## Welcome to H3 Maths

Blog Support for Growing Mathematicians

### Bletchley Park – Home of the Codebreakers

March10

It was amazing. We had someone visit us recently who’s mother worked at Bletchley Park – home of the famous code breakers of World War II. Of course, her mother never told anyone – not even her husband – about the work she did. She had been a bright, young banker, and was selected for her fast and accurate mind.

Now a historic treasure, Bletchley Park has an excellent collection of online resources for the student fascinated by the exploits of those responsible for cracking (and making their own) secret codes during WWII.

This site is an opportunity to explore the different types of machines and codes that were using during World War Two, and even listen to personal stories of the hard, hot work done at Bletchley – “It was like making butter…” Discover the mathematicians who helped crack the incredible number of combinations they were faced with:

John Herivel was a mathematician who made a breakthrough in the search for a way to get into the German Enigma codes. “At the time that Herivel started work at Bletchley Park, Hut 6 was having only limited success with Enigma-enciphered messages, mostly from the Luftwaffe Enigma network known as “Red”. He was working alongside David Rees, another Cambridge mathematician recruited by Welchman, in nearby Elmers School, testing candidate solutions and working out plugboard settings. The process was slow, however, Herivel was determined to find a method to improve their attack, and he would spend his evenings trying to think up ways to do so.” Read more details about Herivel’s work on Wikipedia.

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### When will I ever use this Mathematics stuff???

February15

I was asked a very good question in class recently. “When will I ever use this Mathematics that I am doing?” one student asked me, and it was a genuine question, not an attempt to get out of classwork! My reply was something like this – “You probably never will use it – at least not the exact work we are doing. However, because you make a real effort to solve these problems in class, you are more likely to want to work through problems in life – in your home and in your work.” Of course, there are so many different careers that require a foundation of mathematical skills. For example, this one – a Psychometrician. “A what?” you might be asking?
A psychometrician is a person who practices the science of measurement, or psychometrics. Youo can check out the math requirements for this rather strange career, as well as a host more, at this great Math Careers site.

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### At last – a picture of an atom!

February14

Well, dear bloggers, it has been a while since I posted – excuse after excuse (Christmas and New Year and teaching, etc.). So, here is something to get your intellectual taste buds going – a picture of an atom. This strontium atom (the little dot in the center) is emitting light after being excited by a laser, and it’s the winner of the UK’s Engineering and Physical Sciences Research Council (EPSRC) photography award. The EPSRC announced the winners of its fifth annual contest yesterday. Winning photographer David Nadlinger, graduate student at the University of Oxford, was just excited to be able to show off his research.

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### A Baby is Born 1 min into the New Year – What are the odds?

January1

As fireworks were fired off from Auckland’s Sky Tower to welcome 2018, the country welcomed the first baby of the new year. The healthy baby boy, weighing 3.5kg, was born at Auckland Hospital at 12.01am to a Howick couple on the first day of the new year.

There are 526,600 minutes in a year, meaning the odds of giving birth at 12.01 on January 1 are one in 526,600.

The parents (above) have named their precious bundle Rex — which has nothing to do with being a new year baby, but because they say he looked like a T-Rex at his first ultrasound. (source: NZ Herald)

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### Algebra Booklet for Junior Algebra

December31

The New Zealand Centre of Mathematics offers a free Algebra Booklet for teachers and students. This is an excellent resource and should help improve algebraic skills for Year 9-11 (14-16 yr old) students. This 82 page booklet contains work on simplifying expressions, exponents, solving simultaneous equations using algebra and graphs. This website also includes videos, etc. to assist across a range of topics. [H3 recommends this website as a rich resource for keen math students to explore and revise.]

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### Best 10 Math Movies for Middle+ School Students

December20

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### Back to the Future Pt2

December16

Ben Tippett, a mathematics and physics instructor at UBC’s Okanagan campus, recently published a study about the feasibility of time travel. Tippett, whose field of expertise is Einstein’s theory of general relativity, studies black holes and science fiction when he’s not teaching. Using math and physics, he has created a formula that describes a method for time travel.

People think of time travel as something as fiction,” says Tippett. “And we tend to think it’s not possible because we don’t actually do it. But, mathematically, it is possible.”

Ever since HG Wells published his book Time Machine in 1885, people have been curious about time travel–and scientists have worked to solve or disprove the theory, he says. In 1915 Albert Einstein announced his theory of general relativity, stating that gravitational fields are caused by distortions in the fabric of space and time. More than 100 years later, the LIGO Scientific Collaboration–an international team of physics institutes and research groups–announced the detection of gravitational waves generated by colliding black holes billions of light years away, confirming Einstein’s theory.

