Integers

Lets talk a little about integers.  An integer is

• the natural numbers {1, 2, 3, 4, 5, ...} or positive integers
• the negative integers {....-5, -4, -3, -2, -1}
• the number 0

Basically an integer is any whole number, positive, negative or zero. So, what happens when we use positive and negative integers with adding, subtracting, multiplying and dividing?  Lets discuss some basic rules.

Adding Integers

• Adding two positive integers will leave you with a positive answer.
•  7 + 6 = 13
• Adding two negative integers will leave you with a negative answer.
•  (-3) + (-9) =  -12
• When adding a positive integers to a negative integer, subtract the smaller number from the larger number and use the sign of the integer with the larger absolute value.
• 9 + (-5) = 9 - 5 = 4 (using the sign from the larger integer which is positive)
• (-9) + 5 = 9 - 5 = 4 = -4 (using the sign of the larger number which is negative)

 Subtracting Integers

• When subtracting two positive integers
• Take the smaller integer away from the larger integer and you will get a positive integer.
• 13 - 4 = 9 
• Take the larger integer away from the smaller integer and you will get a negative integer.  Subtract the larger integer from the smaller integer and add a negative sign.
• 5 - 10 = 10 - 5 = 5 = -5
• Subtracting two negative integers, change the minus and negative sign to a plus and follow the addition rules for adding a positive and negative number.
• (-7) - (-6) = (-7) + 6 = 7 - 6 = 1 = -1
• (-4) - (-8) = (-4) + 8 = 8 - 4 = 4
• Subtracting a positive integer and a negative integer, add the two integers together and use the sign from the integer with the largest absolute.
•  19 - (-4) = 19 + 4 = 23
• (-19) - 4 = 19 + 4 = 23 = -23

Here is a great video explaining how easy it is to add and subtract integers using the rules previously described. It also shows a few other methods for adding and subtracting integers.

Multiplying Integers

• Multiplying two positive integers leaves you with a positive answer.
• 4 x 6 = 24
• Multiplying two negative integers leaves you with a negative answer.
•  (-5) x (-3) = -15
• Multiplying a positive and a negative integer leaves you with a negative answer.
• (-4) x 7 = -28

Dividing Integers

• Dividing two positive integers leaves you with a positive answer
• 24/6 = 4
• Dividing two negative integers leaves you with a positive answer
•  (-9)/(-3) = 3
• Dividing a positive and a negative integer leaves you with a negative answer
• (-25)/5 = -5
• 21/(-3) = -7

Here is another great video explaining how to multiply and divide integers.

I hope this helped you learned a little bit about adding, subtracting, multiplying and dividing positive and negative integer.s

Greatest Common Divisors

What is the greatest common divisor?  By definition, the greatest common divisor is the largest integer that divides without remainder into a set of integers (at least one of which is not zero). An integer is a positive or negative whole number or zero.  Sometimes a divisor is also called a factor so you may hear greatest common factor in place of greatest common divisor.  We write this as GDF or GCD.  Watch this youtube video from Khan Academy to see how to find the greatest common divisor.

Example 1 - Intersection of Sets

The video above is a great example of how to find the GCD using the intersection of sets.  Lets use this information and do an example on our own. Find the GCD of the integers 54 and 81.

First we need to find the divisors, or factors, of 54.

D54 = {1, 2, 3, 6, 9, 18, 27, 54}

and the divisors of 81.

D81= {1, 3, 9, 27, 81}

Next we need to look at the common divisors between 54 and 81.

D54 and D81 = {1, 3, 9, 27}

Looking at the list of common divisors we can see that 27 is the greatest number.  Therefore, the

GCD (54, 81) = 27

Example 2 - Prime Factorization

Another way to find the greatest common divisor is by prime factorization.  Prime Factorization is finding which prime numbers multiply together to make the original number.  Using the same integers above, find the GCD for the integers 54 and 81 using prime factorization.To do this we first need to find the prime factorization for each integers.

