This article is focused on a 10th grade math classroom and how the interactive notebook is used. I really like the way the teacher implements the notebooks. The teacher uses the notebooks as a way for students to reflect after taking a test as well as a way to assess the students. I would like to be able to find topics I could have my students write about that I could use as an assessment. This is something I am going to continue to think about.

Reference:

Mason, R., & McFeetors, J. (2002). Interactive writing in mathematics class: getting started. *Mathematics Teacher*, 95(7), 532- 536.

Also below is a photo I snapped from Anton’s workshop in which we looked at the geometries of parallel parking. Oh, what fun! Other highlights included the mathematical art display, hearing and meeting the speakers on Friday and Saturday, and a rising star (or perhaps a growing snowflake), Jessica Li, who ran a fantastic workshop on modeling snowflake growth.

Thanks, Champaign and Bridges, you’re great!

]]>So using someone else’s awesome lesson plan, here’s my crazy kindergarten idea.

The lesson plan revolves around food! I’ve learned in my work with kiddos that they can be bribed into most any task with cookies, pizza, or candy. The fractions are referred to in the realm of halves or thirds, rather than in specific numeric figures. Actually, according to the National Council of Teachers of Mathematics (NCTM), at this level it is more important for students to recognize when things are divided into equal parts than to focus on fraction notation. So, there’s that. Anyway, after a couple of hours of sharing fun foods with friends and getting to do the divvying-up themselves (a thrill that shouldn’t be over-looked,) what will be their take away?

I have no doubt that the next time fractions are mentioned to those kindergarteners, they will immediately associate the word/concept with good feelings. Shoot, they may even be excited to learn further? Regardless, a mathematical concept that has historically encapsulated dread for so many, could now be anticipated and recalled with fondness. Not to minimize the task or importance of teaching the standards, but it seems that in situations such as these, the future dividends of a Friday afternoon dedicated to allaying future fears about math concepts, would be well spent.

Then, perhaps we could take it a step further and touch on fractional concepts a few times in the year. This fun art project is really a math puzzle in its most basic form.

Inspired by the art of Swiss-born artist Paul Klee, this interdisciplinary lesson plan mixes art and fractions. True, my kindergarteners might not understand it on a fractional level, but I will be planting the seeds for their future foray into fractions. Also, visual and tactile learners will surely retain the basics of the “parts of a whole” concept readily.

“More!,” you say? How about a literary component. As a parting gift, check out this dramatization of the fun intro to fractions book, “The Doorbell Rang”.

]]>

So here’s another geometry-based approach to finding what the radius of the circle that just fits inside the triangle has to be. We started off with the right triangle, and sides a and b and c; and there’s a circle inscribed in it. This is the biggest circle that’ll fit within the triangle. The circle has some radius, and we’ll just be a little daring and original and use the symbol “r” to stand for that radius. We can draw a line from the center of the circle to the point where the circle touches each of the legs, and that line is going to be of length r, because that’s the way circles work. My drawing, Figure 1, looks a little bit off because I was sketching this out on my iPad and being more exact about all this was just so, so much work.

The next step is to add three *more* lines to the figure, and this is going to make it easier to see what we want. What we’re adding are liens that go from the center of the circle to each of the corners of the original triangle. This divides the original triangle into three smaller ones, which I’ve lightly colored in as amber (on the upper left), green (on the upper right), and blue (on the bottom). The coloring is just to highlight the new triangles. I know the figure is looking even sketchier; take it up with how there’s no good mathematics-diagram sketching programs for a first-generation iPad, okay?

If we can accept my drawings for what they are already, then, there’s the question of why I did all this subdividing, anyway? The good answer is: looking at this Figure 2, do you see what the areas of the amber, green, and blue triangles have to be? Well, the area of a triangle generally is half its base times its height. A base is the line connecting two of the vertices, and the height is the perpendicular distane between the third vertex and that base. So, for the amber triangle, “a” is obviously a base, and … say, now, isn’t “r” the height?

It is: the radius line is perpendicular to the triangle leg. That’s how inscribed circles work. You can prove this, although you might *convince* yourself of it more quickly by taking the lid of, say, a mayonnaise jar and a couple of straws. Try laying down the straws so they just touch the jar at one point, and so they cross one another (forming a triangle), and try to form a triangle where the straw isn’t perpendicular to the lid’s radius. That’s not proof, but, it’ll probably leave you confident it could be proven.