The division of space into three dimensions, with time in a separate dimension by itself, is incorrect, says Tippett. The four dimensions should be imagined simultaneously, where different directions are connected, as a space-time continuum. Using Einstein’s theory, Tippett says that the curvature of space-time accounts for the curved orbits of the planets.

“The time direction of the space-time surface also shows curvature. There is evidence showing the closer to a black hole we get, time moves slower,” says Tippett. “My model of a time machine uses the curved space-time — to bend time into a circle for the passengers, not in a straight line. That circle takes us back in time.”

While it is possible to describe this type of time travel using a mathematical equation, Tippett doubts that anyone will ever build a machine to make it work.

“HG Wells (seen here) popularized the term ‘time machine’ and he left people with the thought that an explorer would need a ‘machine or special box’ to actually accomplish time travel,” Tippett says. “While is it mathematically feasible, it is not yet possible to build a space-time machine because we need materials–which we call exotic matter–to bend space-time in these impossible ways, but they have yet to be discovered.”

For his research, Tippett created a mathematical model of a Traversable Acausal Retrograde Domain in Space-time (TARDIS). He describes it as a bubble of space-time geometry which carries its contents backward and forwards through space and time as it tours a large circular path. The bubble moves through space-time at speeds greater than the speed of light at times, allowing it to move backward in time.

“Studying space-time is both fascinating and problematic. And it’s also a fun way to use math and physics,” says Tippett. “Experts in my field have been exploring the possibility of mathematical time machines since 1949. And my research presents a new method for doing it.” See also “Back to the Future 1” [Note: HG Wells was friends with Einstein] from: Science Daily

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### An Odd bit of Math

December10

ODD and EVEN NUMBERS. The Pythagoreans knew of the distinction between odd and even numbers. The Pythagoreans used the term gnomon for the odd number.

A fragment of Philolaus (c. 425 B. C.) says that “numbers are of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two.” [I know that sounds odd]

Euclid, Book 7, definition 6 says:An even number is that which is divisible into two parts.”

So the ancient Greeks had a word for “odd” that was the word they used for this kind of shape:

An “odd” number is one that makes that shape (with a bit sticking out) when you try to arrange it in two rows:

o o o o
o o o

There is an “odd” object left over that doesn’t line up. An even
number makes an “even” rectangle:

o o o o
o o o o

You can see in the quotation above that the words “odd” and “even” were used of numbers in English by the 1400’s; the words were used in other ways earlier. [from Mathword here]

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December7

# What are the Odds?

Professor of mathematics, Peter Stoner (June 16, 1888 – March 21, 1980), gave 600 students a math probability problem that would determine the odds for one person fulfilling eight specific prophecies. (This is not the same as flipping a coin eight times in a row and getting heads each time.) First the students calculated the odds of one person fulfilling all the conditions of one specific prophecy, such as being betrayed by a friend for 30 pieces of silver. Then the students did their best to estimate the odds for all of the eight prophecies combined.

According to the Hebrew requirement that a prophecy must have a 100 percent rate of accuracy, the true Messiah of Israel must fulfill them all or else he is not the Messiah. So the question that either vindicates Jesus or makes him culpable for the world’s greatest hoax is, did he fit and fulfill these Old Testament prophecies?

Let’s look at two of the specific prophecies about the Messiah in the Old Testament.

1. “You, O Bethlehem Ephrathah, are only a small village in Judah. Yet a ruler of Israel will come from you, one whose origins are from the distant past.” (Micah 5:2, NLT)

2. “The Lord himself will choose [a] sign. Look! The virgin will conceive a child! She will give birth to a son and will call him Immanuel-‘God is with us.’” (Isaiah 7:14, NLT)

The students calculated that the odds against one person fulfilling all eight prophecies are astronomical-one in ten to the 21st power (1021). To illustrate that number, Stoner gave the following example: “First, blanket the entire Earth land mass with silver dollars 120 feet high. Second, specially mark one of those dollars and randomly bury it. Third, ask a person to travel the Earth and select the marked dollar, while blindfolded, from the trillions of other dollars.”¹

It’s important to note that Stoner’s work was reviewed by the American Scientific Association, which stated, “The mathematical analysis … is based upon principles of probability which are thoroughly sound, and Professor Stoner has applied these principles in a proper and convincing way.” ²

Bible scholars tell us that nearly 300 references to 61 specific prophecies of the Messiah were fulfilled by Jesus Christ. The odds against one person fulfilling that many prophecies would be beyond all mathematical possibility. It could never happen, no matter how much time was allotted. One mathematician’s estimate of those impossible odds is “one chance in a trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion, trillion.”³

Source: from http://y-jesus.com/what-are-the-odds/

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### Christmas by the Numbers

December6

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#### Post Support

The pink triangle is one third of the area of the square. Use half base x height formula for area of the triangle, etc.

