Pearson Adult Learning Centre

Math Tip Archive 2

Math Tip #40

Characteristics of Good Problem Solvers

Are you a good problem-solver? Check yourself against the following list.

Good Problem-solvers 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 step-by-step 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 0-13-180357-3.

Math Tip #39

Steps for Solving Word Problems

The following tips can help you be more successful at solving word problems:
bulletRead the problem carefully. If necessary, read it again. If necessary, read it yet again.
bulletFocus 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.
bulletDecide what operation or operations you will need to find the answer. · Draw a picture of the problem. 
bulletGo back through the problem and pick out the facts you'll need to solve the problem. Ignore any unnecessary information. 
bulletSupply any missing facts. 
bulletWork the problem out accurately. 
bulletDouble-check your work. 
bulletAsk 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 0-13-180357-3.

Math Tip #38

Test-taking Strategies for Math Tests

When you have a math test, you often feel anxious, but the following test-taking strategies can reduce your nervousness and boost your marks.

Before the Test

bulletPrepare by studying all the types of problems that might be on the test. 
bulletTry to anticipate new or trick problems; create some of your own examples. 
bulletIf you are having trouble with some of the material, ask your teacher for help a few days before the test. 
bulletIf you find it useful, study with a friend. 
bulletThink positively about the test. 
bulletRecognize that most people are anxious about tests. These feelings usually pass once the test starts. 
bulletUse visualization. Imagine yourself doing well on the test. Picture yourself solving the problems and getting a high score. 
bulletPromise yourself a reward for doing well on the test. 
bulletPrepare yourself physically. Get a good night's sleep, eat a good meal and arrive at class on time. 
bulletGo to class ready to take the test. Make sure you have everything you need-pencils, erasers, calculator, etc.

During the Test 

bulletListen carefully to any instructions. If you have questions, ask. 
bulletIf you feel nervous, take a few deep breaths and think of a favourite place or activity you enjoy. 
bulletRead directions carefully. 
bulletPace yourself. Work quickly but accurately. Make sure your answers are clear. 
bulletDon'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. 
bulletBudget your time; spend the most time on parts of the test that are worth the most points. 
bulletIf time remains after completing the test, go over it and double-check your answers.

After the Test 

bulletIf you did well, reward yourself. 
bulletIf 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 0-13-180357-3.

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.
bulletRealize that men and women can do math equally well. There is no one type of person who is predestined for math greatness. 
bulletKeep 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." 
bulletBe persistent in working on your math. Everybody makes mistakes. Don't worry about them; just learn from them. 
bulletGet 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. 
bulletWrite 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.") 
bulletMake a commitment to study math everyday. Being prepared is one of the best ways to reduce worry. 
bulletView overcoming math anxiety as just another problem you can solve. 
bulletWhen you don't understand, don't be afraid to ask your teacher.
bulletKeep notes and review your notes.
bulletStudy with a friend.
bulletKeep 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 0-13-180357-3.

Math Tip #36

The Math Student's Responsibilities

The following will help you succeed not only in math, but your other classes too.
bulletCome to school each day ready to work and learn. 
bulletBring what you need to work on your math: your textbooks, notebooks, binders, pens, pencils, calculators, etc. 
bulletBe curious and inquisitive about numbers. Look for relationships and patterns that will help you understand. · 
bulletBe diligent in the completion of your math work. 
bulletBe persistent and determined in your work. 
bulletBe willing to try various strategies in solving problems. 
bulletAsk questions when you don't understand something. 
bulletTake 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. 
bulletGood luck and have fun.

From The Math Teacher's Book of Lists by Judith A. Muschla and Gary Robert Muschla ISBN 0-13-180357-3.

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 0-13-180357-3.

Math Tip #34

Scientific Notation Part Two

This week you are going to learn the rules for writing very small numbers in scientific notation.
bulletThe first factor is greater than or equal to 1 and is less than 10. (Same rule as for large numbers) 
bulletThe second factor is a negative power of 10 in exponential form.
bulletTo 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: · 
bullet0. 00079 = 7.9 X 10-4

Let's try another one: Write 0.06435 in scientific notation. · 

bulletCount 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.

bulletAnswer: 9023 X 10-6

From The Math Teacher's Book of Lists by Judith A. Muschla and Gary Robert Muschla ISBN 0-13-180357-3.

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 109.

Rule for writing large numbers in scientific notation:

bulletThe first factor is greater than or equal to 1 and is less than 10.
bulletThe second factor is a power of 10 in exponential form. 
bulletTo 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 109

Let's try another one: Write 8 955 456 000 000 in scientific notation.

bulletThe 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 1012

You try one. Write 1 340 000 in scientific notation.

Answer: 1.34 X 106

From The Math Teacher's Book of Lists by Judith A. Muschla and Gary Robert Muschla ISBN 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 people-that'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              1036
One duodecillion            1039
One tredecillion             1042
One quattuordecillion     1045
One quindecillion            1048
One sexdecillion             1051
One septendecillion         1054
One octodecillion            1057
One novemdecillion          1060
One vigintillion               1063
One googol                      10100
One googolplex                10googol

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

Math Tip #25

Rules for Operations with Integers Part 2

Here are a couple more mathematical idioms.

"Point of no return..."
"One-way 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: 

bulletLike signs are positive +8 X +7 = +56 -8 X -7 = +56
bulletUnlike signs are negative -8 X (+7) = -56 +8 X (-7) = -56

Dividing Integers

bulletDivide 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 0-13-180357-3.

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

bulletWhen the integers are positive, add them +4 + +3 = +7 and the sign remains positive. · 
bulletWhen the integers are negative, add the absolute values. The sign is negative. 
-4 + -3 = -7 
bulletWhen the signs of the integers are different, subtract the absolute values and keep the sign of the larger number. -4 + +9 = +5

Subtracting Integers

bulletRewrite 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 0-13-180357-3.

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 0-13-180357-3.

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.

One-half ½ 0.50 50% 
One-fourth ¼ 0.25 25% 
Three-fourths ¾ 0.75 75% 
One-third 1/3 0.33 33.3% 
Two-thirds 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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

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 0-13-180357-3.

 

 

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