So coming back to this: the area of the amber triangle has to be one-half times a times r. And the area of the green triangle has to be one-half times b times r. The area of the blue triangle, yeah, one-half times c times r. This is great except that we have no idea what r is.

But we do know this: the amber triangle, green triangle, and blue triangle together make up the original triangle we started with. So the areas of the amber, green, and blue triangles added together have to equal the area of the original triangle, and we know that. Well, we can calculate that anyway. Call that area “A”. So we have this equation:

Where a, b, and c we know because those are the legs of the triangle, and A we may not have offhand but we can calculate it right away. The radius has to be twice the area of the original triangle divided by the sum of a, b, and c. If it strikes you that this is twice the area of the circle divided by its perimeter, yeah, that it is.

Incidentally, we haven’t actually used the fact that this is a right triangle. All the reasoning done would work if the original triangle were anything — equilateral, isosceles, scalene, whatever you like. If the triangle *is* a right angle, the area is easy to work out — it’s one-half times a times b — but Heron’s Formula tells us the area of a triangle knowing nothing but the lengths of its three legs. So we have this:

(Right triangle)

(Arbitrary triangle)

.

Since we started out with a Pythagorean right triangle, with sides 5, 12, and 13, then: a = 5, b = 12, c = 13; a times b is 60; a plus b plus c is 30; and therefore the radius of the inscribed circle is 60 divided by 30, or, 2.

]]>The Pumpkins Preschool for kids activities for Wraysbury and Horton have been as full on as ever for the past two weeks and our apologies to all parents for missing our blog post last week.

Our topic for the week has been Winter at both Wraysbury and Horton and as you can guess Christmas has been a big feature at this exciting time of the year for our Pumpkins children.

Our Christmas Tree for ‘The Tree Festival’ being held at St Andrew’s Church this coming Friday and Saturday is almost finished and we’re delighted with how it looks.

Our Pumpkins children at Horton will be busy making the finishing touches in the coming days and Pumpkins Wraysbury for crafts this week made decorations for the tree last week.

Let’s all hope we can win for the 3rd year running!

We would like to say a big thank you to all of you that have supported The Outfit’s Advent Calendar 2014 appeal ‘Make a child smile this Christmas’.

The Food Bank collected what has been donated so far including 164 advent calendars, 66 selection boxes, 5 packets of chocolate Father Christmas’s and 5 large tubs of sweets! Well done The Outfit Datchet.

If you would like to support this good cause the final day for donations is Tuesday 25th November – please bring donations to Pumpkins Wraysbury or Horton :-)

This week Pumpkins Horton fancied their chances as budding hairdressers – if you think you’re having a bad hair day look at these poor heads!

At Pumpkins Horton we put some physical development beginning with block balancing and building small towers which for small hands is a challenge and takes practice. With plenty of patience from our team, the children demonstrated impressive levels of perseverance – well done children!

Pumpkins children Wraysbury also practised writing their names this week in preparation for when they start Wraysbury Primary School next year. They enjoyed pasta weighing and measuring to learn mathematics basics at Wraysbury.

Our board at Pumpkins Horton with all the children’s stockings are all ready for Christmas and looks just great! We had a lovely time this week decorating our Christmas tree too.

Pumpkins Horton have made lovely hand printed mittens – just have a look at the lovely results here!

A reminder that next week and and until the end of term, the topic at both settings will be Christmas. Pumpkins concludes 2014 for the Christmas and New Year break on Friday 19th December. Can we ask parents of Pumpkins Wraysbury children to please bring in a long piece of tinsel and a metal coat hangar for a Christmas craft activity – thanks for your help.

Pumpkins Preschool Facebook is constantly updated and we’re grateful to those parents who have been kind enough to leave some great Reviews for us :-)

As we near the end of 2014 can I thank all parents for all your support and co-operation and making sure your child’s learning experience at Pumpkins is such organised fun! A big thank you to my great team of helpers who continue to be committed and passionate in all that they do at Pumpkins Preschool.

If you are considering registering your child for a place at Pumpkins please call directly on07500 224115 or for out of hours enquiries please get in touch HERE.

Best wishes

Kellie Fairhall :-)

]]>Still curious, I next examined this polyhedron’s dual. The result was an unusual 36-faced polyhedron, with a dozen irregular heptagons, and two different sets of a dozen irregular pentagons.

*Stella 4d*, which is available at http://www.software3d.com/Stella.php, has a “try to make faces regular” function, and I tried to use it on this 36-faced polyhedron. When making the faces regular is not possible, as was the case this time, it sometimes produce surprising results — and this turned out to be one of these times.

