As March is set to be a month full of hard work for me, given I have my Maths and English exams just around the corner, and with the second to last unit of my UAL Level 2 Art & Design Course to go before I start my final unit in April, this month will be full of hard work, as well as extensive revision that goes towards my English and Maths exams this month.

As well as this, March is also set to be a month surrounding simplicity, as the featured image suggests, as I want to make sure that this month is based around simplicity, as well as the design of Mother Nature overall, and also, this month is set to be full of relaxing times, and being able to enjoy yourself for just how valuable life really is.

Over the course of this month, I will be focusing more on my exams that are coming up, as well as meeting deadlines for my UAL Level 2 Art & Design Course work. This means that I will, in some or most cases, have some trouble in finding the time to publish articles on Mother Nature. However, I will try my best to fit at least a few articles in when I have the free time available.

Until then, welcome to March on Mother Nature!!! A Relaxing Journey Awaits!!!

Alex Smithson

]]>- Obviously 64 of size 1
- 1 of size 8
- Horizontally from top left corner or from top right corner, you can build a side of length 7. Identically it could be done vertically, meaning that 4 squares of length 7 are fitting in the chess board
- Same reasoning applies to the other size

diverges. Such sets are said to be *large*. For example, , the set of all the positive integers, is large, because the harmonic series diverges. On the other hand, for some infinite sets the sum of the reciprocals of the members of converges instead; such sets are said to be *small*. For example, for every integer greater than 1, the set of all the positive integral th powers is small, because the series

converges. There is thus an interesting distinction between two different sizes of infinite sets of positive integers. Let’s have a look at some other infinite sets of positive integers and see whether they are large or small.

For every positive integer , the set of all the positive integral multiples of is easily seen to be large, due to the fact that the harmonic series diverges, and dividing the harmonic series by yields the series

In fact, this proof can easily be generalised to show that for every set of positive integers and every positive integer , is large if and only if is large.

What about the set of all the positive integers which are *not* multiples of ? The sum of the reciprocals of the members of can be written as the difference

The minuend and subtrahend of this difference are both equal to , so the sum of the reciprocals of the members of cannot be immediately seen from writing it like this. However, if we take out the factor of from the subtrahend, we get

and then we can remove the common factor of to get

It is easy to see that the value of this expression is also . So is large, as well.

In fact, for every finite set of pairwise coprime positive integers , , … and , the set of all the positive integers which are not multiples of any of these integers is large, too. This is because, if we let be the value of the harmonic series, then, using the inclusion-exclusion principle, the sum of the reciprocals of the members of can be written as

in which, for every integer such that , the th term is a sum over all the finite sets of integers such that . By factorising this expression, we can rewrite it as

and it is easy to see that the value of this expression is .

Generalising from this, one might expect that if we let be the prime numbers, so that the members of are pairwise coprime and every integer is a multiple of a member of , then

To prove this rigorously, all that is needed is to note that for every positive integer ,

where is the sum of the reciprocals of the integers greater than 1 which are not multiples of any of the first prime numbers. The least such integer is , so is smaller than the sum of the reciprocals of the integers greater than or equal to . Therefore,

As tends towards positive infinity, the sum of the reciprocals of the integers greater than or equal to tends towards 0, because it can be written as , which is less than , and by definition tends towards as tends towards positive infinity. So the right-hand side of the inequality above tends towards 1, and an application of the squeeze theorem completes the proof.

Rearranging (1) to solve for yields

so we now have a way of expressing the harmonic series as an infinite product. This is quite interesting in its own right, but it can also be used to prove a rather surprising result. By taking the natural logarithm of both sides, we see that

The natural logarithm of is , so this means the series on the right-hand side diverges. For every positive integer , the th term of the series on the right-hand side, , can also be written as , and this number is less than or equal to (in general, for every real number greater than , ). If is greater than 1, then this number, in turn, is less than or equal to , because is greater than or equal to . And , i.e. 1, is less than or equal to 1. So, by the comparison test, the series

diverges, too. This shows that the set consisting of 1 as well as all the prime numbers is large, but it is then an immediate consequence that the set of all the prime numbers, not including 1, is large (since we can subtract 1 from the above series and the value of the result will still have to be ).

