how to teach your child number arithmetic mathematics - equality and equations | sums will set you free at abelard.org
latest changes & additions at abelard.org link to document abstracts quotations at abelard.org, with source document where relevant      latest news headlines at abelard's news and comment zone Children and television violence interesting site links at abelard's news and comment zone about abelard and abelard.org
link to short briefings documents France zone at abelard.org - another France economics and money zone at abelard.org - government swindles and how to transfer money on the net   Energy - beyond fossil fuels click for abelard's child education zone visit abelard's gallery technology zone at abelard.org: how to survive and thrive on the web

back to abelard's front page

 

sums will set you free

how to teach your child numbers arithmetic mathematics

equality and equations

 



site map

 
Google
 
Web abelard.org
 
how to teach your child number arithmetic mathematics - equality and equations is part of the series of documents about fundamental education at abelard.org. These pages are a sub-set of sums will set you free
how to teach a person number, arithmetic, mathematics on teaching reading

Before proceeding with this page, make sure that you are familiar with ‘equality’ or ‘same as’, the first page on equality. When dealing with equations, the essential and vital factor is to keep both sides balanced.

equilibrium

Two blocks equals two blocks:

Two blocks equals two blocks

Whatever is done to one side, must be done to the other side - add, subtract, multiply, divide, double, square, or turn inside out - must be done to the other side.

adding - addition - in balance

four blocks and txo blocks = imbalance
Adding two blocks to just one side makes for imbalance
four blocks and four blocks is balanced
Adding two more blocks to the other side maintains the balance

Marker at abelard.org

taking away- subtraction - in balance

In equilibrium
In equilibrium - two blocks on each side

two blocks and one block = imbalance
Take away a block from only one side and the balance is broken
one block and one block = balance
Take away a block from both sides and the balance is maintained

Marker at abelard.org

multiplying - multiplication - in balance

one block and one block = balance
In equilibrium - one block on each side

both sides multiplied by three

if only one side is multiplied by three, the equation becomes unbalanced

multiplying each side by three, there is equilibrium

 

  Follow abelard.org      
   

 

 

 

 

 

 

 

 

 


advertising disclaimer

 

 

 

Marker at abelard.org

dividing - division - in balance

five blocks and five blocks = balance
In equilibrium - five blocks on each side

dividing only ones ide by five leads to imbalance both sides divided by five
if only one side is divided by five, the equation becomes unbalanced dividing each side by five, there is equilibrium

At times, making sure that the two sides are in balance can be complicated, but you always have a remedy - check what you are doing with small numbers and see whether the two sides are, in fact, still in balance after your actions.

 

constants and variables

Numbers can be referred to as constants. When we don’t know what a number is, a variable can be used. That is, we know there is a number, but we don’t yet know what it is. All manner of symbols can be used for variables, usually starting with the lower case alphabet, but eventually you may come across Greek letters, German gothic letters, made-up symbols, or even words. So x, y, teacup, or pink balloon could all be used as variables.

While it is common to talk of constants and variables, it is important to realise that constants are also a form of variable. ‘One’ can mean one elephant, one galaxy, or the one drawing pin you just sat on. Always remember, there is nothing ‘special’ about numbers. They are just more words used by humans as they seek to commmunicate about, and to become masters of, the known universe.

 

BODMAS and simplifying, or rationalising, equations

Before immersing yourself in this section, you might want to study, or re-study, any or all of the following pages:

Equations often will include several variables and operations. (Operations are actions done to numbers and include adding, subtracting, dividing, squaring - multiplying a number by itself.)

Supposing you have the following equation, (2 + 5² - 1/4) ÷ 3 = x, and you want to find the value of x. Yes, you can work it out using lots of blocks but that takes quite a while and isn't very convenient a lot of the time. So what to do, where to start with teasing out a value for x.

This is where BODMAS comes in, an acronym for
Brackets, Orders, Divide and Multiply, Add and Subtract.
This is the order of actions to take when resolving a more complicated equation. Other than this order, you go from left to right along the equation/sum. Note that division and multiplication rank equally, as do addition and subtract.

