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Blog Support for Growing Mathematicians

Apple can buy the whole orchard!


This week Apple announced that its cash holdings have surged to $US256.8 billion. That’s whole lot of folding stuff and is really difficult for any student grasp, especially if they are not getting much pocket money! So, it raises the question – what could Apple buy with all that spare change? Here are some examples (from an article in the Daily Telegraph);

1. If Apple split up its US$256.8b among the world’s population, that would work out at around US$34.70 each.
2. Apple has around 115,000 employees across its head office, international offices, stores and so on, so its cash pile could be divided up to give them all around US$2.2m. That’s easily enough to buy everyone a luxury yacht.
3. All these tech companies!
So, just how big is a billion? Check out this earlier post.

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Max Martin hits Lorde with “Melodic Math”


This just in from the Times: “When Max Martin heard “Green Light” shortly before its release, she told me, “he had a very specific opinion, which had to do with the melodic math — shortening a part” … Martin described “Green Light” as a case of “incorrect songwriting,” Lorde said, clarifying that this “wasn’t an insult, just a statement of fact,” and one, furthermore, that she agreed with: “It’s a strange piece of music. On top of the left-field key change, “the drums don’t show up on the chorus until halfway through, which creates this other, bizarre part.””

Incidentally, Max Martin is “music’s magic melody man, responsible for twenty-one No. 1 Billboard hits—five fewer than John Lennon, and eleven behind Paul McCartney, on the all-time list. But, few outside the music business have heard of Max Martin. Presumably this is because Martin writes all of his songs for other people to sing. He is the poet hiding under the balcony of popular song, whispering the tunes that have become career-making records, such as “… Baby One More Time,” for Britney Spears, “Since U Been Gone,” for Kelly Clarkson, and “I Kissed a Girl,” for Katy Perry. The songs he co-wrote or co-produced for Taylor Swift, which include her past eight hits (three from “Red” and five from “1989”), transformed her from a popular singer-songwriter into a stadium-filling global pop star. (The “1989” tour recently passed the hundred-and-fifty-million-dollar mark.). Read Max’s full bio in the New Yorker here.

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Each Number is Significant!


Every number is significant, even small ones that seem insignificant! For example, I was talking with a parent recently who told me that if they give a dose of medicine to an animal and it is incorrectly measured by 2% then it will kill the animal. I must admit that I was quite stunned by that small difference. When I was a Junior in High School (6th form for those English and New Zealand students) my math teacher told me of a hike he did with a group. For some reason he and a friend made a slightly small deviation from the others and, after a couple of hours, discovered that they were several hundred meters apart – on opposite sides of a large ravine! Yes, small numbers or amounts do matter. Anyway, this month’s Number of the Month is 45, sourced from bestmaths.net:

45 – Number of the Month (April 2017)

  • 45 is a triangular number, and in particular the sum of all the decimal digits
  • (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45)
  • The factors of 45 are 1, 3, 5, 9, 15 and 45
  • 45 in Roman numerals is expressed as XLV
  • 45 is 101101 in binary.
  • 45 is the atomic number of Rhodium.
  • There are 45 degrees in half of a right angle.
  • +45 is the telephone dialing code for Denmark.
  • World War Two ended in 1945.

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The Leaning Power of Pisa


In the latest international PISA tests, some 500,000 students applied their skills to common tests in order to track their relative academic abilities. Asian countries have topped scorers in maths and science and, in 2015, Singapore became the first nation to top in all three subjects. The three nations that have fallen furthest since Pisa began are all Anglo-Saxon: in order, Britain, Australia and New Zealand. The almost identical tracks of Australia and NZ suggest that there may be common factors driving them both down.The biggest gains, among those who have been in Pisa since it started, have all been in Europe: Luxembourg, Portugal, Poland and Germany. Concerns about the dumbing down of education in places like Australia and New Zealand have recently hit the headlines – as in this article. Some have suggested that the reason for lowered performance from New Zealand is the result of classes being streamed, as in this link. However, some of New Zealand’s most successful academic schools have rigid streaming, so this may not be a major factor? Read more on the PISA results here.

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Largest Prime Number for 2016


As the new year approaches, you may be (or not be) interested to learn that the largest known prime number is 274,207,281– 1.

This number is an incredible 22 million digits long and 5 million digits longer than the second largest prime number. Hang on to your primes you might ask, “Why are prime numbers important? Are they of any practical use in real life?

Barak Shoshany, Graduate Student at Perimeter Institute for Theoretical Physics, helps answer your question. “The most notable practical use of prime numbers is in cryptography. Many popular algorithms used in public-key cryptography, which has numerous and extremely important security applications (your computer is probably using several of these algorithms at this very moment), are based on the fact that integer factorization is a “very hard” problem.

What this means is that the time required to factorize integers into their prime factors grows (roughly) exponentially with the number of bits in the integer. So if the encryption uses very large integers, it would take an unrealistic amount of time to “crack” it.

If (or when) quantum computers become a reality, they would have the potential to make all of these algorithms obsolete, since there are quantum algorithms (in particular Shor’s algorithm) that can factor arbitrarily large integers much faster than any known classical algorithm. This has led to the very important field of post-quantum cryptography.”

