Passy's World of Mathematics

Fractions Equations

Posted on October 16, 2011 by Passy

$Pic of wooden dude with fractions$
Image Source: http://sp.life123.com

Fractions Equations usually involve dividing, that is represented using numerator/denominator notation, instead of using a divided by sign.

We can solve Fraction Equations using Back Tracking Flowcharts.

In our example equation: n/3 + 4 = 7 we need to do the opposites of /3 and + 4 in SAMDOB or SADMEP order.

This gives the solution of n = 9 as shown below.

We need to be very careful doing Fractions equations.

Sometimes there are minor differences between questions, and these differences are very important.

Let’s look at how n + 4 / 3 = 7 is different to n/4 + 3 = 7

The equation actually does follow BODMAS order, it is just that this is a bit hidden from view.

Here is how we solve n + 4 / 3 = 7 using a Back tracking Flowchart:

We can see two fraction equations that looked almost the same, are in fact quite different.

Fractions With Negative Values

Solving Fraction Equations which contain negative values requires a certain type of understanding.

We need to understand that a subtraction sign in front of an Algebra fraction is the same as dividing by a negative number.

The following example shows how to do a typical negative subtraction equation that involves an Algebra Fraction.

Here is the Back Tracking Flowchart solution for our equation.

That’s it for Fraction Equations.

As long as you are careful and follow the required steps, getting the correct answers for these should be achievable. Remember that “Practice makes Perfect”, and so make sure you do quite a few of these questions so you can get good at them.

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Equivalent Equations

Posted on October 15, 2011 by Passy

Image Source: http://static.guim.co.uk

The above containers are not all equal, but they are “Equivalent”.

They are “equivalent”, because they all contain milk !

In Algebra, we can have “Equivalent” Equations, that look different in size and complexity, but all contain the same number answer for their unknown variable letter.

Equations which have the same solution value are said to be Equivalent.

2N = 6,
2N + 1 = 7
8N = 24
2N/3 = 2
2N – 5 = 1

all have the same solution of N=3.

These five equations are Equivalent.

We can create each of the above equivalent equations, by starting with the basic equation 2N = 6.

Equation 1: 2N + 1 = 7

2N = 6 (+1 both sides)

2N + 1 = 7

Equation 2: 8N = 24

2N = 6 (x 4 both sides)

8N = 24

Equation 3: 2N/3 = 2

2N = 6 (Divide by 3 both sides)

2N/3 = 2

Equation 4: 2N – 5 = 1

2N = 6 ( -5 both sides)

2N – 5 = 1

Equivalent Fraction Equation

A Fraction Equation that has a solution of N=3
is the equation (N + 5) / 2 = 4

Because we are building an equivalent Fraction equation, the mathematics is a little bit more involved.

The fraction equation could be built from our basic starting equation of 2N = 6, by applying the following two operations:
+ 10 both sides, then divide both sides by 4

2N = 6 (+10 both sides)

2N + 10 = 16 (now /4 both sides)

2N/4 + 10/4 = 16/4 (Reduce down the fractions)

n/2 + 5/2 = 4

(N + 5)/2 = 4

Why Make Equivalent Equations ?

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

Equivalent Equations are useful when we want to “scale up” or “Scale down” a situation.

Let’s have a little “Pizza Party”.

We want each person to have three pieces of Pizza, because this should be the “perfect” amount for them without them over-eating, or going hungry.

Let “N” = the number of pieces per person.

Our solution for each person is N = 3 pieces of Pizza.

How many Pizza pieces do we need for different numbers of people at our little “Pizza Party” ?

If we only have two people, then the equation is 2 x N = 2 x 3 = 6 pieces total.

If we have 8 people, then the equation is 8 x N = 8 x 3 = 24 pieces total.

What if we have two people, but we would like an extra piece left over to give to our older sister (who is on a diet!) .

Then our equation is 2N + 1 = 2 x 3 + 1 = 7 pieces total.

Notice how these situations reflect our previous example Algebra Equations:

2N = 6, 8N = 24, and 2N + 1 = 7.

Equivalent Equations in the Real World

Catering Foods
Image Source: http://www.mycaterer.com.au

Extending our “Pizza Party” example, we could have someone with a Catering business who needs to know how many food items to have avaialble for a function.

For example, they might be supplying “finger food” consisting of mini pizza slices, spring rolls, mini quiches, and so on. For each item they would need to have an estimate of how many pieces to supply for person.

They do not want to supply to little food and leave patrons hungry, but they do not want to have too much food left over at the end that is then wasted.

A set of Equivalent Equations programmed into an Excel Spreadsheet, or even into an iPhone App, could help them work out exactly how much food to provide for 40 people, versus 400 people, and so on.

Something to always keep in mind is it might be useful to add a 10% extra “contingency factor”.

Eg. Have some extra food available to serve, just in case a tray of Spring Rolls all got accidentally burned in the oven, or dropped by a waiter, etc.

