math Concepts 44
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
+ >+ >+ >
National PASS Center
Module 5 - Fractions
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
MATEMÁTICA August 2006
Integrated Math Concepts 46
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