Category Archives: education

Geometry and Numbers – the theory


Multiplication, the theory – by Thales’ theorem

mult pic real theory 2

The diagram can be simplified by using an acute triangle.

mult pic real theory 3   Thales’ theorem

Proof of Thales theorem :
If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.
Given : In ∆ABC , DE || BC and intersects AB in D and AC in E.
Prove that : AD / DB = AE / EC
Construction : Join BC,CD and draw EF ┴ BA and DG ┴ CA.
Statements                                                    Reasons
1) EF ┴ BA                                                      1) Construction
2) EF is the height of ∆ADE and ∆DBE     2) Definition of perpendicular
3)Area(ADE) = (AD.EF)/2                             3)Area = (Base .height)/2
4)Area(DBE) =(DB.EF)/2                               4) Area = (Base .height)/2
5)(Area(ADE))/(Area(DBE)) = AD/DB         5) Divide (3) by (4)
6) (Area(ADE))/(Area(DEC)) = AE/EC         6) Divide (3) by Area(DEC)
7) ∆DBE ~∆DEC                                             7) Both the ∆s are on the same base and
between the same || lines.
8) Area(∆DBE)=area(∆DEC)                        8) So the two triangles have equal areas
9) AD/DB =AE/EC                                           9) From (5) and (6) and (7)

Not only this but also AD/AB = DE/BC

I borrowed this from

Some adjustments, but the Thales theorem is well done. I liked it.

Leave a comment

Filed under arithmetic, construction, definitions, education, geometrical, geometry, geostruct, Uncategorized

Geometry and Numbers – negative ones – “a minus times a minus is a plus”

To accommodate positive and negative numbers we need two extended number lines, with their zeros at the same place

Then multiplication of two negative numbers will always give a positive result, following the same geometrical structure.

The start, where the multiplier begins at 1

mult pic negative 0

Now the 1 connects with the multiplicand -3

mult pic negative 1

The multiplier is now placed at -2

mult pic negative 2 6

And the parallel line from -2 connects to the 6 on the target line

mult pic negative 3

This is so geometrical, and there is no “funny business”. None of the “ought to be 6”. No stuff about the distributive law.

The only geometry needs the Pythagoras theorem, and this will be the next post.

Leave a comment

Filed under arithmetic, education, geometry, geostruct, negative numbers, Number systems, Uncategorized

Geometry and Numbers, not the counting sort

A number line is generally a piece of straight line with a starting point, labeled 0, and equally spaced points labeled 1 and 2 and 3 and 4 and so on till the paper runs out.

The value of a number is the distance from the zero point to the numbered point, in units of the equal spacing.

It is really much easier to draw one of these !


Two parallel number lines, same scale.
add pic 1

add pic 1a

add pic 1bNotice that the zero points do not have to be in the same vertical line.


To get the symbolic form 7 – 2 = 5 we start with 0 on the target line (now the upper line) and join it to the 2 on the subtrahend line. (arrow down) (needs a nicer word here)

Then from the 7 on the subtrahend line we produce the line from 7 parallel to the 0 to 2 line. Then “arrow up” to the target line

Magic ! The result is 5 on the target line.

I like the picture, but the subtraction words are a mess.


We need two number lines, but since multiplication is  “proportional” they will now be crossing, and the common point is labeled 0.

Also, the labels are “target” and “multiplier” and each line has its own scale.

mult pic 1

mult pic 1a

mult pic 1b


mult pic 1c

Bonus: Nomograms, with lines.

The first is a simple calculator, with A + B = Sum

nomogram 1

The second one calculates parallel resistances

nomogram - resistors




Filed under arithmetic, education, geometry, math, nomogram, Number systems, Uncategorized

sin(2x) less than or equal to 2sin(x), for smallish x (!)

Once upon a time, when I was deep into trigonometry, I followed the trail of sine and cosine sums, with sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Then two more:
sin(A) + sin(B) = 2sin((A + B)/2)cos((A – B)/2)
and 2sin(A)cos(B) = sin(A + B) + sin(A – B)

Being unable to remember these last formulae (there are 8 altogether) I learned the basic one and derived, over and over again, the others.
In all of this I was never able to derive the basic formula until de Moivre’s theorem appeared.
So I had another go with trig and the simple version, and this is the result:

Aim of the game : sin(2x) = 2sin(x)cos(x)
First diagram:This does not look promising !
sine pic 1But what about sensible labelling –
Second diagram:
sine pic 2

The vertical from F meets AD in J
and the line from F at right angles to AB meets AB in K.
So we have two pairs of congruent triangles, BH and HD are both equal to sin(x),
Final diagram:
sine pic 3Now the vertical from B is sin(2x)

so sin(2x)/(2sin(x)) is the ratio which should be cos(x)

Fix it

Leave a comment

Filed under education, geometrical, geometry, geometry app, triangle, Uncategorized

Matrix multiplication without tears!

