Math Tip #40
Characteristics of Good Problem Solvers
Are you a good problemsolver?
Check yourself against the following list.
Good Problemsolvers Do Many of the Following

Work
actively toward solutions. 

Search
for key ideas. 

Identify
important information. 

Ignore
unimportant information. 

Can
supply missing information. 

Recognize
hidden questions. 

Work
carefully. 

Follow
a stepbystep method. 

Recognize
relationships. 

Try
various strategies. 

Look
at a problem from various angles. 

Are
open to new ideas. 

Keep
notes of their attempts at solutions. 

Often
recheck their facts. 

Use
logic. 

Rely
on past experience in solving problems. 

Are
persistent. 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #39
Steps for Solving Word Problems
The following tips can help you be
more successful at solving word problems:
 Read the problem carefully. If necessary, read it again. If
necessary, read it yet again. 
 Focus your attention on the question. Identify the problem. What is
it really asking? Look for key words like sum, total, all together,
about, difference, and how much more or less than. 
 Decide what operation or operations you will need to find the
answer. · Draw a picture of the problem. 
 Go back through the problem and pick out the facts you'll need to
solve the problem. Ignore any unnecessary information. 
 Supply any missing facts. 
 Work the problem out accurately. 
 Doublecheck your work. 
 Ask yourself it your answer satisfies the question. Is your answer
logical? Does it make sense? 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #38
Testtaking Strategies for Math Tests
When you have a math test, you
often feel anxious, but the following testtaking strategies can reduce
your nervousness and boost your marks.
Before the Test
 Prepare by studying all the types of problems that might be on the
test. 
 Try to anticipate new or trick problems; create some of your own
examples. 
 If you are having trouble with some of the material, ask your
teacher for help a few days before the test. 
 If you find it useful, study with a friend. 
 Think positively about the test. 
 Recognize that most people are anxious about tests. These feelings
usually pass once the test starts. 
 Use visualization. Imagine yourself doing well on the test. Picture
yourself solving the problems and getting a high score. 
 Promise yourself a reward for doing well on the test. 
 Prepare yourself physically. Get a good night's sleep, eat a good
meal and arrive at class on time. 
 Go to class ready to take the test. Make sure you have everything
you needpencils, erasers, calculator, etc. 
During the Test
 Listen carefully to any instructions. If you have questions,
ask. 
 If you feel nervous, take a few deep breaths and think of a
favourite place or activity you enjoy. 
 Read directions carefully. 
 Pace yourself. Work quickly but accurately. Make sure your answers
are clear. 
 Don't waste time on hard problems. Move on to easier ones, but
remember to go back. Place a mark on the questions you must return
to. 
 Budget your time; spend the most time on parts of the test that are
worth the most points. 
 If time remains after completing the test, go over it and
doublecheck your answers. 
After the Test
 If you did well, reward yourself. 
 If you didn't do as well as you believe you could have, reward
yourself for trying and learn from your mistakes. Resolve to do better
next time. 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #37
Overcoming Math Anxiety
Some people believe because of
past experience that they can't do math and they don't have the
"knack" for it. They begin to dread math and anything to do with
it. If you feel like this about math, read over this list for some ideas
for helping you relax. Be sure to tell your teacher that you are anxious
about math and he or she can work with you to help you overcome your
worries. Remember, you can do math.
 Realize that men and women can do math equally well. There is no one
type of person who is predestined for math greatness. 
 Keep your mind open and your emotions down. Don't let yourself think
that math is hard or impossible. Concentrate on saying to yourself,
"I can do math." 
 Be persistent in working on your math. Everybody makes mistakes.
Don't worry about them; just learn from them. 
 Get in touch with your math feelings. When your mind starts to flood
with math worries, stop, take a deep breath and clear your thoughts.
Get up and walk around. Then redirect yourself to the problem your
working on and tell yourself that you can do it. 
 Write down your math worries on a sheet of paper. (For example,
"I can't do word problems.") Write down possible solutions.
(For example, "I can take better notes, ask my teacher for help,
work on more problems for practice.") 
 Make a commitment to study math everyday. Being prepared is one of
the best ways to reduce worry. 
 View overcoming math anxiety as just another problem you can
solve. 
 When you don't understand, don't be afraid to ask your teacher. 
 Keep notes and review your notes. 
 Study with a friend. 
 Keep a sense of humour. So you missed a problem. It's not the end of
the world. you'll try again tomorrow and probably be successful. 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #36
The Math Student's Responsibilities
The following will help you
