Passy's World of Mathematics

One Step Equations

Posted on September 30, 2011 by Passy

Image Source: http://www.ibspro.net

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.

Solving Addition and Subtraction Equations

These have already been covered in separate lessons on our blog.

Click the links below to visit these lessons.

One Step Addition Equations Lesson

One Step Subtraction Equations Lesson

Here are some reminder examples of how to do Addition and Subtraction Equations.

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 Equation.

The following short video shows how to do a Multiplication Equation.

[youtube http://www.youtube.com/watch?v=bjlyUV9yK6E?rel=0]

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

[youtube http://www.youtube.com/watch?v=bnzZJJ6Wfr0?rel=0]

Here is a detailed video from Khan Academy about how multiplication by grouping actually works.

[youtube http://www.youtube.com/watch?v=9Ek61w1LxSc?rel=0]

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.

Here is a great video all about Division Equations.

[youtube http://www.youtube.com/watch?v=kp2a-PaYOWo?rel=0]

The following video covers both Division and Multiplication Equations.

[youtube http://www.youtube.com/watch?v=58mHEQR8GFs?rel=0]

Solving Reciprocal Equations

Equations like 5n/4 = 10 can be solved by multiplying both sides by the reciprocal of 5/4.

The “Reciprocal of a fraction is that fraction turned “upside down”.

Eg. 5/4 has a reciprocal of 4/5.

The following video shows how to do Reciprocal type equations.

[youtube http://www.youtube.com/watch?v=aYJW2BXaXHM?rel=0]

Equations like 5n/4 = 10, can also be solved using two opposite steps such as x 4 then divide by 5.

We will be looking at two step equations in a separate article on this blog.

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

Interactive Online Activity

The following online 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

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Subtraction Equations

Posted on September 29, 2011 by Passy

Image Source: http://upload.wikimedia.org

In the image above, we have photographically reversed the picture of the basketballs. We did this to emphasize that working with equations involves making things the opposite to what they are currently.

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”.

Variables

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

“Variables” are Algebra letters like x, y, n, T, P, A, etc, that represent an unknown value in a word problem.

Variables are also used when writing computer programs. A variable letter can only represent one item, which means variables like “Cats&Dogs” are not allowed.

Typical Variables in maths questions might be values such as:

c = number of cats
d = number of dogs
T = the total number of animals
P = Price after discount has been applied
d = Dollar amount spent
I = Interest on a credit card
b = the number of boys in a class
A = Cheryl’s current age
n = the number of pizzas for 20 people

and so on.

We use letter values (or variables) in our maths sums, because it is much easier than writing out long groups of words.

Variables are letters that are used to represent unknown items in our maths questions.

We solve Algebra Equations so that we can find out the number values of these variables.

One Step Subtraction Equation

Image Source: http://covers.openlibrary.org

A typical example involving a subtraction equation is the following situation.

In our Sports Store we sold 27 basketballs today, and we now have 11 basketballs left on the shelf.

How many basketballs did we have at the start of the day?

First we write a word equation to represent the situation.

Total Basketballs – Basketballs Sold = Remaining Basketballs.

Let “T” equal the Total Basketballs at the start of the day.

We know that:

Basketballs Sold = 27

Basketballs Remaining = 11

By substituting the above three values into our original word equation, we can create the following Algebra equation:

T – 27 = 11

Our next step is to investigate what is happening to the the unknown letter value “T”.

In the equation, T has 27 SUBTRACTED from it.

To solve our equation we do the opposite of Subtracting, (which is Adding), to both sides of the equals sign.

But how much do we add to each side of the equation ?

T – 27 = 11

To obtain our final answer, we need to get “T” on its own, by eliminating the – 27 that is with it.

We can get rid of the – 27 by ADDING 27, so we need to add 27 to both sides of the equation.

T – 27 = 11
T – 27 + 27 = 11 + 27
————-
T + 0 = 38
T = 38
————-

At the start of the day we had 38 basketballs in our shop.