This is a rather interesting and surprising result. The set of all the prime numbers is large, but for every positive integer , the set of all the positive integral th powers is small, which means that, in a certain sense, there are more prime numbers than positive integral th powers.

]]>Finished? Great. Did you also play in the sandbox at the end? I did. My favorite scenario is one where there is a clear majority of one of the shapes, and plenty of free space.

We start with the setting that our shapes are happy if at least 34% of their neighbors are of the same type. It takes a while for the simulation to come to a halt, but in the end we see that the minority has retreated to a small number of ghettos.

Now we give the shapes a small longing for diversity. They are no longer happy if *all* their neighbors are of the same type. Now the simulation does not stop within a reasonable time, but the general picture stabilizes quite quickly. The former ghettos keep a significant population of triangles, but now they are also popular among squares. They have turned into flourishing multicultural neighborhoods.

Of course reality is more complicated, but still there are clear parallels between this game and the real world. Take the recent history of some parts of Berlin for example. When Berlin was split, West-Berlin neighborhoods like Kreuzberg where unpopular because they bordered the East on several sides. After the wall came down, several neighborhoods in East-Berlin were abandoned by a large part of their population. In both cases that made for very low rents, so these neighborhoods became populated largely by poor people, many of them immigrants. The situation was a bit like the second image in this post.

Nowadays, many of those former ghettos have become so hip that not only many middle class people have moved in, but rent has increased so much that many of the poorer inhabitants have been forced to move out. The process went further than in our little simulation, but it is highly debatable whether that is a good thing. Those of you who understand German might be interested to read this article. It discusses the gentrification of the “Kotti”-area in Kreuzberg, where I lived myself until a few weeks ago.

Both my former neighborhood and my new one in the Moabit district have a cosy mix of triangles and squares. Apart from the German one, mainly the Turkish community is very large. Not only do Döner shops and Turkish supermarkets provide extra culinary choices, but is simply beautiful to hear not just *hallo*‘s in daily life, but also the occasional double-pass between *selamun aleyküm* and *aleyküm selam.*

The Parable of the Polygons was written by Vi Hart and Nicky Case. Nicky has made several other *playables*. I believe this medium has a huge potential, for non-fiction as well as fiction. If you are interested in more easily digestible maths, then you should definitely check out Vi’s youtube channel.

From Alex Bellos: the results of his global online poll to find the world’s favorite number…

The winner? Seven– and it wasn’t even close…

###

**As we settle for anything but snake eyes,** we might send symbolic birthday greetings to John Pell; he was born on this date in 1611. An English mathematician of accomplishment, he is perhaps best remembered for having introduced the “division sign”– the “obelus”: a short line with dots above and below– into use in English. It was first used in German by Johann Rahn in 1659 in *Teutsche Algebra*; Pell’s translation brought the symbol to English-speaking mathematicians. But Pell was an important influence on Rahn, and edited his book– so may well have been, many scholars believe, the originator of the symbol for this use. (In any case the symbol wasn’t new to them: the obelus [derived from the word for “roasting spit” in Greek] had already been used to mark passages in writings that were considered dubious, corrupt or spurious…. a use that surely seems only too appropriate to legions of second and third grade math students.)

]]>

if we plot X, like west to east, and Y, like south to north, zero in the lower left corner, and plot the coordinates of X = 1 then Y = 2, and X = 2 then Y = 5, and then X = 3 then Y = 10, we discover a line could be drawn roughly linking these coordinates that resembles an increasingly northerly sloping graph.

obviously, the area if we consider that, depends on the line and its distance to the X axis, so one could say that, the rate of change of area depends on the line drawn. logically, to continue, the rate of change of the line depends on the gradient at any point on the line, which is, a line drawn as parallel to the graph as possible.

if we examine the graph at X = 1 and Y = 2 the gradient looks steep but is obviously going to become much more steep as X goes on. Y/X = 2 and this seems like the gradient depending on how the line is drawn. notice that, ignoring in the equation, 1, that if we multiply the power by X we could write dY/dX = 2X, and that, if X is still 1, that dY/dX = 2. is that the gradient of the line?

if we, so to speak, if you like, reverse engineer that, to examine, how to obtain the area under the graph and, call it integration, we discover that the equation could be like X + 1/3 X power 3. think about it thus, reduce the power, multiply by the power, ignore that without an X after it. and thus in reverse, call it integration. if we let X = 1 then the integral would be 1 + 1/3 which equals about 4/3 which seems about right too.

can anyone prove that there are 3 dimensions to space? even the Universe contains a virtual reality game concept. and i’ve imagined how that works. imagine turning right, the empty grid of the visuals in front of you shifting left and so on.