The coffee cup logic puzzle – Answer is Cup 5 as all the others have blocked pipes. 🙂

6×6 for the maximum dog pen area of 36 sq meters.

Oxford Exam Answer: According to Rebecca Cotton-Barratt, of Christ Church, this maths question tests abstract thinking”

“I’d initially ask the candidate what shape they think will be formed, and then ask them how they can test this hypothesis,” Cotton-Barratt says.

“They might initially try sketching the ladder at different stages – but ultimately what we want is something that we can generalise and that is accurate (you can’t be sure that your drawing is that accurate, particularly when you’re making a sketch on a whiteboard and don’t have a ruler). So eventually they will fall back on maths, and try to model the situation using equations.

“If they get stuck we would ask them what shape the ladder makes with the wall and floor, and they’ll eventually spot that at each stage the ladder is forming a right-angled triangle. Some might then immediately leap to Pythagoras’ Theorem and use that to find the answer (which is that it forms a quarter circle centred on the point where the floor meets the wall).Of course, Pythagoras could easily find the hypotenuse – it is the green line along the water! (Hint: the hypotenuse is always opposite the right angle!)
………………

Frustratingly there is no definitive answer to the riddle, leaving guessers with no choice but to continue scratching their heads.

Dr Kevin Bowman, course leader for Mathematics at the University of Central Lancashire said: ‘You can interpret it in many ways; one way is no more correct than another.

“There’s no ambiguity in the first equation; 3 apples is 30, so one apple is worth 10.
The Fruit Puzzle…
This isn’t the first mind-bending puzzle to sweep the internet in recent months. Earlier this year, National Geographic’s puzzle asking you to identify which direction a bus is travelling in left thousands of adults scratching their heads (see earlier post). One person suggests that, “because all the bananas aren’t the same, you could say that they all represent different amounts. You might even say that the two coconut pieces in the third equation are different sizes, and therefore add up to three quarters or even seven eighths when put together. In that sense, there are an infinite amount of possible answers.”

Dr Kevin Bowman, course leader for Mathematics at the University of Central Lancashire said: ‘You can interpret it in many ways; one way is no more correct than another.

“There’s no ambiguity in the first equation; 3 apples is 30, so one apple is worth 10.”

Another said, “1 apple equals 10, coconut equals 6 and banana bunch equals 4 so your answer is 20.”

————————————————————————————-
All exterior angles of one coin add up to 360 degrees. Since a coin has 12 sides, each exterior angle = 30 degrees. Two angles are formed between the two coins. Therefore, the angle formed is 60 degrees.

Quite an easy pattern in the Oct 10-11 Post. Subtract the first two numbers to get the first number in the right column; add the first two numbers in the left column to get the last two of the right column!

Parking Lot Puzzle: Turn your computer screen upside down (or stand on your head), then it becomes easy 🙂

In each row, adding gives the last 2 digits and subtracting gives the first.

The blue cherry picker has an extension arm that can’t be seen very well. This has placed the workers closer to the camera and created a strong false sense of scale simply because your eye assumes that the workers should be on the same plane as the base of the cherry picker!

Yes, it was Major General Stanley in the “Pirates of Penzance!” Check out the link in the picture.

The extra rope needed is exactly 2 x pi or 6.28m!

Christmas Teaser: Today is the 1st of January. Bill’s 8th birthday was yesterday, so the day before (December 30) he was still 7 years old. This December he will turn 9 and, next year, will be 10!

What did the math mother feed her new baby? Formula Milk!

What is a bubble? It is a thin sphere of liquid enclosing air (in most cases) or another gas.

Number of toes = 5170

How many Mathematicians to change a light bulb? Why, n+1 of course (one to hold the light)!

Jan 24, 2014: Assuming a free fall rate of 9.8m/sec/sec it would take just 4.06sec to fall 81m.