My dad figured this out years ago.


The string method work for all matrices, and it is at least ten times quicker to “do” than to “write about”.

1 Comment

Filed under algebra, education, math, teaching, transformations, Uncategorized

Pokemon Go – a new purpose for Edtech

I found this on Medium

Gracias to Junaid Mubeen

Oxford Mathematician turned educator. @HGSE ’12. Head of Product @MathsWhizzTutor. Long-distance runner. Anagrams.
18 hrs ago6 min read

View story at

1 Comment

Filed under education, math apps, Uncategorized

A minus times a minus is a plus -Are you sure you know why?

What exactly are negative numbers?
A reference , from Wikipedia:
In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”.
He came to the conclusion that negative numbers were nonsensical.[25]

A minus times a minus is a plus
Two minuses make a plus
Dividing by a negative, especially a negative fraction !!!!
(10 – 2) x (7 – 3) = 10 x 7 – 2 x 7 + 10 x -3 + 2 x 3, really? How do we know?
Or we use “the area model”, or some hand waving with the number line.

It’s time for some clear thinking about this stuff.

Mathematically speaking, the only place that requires troublesome calculations with negative numbers is in algebra, either in evaluation or in rearrangement, but what about the real world ?
Where in the real world does one encounter negative x negative ?
I found two situations, in electricity and in mechanics:

1: “volts x amps = watts”, as it it popularly remembered really means “voltage drop x current flowing = power”
It is sensible to choose a measurement system (scale) for each of these so that a current flowing from a higher to a lower potential point is treated as positive, as is the voltage drop.

Part of simple circuit A———–[resistors etc in here]————–B
Choosing point A, at potential a, as the reference, and point B, at potential b, as the “other” point, then the potential drop from A to B is a – b
If b<a then a current flows from A to B, and its value is positive, just as a – b is positive
If b>a then a current flows from B to A, and its value is negative, just as a – b is negative

In each case the formula for power, voltage drop x current flowing = power, must yield an unsigned number, as negative power is a nonsense. Power is an “amount”.
So when dealing with reality minus times minus is plus (in this case nosign at all).

The mechanics example is about the formula “force times distance = work done”
You can fill in the details.

Now let’s do multiplication on the number line, or to be more precise, two number lines:
Draw two number lines, different directions, starting together at the zero. The scales do not have to be the same.
To multiply 2 by three (3 times 2):
1: Draw a line from the 1 on line A to the 2 on line B
2: Draw a line from the 3 on line A parallel to the first line.
3: It meets line B at the point 6
4: Done: 3 times 2 is 6
numberlines mult pospos
Number line A holds the multipliers, number line B holds the numbers being multiplied.

To multiply a negative number by a positive number we need a pair of signed number lines, crossing at their zero points.

So to multiply -2 by 3 (3 times -2) we do the same as above, but the number being multiplied is now -2, so 1 on line A is joined to -2 on line B

numberlines mult posneg
The diagram below is for -2 times 3. Wow, it ends in the same place.
numberlines mult posneg

Finally, and you can see where this is going, we do -2 times -3.

Join the 1 on line A to the -3 on line B, and then the parallel to this line passing through the -2 on line A:

numberlines mult negneg

and as hoped for, this line passes through the point 6 on the number line B.

Does this “prove” the general case? Only in the proverbial sense. The reason is that we do not have a proper definition of signed numbers. (There is one).

Incidentally, the numbering on the scales above is very poor. The positive numbers are NOT NOT NOT the same things as the unsigned numbers 1, 1.986, 234.5 etc

Each of them should have a + in front, but mathematicians are Lazy. More on this another day.

Problem for you: Show that (a-b)(c-d) = ac – bc – ad + bd without using anything to do with “negative numbers”


Reference direction for current
Since the current in a wire or component can flow in either direction, when a variable I is defined to represent
that current, the direction representing positive current must be specified, usually by an arrow on the circuit
schematic diagram. This is called the reference direction of current I. If the current flows in the opposite
direction, the variable I has a negative value.