The next thing I did was to examine the dual of this latest polyhedron. The result, a cluster of tetrahedra and triangles, was completely unexpected.

The next alteration I performed was to create the convex hull of this cluster of triangles and tetrahedra.

Having seen that, I wanted to see its dual, so I made it. It turned out to have a dozen faces which are kites, plus another dozen which are irregaular pentagons.

Next, I tried the “try to make faces regular” function again — and, once more, was surprised by the result.

Out of curiosity, I then created this latest polyhedron’s convex hull. It turned out to have four faces which are equilateral triangles, a dozen other faces which are isosceles triangles, and a dozen faces which are irregular pentagons.

Next, I created the dual of this polyhedron, and it turns out to have faces which, while not identical, can be described the same way: four equilateral triangles, a dozen other isosceles triangles, and a dozen irregular pentagons — again. To find such similarity between a polyhedron and its dual is quite uncommon.

I next attempted the “try to make faces regular” function, once more. *Stella 4d*, this time, was able to make the pentagons regular, and the triangles which were already regular stayed that way, as well. However, to accomplish this, the twelve other isosceles triangles not only changed shape a bit, but also shifted their orientation inward, making the overall result a non-convex polyhedron.

Having a non-convex polyhedron on my hands, the next step was obvious: create its convex hull. One more, I saw a polyhedron with faces which were four equilateral triangles, a dozen other isosceles triangles, and a dozen regular pentagons.

I then created the dual of this polyhedron, and, again, found myself looking at a polyhedron with, as faces, a dozen irregaular pentagons, a dozen identical isosceles triangles, and four regular triangles. However, the arrangement of these faces was noticeably different than before.

Given this difference in face-arrangement, I decided, once more, to use the “try to make faces regular” function of *Stella 4d*. The results were, as before, unexpected.

Next, I created this latest polyhedron’s dual.

At no point in this particular “polyhedral journey,” as I call them, had I used stellation — so I decided to make that my next step. After stellating this last polyhedron 109 times, I found this:

I then created the dual of this polyhedron. The result, unexpectedly, had a cuboctahedral appearance.

A single stellation of this latest polyhedron radically altered its appearance.

My next step was to create the dual of this polyhedron.

This seemed like a good place to stop, and so I did.

]]>Though it originally appeared in Indian mathematics, in Sanskrit, Leonardo of Pisa (known as Fibonacci) wrote about it in his book *Liber Abaci *in 1202.

*
*The graphic above shows the sequence. It is used in computer programs and has many other applications.

But the one thing that draws me to Fibonacci is the way you can find it in nature. You can see the sequence in everything from flower petals to fruits and vegetables and my favorite, the shell of the Nautilus.

I usually stink at math but in this case I find it to be quite beautiful. Poetic, even.

If you want to learn more about Fibonacci numbers and their applications here is a great website that explains it all.

I hope you enjoy your Fibonacci Day!

]]>If, on the other hand, this augmentation is performed only on the blue faces of the central icosahedron, the result is a tetrahedral cluster of five icosahedra:

The next augmentation I performed started with this tetrahedral cluster of five icosahedra, and added twelve more of these icosahedra, one on each of the blue faces of the four outer icosahedra. The result is a cluster of 17 icosahedra, with an overall icosahedral shape.

All of these images were made using *Stella 4d*, which is available at http://www.software3d.com/Stella.php.

When I was writing the post on Rough Mandelbrot Sets I tried out some variations on the rough set. One variation was to measure the generated M-set against a previously calculated master M-set of high precision (100000 iterations of ). In the image below the master M-set is in white and the generated M-sets are in green (increasing in accuracy):

Here *-instead of approximating with tiles-* I measured the accuracy of the generated sets against the master set by pixel count. Where the ratio of produced something that threw me, the generated sets made sudden but periodic jumps in accuracy:

Looking at the data I saw the jumps were, very roughly, at multiples of 256. The size of the image being generated was 256 by 256 pixels so I changed it to **N by N** for *N = {120, 360, 680}* and the increment was still every ~256. So I’m not really sure why, it might be obvious, if you know tell me in the comments!

