## Welcome to H3 Maths

Blog Support for Growing Mathematicians

### A billion trees? Let’s break that down to human levels

November24

In a recent news item a comment was made about reaching a tree-planting target of 1 billion – “With the right incentives and the right conditions, one person could comfortably plant 400,000 trees in a year. So you would need just 250 people planting that amount every year for a decade (10 years) to reach a billion.”

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### Let’s get two negative about Integers!

November14

A fun skit (source unknown) to help you understand that two negatives together = a positive answer…

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### Common Maths Mistakes – Summary

November14

It would be to simplistic to have a few summary mistakes that are commonly made by Math students, but I will try!

1. Don’t know my times tables (or have them handy). Yes, I know that I have calculators on my devices, but my brain is a quicker one and I need to use it more often – it gets me through the work quicker, helps my confidence and also exercises my thinking muscles! Perhaps if I really tried to learn my tables and math formulae I might prevent an early case of alzheimer’s??

2. I didn’t read the question properly (use highlighter to show each step required).

3. I forgot to plug my answer back into the question to see if it actually worked!!

4. Darn it – I hurried the working and forgot to write down “x = answer” and I left out the units required (e.g. cms or ft). DOH!

5. Heck, I thought I would remember all this math stuff so I forgot to write any notes on the topic the teacher was talking about. Then I never spent any time looking through the textbook (and copying the worked examples that I needed) and was too busy on social media to check out the online resources and similar problems found on (e.g.) khan academy!

6. Sport is toooo important for me to miss, so I skipped checking over the answers and making corrections. By the time I got home it was too late to do any math.

7. Someone (I forgot who) said that it is impossible to revise for math exams. Of course, they must be right?

8. “If at first you don’t succeed, give up.” is a good motto? Well, perhaps I should ask for help from the teacher or TA when I need it – it might actually improve my understanding of new math concepts and how to use the formulas correctly after all.

9. Always sit next to someone who will disturb my working! NOPE – a DUMB idea!! I say, “sorry but I have to work on this alone now, but will see you after class.” They respect my attempts to do well in THE, repeat THE most important subject for my future income and career opportunities.

10. So, make friends with those kids who do well in Mathematics – you need them! Otherwise, use the oldest math excuse in the universe – “My dog ate it!

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### Is there a difference between Logic and Mathematics?

November13

Yes, there is a difference between Logic and Mathematics.

, Professor of Mathematics at Clark University makes the distinction like this – “Mathematics studies numbers, geometry, and form.  Those require definitions and axioms to characterize the subjects of study.  Logic is used to prove properties based on those definitions and axioms. Logic can also be used in ordinary discourse of non-mathematical topics. Typically, the logic needed for mathematical investigations is great while that for non-mathematical investigations is small.”
Try this puzzle for example – it is based on pure logic, rather than Mathematics. Which if the following cups will fill with coffee first (answer will appear on the post support column soon)?
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### Can’t solve it? Sketch it first!

November7

When learning how to calculate surface area and volume of 3D shapes, students are often given a list of formulas without any explanation of the formulas. So when given the following problem,

Find the total volume of a rectangular prism with base side lengths of 6 cm and whose height is 8 cm.

Knowing the formula is V=BxH, students will often solve this by simply multiplying 6×8. Even if they multiply correctly, this shows that they do not understand what volume is measuring, or that they have to find the area of the base. Therefore, I prefer to teach “Volume = any end area x length.” But first, use the following steps to work towards a solution (highlight key terms–>find shape (or equation, etc.)–>put in values–>use formula (or equations steps) –> solution (add units if required). This is what it might look like for the earlier volume problem:

Hint: When you have the correct shape (Google it if not sure) and have added the dimensions, shade the ‘end area’ that has two measurements for finding the area, then times by the height (or length). Your diagram with shaded end might look like this:

Try this: Assuming your classroom is also a rectangular prism (aka ‘box’) you can find the volume by using the area of one end and ‘pushing it through’ the length of the room. That is, ‘End Area x Length.’ Or, you could find the area of the ceiling and ‘drop it down’ the height of the room – again this is ‘End Area x Height.’ So, the rule of ‘End Area x Length‘ is a really good formula to find volumes.

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### Some Common Math Errors

November6

Errors #1:

This image shows a student’s working. What went wrong?

Well, obviously x should = 17, not the square root of 17. Perhaps a better way to explain the error is like this:

When solving equations (expressions with an equal sign) like this, remember the algebraic rule:

“Whatever you do to one side, do to the other!”

So, here the student undid the sq. root on the LHS but did not undo it on the RHS. So, there should be no root sign on the RHS. “If you sq. root one side, sq. root the other too!  (from mathmistakes.org)

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### “A Big Number % Sure!”

November6

Two young boys were riding past our house today and debating whether they had come the right way. One said to the other, “I’m a big number percent sure!

A great comment although, as it turned out, they were going up the wrong road, so I wonder how large the ‘big number’ was?

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### Getting better at Math is spending more time!!

November3

Cathy O’Neil makes the following helpful comment on her blog ‘MathBabe‘ – “In reality, mostly of being good at math is really about how much you want to spend your time doing math. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine. Plus, thinking about things overnight always helps me. So sleeping about math counts as time spent doing math.

[As an aside, I have figured things out so often in my sleep that it’s become my preferred way of working on problems. I often wonder if there’s a “math part” of my brain which I don’t have normal access to but which furiously works on questions during the night. That is, if I’ve spent the requisite time during the day trying to figure it out. In any case, when it works, I wake up the next morning just simply knowing the proof and it actually seems obvious. It’s just like magic.]

So here’s my advice to you, high school kid(s). Ignore your surroundings, ignore the math competitions, and especially ignore the annoying kids who care about doing fast math. They will slowly recede as you go to college and as high school algebra gives way to college algebra and then Galois Theory. As the math gets awesomer, the speed gets slower.” Great words!

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### Math Freedom with YouCubed at Stanford

November2

Don’t forget to check out the great resources at youcubed. This popular initiative at Stanford University aims to inspire, educate and empower teachers of Mathematics, transforming the latest research into accessible and practical forms. “We know from research how to teach math well and how to bring about high levels of student engagement and achievement but research has not previously been made accessible to teachers.” Youcubed places an emphasis on Mathematical Thinking in an environment free (check out this video) from the constraints of a tightly followed textbook. Also check out some of their resources and try to build them into your classroom calendar.

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November1

### NZ dollar spikes on prospect of wage inflation …

So read the headline (source here), followed by this brief explanation: “ANZ senior economist Sharon Zollner noted the New Zealand dollar had taken a pounding, post-election, but shot up from US68.49c before the data’s release to a peak of US69.14c on the back of the data, later settling back to about US69c.” Wow, it sounds like a dramatic statistical change, especially when supported by this impressive graph:

But, discerning mathematicians, check out the actual change. The dollar “shot up from 68.49c to 69.14 before finishing at 69c” a change of only about half a cent – hardly a spike? Of course, the graph has a scale that is vertically exaggerated to a considerable degree. In fact, a rise in the Kiwi to US exchange rate of just a few cents would not fit on the graph at all.

This is a great example of “fake statistical news” or, at least, poor use of data and graphs to “sell” a weak story. Of course, it would not get past the eagle eyes of our junior mathematicians, would it?

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#### Post Support

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.