We can check our answer by substituting all our values into the original equation, and seeing if the equation ends up the same on both sides.

Total Basketballs – Basketballs Sold = Remaining Basketballs.

38 – 27 = 11

11 = 11

By doing the substitution check shown above, we can be completely certain that the start of day total of 38 is the correct answer to our maths question.

Video Lessons on One Step Subtraction Equations

Here is a short video by “Your Teacher” which shows the steps required to solve a one step subtraction linear equation.

[youtube http://www.youtube.com/watch?v=xCs5uVrHsGo?rel=0]

Here is a more comprehensive video about solving Single Step Addition and Subtraction equations.

Note in this video that the Americans call a “see-saw” balancing toy, a “teeter-totter” !

[youtube http://www.youtube.com/watch?v=jVLrsyLqEL0?rel=0]

The following video covers doing one step Add and Subtract equations that involve Integers.

[youtube http://www.youtube.com/watch?v=xGfOlCluPDo?rel=0]

Here is another video lesson on equations that involve Integers.

[youtube http://www.youtube.com/watch?v=jsBXxCPePI8?rel=0]

One Step Subtraction Equations Worksheets

The following worksheets have questions to do on Subtraction Equations. Answers are available when we scroll down to page 2 on each sheet.

The first sheet is very easy, the second and third sheets have negative numbers, the fourth sheet has Year 8 type questions.

Subtraction Equations Worksheet 1

Subtraction Equations Worksheet 2

Subtraction Equations Worksheet 3

Subtraction Equations Worksheet 4

Here is a Speed Test Video that you can do to get fast at solving one step equations.

[youtube http://www.youtube.com/watch?v=ePOZc-Ar42Q?rel=0]

That’s it for one step Subtraction equations.

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Enjoy,
Passy

Posted in Algebra, Equations, Equations One Step, Word Problems | Tagged Algebra Equations, Equations, how to do equations, how to solve equations, math equations, one step equations, single step equations, solving algebra equations, subtraction equations | 15 Comments

Addition Equations

Posted on September 29, 2011 by Passy

Image made at Passy World using Clker.com Clipart

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.

We will begin with the easiest type of Algebra Equations which are “One Step Addition Equations”.

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

[slideshare id=9463777&doc=onestepaddeqnsppt-110928174318-phpapp01]

If you cannot view the above presentation, (because you are using an Apple device), then click below view the following PDF version of the slideshow.

http://www.passyworld.com/passyPDFs/OneStepAddEqnsPDF.pdf

Video Lessons on One Step Addition Equations

Here is a video from “Mahalo Math” on solving one step addition equations.

[youtube http://www.youtube.com/watch?v=hiqwzRYK27c?rel=0]

This is a similar short video by “Your Teacher”.

[youtube http://www.youtube.com/watch?v=3gfZ8vLu4TY?rel=0]

Here is a video from “Brightstorm” that explains the difference between an Expression and an Equation.

[youtube http://www.youtube.com/watch?v=3j-fa18Y9a8?rel=0]

One Step Addition Equations Worksheets

The following worksheets have questions to do on Addition Equations. Answers are available when we scroll down to page 2 on each sheet.

The first sheet is very easy, the second and third sheets have negative numbers, the fourth sheet has Year 8 type questions.

That’s it for one step addition equations.

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Passy

Posted in Algebra, Equations | Tagged addition equations, Equations, how to do equations, how to solve equations, math equations, one step equations, single step equations | 19 Comments

Translate Word Equations

Posted on September 28, 2011 by Passy

$Guy with Math Op symbols clipart$
Image Source: http://www.martincwiner.com

Translating word problems into Algebra requires us to know how various words translate into mathematical symbols such as + – X and / .

If the word problems only have numbers, then it is easy to translate them, once we have practiced doing a few word problems.