I am reminded of the images generated from Fractal Binary and other Complex Bases where large geometric entities can be represented on a plane by iteration through a number system. I’d really like to know what the ** Mandelbrot Number System** is…

Below is a table of the jumps and their iteration index:

Iterations | Accuracy measure |

255 256 |
0.241929 0.397073 |

510 511 |
0.395135 0.510806 |

765 766 |
0.510157 0.579283 |

1020 1021 |
0.578861 0.644919 |

1275 1276 |
0.644919 0.679819 |

1530 1531 |
0.679696 0.718911 |

In this icosahedron, the four blue faces are positioned in such a way as to demonstrate tetrahedral symmetry. The same is true of the four red faces. The remaining twelve faces demonstrate pyritohedral symmetry, which is much less well-known. It was these twelve faces that I once distorted to form what I named the “golden icosahedron” (right here: http://robertlovespi.wordpress.com/2013/02/08/the-golden-icosahedron/), but, at that point, I had not yet learned the term for this unusual symmetry-type.

To most people, the most familiar object with pyritohedral symmetry is a volleyball. Here is a diagram of a volleyball’s seams, found on Wikipedia.

Besides the golden icosahedron I found, back in 2013, there is another, better-known, alteration of the icosahedron which has pyritohedral symmetry, and it is called Jessen’s icosahedron. Here’s what it looks like, in this image, which I found at http://en.wikipedia.org/wiki/Jessen%27s_icosahedron.

The rotating icosahedron at the top of this post was made using *Stella 4d*, a program which may be purchased, or tried for free (as a trial version) at http://www.software3d.com/Stella.php.

The natural numbers are the basis on which all other numbers are built: the integers, the rational numbers, the real numbers, the complex numbers, the hyper-real numbers, the nonstandard integers, and so on. The natural numbers are included in most other number systems.

Properties of the natural numbers such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

In common language, for example in grad school, natural numbers may be called counting numbers to distinguish them from the real numbers which are used for measurement.

**Properties**

**Addition**

One can recursively define an addition on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as “successor”. This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word group). The smallest group containing the natural numbers is the integers.

If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.

**Multiplication**

Analogously, given that addition has been defined, a multiplication × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (N*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.

**Relationship between addition and multiplication**

Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N is not closed under subtraction, means that N is not a ring; instead it is a semiring (also known as a rig).

If the natural numbers are taken as “excluding 0″, and “starting at 1″, the definitions of + and × are as above, except that they begin with a + 1 = S(a) and a × 1 = a.

**Order**

In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.

A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c with a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc. An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers this is expressed as ω.

**Division**

In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.

While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such that

a = bq + r and r < b.

The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

**Algebraic properties satisfied by the natural numbers**

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:

Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.

Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.

Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.

Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).

No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0.

And while this weeks post is late. It was worth it.

So here we go.

For now, we cannot build such a tool, for different reasons. Quantum computers could help.

2. http://physics.stackexchange.com/questions/143942/force-carrying-particles-instead-of-forces

Quantum field theories exploit such models.

3. http://physics.stackexchange.com/questions/110403/mathematical-description-of-electron-configuration

Schrödinger’s Wave Equation is the answer.

4. http://physics.stackexchange.com/questions/143658/is-the-continuity-discontinuity-one-of-the-intrinsic-properties-of-all-physica

Currently no answer regarding the concept of continuity. Is time continuous?

5. http://chemistry.stackexchange.com/questions/19184/orthogonality-of-chemical-elements

Think about fusion. How to generate high energies?

6. http://chemistry.stackexchange.com/questions/14671/electronic-model-with-highest-prediction-rate

Fantastic overview in the answer.

7. http://chemistry.stackexchange.com/questions/14701/invariants-in-chemistry-used-for-model-selection

Currently no answer regarding the invariants. Would be great to know for feature engineering.

]]>Somewhere out of this movement has emerged a second movement proclaiming that teaching coding will teach people to think – seemingly an insufficient number of people were thinking until programmers started banding together to enlighten us.

Yes, part of my objection is to the slightly condescending way these people relate to the rest of us rather than their actual arguments, but there I still have objections to the content of their argument as well.They mainly stem from the fact that they’re arguing for programming as a way of teaching thinking as though other ways of learning to think were not available. In point of fact, the notion of teaching is at least as old as Socratic philosophy, and exists in a wide variety of forms, from Western and non-Western perspectives.

Sometimes coding proponents go as far as to suggest that coding is an ideal to learn maths or logic. Maybe they have a point about logic – formal logic studies maybe too esoteric for a lot of tastes.

On the other to recommend programming as a way of learning maths is kind of odd. You can only learn maths by learning maths. Natural aptitude for maths is highly correlated with natural aptitude for programming – it’s hard to imagine those weak at maths will have an easy time in programming.The strong ones will learn whichever they spend time on – either way time away from maths coding is just time away from maths.

