Balancing Equations


Algebra Equations are used for working out unknown amounts in many real world situations.

For example we might need to know how much deposit do we need to put on a new car so that we can pay it off at $150 a month in under five years.

We might have an 8 point Basketball Average so far this season, and we want to know what our average would go up to if we scored an incredible 28 points in our next game.

These type of situations require the creation and solving of Algebra equations.

The following BBC Maths Animation interactive gives a good introduction to this concept and shows how simple equations can be solved. The mini movie is followed by questions that you can do


Interactive Online Activity




BBC Balance Scale Link
BBC Balance Scale Link


The following BBC Maths Animation gives a good introduction to this concept. The mini movie is followed by questions that you can do

It has a short introductory video, followed by some equations questions that you can try doing.

http://www.bbc.co.uk/schools/ks3bitesize/maths/algebra/equations1/activity.shtml

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.

Here is a short video about solving a One Step Equation using a Balance Beam.






  • 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?

An equation behaves like a pair of balanced scales. The scales remain balanced as long as we do the same thing to both scales.

  • 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

  • EXERCISES

  • The link below has a series of balancing exercises. Do these exercises in your math book.Here are the answers to the first three questions on the worksheet.
  • Q1) 7, 6, 6, 2, 7, 5.5, 2, 7
  • Q2) 11, 8, 13, 5
  • Q3a) Circle = 6, Rectangle = 8
  • Q3b) Circle = 5, Diamond = 2.

  • Online Activities and Games

  • Try the following games to get a better idea of this concept.Balancing game
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  • Cyberchase
    Cyberchase
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  • Poodle Weigh In
  • Balance Beam Poodles
    Balance Beam Poodles
  • 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


The following powerpoint has an explanation of simple Equations.

This suggests that to solve an equation, we can do the same thing to both sides of an equation. That is:

  • 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.



But lets begin on something not so hard.

The following Slideshare presentation goes through everything we need to know about setting up and solving One Step Addition Equations.

  • This is further explained in the following video

  • So what defines an equation in math anyway? What is an expression? Watch this you tube presentation for a full explanation.



Try any of the following worksheets on addition equations



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 opposite of addition is subtraction
The opposite of subtraction is addition
The opposite of multiplication is division
The opposite of division is multiplication.


The following video explains the steps involved in addition and subtraction equations.

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

Add Example PPT Slide
Add Example PPT Slide

Subtract Example PPT Slide
Subtract Example PPT Slide


Try the following subtraction exercises.
  • This PDF file has a mixture of one step addition and subtraction equations to do.


One Step Equations Online Tests

The following question generator from Cool Math enables you to do as many practice questions as you like, and supplies answers at the click of a button.

http://www.coolmath.com/crunchers/algebra-problems-solving-equations-1.htm

http://www.coolmath.com/crunchers/algebra-problems-solving-equations-2.htm

Here is an online Quiz from BBC Maths on One Step Equations.

BBC One Step Equations Online Quiz

Here is an online quiz from Khan Academy that involves working out fraction answers for multiplication equations.

http://www.khanacademy.org/exercises?exid=linear_equations_1



Equations involve using backtracking. Watch the youtube video for an initial overview of the process of backtracking.


  • 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
    Multiply Example PPT Slide
    Multiply Example PPT Slide


    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.
  • Here is a more comprehensive video, that includes using a balance beam to represent the equation and solve it.
  • Example


    Solve the equation 8x = 72.

    Solution:

    external image Image12986.gif


  • 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.

    Division Example PPT Slide
    Division Example PPT Slide

    Here is a great video all about Division Equations.
  • 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 external image Image12988.gif Solution: external image Image12989.gif

    Remember:

    • 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.
BBC Introduction to equations

  • 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

Equations Games


BBall Game
BBall Game

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
Battleship One Step Equations
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
Algebra Planet Blaster
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
Ducks Balanced Equations
Ducks 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
Equation Match Picture Puzzle
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 Basketball Game
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
Equation Buster Game
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
Equation Millionaire

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
Equation Solver
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
Equation Substitution Match
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
Interactive Equation Balancer
Interactive Equation Balancer

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

Solve Equations Time Trial
XP Eqns Online Time Trial
XP Eqns Online 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
Softschools add numbers
Softschools add numbers

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.

Now look at the following power point . Complete each activity.


  • Math Game: Guess My Number


  • Objective: To undo a sum to solve for the starting number.

  • Number of Players: 2.

  • Materials required: Two pens, two sheets of paper.

  • How to Play: The two players take it in turn to state a sum. One player is stating the sum, the other is undoing it to solve it.

  • Step 1:
  • Pick a number between 1 and 10.
  • Say out loud to the other player "I pick a number...but do not tell it. Write that number on a sheet of paper.
  • Step 2:
  • Decide what to do to with it. You could decide to double it, "then say I double it". Write the double of your previous number on your page (unseen by your partner) so you do not forget. You may then choose to add or subtract another number. For example you decide to add 5. Say I now add 5.Work out the answer and write it down. Your partner writes down the operation performed on the number eg +5. Do 5 separate math functions to your original number. Call out each one as you go so your partner can write it down.
  • Your partner will have on their sheet of paper an equation without the starting number. You will have an equation with both a starting number and the solution.
  • Step 3:
  • Once you have finished performing all the operations on the original number. Say "What was my starting number?" Tell your partner the final number.
  • The job of your partner is to UNDO the equation until the starting number is revealed.

  • The Winner: Score can be kept in this game. You can devise pretty much any system you like, but a simple point for each correct guess is the easiest way.

  • The process being employed here is called backtracking and is an essential foundation for equation solving, which is also heavily linked to algebra.