    So for (2 + 5² - 1/4) ÷ 3 = x
  • First work out the sum within any brackets, here (2 + 5² - ¼).
  • Next come orders: these are powers or exponents. Within the brackets, checking through the sum you see that there is a number that has a small 2 by it. 5² is five to the power of two, the 2 is an exponent, and 5² means five squared, or five multiplied by itself: 5 x 5. 5² = 25.
  • There are no divisions or multiplications with in the brackets, so going from left to right, 2 + 25 = 27; 27- ¼ = 26 ¾.
  • Next there is the part of the equation (or sum) outside the brackets: ÷ 3. So 26 ¾ ÷ 3 = x.
  • Before going further, it is more usual to put the unknown variable x on the left side of the equation, x is the thing we are really concerned with so, as in an sentence, ‘the cat is big’ not ‘big is the cat’, we rearrange the equation and write x = 26 ¾ ÷ 3.
  • Now back to calculating x,
  • Thus x = 8 11/12.

 

the beginnings of algebra

No, algebra is not particularly complicated. Instead of using numbers, you merely use letters or combinations of letters and numbers, or should that be numbers and letters.

Note that, in algebra, 2 x a is written as 2a. Another example is 3 x a is written as 3a. 7b is the same as 7 x b.

Thus two blocks equals two blocks, and a = a, and 2b = 2b.
So if a = 2, you will immediately know that 2 = a, and 2a must equal 4. That is 2a = 4.
But you may start off not knowing what a is. a may equal 6 - 3, when it is obvious that a = 3.

Or you may have two equations, one telling you that b = 3c, and the other that c = 4.
So you can see that 3c = 12, and therefore b must equal 12.
Putting the value of c into the equation b = 3c is called substituting, the value of c is substituted into the equation.

Believe it or not, by this time you are already doing simultaneous equations (two equations and one ‘unknown’).

Here are some simple algebraic sums:

  • 2 + b = 7
    b
    = 7 - 2 [2 is subtracted from each side of the equation]
       = 5

  • 2 - b = 7
    2 - b + b = 7 + b
    [b is added to each side]
    2 = b + 7 [ - b + b leaves nothing, or zero]
    2 - 7 = b

    b
    = 2 - 7 [the equation is reorganised so the variable b is on the left]
    b
    = -5 [the last two steps could have been done together]

  • 2 x b = 7 [or 2b = 7]
    b = 7/2 [divide both sides by 2]
    b = 3 1/2

  • 2 ÷ b = 7
    2 = 7 b [multiply both sides by b]
    2/7 = b [divide both sides by 7]
    b = 2/7 [the equation is reorganised so the variable b is on the left]

They call this sort of thing simplifying equations. The prime objective is usually to isolate one element of the equation and express [describe it] it in terms of the rest of the equation. As has already been discussed, the important thing is to keep the equation in balance.

 

isolating elements and nicely complicated equations

The next section takes two nicely complicated equations and shows you how to isolate one element. This is called transformation - you tranform an equation from say, a +b = c, into a = c - b.

You might think that I would build up through lots of simple equations, but this way, you will see just about every trick in the book applied to isolating an element, while maintaining the balance of the equation. If you can tackle these two, you can approach almost anything with confidence and fun. This is the way I go about such things, but some people may prefer to go slowly through myriads of graded examples, climbing step by step up the stairs. Here, we go up the lift and balance out of the high wire!

Remember, it’s all about balance. What you must do is to keep rigorously to the rule that, “what you do to one side, you must do to the other side” - and not to part of one side, but to the whole of the side.

 

how to transform equations 1 - the pendulum equation

An equation can be thought as, for instance, a shorthand question about a particular (scientific) situation. For instance, if a string is 10 cm long, how quickly will a pendulum swing? But sometimes the equation, or shorthand question, has the wrong object as the subject of the sentence. Maybe you know how fast the pendulum is swinging, and you want to know the length of the string. Here the question has been rearranged. But to calculate the value of one variable (string length, pendulum swing time), the mathematical sentence, or equation, will have to be rearranged. Finding the answer you want is called ‘transforming the equation’.