54 = 2 x 3 x 3 x 3
81 = 3 x 3 x 3 x 3
 
Next, we write the factorizations, aligning each repeated factor vertically. The GCD is the product of the prime factors that appear in both factorizations.

                   54 = 2 x 3 x 3 x 3
                   81 =       3 x 3 x 3 x 3
GCD (54, 81) =          3 x 3 x 3

so, GCD (54, 81) = 3 x 3 x 3 = 27, which is the same number we found using the intersection of sets method.

A really useful tool for checking your work is the prime factorization calculator.

Example 3 - Euclidean Algorithm

The theorem of the Euclidean Algorithm states,

"Let a and b be any two natural number, with a > b. Divide a by b to get a remainder r.  If r = 0, then b = GCD(a,b), but if r > 0, divide b by r to get a remainder s.  If s = 0, then r = GCD(a,b), but if s > 0, divide r by s to get a remainder t.  Continue the division-with-remainder process until a remainder of 0 results.  Then the last nonzero remainder is GCD(a,b)."

The Euclidean algorithm works well for finding the GCD of larger numbers.  Find the GCD of 5672 and 28468.

28468/5672 = 5 R 108

5672/108 = 52 R 56

108/56 = 1 R 52

56/52 = 1 R 4

52/4 = 13

Therefore, the GCD(5672, 28468) = 4

Algorithms for Adding Whole Numbers

When you were young you were probably introduced to adding by using something concrete that you could see, touch and manipulate, called a manipulative.  One such manipulative is base-ten blocks consisting of units, strips and mats.

Lets try adding 125 and 137.  We can show that with the following mats, strips and units.

1 mat, 2 strips, 5 units





1 mat, 3 strips, 7 units



We could then show this as

 2 mats, 5 strips, 12 units

 

We can simplify the units

 2 mats, 6 strips, 2 units








Looking at the number of mats strips and units we can determine that our answer to 125 + 137 = 262.

Another approach is to use place-value cards.  Each card is labeled ones, tens, hundreds and so on placing the appropriate number of dots under each place value.  A mark in under ones is worth 1, a mark under tens is worth 10 ones, a mark under hundreds is worth 10 tens or 100 ones and so on.  Lets look at the same math problem we used previously.  125 + 137.  This can be shown using place-value cards like this...

Notice that there was a total of 12 dots in the ones column.  We can simplify by moving one dot into the 10s column leaving us with 2 dots in the hundreds, 6 dots in the tens and 2 dots in the ones for an answer of 262.

The instructional algorithm for adding looks like this

   125
+ 137
     12       5 + 7
     50       20 + 30
   200      100 + 100

   262

Which leads us to the final algorithm which we are used to using.

     1
   125
+ 137
   262   

All examples lead us to the same answer, 262.  Young learners start off using manipulative to gain understanding of adding and place-value and move onto the instructional algorithm and then the final algorithm. 

Numeration Systems

The numeration system we are most familiar with is called the Indo-Arabic (or Hindu-Arabic) numeration system, or more commonly known as the decimal system.  The Decimal System is a base ten system.  One of the reason we use a base ten system is because we have ten fingers.  

We write the number two hundred thirty four as 234. 

Tallies (marks on bone, marks on stone, stones in a bag) were the earliest means of keeping count. 1, 2, 3, and 4 are represented by |, ||, |||, and ||||.  Five is represented by four lines and a diagonal line through the four lines.  This chart represents tally marks for the numbers 1 thru 10.  This pattern continues, marking each four lines with a diagonal line to mark another five tallies.

The number 234 would be represented by

The Egyptian System is another numeration system.  This system is a base-10 system. The powers of the base are 1, 10 , 10^2, 10^3….. It is written left to right.

The number 234 would be represented by

The Roman system is a numeric system, called Roman numerals, used in ancient Rome.  Letters from the Latin alphabet are used for number values. You may already be familiar with some Roman numerals since they can be seen on some analogue clocks and watches, the preliminary pages of books before the main page numbering, sporting events such as the Olympics and Super Bowl, or Monarchs such as King Edward VII.