Another example of Equivalent Equations might be if you were working for the “World Health Organisation” or the “Red Cross” and need to supply Vaccine to villages and towns to fight a disease epidemic.

Image Source: http://www.todaymednews.org

It might be that Children require and injection of 1ml of vaccine, while Adults need 2ml each.

Again a set of equations in a Spreadsheet or computer App could help work out how much Vaccine needs to be taken to a small Village of 20 adults and 8 children, and how much needs to be taken to a Town of 350 Adults, and 100 children.

Any activity involving dividing up items and distributing them could use Equivalent Equations.

Activities where items are allocated to jobs, such as Building, Plumbing, and Electrical work also require scaling up or scaling down depending on the job.

Trades persons can either use some Equivalent Equations mathematics, or else end up with a big shed full of expensive leftovers.

Image Source: http://thomassawmill.com

That’s it for Equivalent Equations.

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Passy

Posted in Algebra, Equations, Math in the Real World | Tagged Algebra, Algebra Equations, balancing equations, Equations, equivalent equations, how to do equations, how to solve equations, math equations, solving equations, solving linear equations, solving two step equations | 12 Comments

Balance Beam Equations

Posted on October 14, 2011 by Passy

$Colorful maths balance beam$
Image Source: http://www.eduplace.com

Balance Beams are often used as a physical representation of Algebra Equations.

Balancing Equations Video

The following video does a great job of showing how to balance equations:

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

Interactive Online Activity

This interactive activity from BBC shows how simple equations can be solved using a balance beam scale.

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

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

Balancing Equations Videos

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

[youtube http://www.youtube.com/watch?v=GSaA358cnpE]

Here is a video about using the Balance Beam concept to solve a basic Two Step Equation.

[youtube http://www.youtube.com/watch?v=NbrH8VkFGUk]

Here is a similar video that involves negative numbers in equations.

[youtube http://www.youtube.com/watch?v=Kju3n32PYpU]

Here is a slightly different kind of balancing problem, called a “Pan Balance” problem.

[youtube http://www.youtube.com/watch?v=vbX83p0xJ9c]

Balance Beam Worksheet

There is a good worksheet at the link below. It starts off with simple questions, and then becomes progressively more challenging.

Click Here for Balance Beam Questions Worksheet

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

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/math-games/poddle-weigh-in/

Algebraic Reasoning

Here is an online puzzle Game that involves working out how much one item is, and then using this information to work out a second item.

The game can be played at the following link.

http://www.mathplayground.com/algebraic_reasoning.html

That’s it for Balance Beams and Equations.

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Posted in Algebra, Equations | Tagged Algebra, Algebra Equations, balance beam equations, balance pan equations, balancing equations, balancing equations games, Equations, how to do equations, how to solve equations, math equations, solving equations, solving linear equations, solving two step equations | 16 Comments

Two Step Equations II

Posted on October 7, 2011 by Passy

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.

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

When we get dressed for school we put on our socks and then our shoes.

When we change into our Sports uniform, we reverse this process and take off our shoes and then take of our socks.

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.

Solving Two Step Equations Using Opposites

Using this concept, We can solve two step equations without drawing Flowcharts.

We still follow the same basic mathematical method, which involves these steps.

1) Work out what operations have been done to the variable letter
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

Here is a summary of the six step process.

Step 1 – Operations on Variable Letter

The first step involves working out the the operations that are applied to our letter variable.

Let’s do an example of 2N + 5 = 11 as our equation to solve.

1) The operations on N are + 5 and x2.

Step 2 – Put Operations into Order

The “Order of Operations” is called “PEMDAS” in the USA, and “BODMAS” in Australia. Other countries have very similar sets of rules.

For our example: 2N + 5 = 11

1) Operations are + 5 and x2
2) BODMAS or PEMDAS for these is x2 then + 5

Step 3 – Determine Opposite Operations

The third step involves working out the “Opposite” operations.

We can do this using the following table.

For our example: 2N + 5 = 11

1) Operations are + 5 and x2
2) BODMAS or PEMDAS for these is x2 then + 5
3) Opposites are /2 and -5

Step 4 – Opposite Operations in Reverse Order

We take our opposites from the previous step, and put them into reverse BODMAS or PEMDAS order, which means they need to go into SAMDOB or SADMEP order.

For our example: 2N + 5 = 11

1) Operations are + 5 and x2
2) BODMAS or PEMDAS for these is x2 then + 5
3) Opposites are /2 and -5
4) SAMDOB or SADMEP order is -5 then /2

The first four steps we have done can be summarised as follows:

Step 5 – Apply Opposites to the Equation

In this step we apply each opposite, in SAMDOB or SADMEP order, one by one to both sides of the equation.

On the left side, operations should cancel out, and eventually simplify to give the letter on its own.