CLEARCHARGE

]]>The OLS estimator is consistent when the regressors are exogenous and there is no perfect multicollinearity, and optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors be normally distributed, OLS is the maximum likelihood estimator. OLS is used in economics (econometrics), political science and electrical engineering (control theory and signal processing), among many areas of application.

]]>One of the first things I would recommend would be to invest in a small range of materials that allow you to implement some simple tasks that could then be expanded into interesting and worthwhile mathematical investigations. For example, if you purchase around ten sets of **playing cards** (go to a cheap two dollar store), you could learn a few basic games (Snap, Making 10, Playing with Place Value – see my book *Engaging Maths: Exploring Number)* that could then be differentiated according to the students you are teaching. A simple game of Making 10 could be used from Grade 1 all the way to Grade 6 by simply changing the rules.

Other materials that are a ‘must have’ for beginning teachers are **dice** and **dominoes**. There are many simple investigations that could lead from simple explorations with these materials. For example, use the dice to explore probability or play a game of *Greedy Pig. *Play a traditional game of dominoes before adding a twist to it, or simply ask students to sort the dominoes (students have to select their own criteria for sorting)– an interesting way to gain insight into students’ mathematical thinking and a great opportunity for using mathematical language. Once students have sorted the dominoes conduct an ‘art gallery tour’ and ask other students to see if they can work out how others have sorted out their dominoes. Photograph the sorting and display then on an Interactive Whiteboard for a whole class discussion and reflection…the list goes on!

Another ‘must have’ for beginning teachers is a bank of good quality resource books. Don’t fall into the trap of purchasing Black Line Masters or books full of worksheets to photocopy. You don’t want your students to be disengaged and you want to be called back for more work! Books such as my *Engaging Maths* series (http://engagingmaths.co/teaching-resources/books/ ), or any of Paul Swan’s books or resources (http://www.drpaulswan.com.au/resources/) are a great place to start. Explore some of the excellent free resources available online such as http://nrich.maths.org/teacher-primary and http://illuminations.nctm.org/, but do be aware that some resources produced outside of Australia will need to adapted for the Australian Curriculum: Mathematics.

In my early research on student engagement, I found that students would remember what they would recall as a ‘good’ mathematics lesson for a very long period of time. In fact, some of the students in my PhD study talked about a ‘good’ mathematics lesson two years after it had taken place. Although you might only be in a classroom for a very short time while you begin your career as a relief teacher, you can make an impact on the students in your care and the way the view mathematics by being prepared with your ‘toolbox’ of engaging and worthwhile activities.

]]>March 2-7, 2015

S – Science

T – Technology

E – Engineering

[A – Arts]

M – Mathematics

So we’ve all heard of STEM, and many of us have heard of STEAM, but do we actually do this in the classroom? I’m a 7th grade science teacher, but I mostly focus on science, with a little bit of technology thrown in. We look at data tables and make graphs, but admittedly, many of my students still have no clue the difference between the x-axis and the y-axis. Occasionally, I’ll spice it up a lesson with an engineering task. Arts? Well, sometimes we draw stick figures, does that count?!

I’ll be the first to admit that I’m not giving STEAM it’s proper place in my classroom. One of my biggest worries for the future of our students is that we will have a country filled with science-illiterate citizens making big decisions. As a teacher in California, I see a heavy emphasis on English and math, while all the other subjects are becoming secondary. I assume it is similar in other parts of the country. While literacy and math skills are important, we can’t discredit science, history, and the arts because they are what make many students want to show up to school. We all need to work together to encourage students to seek out STEM-related opportunities and careers. How will you help?