For example, Cyana and Larissa go to the shops and they each have fifty dollars.

Cyana spends 24 dollars and Larissa spends twice as much as Cyana.

How much money do they each have left after their shopping spree ?

We know that spending money is going to reduce or subtract from the original total of 50 dollars.

We also know that “twice” means two times, and thus refers to multiplication.

Cyana = (50 – 24) = $26

Larissa = (50 – 2×24) = (50 – 48) = $2

“Spend” means subtraction, and “twice” means multiply by two.

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

Sometimes word problems have missing information that we call “variables” and we use an Algebra letter instead of a number to represent these values.

For example, our previous word problem can be written as follows, for a situation where we do not know exactly how much money the girls spent.

Cyana and Larissa go to the shops and they each have fifty dollars.
Cyana spends “M” dollars and Larissa spends twice as much as Cyana.

(“M” is the unknown amount of money that Cyana spent).

How much money do they each have left after their shopping spree ?

Cyana = 50 – M

Larissa = (50 – 2 x M) = 50 – 2M

This time the answer cannot be worked out to an exact numeric dollar amount, and we have to leave the answer with the unknown value of “M” in it.

“M” is a variable that represents how much money Cyana spent. (“M” = money spent).

If we are later told that “M” = 18 dollars then we can use “substitution” to work out exact answers.

Cyana = 50 – M (but M = 18)
Cyana = 50 – 18 = $32

Larissa = 50 – 2 x 18
Larissa = 50 – 36 = $14

Translation Tips for Word Problems

The following translation tables help convert word problems into Algebra expressions and equations.

Look for the following key words and phrases when reading through word problems.

Whenever you find one of these words, translate the word or phrase into +, -, x, / , or = .

This will help form the Algebra equation for the word problem.

The following words in a sentence indicate that Adding is taking place.

The following words in a sentence indicate that Subtraction is taking place.

The following words in a sentence indicate that Dividing is taking place.

The following words in a sentence indicate that Multiplying is taking place.

The following words in a sentence need to be translated into an equals sign.

Whenever we get a word problem to convert into Algebra, we can use these translation tables to help us work out what mathematical symbols we need to use to replace the words.

Here are some simple examples.

Video Lessons

This first video shows how to translate word statements into mathematics for Addition and Subtraction.

[youtube http://www.youtube.com/watch?v=ubaBrD-NQ_w?rel=0]

In this follow up video, Professor Perez includes division and multiplication word problems.

Note that Americans use a Dot to represent a multiply sign, whereas other countries like Australia use an “x” to represent multiplication.

[youtube http://www.youtube.com/watch?v=m72bUX3bi_c?rel=0]

This next video takes things a step further, and shows how we can make “Function Equations” for word problems.

[youtube http://www.youtube.com/watch?v=7OFL3uHvvlQ?rel=0]

Tips and Traps for Word Problems

Word problems involving subtraction often cause difficulty.

For example “five more than twenty” translates to 5 + 20

But “five less than twenty” does NOT translate to 5 – 20

and “five subtracted from 20” does NOT translate to 5 – 20

and “five taken away from 20” does NOT translate to 5 – 20

Think of the following money example to help make sense of this.

Subtraction words like “less than”, “subtracted from”, and “taken away from” all require that we swap the order of the numbers in the sentence.

“Five less than twenty” = 20 – 5

“Five subtracted from 20” = 20 -5

“Five taken away from 20” = 20 -5

Note that the Algebra numbers end up in reverse order to what they were in the original word sentence.

Eg. In the original word sentence the five came first and then the 20. But in the Algebra sentence the twenty comes first, and the five comes last.

The following video is a very comprehensive narrated slideshow about how to translate equations. This video includes examples of “Reversing the Order” subtraction phrases in word problems.

[youtube http://www.youtube.com/watch?v=29uM12GMLA8?rel=0]
Translating Equations Slideshare Presentation

Here is a Slideshare Presentation all about writing equations for word problems.