succeed not only in math, but your other classes too.
 Come to school each day ready to work and learn. 
 Bring what you need to work on your math: your textbooks, notebooks,
binders, pens, pencils, calculators, etc. 
 Be curious and inquisitive about numbers. Look for relationships and
patterns that will help you understand. · 
 Be diligent in the completion of your math work. 
 Be persistent and determined in your work. 
 Be willing to try various strategies in solving problems. 
 Ask questions when you don't understand something. 
 Take pride in your work and never let yourself fall into the trap of
believing you can't do math. Everybody can, if he or she is willing to
put in the work. 
 Good luck and have fun. 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #35
Roman Numerals
Roman Numerals
The ancient Romans were master politicians, soldiers and engineers.
They conquered the world from the British Isles to North Africa and Asia
Minor, spreading Roman civilization everywhere they went. Despite their
achievements and lasting legacies, the Roman's number system was not very
practical. When the Roman empire fell, their number system was replaced by
the Arabic system, which we use today. Roman numerals are still used
today, however, for some clocks, watches, book pages, outlines and
decorations. Roman numerals are not hard to learn, but they are a little
tedious to use. Here are some Roman numerals:
Arabic 
Roman 
Arabic 
Roman 
1 
I 
21 
XXI 
2 
II 
22 
XXII 
3 
III 
23 
XXIII 
4 
IV 
24 
XXIV 
5 
V 
25 
XXV 
6 
VI 
26 
XXVI 
7 
VII 
27 
XXVII 
8 
VIII 
28 
XXVIII 
9 
IX 
29 
XXIX 
10 
X 
30 
XXX 
11 
XI 
40 
XL 
12 
XII 
50 
L 
13 
XIII 
60 
LX 
14 
XIV 
70 
LXX 
15 
XV 
80 
LXXX 
16 
XVI 
90 
XC 
17 
XVII 
100 
C 
18 
XVIII 
200 
CC 
19 
XIX 
500 
D 
20 
XX 
1 000 
M 
For large numbers, the Romans sometimes used a bar (called a vinculum)
over the number to multiply it by 1000.
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #34
Scientific Notation Part Two
This week you are going to learn
the rules for writing very small numbers in scientific notation.
 The first factor is greater than or equal to 1 and is less than 10.
(Same rule as for large numbers) 
 The second factor is a negative power of 10 in exponential
form. 
 To write numbers with exponents, count the number of places to the
right of the decimal point, up to and including the first nonzero
number. That number of places becomes the negative exponent. In
the case of 0.00079, count 4 places to the right of the decimal point
up to and including the 7. Therefore: · 
 0. 00079 = 7.9 X 10^{4} 
Let's try another one: Write 0.06435 in scientific notation. ·
 Count 2 places to the right of the decimal point up to and including
the 6. Therefore: 0. 06435 = 6.435 X 10^{2} 
Try one on your own: Write 0.00000923 in scientific notation.
 Answer: 9023 X 10^{6} 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #33
Scientific Notation Part One
Scientific notation is useful for
expressing very small or very large numbers. For example, the average
distance from the sun to Pluto is about 3 670 000 000 miles. That's too
long to conveniently write, so mathematicians and scientists prefer to use
scientific notation: 3.67 X 10^{9}.
Rule for writing large numbers in scientific notation:
 The first factor is greater than or equal to 1 and is less than 10. 
 The second factor is a power of 10 in exponential form. 
 To write numbers with exponents, count the number of places to the
right of the first nonzero number in standard form. That number
becomes the exponent. In the case of 3 670 000 000, the first nonzero
number is 3. There are 9 digits to the right of the 3 so 9 is the
exponent. Therefore: 3 670 000 000 = 3.67 X 10^{9} 
Let's try another one: Write 8 955 456 000 000 in scientific notation.
 The first nonzero number is 8. There are 12 digits to the right of
the 8 so 8 is the exponent. Therefore: 8 955 456 000 000 = 8.955 456 X
10^{12} 
You try one. Write 1 340 000 in scientific notation.
Answer: 1.34 X 10^{6}
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #32
Rules for Finding the Average (Mean)
You have written 3 English tests
and you are wondering what your average (mean) mark is. The following will
show you how to do that.
1. Add up all the items you need to average. You received 93 % on
your first test, 84% on the second and 87% on the last. 93 + 84 + 87 =
264
2. Divide the sum by the total number of items you added. You wrote 3
English tests so you divide 264 by 3. 264 divided by 3 = 88 %
3. Your average (mean) mark in English is 88%.
There are other practical applications for averages: batting averages
in baseball, finding average incomes, finding average fat content in
hamburgers, etc.
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #31
Properties of Integers
Integers (positive and negative
whole numbers) have special properties. These are fundamental to doing
computations and can actually make them easier. Below, a, b, and c
represent integers, but just substitute numbers in for a, b and c to make
the properties more concrete.
Property 
Addition 
Multiplication 
Closure Property 
a + b is an integer 
(a)(b) is an integer 
Commutative Property 
a+ b =b + a 
ab = ba 
Associative Property 
(a + b) + c = a + (b + c) 
(ab)c = a(bc) 
Identity Property 
a + 0 = a 
1(a) = a 
Inverse Property 
a + a = 0 