While considering STEAM, it is important to consider two underrepresented groups in STEM-related careers: minorities and women.

Minorities in STEM:

Image source: http://www.transportationyou.org/infographic-10-startling-stem-stats/

Women in STEM:

Image source: http://www.forbes.com/sites/bonniemarcus/2014/03/28/mentors-help-create-a-sustainable-pipeline-for-women-in-stem/

**More Resources**:

TED Talk “Growing up in STEM as a girl”:

http://tedxtalks.ted.com/video/Growing-up-in-STEM-as-a-girl-Ca

Article on STEAM from US News:

http://www.usnews.com/news/stem-solutions/articles/2014/02/13/gaining-steam-teaching-science-though-art

I look forward to chatting with you all this week. Remember, one question per day Monday through Saturday!

-Mari Venturino

@msventurino

**PS. If you’re like me, you get anxious when you don’t get a preview of the questions. Here they are! **

Q1: How do you incorporate STEAM into your lessons? #slowchated

Q2: How can we break down barriers to incorporating STEAM into our classrooms? #slowchated

Q3: Share your favorite STEM/STEAM resources! #slowchated

Q4: Why do you think minority students feel discouraged from pursuing STEAM-related careers? #slowchated

Q5: How do you encourage girls to get involved in STEM-related fields, especially in MS and beyond? #slowchated

Q6: What action step are you going to take next week to add more STEAM-related fun into your classroom? #slowchated

]]>

Mathematics educators and policymakers have mandated that problem solving and proof must hold an honored place in mathematics teaching and learning. In particular, mathematics educators should provide students with opportunities to experience what it is like to be a mathematician. However, problem solving and proof, especially as they pertain to the work of a mathematician, are under-researched owing to their elusive nature and resistance to easy analysis and quantification.

Investigations of problem solving and proof must inevitably account for the co-development of individuals’ mathematical concepts, higher order thinking, dispositions, beliefs, and other factors that are conducive to productive learning. Many researchers use the term “problem solving persona” to refer to this complex bundle of co-developing attributes that is attached to and intertwined with problem solving and proof. They emphasize that problem solving personas are always in flux.

My research aims to provide an informative description of the development of problem solving personas among pre-service middle school and high school teachers who are working on an unsolved problem in mathematics. Participants will be given the opportunity to work on obtaining partial results to the rational distance problem: Is there an interior point located at rational distance from each of the four corners of the unit square? Participants will also receive explicit instruction on metacognition via reading and reflection assignments that use Burger’s and Starbird’s popular book, *The 5 Elements of Effective Thinking*.

Studies have suggested that the upper limit of meaningful human relationships that a person can comfortably sustain with stability to be around 150. Since the objective of a project is ultimately success and credibility, it would stand to reason that the interactions between those involved in the project also be comfortably sustainable. We would then redefine the population of individual human relationships to individual interactions, and thus also desire to limit the number of interactions similarly to less than 150 in the interests of sustainability and project success.

In a two-person collaboration, the simplistic nature of the interactions allows for both parties to know and understand each others’ intentions and create consensus (though not always with agreement) with relative ease, as it requires a simplistic bilateral communication model. Once another project member is added to the group however, the complexity is increased exponentially, as the number of interactions needed for consensus becomes a product of all possible combinations, each resulting in it’s own bilateral understanding of a pair’s intentions.

Interactions can occur concurrently, however, symbolically in a person’s mind the intentions conveyed in each interaction are organized individually for each member of the group. Thus, in a 3-person group, the number of interactions on a given topic are characterized as the following interactions.

Person 1 & Person 2

Person 1 & Person 3

Person 2 & Person 1

Person 2 & Person 3

Person 3 & Person 1

Person 3 & Person 2

Mathematically, this would be represented using the factorial sum:

3! = 3 x 2 x 1 = 6 interactions

Thus, in order to keep the number of interactions between project members within the range of comfort and stability, a project team of 5 members is the maximum sustainable, as the factorial sum shows the number of interactions to be:

5! = 5 x 4 x 3 x 2 x 1 = 120 interactions

Additional decision makers introduced into a project will exponentially increase complexity of the project in terms of organization of interactions to produce understanding. There are further effects on complexity introduced within concepts of Game Theory which will not be discussed here, but it can thus be seen that simply adding one more decision maker to the project brings the total number of interactions to 720 for a given topic. Then doubling the number of decision makers to 10 introduces complexity bringing the total number of interactions to over 3.6 million, too far beyond what any person can reasonably comprehend or keep track of.