This last is the crux of it – the proponents of coding in schools discuss the idea as though they are several hours per week of fallow time up for grabs. There are not. Something else has to go to make room for time spent coding. My personal guess is that most of the proponents of the coding in school idea are thinking of something in the humanities rather than a science or maths subject (although at least one self-identified software developer commenting on another blog wanted to reduce arithmetic teaching in schools – as if our society wasn’t innumerate enough!). I’ve read a number of data scientist cvs – if I could change the education of that group of people, I’d be taking coding out and putting English lit in.

]]>*S*ometimes when you see what happens around the world and see how much mankind have moved forward since, since I don’t know – perhaps since he climbed down from the trees and learned to walk on his own 2 feet hundred thousands of years ago, you get this amazing, proud happy feeling.

One recent example is this:-

Many congratulations to the Rosetta team at **ESA** for successfully landing the lander module on a comet. It was not an easy thing to do, after all, the comet is flying past in space at more than 130,000 km/h. It will be interesting to wait and see what discoveries that we will find from this mission on a comet. Perhaps **confirming or disapproving the theory** that all water on this earth was brought down by a string of comets striking the earth millions years ago.

And last month, it was the India’s Mars Orbiter which made serious news and it was for a good reason too:-

The

Mars Orbiter Missioncost Rs. 450 crore (£46 million) in comparison toNASA’s Maven orbiter costing £413 million, which also successfully inserted itself into the Martian orbit on Monday 22nd September 2014.The Indian Prime Minister,

Narendra Modi, even stated: “Our program stands out as the most cost-effective. “There is this story of our Mars mission costing less than the Hollywood movie Gravity. Our scientists have shown the world a new paradigm of engineering and the power of imagination.”(

Source)

Whether you like it or not, the world have become very technologically advance. It has become small too – it is possible to go to any country in the world within a day (compared to months or years 200 – 300 years ago). Information these days is at one’s finger tips literally and connected to the world wide web 24 hours, 7 days a week.

**Sir Ken Robinson** in his talk in TED once said that as early as the 17th century, the industrial revolution drove the education blueprints of many nations towards science, engineering and mathematics. And that had given birth to some of the greatest minds that the world have seen. That industrial revolution plus two world wars that came later however had not stopped the advancement of mankind towards science and technology and whoever who do not embrace it at this age would be left out high and dry. Same thing happened to many companies in the 1990s who failed to embrace the digital age. Still remember Polaroid? And even Nokia, once a world leader in the telecommunication sector **is no longer is in existence** (after it was taken over by Microsoft).

That is why, the country as whole should emphasize more on science and technology instead of religion. Religion which have always been the scourge of science (or the other way around) should be left to individuals and should not form the backbone of a country. Think about it and take a good look of events around the world. There have been more people died and suffered in history due to religion indifference compared to people who died from say science experiments. There have been more dark ages brought by religion than by science. Some of the biggest tragedies in the recent times were done by twisted culprits claiming supremacy of religion and they continued to make an ass of themselves and the better aspect of a religion – such as **Boko Haram** in Nigeria and ISIS in Syria & Iraq.

And back home in Malaysia, we too been taking all the wrong steps. First we decided to teach **Science and Mathematics in Bahasa** instead of the accepted language of science and mathematics which is English. The lame excuse that was given was that the country needed to promote the national language and there was not enough teachers who can **speak proper English**. Unfortunately, despite the obvious reasons and calls from many quarters including the former Prime Minister, this decision have not been reversed to this day and the damage to the nation continues to this day.

But instead, we are entrenching ourselves with trivial issues like this:-

Non-Muslims in Kedah need not be worried or confused over the recent amendment to an 1988 enactment that bars non-Muslims from using Islamic religious words and terminology. State exco member Mohd Rawi Abd Hamid said no non-Muslim had been arrested in the state under the enactment for using terms that are exclusive to Islam and Muslims.

Mohd Rawi said non-Muslims could still use the words in their daily conversations, but not in their own prayers, public speeches or in religious publications.

“If you say you want to go to a masjid (mosque), why not? If you ask me where that masjid is, there’s no problem with the usage of that term,” he said yesterday

(

Source)

Earlier, the whole nation seemed to be busied itself with someone who had organized a “**want to touch a dog**” event and after that, of **an image on a water bottle**.

