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Blog Support for Growing Mathematicians

Oxford Entrance Exam Math Q


Q: Imagine a ladder leaning against a vertical wall with its feet on the ground. The middle rung of the ladder has been painted a different colour on the side, so that we can see it when we look at the ladder from the side on. What shape does that middle rung trace out as the ladder falls to the floor?

(see answer in post support)

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Mathematical Ilusion


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Future job? Don’t bank on it!



bank changes

Squeezed by low interest rates, shrinking trading revenue, and nimbler technology-based competitors, banks are preparing for the day that machines made by men and women take over more of what used to be the sole province of humans: knowledge work.

The human brain is a wondrous machine, but it isn’t changing. The pace of technological advancement is accelerating, and artificial intelligence (AI) may one day make many forms of work extinct.

Both Bank of America and Morgan Stanley, which together employ more than 32,000 human financial advisers, are developing automated robo-advisers. More than 40 global banks have joined forces with startup R3 to develop standards to use blockchain – software that allows assets to be managed and recorded through a distributed ledger, to overhaul how assets are tracked and transferred.

The universal theme of banking’s tech strategy is to make sure that, internally and in dealing with clients, ones and zeros flow seamlessly without messy human interference.

Machine learning, where the decision-making power of algorithms improves as more data are raked in, can replace people in some instances, say finance executives including Daniel Pinto, head of JPMorgan’s investment and corporate bank. Algorithms already tackle tasks such as vetting banking clients, pricing assets, and hedging some orders without human intervention. “As we make those processes more and more efficient, you will need less people to do what we do today,” Pinto says. Read more here


Andrew Ng, chief scientist at Chinese Internet giant Baidu, on how AI will impact what we do for a living

Truck driving is one of the most common occupations in America today: Millions of men and women make their living moving freight from coast to coast. Very soon, however, all those jobs could disappear. Autonomous vehicles will cover those same routes in a faster, safer and more efficient manner. What company, faced with that choice, would choose expensive, error-prone human drivers?

There’s a historical precedent for this kind of labor upheaval. Before the Industrial Revolution, 90% of Americans worked on farms.”

help with math

Read more on the future of Artificial Intelligence here

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New Algorithm to predict Tsunamis


Scientists at the Australian National University have a plan to improve tsunami warning systems around the globe: they’ve built an algorithm.

Using data from monitors in the ocean and modeling what a tsunami looked like when it was birthed, the team of researchers can better predict how big it is, where it’s going, and who’s at risk. This is a big step beyond existing tsunami warning systems, because it uses the actual data to generate predictions, rather than scenarios for tsunami risk that scientists have previously calculated, says Jan Dettmer, a seismologist at the university. (read the full article here).

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The leaning power of PISA


When I was at university one of our senior lecturers was a widely recognized Coastal Geomorphologist. In other words, he was an expert on coastal landforms. I don’t remember anything he taught but I do remember vividly a story he told – which in itself is food for thought for how we teach Mathematics? The lecturer recounted how he attended a lecturer on rising sea levels. The data collected from a local wharf, said the presenter, showed that the sea level was rising at quite rapid rates. My lecturer, being the thoughtful academic he was, asked the question, “Could the data show that the wharf is sinking?!”

Decline standards Oz

Every three years, international data is released on how well students are scoring in a few key subjects, including Mathematics and Science. The graph below indicates a big drop in these standards for 15 year old students in Australia. Now, of course, the data could also show that scores for other countries have gone up and that, in fact, there is no actual decline in the standards for our Australian students!

An article on the TES website ask, “What if there are “serious problems” with the Pisa data? What if the statistical techniques used to compile it are “utterly wrong” and based on a “profound conceptual error”? Suppose the whole idea of being able to accurately rank such diverse education systems is “meaningless”, “madness”? What if you learned that Pisa’s comparisons are not based on a common test, but on different students answering different questions? And what if switching these questions around leads to huge variations in the all- important Pisa rankings, with the UK finishing anywhere between 14th and 30th and Denmark between fifth and 37th? What if these rankings – that so many reputations and billions of pounds depend on, that have so much impact on students and teachers around the world – are in fact “useless”?”

Yes, this is a clear case of the seriously Leaning Power of Pisa!