= 1 (see first line in the post)

Yes, the TV show with hints of Mathematics and Physics (along with the usual tensions of flatmates?) – did you choose 79?

Leonhard Euler (1707-1783) was an incredibly productive mathematician who published almost 900 books! He took an interest in Latin Squares – grids where each row and column each contains a member of a set of numbers. This forms the basis for Sudoku!

Trig Ratios post: yes, the Sine and Cosine ratios are the same when their angles add up to 90 degrees! This relationship can be expressed as: Sine A = Cosine (90-A) or Cosine A = Sine (90-A)
Good work in identifying this trig pattern. Now, here is a follow up questions which we will address in the next post. Does this pattern suggest that there is a link between Sine and Cosine ratios? Come on, come on… be quick with your answer…Yes, well done – of course there must be!!

Yes, zero is an Integer (which keeps to negative and positive integers apart).

Sam had to position himself to make sure that he 8 the chocolate!

There are 7 days in a week (i.e. Modulo 7). 490 days will be the same day that you chose, so the 491st day will be tomorrow!

Yes, C is the missing section – giving the same difference between numbers in the rows and columns.

That’s a mean looking crocodile! Unless, of course, you knew that it measured just 40cm – yes, just over a foot long!! The camera’s wide angle lens has distorted the image and this makes tiny croc look menacing!

Yes, the 100m time for Bolt works out to be 37kms/hr or 22mls/hr. Of course that is just the average time, not the max speed he reached!

Category 3 climbs last approximately 5 kilometres (3.1 miles), have an average grade of 5 percent, and ascend 150 metres (500 feet).

Category 2 climbs are the same length or longer at an 8 percent grade and ascend 500 metres (1,600 feet).

Category 1 climbs last 20 kilometres (12.4 miles) with an average 6 percent grade and ascend 1,500 metres (4,900 feet).

Category H climbs are the hardest including an altitude difference of at least 1,000 metres (3,280 feet) from start to finish and have an average grade of at least 7 percent.
….
Finding missing numbers is great fun and many readers are regular users of Sudoku. In the recent post (July 13) we find that the sum of the numbers in each row and column is 6, 12, ? Therefore, we need to get 18 as the sum in the final row and column. So, 9 is the missing number in order to complete the puzzle.

Great to see some recent posts on Calculus and we hope that some of our junior students (Years 6+) have a close look at these and develop an interest in this (more advanced) Mathematics.

Trend lines are a practical way to analyse the patterns of data over time and are particularly helpful in population, commerce and environmental change, such as the arctic ice post. The best way to find the answer to the question posed in this post is to click on the original article, copy the graph and paste into (e.g.) Word, using the landscape format. Then, using a ruler, carefully draw the same lines that I have shown in the post. This will help arrive at a more accurate answer. When you have the answer, post a comment to the blog and we can check it out to see if you are right (or close). Good luck Junior Mathematician!

1 year = 31 556 926 seconds

1729 – A rather dull number?
The mathematician G. H. Hardy was visiting the Indian mathematician Ramanujan while he was ill in hospital. Hardy was making small talk and remarked that 1729, the number of the taxi that brought him to the hospital, was a rather dull number. “No Hardy!” repled Ramanujan, “It is a very interesting number. It is the smallest number which can be expressed as the sum of two cubes in two different ways!” You see, even “dull” numbers have special properties!

#### Blog Diary

Dear Blog Diary,

Our night sky has always fascinated H3, and there have been some recent releases of amazing images from our nearby galaxies. The size and sheer complexity of our solar system is staggering and, mathematically, quite difficult to describe because the numbers are simply so big!

The fireworks background gives readers some idea of how students feel when they suddenly get a mathematical concept and can apply it with success. This is what excites learners to do well in their math studies. This is also what inspires teachers to want to help students have these "aha" moments! As the famous Winston Churchill said, "Never, Never, Never, Never, Never give up on your maths!" (Well, he almost said that).

The "x" factor - it was intriguing to see the TED talk post that explained why we use x to indicate an unknown quantity in Algebra. Hope our readers also enjoyed this view on what we take for granted in our everyday Mathematics.

Lewis and Clark explored routes to the American west...all the way to Oregon City where, today, there is a great museum to herald this famous migration period (see link in the post). So, the header image show canoes heading in which direction? East? How do you know? Should mathematicians expect every picture or drawing to point north? NO, of course not! So, to answer the post question - the canoes could be heading in ANY direction!

