# Tag Archives: despair

## “Observe and make use of structure”. Observe would be a start. A tale from the chalkface.

Here’s my little story:
It was a class of day release students on a Higher National Certificate course in engineering. I reached a point in one class with a relationship between p and q, p = kq, with k a constant. “What’s its graph look like”, I asked. Deathly silence. “Ok, let’s try x and y”. Result y = kx. Same question, same response. “Well, what about y = 3x ?”. Same question, same response. So I wrote y = 3x + 2. Their eyes lit up, and they unanimously shouted “It’s a straight line!”.

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Filed under algebra, education, humor

## Commutative, associative, distributive – These are THE LAWS

Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.

I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c

But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.

The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.

It’s not surprising that kids have trouble with negative numbers!

Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?

Filed under abstract, algebra, arithmetic, education, language in math, teaching

## What is an equation? . . . . .What is NOT an equation?

Before starting, a definition: Any combination of numbers and letters and arithmetical operations (including   =   <   >   <=   >=) with more than three symbols is an algebra “thing”.  So, in passing, observe that a “number sentence” is an algebra “thing”.

Equations are neither true nor false: Some examples –

x + 10 = 45
x + 2y = 8
x^2 + y^2 = 4
y = x^2 + 5x + 7
x^2 + 5x + 7 = 0
x = 35
ax + by + c = 0

In each case the equation specifies the value or values of the letter quantities

x + 10 = 45
The value of x is such that if I add 10 to it I get 45

x + 2y = 8
The values of x and y are such that twice the y value added to the x value gives me 8

x^2 + y^2 = 4
The values of x and y are such that the square of the x value added to the square of the y value is equal to 4. This and the one above specify pairs of values.

y = x^2 + 5x + 7      You do these two
x^2 + 5x + 7 = 0

x = 35
The value of x is specified to be 35
and lastly,   ax + by + c = 0
————————————-
Then we have identities, sometimes called equivalence statements.
These are ALWAYS true.
Examples:

3 + 2 = 5
4 = 1 + 3
8 = 11 – 3
(x + 1)^2 = x^2 + 2x + 1
2x(4 + 7) = 2×4 + 2×7 (where x is multiplied by)
6/8 = 3/4
————————————-
Later on, in algebra, we get definitions:

f(x) = 3x + 2
This means “The rule for the function whose name is f and whose input is x is multiplythevalueofhteinputxbythreeandaddtwotoit”
or “The value of the output of the function f for input x is the value of 3x + 2”.

y = f(x)
This means “The value of the output of the function f for input x is to be given the name y”.
These are NOT equations and they are NOT identities.
————————————-
The whole current mess arises from the use of the equals sign for “gives” or “makes”, or “we get”, as in “3 + 5 makes 8”, or  “If we multiply 4 by 6 we get 24”, and we write

3 + 5 = 8, and 4 x 6 = 24
3 + 5 is 8 and 4 x 6 is 24 would be better.

The newfangled term “number sentence” appears to have been invented in order to avoid dealing with the correct mathematical jargon, but I see it as making things EVEN worse,