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Math Equations Delay Flight


An Italian maths professor was recently escorted off an American Airlines flight after a fellow passenger Guidofeared his mysterious scribbling on a notepad was evidence that he was a terrorist. In fact Guido Menzio was working on an equation connected with a presentation on price-setting he was due to present. He was flying from Philadelphia to Syracuse to give a talk at Queen’s University in Ontario, Canada. He was solving a differential equation, but said he was told the woman thought he might be a terrorist because of what he was writing. But the sight of a slightly scruffy, curly-haired man scrawling odd symbols on a notepad was enough to alarm the woman who was sitting next to him. Mr Menzio, a highly respected academic who has also had spells at Princeton and Stanford universities, succeeded in convincing the authorities that his doodles were an equation. (source: Daily Telegraph)

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How to fix decline in Math Scores


primary math

Recent data released in New Zealand paints an alarming picture of decline in Mathematics scores for Primary School leavers. These students then enter High School without the math skills to succeed. Here are some of the results:

“Maths scores have been declining since 2002, according to OECD data. National Standards figures show a slight climb in the past three years, but say one in four are behind in the subject by the time they leave primary school at Year 8. A national monitoring study from 2013 had even lower results, with just 41 per cent of students at the expected level when they leave primary, despite the majority achieving well just four years earlier. The drop-off after Year 4 – when students are aged 9 to 10 – is a trend across all subjects, but in mathematics it is particularly significant…

profpicProfessor John O’Neill, the director of Massey University’s Institute of Education, called the problem a “chicken and egg” scenario. He said because many students dropped maths – and science – part way through high school, teaching students often lacked subject knowledge in those areas.

“In the past, you could fill students’ gaps in learning at teachers’ college. But whereas 10 years ago students would get several hundred hours in a learning area, now they might only get 50. It’s not enough,” he said. Teachers then lacked confidence, their students got a raw deal, ended up not liking maths, and the cycle began again.

Some educators believed the answer was shifting teaching to masters level to raise entry levels, but even then graduates could come through with limited maths and science ability, Mr O’Neill said.

“Part of the problem is with NCEA, in that it allows too much choice. It’s a very good system but it allows students to drop out of science and maths too early.”

He said the Government needed to be “courageous enough” to recognise it would take the country 20 years, a sophisticated policy response and a long-term funding injection to break the cycle.

The Ministry of Education has already added a suite of maths acceleration programmes and professional development to support underachievement in maths, at the cost of $20 million per year.One is a 15-week intervention for struggling students, the other a programme that supports teachers to undergo extra training to become Mathematics Support Teachers over two years, during which time they work with small groups of high-needs pupils. They eventually help other teachers in their classrooms too.” Read the full article here

Editor’s note: My personal view is that, like any student interest, Mathematics needs to be fostered through a variety of different strategies including coaching, challenge and celebration. Parents also play a crucial part in devoting “Math Time” and sharing the excitement of working with numbers, patterns, shapes and puzzles. Whilst it is easy to blame teachers for the decline in mathematical ability, there is some very capable and courageous teaching going on – teaching in Mathematics that inspires and delights numerical discovery among younger students. Some ideas can be found here:



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We can’t go back in time, can we?


Hey, with Daylight Saving coming into effect, we lost an hour last night, but where did it go? I was taught that, in order to go back in time, you would need to travel faster than the speed of light (thanks Einstein). But, the physicists tell us, this is impossible, so we can’t have lost any time at all? No, I think this “daylight saving” is a myth. Check out this time article and see if you can figure it out:

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Academy ups the Math prerequesite in Oz universities


All year 12 students should be made to study intermediate mathematics if they want to enrol in a science, engineering or commerce degree at university, according to a national report by the Australian Academy of Science (AAS). “This is an absolutely critical issue. We’ve hit a pretty low point,”  said the director of the Australian Mathematical Sciences Institute Professor Geoff Prince, who helped develop the report.

A study of almost 50,000 maths students in the 2013 HSC (the NSW Curriculum Assessment), published in the Australian Journal of Education, revealed a scaling advantage for those who took the general mathematics course.

maths decline graph

But the shift by universities to list assumed knowledge rather than strict prerequisites for degree courses had caused fewer high school students to take harder maths courses and resulted in higher drop-out rates, found research by the Australian Mathematical Sciences Institute.

“If it happened tomorrow it would cause a real shock to the education system because many schools don’t have the resources to be able to teach these [intermediate maths] subjects.

“We need to give schools the time to adapt, and they may need some support to do so,” Professor Prince said.

Among its 12 recommendations, the report also pushes for increasing professional development for out-of-field maths teachers.

“The data we have is pretty emphatic: there is a very measurable difference in academic success … between students who have two-unit Mathematics [in Year 12] and those who don’t,” said University of Sydney deputy vice-chancellor Tyrone Carlin.

Education minister Simon Birmingham, who is due to present the report at Parliament House, said it “laid clear path” for this generation of students and into the future.

“Around 75 per cent of Australia’s fastest growing industries require science, technology, engineering and maths skills which is why we have committed $112 million for programs to encourage more students to get engaged in those areas,” Senator Birmingham said.

Read the full article here

[Post Note: The moral for our growing mathematicians is to always take their Math courses as far as they can – it opens more opportunities for future study and for employment! If you undervalue your Mathematics you restrict your future!]
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Pythagoras surfs up a right-angled triangle


Seen on the lake during a school day – 3 surfers making a right-angled triangle! Now, can you identify the hypotenuse (answer in the Post Support column)?
hypot surfers

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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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