Math Tip #14

Types of Fractions Part 1

A fraction is a part of a whole. You use fractions all the time-when you do half (1/2) the housework or eat one-quarter(1/4) of the cake. There are many different kinds of fractions.

Simple Fraction or Common Fraction: a fraction in which the numerator (top number) and the denominator (bottom number) are both integers. (See Math Tip #1)

For example:  ,-

Proper Fraction: a fraction in which the numerator is smaller than the denominator.

For example:  ,-

Improper Fraction: a fraction in which the numerator is equal to or larger than the denominator. Improper fractions are often changed to whole or mixed numbers.

For example:  ,-

Mixed Number: a number that is a combination of an integer and a proper fraction. It is a mix of two kinds of numbers.

For example:  , -1

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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Math Tip #13

Even More on the Least Common Multiple (LCM)

Here is a third way to find LCM. Choose the method that makes the most sense to you. You don’t have to remember all three ways! This method is definitely the longest to write out!

1.      Factor each number into its primes. (See Math Tip #3).

2.      Write each product using exponents.

3.      Write each base.

4.      If the base if a factor of only one number, write the base and the exponent in exponential form.

5.      If the base is a factor of more than 1 number, write the base in exponential form using the larger (or largest) exponents of the bases. If the exponents of a given base are the same, write the base and exponent in exponential form.

6.      The LCM is the product of these numbers.

Example: Find the LCM of 6 and 8.

1.  6 =                   2   x    3    

     8 = 2  x  2  x   2                               

2.  6 = 2¹   x  3¹

     8 = 2³

3. 3¹ x  2³    = 3  x  8 = 24

4.  24 is the LCM of 6 and 8.

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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Math Tip #12

More on the Least Common Multiple (LCM)

Here is another way to find LCM:

1.      Find the product of the two numbers.

2.      Divide this product by the greatest common factor (GCF) of the numbers.

(See Math Tip #10)

Example: Find the LCM of 6 and 8.

1.      The product of 6 and 8 is 6 x 8 = 48.

2.      The GCF of 6 and 8 – the factors of 6 are 1,   2,  3,  6

         the factors of 8 are  1,  2,  4,  8

            The GCF of  6 and 8 is 2.

3.   Divide 48 by 2 = 24.    24 is the LCM of 6 and 8.

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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Math Tip #11

Rules for Finding Least Common Multiple (LCM)

A common multiple is a number that two other numbers will divide into evenly. Therefore, the least common multiple (LCM) is the lowest multiple of  two numbers. You can use these rules for finding common denominators in fractions. I am going to show three ways to find LCM over the next three weeks.

Way One:

1.      Start with the bigger number

2.      List its multiples by multiplying the number by 1, 2, 3, 4, 5, etc.

3.      After each multiplication, check to see if the multiple of the larger number is also a multiple of the smaller number. If it is, you have found the least common multiple (LCM).

Example:  Find the LCM of 6 and 8

8 is the larger number so   1 x 8 = 8.   Is 8 a multiple of 6?   No.

           2 x 8 = 16. Is 16 a multiple of 6? No.

           3 x 8 = 24. Is 24 a multiple of 6? Yes, because 4 x 6 = 24!

Therefore 24 is the LCM of 6 and 8.

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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Math Tip #10

Rules for Finding the Greatest Common Factor (GCF)

A factor is a number that divides into a larger number evenly (no remainder). The greatest common factor of two or more numbers is the largest number that is a factor of each number.  These rules are helpful for reducing fractions and factoring polynomials.

One Way to find GCF-Listing Factors:

1.      List all the factors (the numbers that divide evenly) of the first number.

2.      List all the factors of the second number.

3.      Circle the largest factor that appears in both lists. You have found the GCF.

Example: Find the GCF of  12 and 42.

1.      Factors of 12: 1, 2, 3, 4, 6, 12

2.      Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

3.      GCF of 12 and 42 = 6

Another Way to Find GCF-Expressing the Prime Factorization of Each Number

1.      Factor each number into its primes. (See Math Tip #3)

2.      Circle the factors (by pairs) common to each number.

3.      The GCF is the product of the numbers which are circled

Example: Find the GCF of 12 and 42

12 =  2  x  2  x  3 
42 = 2          x  3  x  7

        2         x   3 =  6

GCF of 12 and 42 = 6

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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Math Tip #9

Divisibility Rules

These rules can help you decide what number divides evenly (no remainder) into another. This is useful when you are finding common denominators or reducing fractions or in algebra when you want to factor polynomials. Some of these rules are short, easy to remember and therefore useful. Some, 7 and 11,  are too long and complicated to be really useful.

A number will divide evenly by:

2, if it is an even number which ends in 2,4,6,8,0

3, if the sum of the digits of the number can be divided by 3

4, if the number is even and the last two digits of the number can be divided by 4.

5, if the number ends in 0 or 5

6, if the number is even and the sum of its digits can be divided by 3

7, this definition is not that useful, but here it is: if you can drop the ones’ digit and      subtract 2 times the ones’ digit from the remaining number. If that answer can be    divided by 7, the original number can be divided by 7.

8, if the number formed by the last 3 digits of the number can be divided by 8

9, if the sum of the digits can divide by 9

10, if the number ends in 0

11, this definition is even less useful than the one for 11: add the alternate digits, starting   with the first digit. Then add the alternate digits starting with the second. Subtract the   two. If their difference can be divided by 11, the original number can be divided by 11.

