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sums will set you freehow to teach your child numbers arithmetic mathematicsequality and equations
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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. equilibriumTwo 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
taking away- subtraction - in balance
multiplying - multiplication - in balance
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dividing - division - in balance
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 variablesNumbers 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, equationsBefore 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
the beginnings of algebraNo, 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. Or you may have two equations, one telling you that
b = 3c,
and the other that c = 4. Believe it or not, by this time you are already doing simultaneous equations (two equations and one ‘unknown’). Here are some simple algebraic sums:
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 equationsThe 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 equationAn 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 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:
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.
And now let’s delve into the nitty-gritty of the
We rearrange the equation. This takes are a number of
steps. 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 equationThe 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. 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.
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 :
connecting in parallel
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. |
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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% |
| 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 | ||
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email abelard at abelard.org © abelard, 2009, 11 october the web address for this page is http://www.abelard.org/sums/teaching_number_arithmetic_mathematics_equality_and_equations.php |
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.
equation.
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: 
.
