Category Archives: teaching

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”.

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

Linear transformations, geometrically


Following a recent blog post relating a transformation of points on a line to points on another line to the graph of the equation relating the input and output I thought it would be interesting to explore the linear and affine mappings of a plane to itself from a geometrical construction perspective.

It was ! (To me anyway)

These linear mappings  (rigid and not so rigid motions) are usually  approached in descriptive and manipulative  ways, but always very specifically. I wanted to go directly from the transformation as equations directly to the transformation as geometry.

Taking an example, (x,y) maps to (X,Y) with the linear equations

X = x + y + 1 and Y = -0.5x +y

it was necessary to construct a point on the x axis with the value of X, and likewise a point on the y axis with the value of Y. The transformed (x,y) is then the point (X,Y) on the plane.

The construction below shows the points and lines needed to establish the point(X,0), which is G in the picture, starting with the point D as the (x,y)


transform of x

The corresponding construction was done for Y, and the resulting point (X,Y) is point J. Point D was then forced to lie on a line, the sloping blue line, and as it is moved along the line the transformed point J moves on another line

gif for lin affine trans1

Now the (x,y) point (B in the picture below, don’t ask why!) is forced to move on the blue circle. What does the transformed point do? It moves on an ellipse, whose size and orientation are determined by the actual transformation. At this point matrix methods become very handy.(though the 2D matrix methods cannot deal with translations)

gif for lin affine trans2

All this was constructed with my geometrical construction program (APP if you like) called GEOSTRUCT and available as a free web based application from

The program produces a listing of all the actions requested, and these are listed below for this application:

Line bb moved to pass through Point A
New line cc created, through points B and C
New Point D
New line dd created, through Point D, at right angles to Line aa
New line ee created, through Point D, at right angles to Line bb
New line ff created, through Point D, parallel to Line cc
New point E created as the intersection of Line ff and Line aa
New line gg created, through Point E, at right angles to Line aa
New line hh created, through Point B, at right angles to Line bb
New point F created as the intersection of Line hh and Line gg
New line ii created, through Point F, parallel to Line cc
New point G created as the intersection of Line ii and Line aa

G is the X coordinate, from X = x + y + 1 (added by me)

New line jj created, through Point G, at right angles to Line aa
New line kk created, through Point D, at right angles to Line cc
New point H created as the intersection of Line kk and Line bb
New point I created, as midpoint of points H and B
New line ll created, through Point I, at right angles to Line bb
New point J created as the intersection of Line ll and Line jj

J is the Y coordinate, from Y = -x/2 + y  (added by me)
and K is the transformed point (X,Y) Point J chosen as the tracking point (added by me)

New Line mm
Point D moved and placed on Line mm


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Filed under algebra, conics, construction, geometrical, geometry app, geostruct, math, ordered pairs, rigid motion, teaching, transformations, Uncategorized

Mathematics in my garden

So there it was, built from one clothesline to another, glistening in the sunlight one morning last week:


A near perfect spiderweb, demanding a better photo:

Here is  the horizontal strip, showing the variation in spacing of the spiral:

and again, with a superimposed scale. The spider is at the centre.spiderweb cropped with dots

And now rotated, to give more detail.spiderweb cropped with dots rotated

Now “Get modelling!”.


Filed under algebra, geometrical, Puerto Rico, series, teaching, tropical garden, Uncategorized

What exactly is Base 10 arithmetic ?

Teacher: “Now we’re going to learn about base 10 arithmetic”.
Wise guy: “Is that where 3 + 4 = 12, or is it where 3 x 4 = 12 ?”.

I did a search on the net and found the term “base 10” all over the place. What does it mean?

An apparently annoying question:
“Does the 1 in 10 stand for the number 10’s in 10?”.

The interpretation of 10 in the system described as “Base 10” depends on the base of the system, so what is it? How do I find out?

We have here a logical problem. The term “Base 10” as a definition is self referential. It is more subtle than this definition of a straight line:

“A straight line is a line which is straight”.