Multiplication Property of Zero 

a(0) = 0 
Distributive Property 
a(b + c) = ab + ac 

From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #30
Math Signs and Symbols: Part Two
Here are some more math
superstitions.
Do these and you'll have bad luck:
Give someone a gift of a knife unless you include a coin.
Start a trip or project on Friday the 13th.
Break a mirror and suffer seven years of misfortune.
But study today's math tip and you'll have good luck all year!
The following list shows some more of the most often used signs and
symbols in math:
$
dollar sign
@
at
#
number or pound
%
percent
∆
triangle
◦
degree
∟
right angle
±
plus or minus
π
pi which is about 3.14
║
is parallel to
√
positive square root
~
is similar to
∩
intersection
U
union
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #29
Math Signs and Symbols: Part One
Here are some math superstitions.
Do these and you’ll have bad luck:
Hang a calendar before January 1 …
Light
three cigarettes on one match…
Plant
seeds the last three days of March…
But study today’s math tip and you’ll have good
luck all year!
The following list shows some of the most often used signs and symbols in math:
+
addition, plus, positive

subtraction, minus, negative, opposite
X
multiplication, multiply by, times
*
multiplication, multiply by, times
÷
division, divided by
y
division
=
is equal to, equals
≈
is approximately equal to
≠
is not equal to
>
is greater than
<
is less than
≤
is less than or equal to
≥
is greater than or equal to
∞
infinity
From The Math
Teacher’s Book of Lists by Judith A. Muschla and Gary Robert Muschla
ISBN 0131803573.
Math Tip #28
Big Numbers
Here are a couple more
mathematical idioms.
"Divide and conquer..."
"Feel like you're ten feet tall..."
And now for the main tip
Numbers over a hundred thousand are pretty difficult for most of us to
really understand. One hundred thousand people fill a big football
stadium. Then, imagine five stadiums that large side by side all filled
with peoplethat's about half a million. Ten of those same stadiums would
be a million, but after that the numbers are too hard to imagine. Here is
a list of really, really big numbers. I have used exponents to show
numbers after 1 decillion. The number up in the air (the exponent) shows
how many zeros I would write after the number 1.
One million 1 000 000
One billion 1 000 000 000
One trillion 1 000 000 000 000
One quadrillion 1 000 000 000 000 000
One quintillion 1 000 000 000 000 000 000
One sextillion 1 000 000 000 000 000 000 000
One septillion 1 000 000 000 000 000 000 000 000
One octillion 1 000 000 000 000 000 000 000 000 000
One nonillion 1 000 000 000 000 000 000 000 000 000 000
One decillion 1 000 000 000 000 000 000 000 000 000 000 000
One undecillion
10^{36
}One duodecillion
10^{39
}One tredecillion
10^{42
}One quattuordecillion
10^{45
}One quindecillion
10^{48
}One sexdecillion
10^{51
}One septendecillion
10^{54
}One octodecillion
10^{57
}One novemdecillion
10^{60
}One vigintillion
10^{63
}One googol
10^{100
}One googolplex
10^{googol}
Are numbers infinite? Can you ever find the largest number?
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #27
Bases
Here are a couple more
mathematical idioms.
"Back to square one..."
"Square deal..."
And now for the main tip:
Bases
The base of any number system is the number of different symbols used.
The Arabic system, which we use, is a base ten system because ten symbols
are used in writing numerals: 0, 1,2,3,4,5,6,7,8,9. The base ten system
probably reflect our eight fingers and two thumbs.
Base ten is not the only way. Number systems can be based on any number of
symbols. Base two (binomial) system, for instance, only has two digits, 0
and 1. Computers use base two to do their calculations. Pulses of