I had a discussion with a fellow teacher the other day that was along the lines of how sad it was that students today have lost a sense of fine craftsmanship when it comes to products and services. For example, old cameras were beautifully crafted and lasted, with regular servicing, for up to one or two generations. Today, with our "instant society" we are surrounding with products that have little permanency. The revival of fine architecture in the Art Deco movement is a recent highlighted post. In the same way, important mathematical proofs are timeless and give us all a better sense of something solid and permanent in our fragile world. I do hope that students who engage in Mathematics at any level also share this passion for numbers, patterns and proofs that are fixed and reliable signposts in a sea of turbulent ideas and rapid change.

Thanks to the positive feedback from Warren in Perth who wrote, "Congrats and good luck in your crusade to bring the joy and beauty of maths back to schools." See the Welcome page for the full comment. It is always great to have helpful ideas and feedback from blog readers. Again, thanks so much for taking the time to read H3 Maths.

It was in the news recently that Apple was looking to spend some \$97 billion - that's 97,000,000,000. At the rate of \$1000 a day, it would take an incredible 265,780 years to spend. That's an insane amount of money and it would be a good exercise to work out how this amount could help fix some of the big issues in the world today, such as the debt crisis in Europe, or Global Warming.

Being able to "roughly" work out an answer in Mathematics is called "Approximation". A good example of using this is in the little test post from the New York Times - looking at the rise in median house prices across a period of time. The answer is lower down in this column... :-)

Above is an algebraic expression with two sets of brackets, -
(x+1)(y-2). The brackets mean "multiply" so each bracket is a factor of an expanded algebraic expression. There are four parts to the bracketed factors, hence the term "quadratic" which comes up often in Year 9 and Year 10 (Freshman and Sophomore) grades. As a growing mathematician you will need to become competent with factorising and expanding algebraic terms.

Great to see so many visitors from 17 different countries - a Prime Number as well! Of course, there are more countries in our Visitor list but they did not show up on the new clustr map.

The blog about maths being all about language is really not entirely true...was just waiting for someone to comment! You see, Mathematics is also very much about shapes, patterns and trends, which were left of the list. In fact, maths is really about everything!! (Answer to median house prices = B)

Welcome to our first visitor from South Africa!

Numbers - they are the DNA of Mathematics and some recent posts will focus on the way that different number groups (called Number Sets) behave - very much like the different groups of people that you mix with (or not) at a party!

Making visual connections is an often forgotten focus in Mathematics yet is integral to most maths testing. I hope you enjoy the challenge of finding the right location for the van on Lombard Street! Your need a sense of orientation and scale but it is really not that difficult.

Welcome to our visitor from San Francisco, just after the San Fran posting! This is a great city, with so much architectural and cultural diversity as well as such a wonderful location.

Patterns - now here's a great subject to get your maths juices boiling! Show me a keen math student and I will guarantee that he or she is into patterns! Of course, the true-blue mathematician is also into random patterns - which we call "chaos" - and that is another great math topic to look at at some other (random) time! Do Zebra stripes count as random patterns? ;-)

The importance of a good breakfast is our focus for the weekmix!

Great to see a recent blog visitor from Gresham, Oregon. Great scenery around the Columbia River Gorge including the second highest waterfall in the USA. Home to some good mathematicians too!

A good friend and wonderful Mathematics teacher (now retired but used to live in Gresham too) send through this kind comment from the USA recently; ".. spent some time on your math blog and was very impressed. I am hoping that students are taking advantage of it. I was particularly impressed with your process of getting students to think mathematically and not just look at math as a hallway that is filled with hurdles called classroom exercises. The most exciting part of math is when you open a side door and explore other rooms that may lead to a maze of interrelated opportunities in math explorations." Many thanks!

A visitor reads our blog from the I-95 (see post). Is this a space-time warp from our Dr Who files or a wonky GPS?

Dear Blog,
Over 100 visitors for January. 100 visitors reminds me of the famous story regarding the great mathematician, Carl Friedrich Gauss. He started primary (elementary) school at age 7 and his genius became apparent when his teacher asked the class to add up (the sum) of all (integer) numbers from 1 through 100. Gauss did this almost instantly by noticing 1+100 = 101; 2+99 = 101, 3 + 98 = 101 for a total of 50 pairs. Therefore the total was 50 x 101 = 5050. He may have reached this mentally by doing 50x100=5000 + 50 = 5050? Whatever method, what a quick mathematical mind at such a young age! Yes, Gauss had a keen interest in how numbers worked and this is a key to doing well in Mathematics.