Yahoo Answers: Reference direction for potential difference
Best Answer: Potential difference can be negative. It depends on which direction you measure the voltage – e.g.
which way round you connect a voltmeter. (if this is the best answer, I hate to think of what the worst answer is)


Filed under algebra, arithmetic, definitions, education, geometrical, math, meaning, negative numbers, Number systems, operations, subtraction, teaching, Uncategorized

I found Ciedie Aech. Now you have.

Well, actually, she commented on a comment of mine on someone else’s blog, which led me to finding her book, here:

Have you seen it yet? Have you read it yet?

It is a brilliant first-hand account of the school “reform” process from the receiving end, with a logically presented sequence of analyses, intertwined with actual happenings and incidents which make your hair stand on end.
The often believed statement “Corporate school reformers were once open about their belief that public education was hopelessly broken” she argues is simply untrue, but that this was what they wanted others to believe. They didn’t have to.
Her story covers the years from 1995 to the present, and shows the full depth of mayhem caused by the “reform” movement.

Her account of the not too imaginary classroom where all the time is taken up following all the edicts and mandates that there is no time to actually do any teaching. It is priceless.

Here is a section on one of the many stupidities encountered:

Flying Blind
Frantically written upon demand by an evidently unbounded
wellspring of young hires, a torrent of suddenly created
district exams gushed up in a manner which soon began to feel
truly magical. And, as was becoming rapidly apparent, actually
understanding many of these precipitately manufactured tests?
Called for just a touch of magic as well.
Pushed repeatedly into the role of test graders, it wasn’t long
before a diversely collected school personnel began to comment
upon, and even argue about, not only the point value attached
to student responses but, more and more frequently, to the
tangible intentions behind the intricately worded test questions
“Help!” I whispered to a grading partner one afternoon.
“Do you have any idea what this means?”
Sliding a test booklet across the table, I pointed to an essay
prompt so convoluted that I could make little sense of it:

“In what way does this story’s diction create foreshadowing while
working sympathetically inside the author’s choice of syntax?”

My students – well, if we were being very optimistic, at
least a couple of them – possibly knew what diction, foreshadowing,
and syntax meant. But even I didn’t know how to combine
these three uniquely discrete elements in a logical response for
this tortuous prompt. I struggled with my conscience, tempted to
give full credit to the student who had written simply, and I thought most reasonably:

“I don’t know what the fuck this is talking about.”

Another student, less inclined to waste words?
Had printed more succinctly: IDK.
I Don’t Know.
Well damn, kid, me neither.
Holding little patience for those old-school processes so
monotonously tied to a methodically careful (and oh-so-tedious)
analysis, as the years bent to the magic of no-waiting transformations
systematically edged out an educator resistance, it was
rapidly determined that a test question ambiguity (up to and including
plainly misleading typos) did not, actually, invalidate
tests. Nor, subsequently, nullify an endlessly collected testing
data. Specifically hired to address issues of examination, testing
experts were ready to advise; expressly versed in party line, assuredly
and absolutely they always knew the answer. Every single
Oh, it was magical.
They could simply walk over and show you. “See?” Here
they could point with an absolute confidence to the official answer
sheet. “It’s right here,” they could tell you. “The answer is: D.”
Or: Two.
Or: No change.
In years now gloriously imbued with the high brilliance of
an instantaneous reformation, all you ever really had to do? Was
close your eyes. And, then, clicking your heels together: Believe.
Believe, as you took your first frightening step over an unknowable
cliff; believe, as anxiously you began to flap your arms; believe,
as apprehensively you started to fly alongside in a blind
Believe, absolutely and without reservation?
In the answer sheet.


Filed under big brother, education, horrors, school, teaching, testing, tests, Uncategorized

Too Close for Comfort, a “must read” link

Forecasting the future of “education”, by Eli Horowitz

Prescient ??????

View story at

View story at

Leave a comment

Filed under education, future, horrors, humor, satire, school, transformations, Uncategorized

CBE is not for me, in verse

“Mathematics, take a break, the doggerel has come back !”

Today I start to fill my pail.
Through the next test I will sail.
Common sense must then prevail:
I’ll clear my brain of what’s now stale
To make some space inside the pail.
For competence, the holy grail,
Ensures that I will never fail,
Though moving forward like a snail.
* * * * * * * * * * * * * * * * * * * * *
The whole damn thing’s to no avail.


Filed under education, testing, tests, Uncategorized, verse