It’s therefore in the best interests of the project success and the credibility of those involved to keep the group of decision-makers for a project limited to a small set of individuals with access to the resources necessary to complete their project. Any more than the aforementioned upper limit of participants will incur such complexity as to create unnecessary difficulties in management of the project for it’s stakeholders.

]]>beyond language

where a soul’s conjecture

curves in Euclidean space, warped

within a matrix trace,

immersed in the possibility

of an infinite grace

she encounters relativity,

enmeshed in incompressible

geodesic loops limning the curvilinear

path of a lexical soul theorem

encapsulated by exotic spheres

spun – expanding and contracting

in a bounded singularity

where superposition exposures

underlie asymptotic entropy

forming polaroid parameters

of memory’s carving blade

two souls bound in isometric equation

meet upon the contrapositive vector field,

unraveling at the edge of imagination’s fire

interior symbols burn the

cartography of her exponential map

upon the topology of his geometry

demarcating areas of profound significance

at the paradoxical barrier

of parallels crossing,

impossibilities colliding,

converging on an ancient solution

Notes: A tribute to Grigori Perelman and dVerse Poets Pub, the only place in the universe that would allow me to share poetry about mathematics. The community at dVerse literally changed the course of my life trailing joy, friendship, and love in its wake.

]]>I used *Stella 4d* to make this. This program’s name, in the last sentence, is a link; if you follow it, you’ll be taken to a site where you can give it a try for free.

That’s why I’m leading this review of comics with Jef Mallet’s **Frazz** (February 27) even if it’s not transparently a mathematics topic. The biggest problem with calendar reform is there really aren’t fully satisfactory ways to do it. If you want every month to be as equal as possible, yeah, 13 months of 28 days each, plus one day (in leap years, two days) that doesn’t belong to any month or week is probably the least obnoxious, if you don’t mind 13 months to the year meaning there’s no good way to make a year-at-a-glance calendar tolerably symmetric. If you don’t want the unlucky, prime number of 13 months, you can go with four blocks of months with 31-30-30 days and toss in a leap day that’s again, not in any month or week. But people don’t seem perfectly comfortable with days that belong to no month — suggest it to folks, see how they get weirded out — and a month that doesn’t belong to any week is *right* out. Ask them. Changing the default map projection in schools is an easier task to complete.

There are several problems with the calendar, starting with the year being more nearly 365 days than a nice, round, supremely divisible 360. Also a factor is that the calendar tries to hack together the moon-based months with the sun-based year, and those don’t fit together on any cycle that’s convenient to human use. Add to that the need for Easter to be close to the vernal equinox without being right at Passover and you have a muddle of requirements, and the best we can hope for is that the system doesn’t get too bad.

The cause of calendar reform isn’t hopeless, if you’re worried about the calendar, but public discussion of ways to make the system better is at a relative low ebb. In the wake of World War I the idea of making a better world at least by making time-keeping better had some currency. The World Calendar Association published a Journal of Calendar Reform from 1931 to 1955. The League of Nations maintained a Committee of Inquiry into Calendar Reform for three years. In 1923 the Eastern Orthodox church adopted modified leap year rules to make its calendar a bit more precisely matched to the seasons; in 1928 Britain passed the Easter Act, which would allow an Order of Council to set the date of Easter to the Sunday after the second Saturday in April, rather than allowing it to drift around the calendar.

I admit I don’t expect a groundswell of calendar reform energy to appear anytime soon, since it seems like the messiness of the Gregorian calendar doesn’t rate the top 2,038 problems people have to face on any given day, but then I go back to doing some database work and remember how nice it would be to have a calendar that wasn’t an awful hack.