For our example: 2N + 5 = 11

1) Operations are + 5 and x2
2) BODMAS or PEMDAS for these is x2 then + 5
3) Opposites are /2 and -5
4) SAMDOB or SADMEP order is -5 then /2

Shown below is the working out we do to complete Step 5.

We have applied the Step 4 operations to both sides of the equation, and thereby obtained the solution answer.

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.

Solving Two Step Equations – Examples

Here are some examples for you to try.

Simply complete the missing items on each of the items below.

Here is how we apply the Step 4 operations to both sides of the equation, and thereby obtain the solution answer.

Here is an example that involves a fraction variable, which means Division is occurring.

Here is how we apply the Step 4 operations to both sides of the equation, and thereby obtain the solution answer.

This final example has Brackets in it, which must be done first, regardless of what operation is inside them.

Here is how we apply the Step 4 operations to both sides of the equation, and thereby obtain the solution answer.

Videos About Solving Two Step Equations

Here is a video about solving two step equations.

[youtube http://www.youtube.com/watch?v=juG-iIuTJQE]

Here is a video which goes all the way through solving equations, from one step equations to two step equations.

[youtube http://www.youtube.com/watch?v=w7WntVJQgEY]

Here is another video that ties one step equations in with two step equations really well.

[youtube http://www.youtube.com/watch?v=pNrBhfAL-iA&w=540&h=396]

The following video involves solving two step equations that have positive and negative Integer numbers in them.

[youtube http://www.youtube.com/watch?v=HyUiYeLee3g]

This final video shows some challenging Equation questions that involve Fractions.

[youtube http://www.youtube.com/watch?v=vk4CP10bys0]

Presentation on Solving Two Step Equations

[slideshare id=808199&doc=solving-a-linear-equation-1228192149892836-8]

Two Step Equation Game

“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 by clicking the following link.

Click here to play Equations Millionaire

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Passy

Posted in Algebra, Equations, Equations Two Step | Tagged Algebra, Algebra Equations, Equations, how to do equations, how to solve equations, linear equations, math equations, solving linear equations, solving two step equations, two step equations | 14 Comments

Two Step Equations I

Posted on October 4, 2011 by Passy

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

In this lesson we look at solving two step equations using back tracking flowcharts.

A typical two step equation is 2N + 5 = 11

We can use “Trial and Error”, (also called “Guess Check and Improve”), to work out the number value of “N” which gives us an answer of 11.

Trial and Error, or guessing answers, is fine for simple equations.

But when we have more difficult equations like 7k – 110 = 2, it could take many guesses until we get the answer.

Consequently, we generally use a different method for solving two step equations which is called “Flowcharting” or “Back Tracking”.

Here are the main steps involved with Back Tracking using Flowcharts:

The first step involves working out the order of the operations that are applied to our letter variable.

The “Order of Operations” is called “PEMDAS” in the USA, and “BODMAS” in Australia. Other countries have very similar sets of rules.

In 2N + 5 = 11 there 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.

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:

Here is a partially completed Flowchart for an equation that has multiplying and subtraction in it.

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.

This equation has “Brackets” (or “Parenthesis”) in it, and whatever operation is inside these must be done first.

The required operations are – 3 then x 2, and not the usual 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).

In every Two Step Equation Forward Flowchart:

The letter variable always goes on its own in the first box

The whole left hand side of the equation always goes in the third and final box.

The Operations that were done on the variable letter go onto the arrows, in BODMAS or PEMDAS order.

Back Tracking Flowcharts

Let’s now look at how we can add more boxes and arrows onto our Forward Flowchart, to make a full “Back Tracking Flowchart”. This flowchart will solve the equation for us.

We add extra boxes under each of our Forward boxes, and also add back tracking arrows onto our diagram.

Whenever we make a Back Tracking Flowchart, we always create the same overall structure.

This may look very complicated at first, but things will become much clearer when we do some number examples.

The following “Opposite” operations are always used for solving equations.

Let’s return to our first example equation: 2N + 5 = 11, and see how its solution of N=3 can be worked out using Back Tracking.

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.

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.

Video About Back Tracking

Here is a video about using Back Tracking to solve for unknown values. It flows along quite quickly, so you may need to pause it every now and then to review what is happening.

[youtube http://www.youtube.com/watch?v=w49gh915Cko]

Back Tracking Two Step Equations Summary

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.

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If you would like to submit an idea for an article, or be a guest writer on our blog, then please email us at the hotmail address shown in the right hand side bar of this page.

Please state in your email that you wish to obtain the free subscriber copy of the “Two Step Equations Flowcharts” PowerPoint.

Enjoy,
Passy

Posted in Algebra, Equations, Equations Two Step | Tagged 2 step equations, Algebra, Equations, equations back tracking, equations flowcharts, how to do equations, how to solve equations, linear equations, solving 2 step equations using Flowcharts, solving algebra equations, solving equations using Flowcharts, solving two step equations using Flowcharts, Solving using back tracking, two step equations | 14 Comments

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