Equations

=EQUATIONS= ** Solving an Equation ** > An equation is a mathematical statement that has two expressions separated by an equal sign. The expression on the left side of the equal sign has the same value as the expression on the right side. > > > ====Solving an equation means manipulating the expressions and finding the value of the unknown. In math this is called a **variable**. Examples of variables are letters such as x,b,c,==== > > An equation might be: x = 4+8. To solve this equation we would add 4 and 8 and find that x = 12.

> An equation has two expressions separated by an equal sign. The expression on the left side of the equal sign has the same value as the expression on the right side. For example 3 + 5 = 8. Both sides equal 8. =The left side always equals the right side.= > In equations, one or both of the expressions may contain variables (a pronumeral). Solving an equation means manipulating the expressions and finding out the value of the variables. > Example : **x - 3 = 5. What is x?**

> To keep an equation equal, we must do exactly the same thing to each side of the equation. If we add (or subtract) a number from one side, we must add (or subtract) that same number from the other side. > To get x on its own in the above equation we need to add +3. To solve the equation above we would add 3 to both sides. The equation would become: > x - 3 + 3 = 5 + 3. This becomes x = 5 + 3 or x = 8 > The following Slideshare presentation goes through everything we need to know about setting up and solving One Step Addition Equations. media type="custom" key="10732108" This is further explained in the following video > media type="youtube" key="hiqwzRYK27c" height="360" width="640" >> >> ==So what defines an equation in math anyway? What is an expression? Watch this you tube presentation for a full explanation.== media type="youtube" key="3j-fa18Y9a8" height="360" width="640"

Try any of the following worksheets on addition equations
[|equations worksheet 1.pdf] [|equations worksheet 1.pdf] [|equations sheet 3.pdf] [|worksheet 4.pdf]

=== For equations which have a number ADDED to a letter, we SUBTRACT away that number to find out the value of the letter “variable”. === === For equations which have a number SUBTRACTED from the letter, we ADD that same number to both sides of the equation. This will allow us to find out the number value of our letter “variable”. ===

One step equations require one “opposite” operation to be performed on them, which then allows us to obtain the value of their unknown variable.

The different types of one step equations and their opposites are shown below.

= Here is a reminder of how to do Addition and Subtraction Equations. =


 * [[image:http://www.passyworld.com/passyImagesTen/AddExample540x407JPG.jpg caption="Add Example PPT Slide" link="@http://www.passyworld.com/passyImagesTen/AddExample540x407JPG.jpg"]] ||


 * [[image:http://www.passyworld.com/passyImagesTen/SubExample540x407JPG.jpg caption="Subtract Example PPT Slide" link="@http://www.passyworld.com/passyImagesTen/SubExample540x407JPG.jpg"]] ||

Try the following subtraction exercises. [|integers-subtraction-1.pdf] [|integers-subtraction-2.pdf]  [|integers-subtraction-3.pdf]  [|integers-subtraction-4.pdf] This PDF file has a mixture of one step addition and subtraction equations to do. [|Equation Problems.pdf]

> Solving Multiplication Equations > If our variable letter has a number directly in front of it, then this means it is multiplied by that number. Eg. 3m means 3 times m or 3 x m or 3.m The Opposite of Multiplication is Division. To solve a multiplication equation, we DIVIDE BOTH SIDES by whatever number is in front of our variable letter. Here is an example of how to solve a typical Multiplication
 * [[image:http://www.passyworld.com/passyImagesTen/MultExample540x404JPG.jpg caption="Multiply Example PPT Slide" link="@http://www.passyworld.com/passyImagesTen/MultExample540x404JPG.jpg"]] ||

The inverse operation of × is ÷. So, to solve an [| equation] involving multiplication, we divide both sides of the equation by the same number. The following short video shows how to do a Multiplication Equation. media type="youtube" key="bjlyUV9yK6E" height="315" width="560"