Among more obviously mathematics-based strips, Mark Anderson’s **Andertoons** (February 25) has (presumably) bad news presented in a cheery light by “rethinking the X axis”. There *can* be fair, non-deceptive reasons behind presenting charts in novel or unusual ways. The most popular kinds of charts are probably line and bar charts, and pie charts. Remarkably, all these methods of data visualization are credited to the same engineer/political-scientist, William Playfair, 1759 – 1823. The various graphs gained popularity at different points — the pie chart, particularly, came into its own after Clara Barton used it to explain to Parliament how people died in the Crimean War — but add the geographic map to that set and you have probably all the most-needed types of graph even to this day. And to point out the sort of tangled, interconnected world Playfair lived in, he had been a draftsman and personal assistant to James Watt, was one of the militia that stormed the Bastille, adapted the (semaphore) telegraph for use in England, and oversaw the 1806 edition of Adam Smith’s The Wealth of Nations.

**Andertoons** pops back into my pages on the 27th with fractions haunting a student. I can’t offhand think of a way that fractions would sneak into an English class, but there’s probably something somewhere. Anyway I like the kid’s look of wide-eyed despair.

Zach Weinersmith’s **Saturday Morning Breakfast Cereal** makes its required appearance around these parts on the 25th of February with a bit of absolute-value humor. Actually, the punch line postscript — “this is *one* reason why [ no one likes mathematicians ]” — is something I do. I think that’s more a grammar-nerd habit, though. The normal form, for me, is to start talking about (say) “the reason why” and realize I can imagine several valid reasons, so I trim the article down to “*a* reason why” instead.

Dave Whamond’s **Reality Check** (February 27) is an anthropomorphic-numerals joke that makes me wonder if the comedian wasn’t drawn sideways by accident. Could go either way, really.

Richard Thompson’s **Richard’s Poor Almanac** (February 27, rerun) includes a correction that “we’re sorry we ran that sudoku with all the transposed numbers. Hope it didn’t ruin your day, ha-ha!” A sudoku would be ruined only if the numbers were incompletely transposed, though; if you swapped out (say) every appearance of 1 for a 2, and 2 for a 1, the puzzle would still be as solvable, and not any harder. Remember that sudoku is a logic puzzle, not arithmetic; all that matters is that the symbols are distinguishable.

The last entry I have for this appearance, and this month, is an unusual one, and I admit it’s marginal in mathematics content but the drawing is hard to ignore. Peter Maresca’s feature **Origins of the Sunday Comics** on February 27 ran a February 1907 page titled **Animaldom**, by the cartoonist, painter, sculptor, and cowboy Joseph Jacinto “Jo” Mora. This entry, titled “The Wise Cat and the Fool”, establishes that you know the wise cat to be learn’d by his study of “French, Astronomy, Mathematics and the Classics”, and if you weren’t sure about his intelligence he’s shown with compass in hand, so you know he does geometry and stuff too. It’s an early-for-the-comics illustration of “does mathematics” as shorthand for “is really smart”. And for all his intelligence this doesn’t lead him to wealth, which matches my experience too, for what that’s worth.

From this bunch of strips I think the funniest is the second of the **Andertoons** and the poor kid finding fractions in all his classes. But the **Animaldom** is by far the best-drawn and the most interesting of them, even if it’s stodgy in that way I associate with children’s books from before Dr Seuss.

408’s factors are listed below the puzzle.

- 408 is a composite number.
- Prime factorization: 408 = 2 x 2 x 2 x 3 x 17, which can be written 408 = (2^3) x 3 x 17
- The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 408 has exactly 16 factors.
- Factors of 408: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408
- Factor pairs: 408 = 1 x 408, 2 x 204, 3 x 136, 4 x 102, 6 x 68, 8 x 51, 12 x 34, or 17 x 24
- Taking the factor pair with the largest square number factor, we get √408 = (√4)(√102) = 2√102 ≈ 20.199

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

]]>You will receive the set papers and marking schemes. After you submit to me your marks you will receive the merit list and certificates for the participating school, best performing school and best performing students (Top 3)

Call 0726537849 or write to danielochich@gmail.com

]]>