electrical energy representing 0 and 1 turn tiny switches on and off in
the microprocessor of the computer.
The following compares some different bases.
Base Ten Base Two Base Five Base Eight 1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 10 5
6 110 11 6
7 111 12 7
8 1000 13 10
9 1001 14 11
10 1010 20 12
11 1011 21 13
12 1100 22 14
13 1101 23 15
14 1110 24 16
15 1111 30 17
16 10000 31 20
17 10001 32 21
18 10010 33 22
19 10011 34 23
20 10100 40 24
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla
ISBN 0131803573.
Math Tip #26
Properties of Integers
Here are a couple more
mathematical idioms.
"Simple as one, two, three..."
"Square meal..."
And now for the main tip:
Integers have special properties. Understanding these properties can
make computation easier.
a, b, and c are integers
Closure Property of Addition a + b is an integer
Closure Property of Multiplication (a)( b) is an integer
Commutative Property of Addition a + b = b + a
Commutative Property of Multiplication ab = ba
Associative Property of Addition (a + b) + c = a + (b + c)
Associative Property of Multiplication (ab)c =a(bc)
Identity Property of Addition a + 0 = a
Identity Property of Multiplication 1(a)=a
Inverse Property of Addition a +a = 0
Multiplication Property of Zero a(0) = 0
Distributive Property a(b +c) = ab + ac
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #25
Rules for Operations with Integers Part 2
Here are a couple more
mathematical idioms.
"Point of no return..."
"Oneway ticket..."
And now for the main tip:
Integers include all positive and negative whole numbers and zero.
Multiplying Integers
Multiply the numbers and follow these rules:
 Like signs are positive +8 X +7 = +56 8 X 7 = +56 
 Unlike signs are negative 8 X (+7) = 56 +8 X (7) = 56 
Dividing Integers
 Divide the numbers and follow the rules for multiplying integers. 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #24
Rules for Operations with Integers Part 1
Here are a couple more
mathematical idioms.
"Sixth sense..."
"Going in circles..."
And now for the main tip:
Integers include all positive and negative whole numbers and zero.
Adding Integers
 When the integers are positive, add them +4 + +3 = +7 and the sign
remains positive. · 
 When the integers are negative, add the absolute values. The sign is
negative.
4 + 3 = 7 
 When the signs of the integers are different, subtract the absolute
values and keep the sign of the larger number. 4 + +9 = +5 
Subtracting Integers
 Rewrite the problem using the definition of subtraction: (a  b) = a
+ (b) ·
Follow the rules for adding integers 3  6 = 3 + (6) = 3 
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #23
Rules for Solving Proportions
Here are a couple more
mathematical idioms.
"Looking out for number one..."
"A third wheel..."
And now for the main tip:
Rules for Solving Proportions
A proportion is a statement that two ratios are equal. Proportions can
be helpful in solving word problems, particularly those involving
percents.
·
Set up the proportion and solve for N.
N = 10
8
13
·
Show the cross products of the proportions.
8 X 10 = 13 X N
·
Find the products.
80 = 13N
·
Divide both sides.
80 = 13N
13 13
80
= N
13
6 2/13 = N
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #22
Percent Equivalents
Here are a couple more
mathematical idioms.
"A penny for your thoughts."
"Take five."
And now for the main tip:
Percent Equivalents
The following shows the relationships between fractions, decimals and
percents. You should memorize these.
Onehalf ½ 0.50 50%
Onefourth ¼ 0.25 25%
Threefourths ¾ 0.75 75%
Onethird 1/3 0.33 33.3%
Twothirds 2/3 0.66 66.6%
One whole 1 1.00 100%
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #21
Types of Decimals: Part Two
Math is such an important part of
our lives that it has found its way into common expressions. Here's a