12, if the number can divide by both 3 and 4.

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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Math Tip #8

Other Types of Numbers

The following types of numbers are less well known, but interesting none the less.

Automorphic Numbers: numbers that when raised to a power end in the original number.  5² = 25;   5³ = 125; 6² = 36

Cute Numbers: numbers that have exactly four factors.  6-factors are 1, 2, 3, 6;
8-1, 2, 4, 8.

Fibonacci Numbers: numbers of the sequence 1, 1, 2, 3, 5 … Do you see the pattern?
Each successive number is the sum of the two preceding numbers. 1 + 1 = 2; 1 + 2 = 3
2 + 3 = 5. What are the next three Fibonacci numbers?

Random Numbers: numbers that are obtained without any pattern. For example, picking numbers from a hat, such as 5, 3, 7, 9, 8, 8, 2, 1, 6.

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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Math Tip #7

Amicable numbers are interesting, but weird. They always come in pairs. Each number of the pair has factors that (not including itself) add up to equal the other number of the pair.

For example, 220 and 284 are amicable numbers. Why? because the factors of 220, excluding 220, are 1,2,4,5,10, 11, 20 22, 24, 55 and 110. 1+ 2 + 4 + 5 + 10 + 11 + 20 + 22 + 24 + 55 + 110 = 284. The factors of 284, excluding 284, are 1, 2, 4, 71 and 142. 1 + 2 + 4 + 71 + 142 = 220.

You can imagine someone stayed up very late at night figuring out this particular category! In fact, so far, 1000 amicable numbers have been found. Can you find another pair?

Here are a few more to help you out:

220 and 284

1 184 and 1 210

2 620 and 2 924

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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Math Tip #6

More on Perfect Numbers

1.      The following formula will calculate a perfect number:

             2^p-1(2^p-1) where p and (2^p-1) are prime numbers.
             For example, to find the perfect number 6, the formula would look like this:
            2¹(2²-1)= 2(4-1) = 2(3) = 6.

2.      Perfect numbers have no practical use, but mathematicians still love them.

3.      No one has found an odd perfect number, and nobody has proven whether or not an odd perfect number exists.

4.      So far only 30 perfect numbers have been found. The thirtieth perfect number has

130 099 digits which won’t fit on this page!

5.      Here are  the first eight perfect numbers:

Perfect Number                 Formula

6                                         2¹(2²-1)

28                                                                                                2²(2³-1)

496                                    2⁴(2⁵-1)

8 128                                 2⁶(2⁷-1)  

33 550 336                        2¹²(2¹³-1)

8 589 869 056                    2¹⁶(2¹⁷-1)

137 438 691 328                  2¹⁸(2¹⁹-1)

2 305 843 008 139 952 128      2³⁰(2³¹-1)

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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Math Tip #5

Abundant, Deficient and Perfect Numbers

This tip is for those of you who want to go beyond usual math courses. The ancient Greeks spent a lot of time thinking about numbers and putting them into categories.

They classified all natural numbers as abundant, deficient or perfect.

 Abundant: a number that is less than the sum (added total) of its factors, not including itself. For example, 12. Its factors (excluding itself,12) are 1,2,3,4,6.

1 + 2 + 3 + 4 +  6 =16.  12 < 16.

Deficient: a number that is greater than the sum of its factors, not including itself. For example, 4. Its factors (excluding itself, 4) are 1, 2. 1+2 = 3.   4>3.

Perfect: a number that is equal to the sum of its factors, not including itself. For example,

6. Its factors (excluding itself) are 1,2,3. 1 + 2 + 3 = 6.  6 = 6.

Can you classify other numbers? Can you find the next perfect number? Watch this space next week for more information on  perfect numbers.

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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Math Tip #4

Squares and Cubes

To square a number, multiply it by itself.
For example, 3 x 3 = 9 and say 3 squared is 9.

To cube a number, multiply it by itself three times.
3 x 3 x 3 = 27 and say 3 cubed is 27.

Here are some examples:

Number            Squared                Cubed

1                                  1                                  1

2                                  4                                  8

3                                  9                                  27

4                                  16                                64

5                                  25                                125

6                                  36                                216

7                                  49                                343

and so on

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

Are prime numbers the best numbers? The most tasty? Are they the expensive numbers-like prime rib? As you have probably noticed math definitions of words are quite different than our everyday meanings of words, and prime numbers are a good example of that difference.  A prime number is an integer (positive or negative whole number) greater than 1 whose only whole number factors are itself and 1. To say it another way, a prime number divides evenly (no remainder) ONLY into itself or 1.

Here are some prime numbers:

2,3,5,7,11,13,17,19,23,29,31,37,41,43 . . .  Can you find more?

And what do you call all the other numbers that have more than two factors?

Composite Numbers:  4,6,8,9,10,12,14,15,16,18 . . .

And what about 1? It is neither prime nor composite. Why?

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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Math Tip #2

Cardinal and Ordinal Numbers

When you count, or talk about a total number of items, you are using cardinal numbers.
For example, six or ten.
You use ordinal numbers to show order. For example, sixth or tenth.
Here are some examples:

Math Tip #1

The Real Numbers

Are those the numbers we can touch? Are they the opposite of the unreal numbers? Real numbers are all the numbers we use. Some of them may seem more familiar than others. For example:

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