The problem arises from the almost universal confusion between the two things:
1: The name of a number, in this case “ten” is supposedly implied
2: The symbols representing a number, in this case 10 in the base ten system”

So the answers to the questions “What is it? How do I find out?” above are “Unknown” and “You can’t”

Writing “Base 10” when you mean “Base ten” is probably the first step in making math meaningless.



Filed under abstract, arithmetic, confusion, definitions, language in math, math, teaching

A. N. Whitehead on negative numbers (1911)

This is really worth reading. It is from his book, “Introduction to Mathematics”, published in 1911.

whitehead intro to math negative nos





Filed under abstract, arithmetic, Number systems, teaching, Uncategorized

“Shear”, the forgotten transformation.

Transformations of the plane are many and various.
The “nice” ones are “rigid motions”, and this term includes rotations, reflections and translations. The shape and the size of a geometrical object are not altered by a rigid motion.
There are also “shape preserving” transformations, called “dilations”, in which an object is stretched or shrunk equally in all directions.
An often overlooked transformation is the “shear”, in which there is a fixed line, and points not on that line are pushed parallel to the line in proportion to their distance from the line. Think of a stack of paper,perfectly stacked, and then pushed sideways so that the side of the pile is still flat. You will see a parallelogram at the front of the pile.
A shear will change the direction of a line, turn a rectangle into a parallelogram and turn a circle into an ellipse,
the area of any closed figure does not change at all.

Here is the static picture of a fixed point J, a fixed line, the x-axis, and a set of points on the horizontal line through A.
Also two triangles, LND and LDF, which are going to be sheared
shear transformation in xy plane
And here is the shearing in action, for varying amounts of shear, determined by the value of k.
gif for shear
Notice that triangle LMN changes a lot, and its area changes, but the areas of triangles LND and LDF do not change at all.
Not shown is a rectangle and a circle, which would change into a parallelogram and an ellipse, but their areas will not change with a shear.

For more on this go back to my Christmas post:

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Filed under geometry, geostruct, rigid motion, teaching, transformations

Euclid and vertical angles

Euclid and angle between two lines

euclid pair of lines

Euclid’s definition of angle:

From Euclid’s Elements, Book 1
Definition 8.
A plane angle is the inclination to one another of two lines in a plane which meet one another and
do not lie in a straight line.
Definition 9.
And when the lines containing the angle are straight, the angle is called rectilinear.

In the diagram we see that angle A can be taken as the inclination, but we can also see that B can be taken as the angle of inclination.

So, which is it?

If the definition is meaningful then the two angles have to be equal in size, regardless of the lack of a measurement system for angles.

My point is that the theorem about vertical angles (Euclid’s Proposition 15) is redundant, and so there is no need to prove it.

This would save students a lot of time and relieve them of the feeling that proof was pointless. This time could be better spent on proving some less obvious things.

Adding angles is a straightforward manipulative activity, but Euclid also uses subtraction of angles, which is not an obvious thing to carry out, and technically requires an additional postulate. See this:

On the formal approach to subtraction

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Filed under definitions, Euclid, geometry, math, teaching

GEOSTRUCT now with Save and Fetch for your constructions

Thanks to intense use of the internet I finally found a simple, understandable way of implementing Save and Fetch operations, enabling the keeping and reusing of any construction.

Here is a reminder of the application (app, program, software, whatever), with the file handling operations:

post example A
The user panel and a simple example of three points on a circle, with the bisectors of the pairs of points.

post example C
The history panel showing the actions that have been carried out

post example D
The Save popup,and, below, the resulting text file.

post example E

There is now a not quite finished Spanish option – just click “ESPANOL”

Also a modified “move object” procedure for use with a tablet,or even a smartphone.

The whole application is constructed as a web page, and to run it just click this link: geostruct

The full url is

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Filed under education, geometry app, geostruct, math, math apps, teaching, web based