Solving Division Equations If our variable letter has a number directly under it as a fraction, then this means it is divided by that number. Eg. k/2 means k is divided by 2. The Opposite of Division is Multiplication. To solve a Division equation, we MULTIPLY BOTH SIDES by the number that our variable letter is being divided by. Here is an example of how to solve a typical Division Equation. The inverse operation of ÷ is ×. So, to solve an [| equation] involving division, we multiply both sides of the equation by the same number. Example 6 Solution: =Remember:= > [|BBC Introduction to equations] >> [|equationsolvingintro_tcm4-123391.ppt] = = =TWO STEP EQUATIONS= when we have more difficult equations like 7k – 110 = 2, we generally use a different method for solving two step equations which is called “Flowcharting” or “Back Tracking”. Back Tracking involves reversing the equation to find out the value of the unknown variable.
 * An equation is a statement that contains an equal sign.
 * To solve an equation, we do the same thing to both sides of the equation.
 * The same number can be **subtracted** from both sides of an equation.
 * The same number can be **added** to both sides of an equation.
 * Both sides of an equation can be **divided** by the same number.
 * Both sides of an equation can be **multiplied** by the same number.
 * Click on the following links. There are various activities and explanations on equations for you to work through.The first activity Introduction to Equations is a great introduction activity.
 * =**REVISION AND OVERVIEW :EQUATION POWER POINT**=
 * Watch the following power point presentation. Complete each of the exercises in the slides.The answers are given for each. There are various types of equations for you to try as well as two step equations. If you are feeling confident continue to work through the other slides



Image Source: http://www.esquire.com

When we park a car, there are two main steps: backing and turning in, then straightening up.

When we leave the parking bay, we REVERSE these steps by turning out, going forward, and entering back onto the roadway.

We do the Opposites in the Reverse Order.

The same concept applies to equations.

To build an equation, operation steps have been done to a letter variable.

To solve an equation we do the opposite steps in reverse order.

This is called BACKTRACKING
Watch this video About Back Tracking media type="youtube" key="w49gh915Cko" height="360" width="640"

=Lets look at this equation: 2N + 5 =11.=

The first step to solving the value of N involves working out the order of the operations that are applied to it.

= In this equation t here is adding of 5, and also multiplying by 2.= The Order of Operations tells us that multiplying by 2 happens before the adding of 5.

N is times 2 as the 2 is in front of the letter. N also has 5 added on.

To make a Forward Flowchart to represent “N”, and the multiplying by 2 followed by the adding of 5, we need to draw three rectangles. In between the rectangles we place two arrows, because this equation has two operations (x2 and +5) performed in it. The resulting Forward Flowchart looks like this: The final result of this equation using backtracking should look as shown below. Checking Solutions Using Substitution We can always check the solution to any equation by substituting the number answer we obtained back into the original equation. Here is how we can check the N=3 solution for the equation 2N + 5 = 11.

= EXAMPLES = Here are some partially completed Flowcharts for equations. See if you can fill in the missing items. (Answer: The Flowchart requires x3 then -2 and a 3k in the middle rectangle box). Here is another example of a two step equation, but this one has dividing and then adding. See if you can fill in the missing items. (Answer: The Flowchart requires on the arrows divided by 5 or /5, followed

by + 2. x/5 needs to go in the middle box). Here is a third example Flowchart for you to complete.

The final result of this equation using backtracking should look as shown below.

Note: This equation has “Brackets” in it, and whatever operation is inside these must be done first.
The required operations are – 3 then x 2, **and not** x 2 then -3. This happens because the brackets force us to do the – 3 first, whereas without brackets we would do – 3 last. (Answer: The Flowchart requires on the arrows -3 then x 2, and the final box needs to contain the left hand side of the original equation which was 2(a-3) = 8 ). In every Two Step Equation Forward Flowchart:The letter variable always goes on its own in the first box



