Integers+equations

=INTEGERS (DIRECTED NUMBERS): the world of positive and negative numbers=

We can work out an overall score for our day by combining the negative scores with the positive scores. We then cancel out each individual positive, with an individual negative to obtain an overall score.

The following Slideshare presentation shows several other interesting applications of Integers in the Real World.



Adding Integers Using a Number Line Many people like to use a number line for adding Integers. On the number line: Negative numbers involve moving to the left, and positive numbers involve moving to the right. We always start our sum at zero in the middle of the number line, and then move to locate the first number on the line. If our second number is NEGATIVE, we move that many places to the LEFT. If our second number is POSITIVE, we move that many places to the RIGHT.


 * [[image:http://www.green-planet-solar-energy.com/images/integer-number-line.gif caption="Pic of a nbr line from the web" link="@http://www.green-planet-solar-energy.com/images/integer-number-line.gif"]] ||


 * ALWAYS START AT ZERO **


 * POSITIVE MEANS MOVE TO THE RIGHT **


 * NEGATIVE MEANS MOVE TO THE LEFT **

The following You Tube video shows how to add integers by moving along a number line as well as shortcut rules for adding Integers. media type="youtube" key="R3ugqwQugVo?feature=player_embedded" height="360" width="640"

In the following example +3 + -5 = ? on our number line we move to +3 places to the right. We do this because our first number in the sum is +3. Next we move five places to the left, because our second number is -5. We finish up at -2 on the number line, and this is our final answer.



Adding Integers Example 2

The sum: +6 + -4=? The resulting answer is +2 or just 2.



Adding Integers Example 3 In this example we do the sum: -3 + -2



=**Lets look at the Rules**= Watch and work through the following PDF for an overview of the mathematical concepts in directed numbers.

Rules of Addition of directed numbers

 * =**Two 'pluses' make a plus**- so if two '+' signs are written next to each other you can replace them with a single '+' sign.=
 * Thus -3 + (+2) = -3 + 2 = -1


 * =**Two 'minuses' make a plus**- so if two '-' signs are written next to each other, you can replace them with a single '+' sign.=
 * Thus 6 - (-2) = 6 + 2 = 8


 * =**A plus and a minus make a minus**- so if one of each sign sit next to each other, then you can replace them with just a '-' sign.=
 * Thus -4 - (+3) = -4 - 3 = -7 and 3 + (-7) = 3 - 7 = -4

media type="youtube" key="kMvT377Gza4" height="315" width="420"

=**Subtracting Integers**=

Subtracting integers is difficult. This is because it can be hard to understand how we can take away things that already have a minus sign in front of them. =RULE. To subtract an integer, add its opposite.= For example. (-8) – (+9) = The opposite of +9 is –9. Change sign to opposite: (-8) + (-9) = -17 RULE 1 examples: 1. (+7) – (+4) = (+7) + (-4) = +3 2. (+5) – (-6) = (+5) + (+6) = +11 3. (-3) – (+8) = (-3) + (-8) = -11

**1) Change the subtraction operation sign to it's reciprocal - the addition operation sign**  (-8) - (+ 7) = ** becomes (- 8) + (+ 7) = ** **2) Change the operation sign of the integer that follows to its reciprocal.** **(- 8) + (+ 7) = becomes (- 8) + (- 7) = **
 * REMEMBER**:

**Overall **



Integer Addition sums are much easier to do than Subtractions, so we always apply the “Add the Opposite” rule to turn every Subtraction question into an Addition sum.

=**AND ALWAYS** Change double negatives to a positive= = =

=** For example:What about ? -5 - - 2 **= So the answer to our original question is: -5 – -2 = -3.
 * ?**= We can work out the answer to this sum using the canceling zeroes method as follows:

Perhaps the following video will help you remember the rules:
media type="youtube" key="Xxi4ItFpUvw" height="315" width="420" Addition and subtraction problems can be simplified by replacing two operation signs that are next to each other with one equivalent sign. For example, (−6) − (−2) simplifies to −6 + 2, because − (−) is equivalent to +.

+ (+) is equivalent to + − (−) is equivalent to + + (−) is equivalent to − − (+) is equivalent to −

➜ When using a number line, start at the position of the first number and then move left or right the number of units indicated by the second number. The final position is the result of the calculation.

➜ When the sign between the two numbers is +, move to the right (positive direction) along the number line.

➜ When the sign between the two numbers is −, move to the left (negative direction) along the number line. =**Overall: This is summarised in the table below: **= +5+6= || + 5+6=11 || positive || right || +5+-6= || - +5-6=-1 || negative || left || -5+6= || - -5+6=1 || negative || left || -5--6= || + -5+6=1 || positive || right ||
 * || ** OPERATIONS ** || ** EQUIVALENT **
 * OPERATION ** || ** MOVE IN A POSTIVE OR NEGATIVE DIRECTION ** || ** MOVE LEFT OR RIGHT ALONG A NUMBER LINE ** ||
 * ** Adding a positive number ** || + (+)
 * ** Adding a negative number ** || + (-)
 * ** Subtracting a positive number ** || -(+)
 * ** Subtracting a negative number ** || -(-)

Multiplying Using Negative Numbers

But what about negative numbers ?