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So you want to work for Apple?


The following question was one that appeared during interviews for potential employees at Apple. How would you answer it?

“There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes have been incorrectly labeled such that no label identifies the actual contents of the box it labels. Opening just one box, and without looking in the box, you take out one piece of fruit. By looking at the fruit, how can you immediately label all of the boxes correctly?”

[Answer shortly in Post Support]

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Secrets to Top Math Scores in Victoria, Australia


“You have to have talent, and you have to be willing to work hard!” This sums up the high scores achieved recently by students in their state-wide math exams.

A recent article highlights the underlying secret to high scores in senior school Mathematics, in the southern state of Victoria, Australia: “In Melbourne’s eastern suburbs, Vermont Secondary College has been celebrating some impressive further maths results. A record three students at the state high school achieved a perfect study score of 50 in the most popular maths subject – and they were all in year 11.

Overall, 18 students at the state school achieved a 40-plus score in further maths.

Director of numeracy Mary Zervos attributed this success to the school’s team of young and experienced maths teachers. She said students worked tirelessly, completed many practice exams and asked questions. “You have to have talent, and you have to be willing to work hard,” she said.

“We are a community school that works with all the students in the neighbourhood to achieve the best we can for them.”

Westall Secondary College was another government school that produced some sharp further maths minds unlikely to be fazed by curly exam questions about 50 cent coins. The relatively small school in Clayton South has just over 50 year 12 students, but managed eight scores of 40 or above in the subject.

Principal Tristan Lanarus said the school had a strong maths culture, and one of its students last year represented Australia at the International Maths Olympiad.”

Recently, there has been much debate around the fact that more money does not equal better academic results. For example, Australia has poured an extra $10 billion of funding into schools over the past decade, but its students are going backwards in international rankings, having been beaten most embarrassingly last year in the Trends in International Mathematics and Science ­report by students in impoverished Kazakhstan. Yet Australia is one of the biggest spenders on education, outlaying twice as much money as a percentage of GDP as top performers Singapore, Hong Kong and Korea. Read more in the Daily Telegraph here.

[Ed: “Culture” features widely in some recent innovative Mathematics programs. For example, the difference between Western and Asian math scores was discussed at length here on cnbc and in this paper from Massey University – developing-mathematical-inquiry]

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Failing Math Standards


This just in from my newsfeed,

“WASHINGTON (AP) American students have a math problem.

The latest global snapshot of student performance shows declining math scores in the U.S. and stagnant performance in science and reading.

Education Secretary John B. King Jr. says this country is losing ground and he finds that to be a troubling reality when, in today’s economy, ‘the best jobs can go anywhere in the world.'”

anythingelseWell, hello hello, the same news story has come out of Australia and New Zealand recently too. In fact, Math scores are falling in most western nations (see: http://www.nzherald.co.nz/nz/news/article.cfm?c_id=1&objectid=11757118). The politicians are placing the blame at the feet of teachers (see: http://www.nzherald.co.nz/opinion/news/article.cfm?c_id=466&objectid=11759955) and teachers are placing the blame at the feet of math-deficient parents and parents who never could do Math are blaming anyone who will listen. It turns out that Australian 15 year olds are now two years behind their international peers (see: http://www.smh.com.au/national/education/australian-school-students-two-years-behind-worlds-best-performing-systems-20161206-gt4w8p.html); and 15 year old math students in the ACT (Australian Capital Territory) are 18 months behind their overseas peers in Mathematics. So, who is to blame – the media, business, test scores themselves, or perhaps…no, it can’t be…the students themselves??! Oh, this debate will rage as long as Mathematics is being taught or, perhaps, trying to be taught? (Note: students themselves hardly ever get blamed for this decline).

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When do miles = kms?


Since this…(for fun)

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Math Graphs prepared for Alien Arrival



In the new film “Arrival,” alien visitors appear all around Earth and humankind scrambles to understand their purpose in visiting. Perhaps they have come to improve their math?

Anyway, the movie’s creators  approached Wolfram Research (creators of Mathematica, the online problem-solver Wolfram Alpha, etc.) to produce some charts for use in the movie. CEO Stephen Wolfram, helped out and his son, Christopher, generated visualizations for use in the movie.

“Wolfram told Space.com, “Who’s to know? Maybe something that I invented for science fiction will turn into some real physics.” [8 Modern Astronomy Mysteries Scientists Still Can’t Explain]

Space.com talked with Wolfram about the way science fits into movies, how aliens are like artificial intelligences and whether math is invented or discovered — and what that would mean for alien mathematics and alien thought.

Space.com: How were you approached to work on the project?

Stephen Wolfram: Because a lot of scientists use our software systems and we produce a lot of interesting graphics, we have a pretty regular stream of requests from movie makers of various kinds saying, “Can we show this graphic in our movie.” This one was kind of amusing, because it was like — we’re about to start shooting this fairly big-budget movie, and we need these screens that should look realistic and can you help us do this?”

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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.

Other answers:
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.

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