The number 234 would be represented by 

CCXXXIV

Another numeration system is the Babylonian system.  It is a base 60 system. There is a basic symbols for 1 through 59, but to write numbers more than 59 you would use the basic symbols for 1-59 and the concpet of place value. Place value is a power of the base. Their place values are 1, 60, 60^2, 60^3...  There is no This video explains a little about the Babylonian system and place value.

 The number 234 would be represented by (where V = 1 or 60 and < = 10)

v v v      < < < < < v v v v

The Mayan system is a base-20 system.  Numbers after 19 were written vertically in powers of twenty .  The numerals are made up only three symbols; zero (shell shape), one (a dot) and five (a bar).  

 The number 234 would be represented by

Of all the numeration systems used we are most familiar with the decimal system and I think writing two hundred thirty four is easiest using the decimal system.

Venn Diagrams

What is a Venn Diagram?  The Venn diagram was introduced by John Venn.  It is a diagram that shows all possible logical relations between a finite collection of sets.  A Venn diagram consists of overlapping circles. Each circle contains all the elements of a set. Where the circles overlap shows the elements that the sets have in common. Generally there are two or three circles.  Sounds a little complicated and can look that way too!

A 5 set Venn diagram

The intersection of the Greek, Latin and Russian alphabet.

Lets start with a couple definitions and some examples to see if we can make this less complicated.
A set is a collection of objects.
A ∩ B is the intersection of two sets or all items common to A and B.

How are Venn diagrams useful in math? Venn diagrams enable students to organize information visually so they are able to see the relationships between sets of items.

 
Venn diagrams can be used for to compare other sets too.  Lets discuss its use in the life of a College Student.
Let A = enough sleep, B = social life and C = good grades.
Then,
A ∩ B = Slacker
B ∩ C = Zombie
A ∩ C = Nerd
and A ∩ B ∩ C = Defies the Laws of Physics

This can be shown in the following Venn Diagram.


















Lets try another example involving food.

If A = Flour, B = Egg and C = Milk
Then, 
A ∩ B = Pasta
A ∩ C = Batter
B ∩ C = Omelette
and
A ∩ B ∩ C = Pancakes

Problem Solving Strategies

Math.  A little word that strikes fear among so many students.  A jumble of numbers and letters in a problem and you don’t know how to solve it.  Here are some strategies you can use when trying to solve a math problem.

1. Guess and check

2. Make an orderly list

3. Draw a diagram

4.    Look for a pattern

5.    Make a table

6.    Consider special cases

7.    Use a variable/Use two variables

8.    Work backwards

9.    Eliminate possibilities

10. The pigeonhole principle

11. Inductive reasoning

12. Deductive reasoning

Most strategies listed seem pretty self-explanatory.  But what is the pigeonhole principle?  I don’t remember learning about that in school.

By definition, the pigeonhole principle state, if m pigeons are placed into n pigeonholes and m > n, then there must be at least two pigeons in one pigeonhole. 

Example 1

If we have 10 pigeons and 9 pigeon holes, then m=10 and n=9.  Therefore 10 > 9 so there must be at least two pigeons in one pigeonhole.

Example 2

If we have 26 pigeons and 25 holes, then m =26 and n=25.  Therefore, 26 > 25 so again there must be at least two pigeons in one pigeonhole.

Example 3

If we have 7 pigeons and 9 pigeonholes, then m=7 and n=9.  Therefore, 7 < 9 and the pigeons can each have their own hole. 

See, math isn’t so hard!  The pigeonhole principle seems pretty simple to understand.  If you don’t have enough space somebody has to share.  Peter Gustav Lejeune Dirichlet is believed to be the first to formulate this idea.  Click on the links to learn more about Peter Gustav Lejeune Dirichelt and the pigeonhole principle.