Take the Pendulum Equation. This equation describes mathematically how long a weight on the end of a string (the pendulum) takes to swing back and forth once (its period of oscillation).

how a pendulum swings
When a pendulum bob is moved from the vertical, the the force of gravity pulls the bob downwards - it tries to drop (remember Galileo dropping weights from the leaning Tower of Pisa?).

But the pendulum cannot go straight down because it is constrained by the pendulum string. When the pendulum reaches its lowest point, where the pendulum line is vertical again, it cannot stop immediately. There is enough energy for the pendulum to keep moving. And it does so until the pull of gravity is equal to the energy in the moving pendulum. At this point, the pendulum falls back earthward. This we see as the pendulum swinging back and forth. If there is no friction where it is attached to its suspension point and there is no wind resistance to the pendulum’s, this would continue forever.

Here is the simple form of the Pendulum Equation: The Pendulum Equation


sorting out equation gobbledygook

Now this equation may look like gobbledygook, but we will translate this equation so it makes sense in English.

The variables for a simple pendulum Each letter stands for one of the variables in how a pendulum moves. In the simple pendulum equation, the bob (the lump at the end of the string that swings) is assumed to have no mass.

T = period in seconds. The period is the time taken during for one full swing, or oscillation, of the bob back and forth. The bob moves away from its starting point and returns to it.

L = length of the pendulum arm [or string] in metres.

π = the constant pi, whose value is 3.14159... and so on. Pi is used in geometry calculations involving circles and arcs.

g = acceleration due to gravity. An average value for g on Earth is 9.81 m/s².

Marker at abelard.org

And now let’s delve into the nitty-gritty of the equation.

 

  • If you look carefully, you will see that the period T depends directly on the length L of the pendulum - as L becomes larger, so does T.
  • On the other hand, because g is at the bottom of the fraction (or division) doing the dividing, as it becomes bigger, the period T becomes shorter. And in the real world, if g is greater, the pendulum will fall faster, and so T will be less.
  • Overall, does not have a linear relationship - that is, it cannot be used to make a straight line graph - because there is a square root in the equation. This information is useful should you want to rearrange the equation.


isolating an element from the apparent mess (transformation)
Now looking at this apparent mess, which is the most difficult element for us to extract? Suppose you’re interested in the length of the pendulum string can be, for it to take a certain time to swing back and forth (to oscillate in a certain time period). To find this out, you will have to rearrange the equation so that the length, L, is to the left of the equals sign and the rest of the equation is on the right. Yes, L looks a good candidate to me. If we can tease out L, we can tease out anything. So how the blazes are we to dig L out of bag of goodies?

We rearrange the equation. This takes are a number of steps.
First the step is described, and then we show how the equation is affected:

And here follows the more orthodox method of rearranging this equation. abelard is of the opinion that it is not so easy to understand:

 

how to transform equations 2 - the resistor equation

The resitor equation is another tricky mathematical animal to tame. But why is there a resistor equation, and why on Earth someone might want to use it? Here is some fairly easy explication, but you can skip it if your head starts to go fizz.


first, what is a resistor?

A resistor is an electrical element found in almost every electronic circuit. Resistors, as the name suggests, impede current flow and they are used to control the way current courses through the circuit. Their resistance is measured in ohms (symbol Ω). Because most resistors are too small to display figures or letters, their ratings are displayed by colour-coded bands.

The behaviour of an ideal resistor is defined by Ohm’s law, “current, I, is directly proportional to voltage, V, for a metal conductor, R, at a constant temperature”, or mathematically: I=V/R [we have rearranged Ohm’s law, V=IR, to match the definition just given]. Ohm’s law will be referred to later on when we explain the resistor equations (yes, there is more than one version).

Resistors can be combined by connecting together in in series or in parallel, or in a combination of these ways. This can be helpful if you do not have a resistor of the exact value needed as the values combine..

Often, when building electronic (or electrical) devices, it is necessary to calculate the voltage, or the current being used, or the resistance in the circuit, so that the electronic elements both work, and do not burn up and fail. The resistor equation is used to find out the combined resistance of several resistors in a circuit.


connecting in series

This is when resistors are linked one after the other, in a series.