Obviously we are getting our priorities all wrong and that is why for reasons like this, we will not going to be a developed country in 6 years time. Forget achieving vision 2020. We are too worried on what we can say and do in the name of religion. We are not worried about building more schools, getting our children to embrace science and mathematics in the most convenient way (by learning in English) and push for greater space and opportunities for citizens to speak aloud and to agree or disagree with the establish norms. One wrong say or act in this country can make one run foul of the dreaded Sedition Act and the authorities. We cannot advance if we confine ourselves to very few options and old rules. We even **banned Darwin** from this country.

And that is why, the country as whole need to wake up to reality of things and what is important for the society survival? And if one still have doubts as to where the nation should be moving, perhaps this will give a food for thought:-

Science is the engine of prosperity. Economists have said that a third to a half of U.S. economic growth has resulted from basic research since World War II. The cars and trains that got us here today, our smart phones, the energy that lights this chamber, the clothes we wear, the food we eat: All of these were developed and improved through research.

And so it is. Science is a system for exploring, and for innovation. It can fuel our nation’s economic growth. It can form a path for our young people in a competitive global marketplace. And it can fire our imagination.

(

Source)

And mankind seems to be heading that direction too and probably in a greater pace due to the wealth of information available on the internet:-

Religion will become extinct by 2041 as the world becomes more developed and wealthier,, proclaims a new study by a noted author and biopsychologist Nigel Barber.

Barber makes the claim in his upcoming e-book, Why Atheism Will Replace Religion, which will be available next month. The Irish author says Christianity, Judaism, Islam, Buddhism and all other types of religious beliefs will be wiped out by atheism.

Barber notes there is a direct correlation between religious or atheist beliefs with economic development and level of education. The more educated and wealthier a society is, the less religious they are and vice versa. Barber adds that religion is most popular in underdeveloped countries.

(

Source)

Religion is important but it should be something personal and it should never mix with the state of a nation. Science on the other hand is going to be the stepping stone for many things to come to make a country strong, capable and flexible. Something for us all to ponder especially for the next generation of Malaysians. Good governance helps too but let’s start with something simple – give more focus on science and mathematics and put this country on the path of it’s own industrial revolution if you may. There must be a new paradigm of vision and a greater power of imagination. As I have said, religion is important and let’s not discard it from our life but not to a point that it drive the state **backwards and into the dark ages**. We just need to look into history and move forward.

Have a good weekend ahead…

*“The ancient Sumerians understood the connection between cycles, time and mathematics. In addition to the pragmatic use of the wheel or circle, they also developed the initial calculations of the circle to be 360 degrees. Their use of base-60 ‘sexagesimal’ math in the systematic measurement of time has carried with humanity to this day…”*

[This *mardukite.com* blog post is officially excerpted from *Liber 51/52*, available in the anthology **“ Mesopotamian Religion“**

The annual year was originally only divided into three seasons: beginning, middle and end. A year in Babylonia was separated into a cycle of 12 periods of 30 degrees or days. These periods, equated to the ‘moon’, were called ‘moonths’ or more appropriately ‘months’. Of course, the sky-wise priests were aware of the actual appearance of 13 lunar cycles in a year, so an additional shortened month was acknowledged to make the cycle fit. In most cases, a ‘new moon’ meant a ‘new month’ and so the days counted in a month are the days counted in the progression of a moon – though naturally the disparities between lunar and solar time had to be accounted for, and with time the ‘Chaldeans’ had perfected it.

The annual cycle was marked distinctly by two primary religious festivals – the spring festival of Akitu and the winter festival of **Zagmuk**. Both appear to be represented or distinguished by the symbol of ‘divine marriage’, later meaning the relationship between the ruling king and his lands.

Originally, however, the more popular fertility interpretation of these festivals, particularly in the spring, were based on land renewal and with the development and spread of these tradition, Akitu became known as *Ostara* – the pagan *Easter* – in dedication to Ishtar (Inanna). Not too surprisingly, the pre-Christian account incorporated into the symbolism of the later Judeo-based traditions also includes the proverbial theme of resurrection – in our case: the infamous story of Ishtar’s ‘descent‘ into the ‘Underworld’, where she is perceived of as ‘dead’ for three days.

Given the way modern calendars are oriented, the start of each ancient month would be considered near the ‘middle’ of current months – much like the seasonal observations. Although the festivals in ancient times were oriented to the naturally occurring solstices and equinoxes, it was often customary to observe them ceremonially during the closest full moon. All of this gave way to a generally ‘fluid’ incorporation of time into society that is varied in its interpretations among modern scholars.