[slideshare id=610434&doc=c2ch0108-1222031873020264-9]

Translating Words into Algebra Lessons

Math Goodies has a good lesson on writing equations for word sentences with a quick five question online quiz at the end of the lesson.

Click the link below to do this lesson.

http://www.mathgoodies.com/lessons/vol7/equations.html

They also have a similar lesson on translating algebra expressions from words.

http://www.mathgoodies.com/lessons/vol7/expressions.html

The following web page has some great hints for teachers and students about Solving math word problems and setting up equations

http://www.homeschoolmath.net/teaching/teach-solve-word-problems.php

Translating Words Into Equations Worksheets

There are several Number Problems PDF worksheets on the following web page.
Write equation then work out the answer, and then page down to the Answers page to check your work.

http://www.algebra4children.com/printables.html

There are eight mix and match problems on this page, but no checking of the answers is provided.

http://www.math.com/school/subject2/practice/S2U1L3/S2U1L3Pract.html

That’s about it for translating world problems into Algebra and Mathematics.

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Passy

Posted in Algebra, Equations, Word Problems | Tagged algebra translation table, Equations, how to do equations, how to solve equations, how to write equations, math equations, translate word problems, translate word to algebra, translating equations, translating word problems, word problems | 27 Comments

Microsoft Mathematics

Posted on September 27, 2011 by Passy

Microsoft Mathematics 4.0 is a free download that provides a set of mathematical tools to help students get mathematics work done quickly and easily.

Students can learn to solve equations and draw graphs, while gaining a better understanding of fundamental concepts in number, algebra, trigonometry, physics, chemistry, and calculus.

Microsoft Mathematics includes a fully-featured graphing calculator that is designed to work just like a handheld calculator.

Additional math tools help evaluate triangles, convert from one system of units to another, and solve systems of equations.

Here is a video introduction to some of the Graphical Calculator features of MS Math 4.0

[youtube http://www.youtube.com/watch?v=LVuaRxTNZR8?rel=0]

MS Mathematics 4.0 Equation Solver

Here at Passy World we have only started using Mathematics 4.0 recently.
So far we have mainly been solving Linear Equations.

On the main ribbon menu, we click onto “Equation Solver” and type in our equation like this.

We can then click on the “Solve” button to obtain our answer.

In Worksheet view, we can click on “Solution Steps” to get the full working out displayed.

The Equation Solver easily handles all types of Linear Equations.

By Clicking on the “Graphing” tab we can get a graphical solution of our answer.

The Graph Plotter is great to use for demonstrations of how y = mx +c graphs, (or American y = mx + b lines), as well as graphical solutions for simultaneous equations.

The plotter also does a range of other graphs including parabolas, cubics, and even three dimensional graphs.

We are planning to explore a lot more of Mathematics 4.0, as there is plenty of useful functionality available.

Detailed Review of MS Mathematics 4.0

Here is a video which shows the primary features of MS Mathematics in detail.

[youtube http://www.youtube.com/watch?v=DyYacuhyPXw?rel=0]

Remo Knops has a comprehensive review of Microsoft Mathematics on his blog.

Click here for Remo Knops MS Math 4.0 Review

MS Mathematics 4.0 Free Download

Microsoft Mathematics 4.0 can be downloaded for free and installed on any Windows computer from the web address given below.

Most people use 32bit Windows and need to click the download button next to MSetup_x86.exe option. But if you are using 64bit Windows, then click the download button next to MSetup_x64.exe option.

http://www.microsoft.com/download/en/details.aspx?id=15702

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Enjoy,
Passy

Posted in Equations, Math Applications, Microsoft Mathematics, Straight Line Graphs | Tagged graph calculator, math 4.0 review, microsoft mathematics, Microsoft Mathematics Review, ms math, ms mathematics, online grapher review, online graphs | 15 Comments

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