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 ?

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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,

 

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Filed under algebra, arithmetic, language in math, teaching

Motherhood now, or should I study math first?

It’s doggerel time again, this time with apologies to Harry Graham, who apparently didn’t write the original “Oh mama dear, what is that mess ……”.  See  allpoetry

         “Oh Mommy dear, what are these sums you can’t do anymore?”
      “Hush,hush my child, just do your best, It’s called the Common Core.”

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Filed under humor, language in math, Uncategorized, verse