couple of mathematical idioms. Can you think of any more?
"Two's company; three's a crowd…"
"It takes two to tango."
And now for the main tip:
Types of Decimals Part 2
Here are some more definitions of decimals.
Finite or Terminating Decimal: a decimal that has a finite number of
digits.
Examples: 0.4, 0.3659, 0.4712058
Infinite or Nonterminating Decimal: a decimal that has an unending
number of digits to the right of the decimal point.
Examples: √3,
π,
23.46725 …
Repeating or Periodic Decimal: Nonterminating decimals in which the
same digit or group of digits repeats. A bar is used to show that a digit
or group of digits repeats. All rational numbers can be written as finite
or repeating decimals. (See Math Tip #1)
Examples:
Nonrepeating or Nonperiodic Decimal: decimals that are nonterminating
and Nonrepeating.
Examples: √3,
π
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #20
Types of Decimals: Part One
A decimal is any numeral in the
base ten number system, but the following are several specific
definitions.
Decimal Fraction : a number that has no digits other than zeros
to the left of the decimal point. Examples: 0.349, 0.65, 0.4021
Mixed Decimal: an integer and a decimal fraction
Examples: 9.462, 35.7, 658.42
Decimal Equivalent of a Proper Fraction: the decimal fraction
that equals the proper fraction.
Examples: 0.25 = ¼ , 0.3 = 3/10
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #19
Place Value
Place value is incredibly
important for understanding arithmetic. Take a minute to review this chart
for 5 762.431
Thousands hundreds tens ones . tenths hundredths thousandths
5 7 6 2 . 4 3 1
Another way of writing this would be to say
5 thousands or 5 000 7 hundreds or 700 6 tens or 60 2 ones or 2 4
tenths or .4 3 hundredths or .03 1 thousandth or .001
From The Math Teacher's Book of Lists by Judith A. Muschla and Gary
Robert Muschla ISBN 0131803573.
Math Tip #18
Multiplication and Division of Fractions
Multiplication of Fractions
·
Multiply the numerators
·
Multiply the denominators
·
Simplify if possible
X
=
=
Division of Fractions
·
After setting up the problem write the reciprocal of the
divisor. (The divisor is the fraction after the division sign.)
·
Rewrite the division sign as multiplication
·
Multiply the numerators
·
Multiply the denominators
·
Simplify if possible
divided by
The reciprocal of
is
divided by
=
X
=
= 1
From The Math
Teacher’s Book of Lists by Judith A. Muschla and Gary Robert Muschla
ISBN 0131803573.
Math Tip #17
Subtraction
of Fractions with Unlike Denominators
· Find the lowest common denominator by finding the least
common multiple of the denominators.
·
Write equivalent fractions with the common denominator.
·
Subtract the numerators. (Do NOT add the denominators.)
·
Simplify if possible.
=

=
x
=

x
=
From The Math
Teacher’s Book of Lists by Judith A. Muschla and Gary Robert Muschla
ISBN 0131803573.
Math Tip #16
Addition
of Fractions with Unlike Denominators
· Find
the lowest common denominator by finding the least common multiple of
the denominators.
·
Write equivalent fractions with the common denominator.
·
Add the numerators. (Do NOT add the denominators.)
·
Simplify if possible.
=
+
=
x
=
+
x
=
=
From The Math Teacher’s Book of
Lists by Judith A. Muschla and Gary Robert Muschla ISBN 0131803573.
Math Tip #15
Addition
of Fractions with Like Denominators
Add the numerators
·
Write the sum over the common denominator. (Do NOT add the
denominators.)
·
Simplify if possible
+
=
Subtraction of Fractions with Like Denominators
·
Subtract the numerators
·
Write the difference over the common denominator. (Do NOT
subtract the denominators.)
·
Simplify if possible

=
From The Math
Teacher’s Book of Lists by Judith A. Muschla and Gary Robert Muschla
ISBN 0131803573.