If we have 2 x -3 then we have two lots of -3 which is -3 + -3 which equals -6.

4 x -3 = ? is four lots of negative 3 = -3 + -3 + -3 + -3 = -12

But what about: -2 x 3 = ? OR: -2 x -3
There is a pattern that Integer multiplications always follow:



Here there are again three simple rules to follow:


 * If two positive numbers are multiplied together the answer is positive.
 * Thus 2 x 4 = 8


 * If two negative numbers are multiplied together the answer is positive.
 * Thus (-2) x (-4) = 8


 * If a positive and a negative number are multiplied the answer is negative.
 * Thus (-2) x 4 = (-8)

So basically, when multiplying numbers, **if your numbers have the same sign the answer is positive, but if the two numbers have different signs the answer is negative**. These Integer Multiplication Rules can be summarised as follows.

The following video will give you an overview of multiplying integers. media type="youtube" key="D3jfoq5HtXc?feature=player_embedded" height="360" width="640"

For an overview watch the video below.

media type="youtube" key="6xKG0E-QwhI" height="315" width="420" = = =Remember:= = When MULTIPLYING : Same Sign Positive, Different Signs Negative. = = = = Dividing Integers =

The rules for dividing integers are similar to the rules for multiplying integers. When you divide two integers with the **same sign**, the result is always **positive**. Just divide the absolute values and make the answer positive.

 **Positive ÷ positive = positive**

 **Negative ÷ negative = positive** When you divide two integers with **different signs**, the result is always **negative**. Just divide the absolute values and make the answer negative.

 **Positive ÷ negative = negative**

 **Negative ÷ positive = negative** These dividing rules can be summarised in the following diagram.



=GAMES: Click on the following links for some games involving integars= [|Speedboat Game] [|Pool Game] = = //Speedboat Negative Numbers// In every trial you will be given two math problems. Click the problem which has a positive result, and your speedboat will be sent to its way. Every such click on a problem with a positive result will accelerate your speedboat more and more.
 * ==Subtracting and Adding Negative Numbers==

Your goal is to pass the other speedboats and finish the track first.

You can choose from three levels of difficilty of problems. ||  || ==Ordering Negative Numbers== //Number Balls// This game will present you with various moving balls which have numbers on them. Some of the numbers are of positive value, and some of negative.

Click the balls in ascending order, from the smallest number to the largest. || Subtracting Integers Car Racing Game This game starts off with us using the left and right arrow keys to drive our car in a race, and then after a little while we have to stop and do an Integer Subtraction question. If we get the question right, we get bonus items like a rocket booster pack to fly our car for a while, until the next question comes. Click on the image below, or the link which follows, to play this game. @http://www.math-play.com/math-racing-subtracting-integers-game/math-racing-subtracting-integers-game.html Subtracting Integers Fish Game In this fun game we have to click the letter that is the correct answer, before the fish swims to this correct answer. Click on the image below, or the link which follows, to play this game. @http://www.slidermath.com/integer/SlamnS.shtml =[|Arithmetic Four]= [|Arithmetic Four]

[] ”Connect 4“ type game for whole number and integer addition, subtraction, multiplication and division. Subtracting and Adding Negative Numbers

=[|Placing numbers on a number line]= [|Placing Numbers]

[] Position numbers on the number line, includes whole numbers, decimals and directed numbers. =[|Positives and Negatives]= [|Positive and Negatives]

[]

This interactive requires you to drag the simplified number to the correct positive/negative calculation in the table. = = =[|XP Math Integers Addition: -9 to +9]= [|Postive Integars]

[] Practise directed number addition.

=[|XP Math Integers Subtraction: -9 to +9]= [|Negative Integars]

[] Practise directed number subtraction.

=[|XP Math Integers Subtraction: -99 to +99]= [|Harder Integars]

[] Practise directed number subtraction, larger numbers.

[|Dice Game]
[] A simple dice game for practising directed number. Option to play the computer or a friend – with offline (paper based) support.

=[|Negative Number Line]= [|Negative Number Line]

[] A numberline that shows negative numbers for the interactive whiteboard. It ranges from -50 to +50 American Football Adding Integers Game This is a fun game that helps us learn how to add Integers. A loss of yards means a negative number, and a gain in yards means a positive number. We click the mouse at the correct position on the number line to show the result of each world problem.

Click on the image below, or the link which follows, to play this game. @http://www.mathgoodies.com/games/integer_game/football.html