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

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.