Back Tracking Examples Here are some examples for you to try. Simply complete the missing items in the Back Tracking Flowchart. You can check your work against the answers that are given in the section which follows these three examples. This next question has divided by in it. Note that we write divided by 5 as /5 when applying division to letter variables. This final example has Brackets in it, which must be done first, regardless of what operation is inside them. Answers for Back Tracking Examples k = (10 + 2) / 3 = 12 / 3 = 4 Middle Box has 12 in it, and left side box has 4 in it. Final Answer is k=4. x = (6-2) x 5 = 4×5 = 20 Middle box has 4 in it and left side box has 20 in it. Final Answer is x=20. a = 8 / 2 + 3 = 4 + 3 = 7 Middle box has (a-3) in it. Top arrow has x 2. Bottom arrows have divided by 2 and + 3 going backwards right to left. Bottom middle box has 4 in it. Final Answer in left bottom box is 7, so a=7. Note that it is okay to replace the / symbol with the “divided by” symbol in your answers for any of these equations, except for letter variables which must be written as x/5, y/3, 2m/5, etc.

We need to set up three double box rectangles, with arrows in between them. The letter variable always goes on its own in the top left hand box The whole left hand “Algebra” side of the equation always goes in the third and final top box. The Right hand side number for the equation always goes in the very right hand bottom box. The Operations that were done on the variable letter go onto the arrows, in BODMAS or PEMDAS order on the top, and in Opposite order on the bottom. The number answer always ends up in the bottom left hand side box. The flowcharts for two step equations are always set up using the structure shown in the following diagram. 1) Work out what operations have been done to the variable letter
 * Back Tracking Two Step Equations Summary **

2) Put these operations into BODMAS or PEMDAS order

3) Work out what the Opposite Operations are

4) Put the Opposite Operations into SAMDOB or SADMEP order

5) Apply the Opposites one by one to both sides of the equation

6) Simplify the final answer

Videos About Solving Two Step Equations

Here is a video about solving two step equations. media type="youtube" key="juG-iIuTJQE" height="360" width="640" Here is a video which goes all the way through solving equations, from one step equations to two step equations. media type="youtube" key="w7WntVJQgEY" height="360" width="640" The following video involves solving two step equations that have positive and negative Integer numbers in them. media type="youtube" key="HyUiYeLee3g" height="360" width="640"

Presentation on Solving Two Step Equations

media type="custom" key="10771342"

=Equations Games=



In this post we present a number of free Algebra Equations Games and Activities that students can use to reinforce their equation solving skills. Simply click on the image of the game, or the provided text link, to open the game in a new window on your web browser. Since most of these games use Flash, Shockwave, or Javascript, they probably will not work on Apple devices. Apple products do not have the functionality to run such applications, but the games should work fine on any normal netbook, laptop, or PC. Battleship One Step Equations This is played just like the classic Battleship game. We click on the opponent’s right hand side grid and get splash circles if there is not a ship on that grid square. However, when there is a ship there, we get given a one step equation to solve. If we get it correct, we get a dot to confirm the hit. If we get it wrong we can try again by clicking back on the dot and re-doing the same equation on our next turn. Note that the game does use negative numbers, and so some questions will look like this: 15 = 5 – x. For this example equation, the correct answer from the multiple choice options will be -10. The game can be played at the following link. @http://www.quia.com/ba/36544.html Algebra Planet Blaster This game will not start unless you first click your mouse into the game area, then the cursor movement and space bar shooter start functioning. The equations are one and two step equations involving both positive and negative numbers. The game only has one level, but restarting the game gives a new set of equations to do. The game can be played at the following link. @http://www.aplusmath.com/Games/PlanetBlast/index.html Balanced Equations In this game, we need to click and drag numbers down from the top and into the right position to create a balanced equation. In a balanced equation, both sides of the equals sign generate the same number. Eg. 10 x 2 = 5 x 4. The game can be played at the following link. @http://funschool.kaboose.com/formula-fusion/games/game_great_equations.html Equation Match Picture Puzzle This game by BBC requires the free Adobe Shockwave player to be installed on your computer. The object of the game is to match up a pair of equations that both have the same Answer. Eg. We could match x-5 = 2 (which has an answer of x=7) with 3x=21 which also has an answer of 7. When we match correctly, two more parts of the underlying image are revealed. The game has levels, where Level 1 appears to only give simple one step equations. Level 3 gives letters both sides and brackets equations. The game can be played at the following link. @http://www.bbc.co.uk/education/mathsfile/shockwave/games/equationmatch.html One Step Basketball Game One-step adding and subtracting game, as well as a one-step multiplication and division game. The equations are challenging, as they use fractions, negative numbers and decimals. If you get a question correct, you get to aim your ball and have a shot at the basket. This game can be played at the following link. @http://www.math-play.com/One-Step-Equation-Game.html There is this exact same game, but as a Two Step Equations Game, at the following link: @http://www.math-play.com/Two-Step-Equations-Game.html Equation Buster Game There are four levels of this game, but each level always has the same equation to solve for that level. Level 1 is always the same single step equation, and Level 4 is always the equation 4w + 2 = 2w – 4. However it is till worthwhile giving this game a go. The idea is to go through the solving steps one by one, and if we reach the answer in the least possible steps we get a double tick on our answer. The main page where levels can be selected is at the following link. @http://www.gamequarium.com/equations.html

