## 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 Multipl****e
****(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 = 2^{1}
x 3^{1
}

8 = 2^{3}

3. 3^{1 }x
2^{3 }=
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 Multipl****e
****(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. 5^{2} = 25;
5^{3 }= 125; 6^{2} = 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:

2^{1}(2^{2}-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 2^{1}(2^{2}-1)

28
2^{2}(2^{3}-1)

496
2^{4}(2^{5}-1)

8 128
2^{6}(2^{7}-1)

33 550 336
2^{12}(2^{13}-1)

8 589 869 056 2^{16}(2^{17}-1)

137 438 691 328
2^{18}(2^{19}-1)

2 305 843 008 139 952 128
2^{30}(2^{31}-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.