The ancients made use of ‘water-clocks’ at night and ‘sun-clocks’ during the day. But more important to the survival of an agricultural society then gauging the minutes of a day for a ‘time-punch’ was the tracking of the annual cycle for planting and harvesting. Quite different than what the remainder of the Western World has familiarity with, the seasonal cycles in the deserts of Babylonia are unique. We have a recognizable summer in June, July and August where there is not rain and nothing grows – as we might expect – but then the region is plunged right into its rainy season in September, and farmers must be ready to plant their barley by October with a harvest necessary before the summer sun returns.

A different system of observation was used to calculate and measure ‘divine time’ in relation to ‘earth time’. This gave rise to what contemporaries call an ‘age’ – such as the current ‘age’ of *Pisces* and the forthcoming ‘age’ of **Aquarius**. Apart from the garbled nonsense of today’s horoscopes, the observation of zodiacal ages and alignments during the year are very real events. For whatever credibility the modern mind might wish to give the ancient astrologie omen tablets, the ability to perfectly chart time over long periods by using verifiable astronomical events, that we can even rely on today as investigators into this ancient culture, is quite impressive by any standards.

We can establish the chronological procession of the ages, but not necessarily a definition of when they have absolute turning points. They are measured in 2,160 year periods (72 x 30), connected to their ‘domain’ of visibility in the ‘Celestial Sphere’. The progression is visible in the stars but the clear boundary line that defines each is in many ways obscure. For example, when specifically does the current age enter ‘Aquarius’? Counting backward, the Mardukite school of thought might have suggestive input to apply.

Following earlier thwarted attempts to solidify global rulership, the real Babylonian Reformation by the Anunnaki god **MARDUK**, with the aid of Nabu, occurred as a result of the ‘Age of Aries’ having arrived and ‘promised power’ not being ‘passed’ to him. This would have to be circa **2150 B.C.**, when the movement became notably public.

How long prior to this, pointing to the turn of the ‘Age of Aries’, would they have waited? If it were only ten years, then the Piscean Age really would have been marked by the birth of Jesus Christ – exactly 2,160 years later – notoriously represented by the fishes. Other ‘scholarly’ dates for the start of the Age of Aries include 2200, 2150, 2000 and 1875 B.C. Based on these figures, this ‘era’ of reportedly ‘new consciousness’ could be in effect now, later in this century, in the year 2150 or even closer to 2600…

Time, as we have found no different today then yesterday, is indeed entirely relative!

]]>From initial concept to a tangible book-in-the-hand is a long, hard journey – ask any author – but the satisfaction of finally holding and perusing the end result is worth it all.

After publishing my first book on motion physics for the layperson four years ago, I was in no way ready to consider beginning yet another book. However, for both authors and imaginative inventors, a good idea is hard to resist, and the theme of America’s students struggling in science and math relative to students in other countries proved too important and interesting to pass up. More important than national test scores and rankings are the frustrations felt by many parents, guardians, and teachers when their students are underperforming in school.

**Why Do So Many Students Struggle with**

**Learning – Especially in Science and Math?**

Student standardized test scores in science and math *are* mediocre at best and falling for America’s students relative to many other countries – a rather shocking development. Once I began to seriously reflect upon why so many students are underperforming in school, the reasons quickly became clear to me.

Diagnosing the problem was the easier half of the drill; finding cures for the ailing performance of so many of our students proved more challenging, yet I am confident in my ultimate RX prescription for healing our students’ academic woes. **The integrated guide and plan I offer as a remedy for parents, guardians, mentors…and teachers, too, is based on common-sense parenting/mentoring and learning principles – many of which have been lost to recent generations.** Today’s ubiquitous technology, while often very helpful and even necessary, is also identified as a significant cause of our problems – but by no means the only one.

As I wrote the book and solicited comments, one that surfaced more than once went like this: *“The parents and guardians who, together with their students, most need the guide and plan you offer in the book, are the least likely to buy it.”* I sadly agree, to an extent, but remain confident that many struggling parents and guardians *will* take advantage of my ideas and suggestions.