Equation Millionaire

“Equation Millionaire” is a game that will challenge your two step equations solving skills.

This game has a mixture of difficulties, ranging from single step with negative numbers, through to brackets equations and fractions. It has a set of three “hints” that are like lifelines, and give clues such as “The answer is not D”. This game can be played at the following link. @http://www.quia.com/rr/4096.html Equation Solver This is more of an interactive online activity, where we can choose the reversing operation to do, type in the value we want to do the operation to and then press enter to get to the next line. Note that we use the red “:” for doing divided by. We can also make up our own equation, type it in, and then solve it. The activity can be found at the following link. @http://www.mathsnet.net/algebra/balance.html Equation Substitution Match This game required us to install the free “Adobe Shockwave Player” add-in to our browser before we could play the game. The game involves substituting into an equation and working out which is the correct answer. It has three levels of difficulty. The game can be played at the following link. @http://www.bbc.co.uk/education/mathsfile/shockwave/games/postie.html Interactive Equation Balancing This activity is really cool. We can click on the purple buttons to add or remove x’s or ones. As we do this, the items are added or removed from both sides of the balance. The idea is to reduce the items on the balance down until we just have one “x” on the balance. The remaining numbers on the other side of the balance tell us what the answer for the value of “x” is. This activity can be found at the following link. @http://www.mathsisfun.com/algebra/add-subtract-balance.html Poodle Weigh In This game involves putting number weights on the balance to match the weight of the strange looking Poodle. Hover the mouse over the bottom right hand corner “Help” button, to get instructions on how to play the game. Hover the mouse over the bottom left hand corner “Hint” button, to reveal the number equation which needs solving. Then click on the number weights to make them go onto the balance and add up to the required answer. To remove a number off the balance, simply click the number on the right hand side of the balance that we want to remove. The game can be played at the following link. @http://pbskids.org/cyberchase/games/algebra/algebra.html Solve Equations Time Trial This game is more of a time trialled Online Test, rather than a game. It focuses on two step equations and includes negative numbers. The game can be played at the following link. @http://www.xpmath.com/forums/arcade.php?do=play&gameid=64 In addition, there are XP Math One Step Equations Time Trials activities at the web pages below. These cover One Step Addition, Subtraction, Multiplication, and Division. @http://www.xpmath.com/forums/arcade.php?do=play&gameid=69 @http://www.xpmath.com/forums/arcade.php?do=play&gameid=68 @http://www.xpmath.com/forums/arcade.php?do=play&gameid=53 @http://www.xpmath.com/forums/arcade.php?do=play&gameid=72 Addition Balance Game This one is really a basic primary school game, and involves working out missing values in an addition sum. However it does the train students to be thinking of the concept of balancing, and is good brain exercise when students push themselves against the timer. The game can be played at the following link. @http://www.softschools.com/math/addition/balance_equations/ That’s it for our selection of Equations Games. These games could be added individually to lessons, or used as a group item when students are revising their work.