.The future: I\(I love this cartoon)
Monthly Archives: October 2015
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
And here is the shearing in action, for varying amounts of shear, determined by the value of k.
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:
Euclid and angle between two lines
Euclid’s definition of angle:
From Euclid’s Elements, Book 1
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.
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
As a former teacher of statistics I have to pass this on.
In February of this year, as communities and schools in New Jersey were awaiting the arrival of PARCC testing, I wrote this opinion piece for the Bergen County Record. In it, I said:
What can be expected? If experiences of other states that have already implemented PARCC- and CCSS-aligned exams are illustrative, New Jersey’s teachers, students and parents can expect steep declines in the percentage of students scoring in the higher levels of achievement. Neighboring New York, for example, has its own Pearson-designed CCSS-aligned exam, and the percentage of students scoring proficient or highly proficient was cut essentially in half to roughly 35 percent for both math and English….
….There is no reason to believe that 11th-graders today are any less skilled than their peers who took the HSPA last year or who took the NAEP in 2013, but there are plenty of reasons to believe that a drop in…
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Idly poking around the net I found this example:
Solve for x
Move all terms that don’t contain x to the right-hand side and solve.
I cried !
Read more from mathway.com
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:
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
I was on the virtually powerless governing body of the local primary school in the UK when the first National Curriculum came out, some time in the early 80’s. Very “New Math”y. Reworked a few years later. Here is some stuff from the UK Dept for Education about the latest rewrite. The old “Back to Basics” brigade are in the ascendant, but at least the UK is not drowning under High Stakes Testing. Have a look:
Key stage 1 and 2 (ages 5 to 10)
Key stage 3 (11 to 13)
key stage 4 (14,15)
ans about assessment
Only the dedicated study math in the last 2 years.
You might find this interesting as well, just look at how little time is spent taking tests, and then only in three of the years.
Then I found this. Looks familiar !
Why the big curriculum change?
The main aim is to raise standards, particularly as the UK is slipping down international student assessment league tables. Inspired by what is taught in the world’s most successful school systems, including Hong Kong, Singapore and Finland, as well as in the best UK schools, it’s designed to produce productive, creative and well educated students.
Although the new curriculum is intended to be more challenging, the content is actually slimmer than the current curriculum, focusing on essential core subject knowledge and skills such as essay writing and computer programming.
In almost all computer languages a string is just a list of characters, like this sentence.
The toString() method and the function String(..) both convert a number to a string.
With x = 12.3 and y=10000000
String(x)*2 gives 24.6
String(x)/String(y) gives 0.00000123
String((100 + 23).toString()-((-1)*String(x))) gives 135.3
(100 + 23).toString()+String(x) gives 12312.3
Work it out !!!!
I found this today. It’s worth a read.
Author: Gary S. Stager
Here’s an extract:
Conrad Wolfram estimates that 20,000 student lifetimes are wasted each year by school children engaged in mechanical (pencil and worksheet) calculations.
Expressed another way, we are spending twelve years educating kids to be a poor facsimile of a $2 calculator.
Forty years after the advent of cheap portable calculators, we are still debating whether children should be allowed to use one.
We are allowing education policy and curriculum to be shaped by the mathematical superstitions of Trump voters.
Educators need to take mathematics back and let Pearson keep “math.”
I found the following article ” The Coddling of the American Mind” while rooting about, after finding what I was looking for.
Read it, or at least some of it, as it is quite long. The fear of being upset is screwing the college sector, and my guess is that the UK won’t be far behind. “You can’t say that. We shouldn’t be reading that. Someone will be traumatised”, and so on. All must be protected. It is the death of critical thinking.