** I envision a very viable market for the book with prospective parents and the parents of preschoolers who wish to be proactive in maximizing school success by providing an early, nurturing environment for their youngsters.** Not everyone is initially equipped by nature with the insight required for effective parenting/mentoring. Good parenting is like so many other ventures in life: The best way to proceed is by working hard

**For a closer look at the book and how to order it, click on “My New Book on Science / Math Education” on the blog header or click on the following link:**

**https://reasonandreflection.wordpress.com/about-my-new-book/**

**To go directly to the book’s dedicated website for still more information and to order, click on the following link:**

**http://reasonandreflection.com/book2/**

**For an excerpt from the book, also see my previous post: “Teaching Children Math…By Example,” in the archives for Sept. 27, 2014. Click the following link:**

**https://reasonandreflection.wordpress.com/2014/09/27/teaching-children-math-by-example/**

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I made a couple of graphics a few weeks ago anticipating my 300th post.

Observations:

- The only number with exactly one factor is one.
- The number of factors of all the black integers on the chart are powers of 2. These integers have irreducible square roots.
- Consecutive integers with the same number of factors can only occur if the number of factors is NOT a prime number.
- 90/300 or 30% of the first 300 integers have 4 factors. (Largest group)
- 62/300 or 20.6666% of the first 300 integers are prime numbers and therefore have 2 factors. (2nd largest group)
- 49/300 or 16.3333% of the first 300 integers have 8 factors. (3rd largest group)
- 40/300 or 13.3333% of the first 300 integers have 6 factors. (4th largest group) All integers with 6 factors have reducible square roots.
- 22/300 or 7.3333% of the first 300 integers have 12 factors. (5th largest group)
- All but 37/300 or 12.3333% of the first 300 integers are in one of those 5 groups.
- 118/300 or 39.3333% of the first 300 integers have square roots that can be simplified.

Warning: the next chart and observations could make your brain hurt:

- How many numbers have exactly 2 factors? Euclid proved that there is an infinite number of prime numbers which means there is an infinite number of integers with exactly 2 factors.
- How many integers have exactly 19 factors? Even though the smallest integer with exactly 19 factors is 262,144, there is still an infinite number of integers with exactly that many factors. The integers in that list are each prime number raised to the 18th power. Since there are an infinite number of prime numbers, there is an infinite number of integers with exactly 19 factors.
- {2^996, 3^996, 5^996, . . . } is the infinite list of integers with exactly 997 factors. Likewise {2^(p-1), 3^(p-1), 5^(p-1), . . . } where p is a prime number is the infinite list of integers with exactly p factors.
- If the number of factors is c, a composite number, then it could be said that there is more than an infinite number of integers with that many factors because the infinite list of integers will include {2^(c-1), 3^(c-1), 5^(c-1), . . . } as well as many other integers.

that arise from common counting techniques known to most who do high school and middle school math competitions.

First, the classic example: how many groups of people can be made from a group of size ? One way of finding this is to use the sum on the left side of the identity, as where results in the number of ways to choose a group of size from a group of size . As for the right side, imagine each person is being asked if they want to be in the group. They have the option to deny or the option to accept. Therefore, with two choices a person, the number of groups becomes .

Second, imagine flipping a coin times. Then the probability of getting exactly heads is

since the binomial expression calculates the number of combinations of coins with heads and the is the total. Summing the probability of getting heads from 0 to yields one because the total probability is always one. Then multiplying both sides by yields the identity above.

Third, imagine an ant on the origin of the coordinate plane lattice grid, meaning the ant can only travel along segments parallel to the axes connecting points with integral coordinates. Think of how many ways this ant can travel away from the origin in steps, with the restriction the ant can only go right and up. Then the ant has the choice of right or up each step, leading to possible ways to travel. However, there is another way of enumerating this. Let U stand for an up move and the total number of up moves and let R stand for a right move and the total number of right moves. Then U + R = n. If you fix U and n, then R is also fixed. Therefore the enumeration becomes the binomial sum stated above.

Finally, simply plug into the binomial theorem.

]]>This image above has only one polyhedron-type hidden from view, in the center: a red truncated cube. Next, more of this pattern I just found will be added.

The next step will be to add another layer of blue octagonal prisms.

This was an accidental discovery I made, just messing around with *Stella 4d*, a program you may try for yourself at http://www.software3d.com/Stella.php. The next cells added will be red truncated cubes.

Next up, I’ll add a set of pink rhombcuboctahedra.

The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

Now I’ll add more of the red truncated cubes.

At this point, more yellow cubes are needed.

The next polyhedra added will be pink rhombcuboctahedra.

And now, more of the blue octagonal prisms.

As long as this pattern is followed, this may be continued without limit, filling space, without leaving any gaps.

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