Their total equivalent resistance is found by the equation, Req = R1 + R2 + ... + Rn . Finding the value of one of the resistors in the circuit is just a matter isolating that resistor by subtraction.

Thus, to find R1 in the equation Req = R1 + R2 + R3, subtract R2 + R3 from each side :
Req - (R2 + R3) = R1, or R1= Req - R2 - R3.

 

connecting in parallel

Resistors which connected together so they are in paralleleach have the same potential difference, or voltage. To find their total equivalent resistance, this equation is used:

Now supposing the circuit has two resistors in parallel, the equation will be .

Find R2, what is R2?

Now this is quite tricky to do, we use several mathematical tools which have already been described on other related pages at abelard.org. The mathematical tools are highlighted here in yellow, if you click on them you will be taken to the relevant section of that page.

 

Google
 
Web abelard.org

 

end notes

  1. Orders are also referred to as indices.

  2. Division can be described using ‘of’. For example, four of five is four fifths, 4/5.

  3. There are more complicated versions of the pendulum equation that use concepts like calculus for more detailed calculations of the behaviours of a pendulum, such as its velocity. More complicated versions may also take into account things like the angle from the perpendicular at which the bob is released.

  4. Pi (π is the Greek letter equivalent to P) is a mathematical constant that can be defined as π = C/d, where C is the circumference of a circle and d is its diameter. Pi is an irrational number, that is it cannot be expressed (or written) exactly as a fraction a/b, where a and b are integers; pi cannot be written as a ratio. Thus the decimal representation of pi, which starts 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510… [pi to the first fifty decimal places], never ends or repeats. Pi has been calculated, so far, to more than a trillion (1012) decimal places. However, for most calculations, just a few decimal places are sufficient. 22/3 is a common fractional approximation for pi.

  5. We are using metric units as values for the different variables - L in metres, T in seconds, g in metres per second squared (m/s²). The imperial equivalent is to describe L in feet and g as 32.2 ft/s.

  6. One ohm is the resistance value through which one volt will maintain a current of one ampere.

    Power is measured in watts, voltage in volts, current in amperes - amps for short, and resistance in ohms.

  7. resistor ratings bands Resistor ratings are usually coded with four bands: bands A and B the significant figures of the resistor’s value, band C is the decimal multiplier, and band D, if it is included, shows a tolerance value as a percentage (no color means a tolerance of 20%). The following is an abbreviated version of the colour coding.
      black brown red orange yellow green blue violet grey white gold silver
    significant figure 0 1 2 3 4 5 6 7 8 9    
    multiplier x 100 x 101 x 102 x 103 x 104 x 105 x 106 x 107 x 108 x 109 x 10-1 x 10-2
    tolerances - 1% 2% - - 0.5% 0.25% 0.1% 0.05% - 5% 10%
    Thus, the resistor illustrated above is rated as 73 x 105 Ω, or 7,300,000 Ω, with a tolerance of 5%.
    Notice that the colours from red to violet are the colours of a rainbow, where red has low energy and violiet higher energy!

    And here are some other resistors, so you can work out their ratings:


  8. In the real world, resistors can heat up, burn out, or have other problems, as well as not having perfectly linear behaviour - their resistance may be consistant with different current, voltages. On the other hand, in a ideal, or perfect, resistor the resistance remains constant regardless of the applied voltage or current flowing through it, or the rate of change of the current - the perfect resistor has linear V vs I behaviour.
sums will set you free includes the series of documents about economics and money at abelard.org.
moneybookers information e-gold information fiat money and inflation
calculating moving averages the arithmetic of fractional banking

You are here: how to teach your child number arithmetic mathematics - equality and equations
< sums will set you free < Home

about abelard memory, paranoia & paradigms francis galton france zone   memory & intelligence loud music & hearing damage children & tv violence


spacing line at abelard.org

 

  information abstracts briefings   news headlines news archives latest

email abelard at abelard.org

© abelard, 2009, 11 october

all rights reserved

the web address for this page is http://www.abelard.org/sums/teaching_number_arithmetic_mathematics_equality_and_equations.php

3010 words
prints as 12 A4 pages (on my printer and set-up)