Multitasking: Good for computers, bad for people.

See if you can read this to the end without answering the phone, noticing a notification, etcetera:

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Area models for completing the square, dynamic approach.

An area model, or a dot array model (same thing really) is one way of illustrating the algebraic completion of a square.

I have used dots as they are easier to create.

The quadratic is viewed initially as the “standard form”, and then rebuilt dynamically line by line into the “square plus a bit over” form, as shown in the following sequence:

area model 1

The odd valued coefficient of x in the original expression can appear as a row and a column of half-dots, or half squares in the area model form.

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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.

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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.

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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




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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

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Room 808

2084 ?
No. Much closer.


Image credit: the Ministry of Education, Airstrip One, Oceania

It was a warm but overcast day in late August and the clocks were striking thirteen.

Mr Winston Smith, Principal of the Victory G+MINDSET Academy (formerly the Bogstannard Comprehensive School), woke to find himself lying on something that felt like a camp bed, except that it was higher off the ground and it seemed that he was fixed down in some way so that he could not move. Light that seemed stronger than usual was falling on his face.

He gasped as he realised that the infamous MiniEd interrogator, “Grammar School” O’Greening, was standing at his side, looking down at him intently. At the other side of him stood a man in a white coat, tapping an iPad.

“Tell me, Winston,” said O’Greening gently, but with a chilling undercurrent of steel in her voice, “how many buckets am I holding up?”


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#SciFi Prompt 100!!! – F.U.D.

Not so optional !

J. Giambrone




We are in dire straits with an all-out assault on science, evidence and reason coming out of the White House all day long, every day now. The direct attack on science is part of a much-wider trend.

Science is imperfect and the truth has no billionaire backer fighting for it. The deck is stacked in the other direction.


Fear Uncertainty Doubt:

Fear, uncertainty and doubt (often shortened to FUD) is a disinformation strategy used in sales, marketing, public relations, talk radio, politics, religious organizations, and propaganda. FUD is generally a strategy to influence perception by disseminating negative and dubious or false information and a manifestation of the appeal to fear.

…By spreading questionable information about the drawbacks of less well known products, an established company can discourage decision-makers from choosing those products over its own, regardless of the relative technical merits.


President FUD is the living…

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Renamed “A bridge too far”.

A Narcissist Writes Letters, To Himself

Wouldn’t it be wonderful
if Democrats and Republicans
set aside their differences
& came together under one roof

& then, due to decades
of neglectful infrastructure funding policy,
erosion collapsed that roof?

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MATHS BITE: The Cantor Set

And they call them “real” numbers!


The Cantor Set is constructed in the following way:

Start with the interval [0,1]. Next, remove the open middle third interval, which gives you two line segments [0,1/3] and [2/3,1]. Again, remove the middle third for each remaining interval, which leaves you now with 4 intervals. Repeat this final step ad infinitum.


The points in [0,1] that do not eventually get removed in the procedure form the Cantor set.

How many points are there in the Cantor Set?

Consider the diagram below:

Screen Shot 2017-02-21 at 8.05.41 PM.png

An interval from each step has been coloured in red, and each red interval (apart from the top one) lies underneath another red interval. This nested sequence shrinks down to a point, which is contained in every one of the red intervals, and hence is a member of the Cantor set. In fact, each point in the Cantor set corresponds to a unique infinite sequence of nested intervals.

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