The name's Bond, Number Bond (Image:Wikipedia)

Parents and students often ask us if we do number bonds, at Maths-Whizz, as though they are mathematical methods exclusive to the initiated few. Nothing could be further from the truth.

Number bonds are, in short, pairs of numbers that combine to make a third. Number bonds help show that every whole number larger than one is made up of other whole numbers. 1 & 4, 2 & 3 both make five. They are the number bonds of five.

Number bonds are vital for mental arithmetic. When you ‘partition’, or ‘decompose’ a number you get number bonds..

For instance, knowing that can be 15 can be partitioned to 10 and 5 helps you perform calculations with 15. You can calculate with the ‘10′ part, and then with the ‘5′ part, and combine their answers. 5 and 10 are number bonds of 15.

Knowing and remembering the number bonds of 100 makes it easier to handle money, or give change. If I know that 70 and 30 are number bonds of 100, then I know instantly how much change to expect from a pound when I buy a 70p packet of sweets.

Number bonds can help you add. Knowing that 5 and 2 are number bonds of 7 makes adding 7 to five much faster. Add five to five to get 10, and then add the remaining two to get 12! And this is just scratching the surface.

In Maths-Whizz we don’t refer to ‘number bonds’, but we do talk about partitioning numbers, about finding pairs of numbers, about mental methods that require an understanding of number bonds, and so forth.

Dozens of our lessons rely on knowledge of number bonds, and at least ten specifically test rapid recall of number pairs, and missing numbers. We even test more advanced students on splitting a whole number into two decimals.

You can try one such number bond speed game on our maths addition page. Scroll down to Year 5, and click the right-hand lesson image.

Click to try a free number bonds speed game

Get busy number bonding!

…ome hapless guy got slammed with a “penhora,” which translates into English as “distrainment”—the seizure of personal property to enforce the payment or discharge of an obligation. In this particular case, the subject of the distrainment had suffered the seizure of 1/6 of his assets. He petitioned the court for relief, claiming that he was suffering grave economic hardship.

The court solemnly pondered the petitioner’s request, noting the necessity of proportionately balancing the petitioner’s well-being against his responsibility to discharge his legal obligations. Upon consideration, the court ruled that the distrainment of 1/6 of the petitioner’s assets had been too severe and ordered a relaxation of the order. The new order instead stipulated a seizure of 1/5 of his assets.

This would be hilarious, if it wasn’t somewhat less than funny for the recipient of the Judge’s generosity.

But it goes to show that maths is necessary for all sorts of careers, not least those in the distinguished legal profession…

Money makes the maths go down...

Money can make the maths go down, to coin a phrase.

Since Britain’s currency was decimalised many of us have had their first introductions to base ten maths and decimal concepts in money.

Most children are likely to have got a basic grasp of the notion of pennies and pounds, and how they relate, from an early age. And children are likely to have understood this long before they are required by national curricula to have understood tenths, hundredths, performing operations on decimal numbers, and the like.

If nothing else, this proves that if there is a need and a daily context in which to learn, we can acquire complex skills without really realising. But it also gives us a useful tool when tackling calculating with decimal places.

Take the question “3.6 divided by 12″.
It sounds pretty tough to the unconfident. After all, 12 is larger than 3.6, which means the answer will be smaller than one, a decimal value.

Now take the question “£3.60 split amongst twelve people”.
Does the question seem easier? Suddenly it’s about a dozen students going Dutch on an order of poppadoms (substitute scenario as appropriate!).

As soon as we add a pound sign in front of a decimal value in a calculation, it can seem easier to visualise, to handle.

0.01 is one hundredth, an abstract concept. Whereas £0.01 is a penny – it’s solid, and there’s a hundred of them in a pound – easy!

If you struggle with decimal calculations, it might just help to shove a pound sign in front of the numbers, turn 0.4 into £0.40 or 40p, etc. One of the commonest pitfalls of multiplying or dividing with decimals is the failure to show the answer in the right order of magnitude.

For instance, taking 3.6/12, we could make the question easier by writing it 36/12 – the answer is 3. But now what do we do to find the answer to 3.6/12? This can stump people, but using the context of money means that we know £3.6 is 36 10p coins, or 360 pennies.

If the answer to 36/12 is 3, that means three 10p coins, or 30p! And how do we write 30p in pounds? £0.30. Easy! We’ve just converted the whole-number answer to our simplified calculation back into the final decimal answer.

This Maths-Whizz tip comes courtesy of Hilary Koll and Steve Mills, the esteemed and award-winning maths boffins who designed all our Maths-Whizz lesson concepts, people to whom even the God of Whizz tugs his forelock.

The number line can be a powerful beast, employed in addition and subtraction, and frequently in concepts of place value. The number line describes the world in the way instantly familiar to most of us, with smaller items on the left (if horizontal) or towards the bottom (vertical).

We could investigate the many ways culture and psychology define our experience of numbers, and why so many (but not all) of us perceive numbers increasing in those two directions, but that would be beyond the scope of this blog – even if the God of Whizz might enjoy the intellectual excursion…

Number lines in addition and subtraction

Instead, we can have a quick look at how to add and subtract using the number line. The line is a great way to visualise the ‘journey’ up and down through the numbers. Just like a ruler, the line can be any length, at any level of detail. Keep zooming in, and the increments on our number line show smaller and smaller amounts.

Number lines can be to scale (like a ruler) or not to scale, like a simple ordered list. In the example maths addition lesson above, we show how you can add a number with Tens and Units digits to a number with Hundreds by counting along the number line, starting with the larger number.

You need to understand the values of digits; for example, 63 is made up of six tens and three units (or 60 + 3). You also need to know that adding numbers means moving to the right along the number line, and subtracting means moving to the left. We make a large rightward ‘hop’ to add six tens and a smaller ‘hop’ to add three units.

For those of us who think in visual terms, it can be very helpful to think of addition and subtraction as movement along the number line. More complex examples show the benefits of this approach. Say we have to answer this question:

800 + 300 – 150 + 220 – 400 = ?

We can imagine starting on the line at 800. We jump forward 300, and we jump backward 150, and so on. If we plot our frog-like adventures we find we’ve gone backwards 550 and forwards 520. This puts us 30 behind where we started. 30 less than 800 is… 770!

Of course, this is a slow and round-about method when your mental maths improves, but visualisation of the process can help enormously. You can do all the forward jumps in one go, and then to all the backward jumps in one go; the answer is the same.

At Maths-Whizz we know that anything that helps reveal the underlying maths is good – because once you can get a mental grasp of the ideas, the numbers suddenly seem to make sense…

This only scratches the surface, of course, so do share any ideas you have about working with number lines on the Whizz Forum.

No, you’re not allowed to use a calculator, or or phone a friend…

Stuck? OK, it’s 2.

“What?” I hear you cry, “But that doesn’t make sense. We’ve divided something small by another small thing, that should make an even smaller thing, surely.”

And, of course, you’d be wrong. If you already knew the answer, or you’ve already seen this February’s Channel 4 documentary The Kids Don’t Count, then read on with a smug smile. If you were genuinely stuck by the question, let us demystify division for you.

First, division isn’t about making things smaller. Sure, that’s often what happens: when you divide 10 by 2 you get 5. But what does it mean to divide?

The division sentence A/B = C can be described in two ways:
- We’re taking A and splitting it into B groups of equal size
- We’re taking A and and finding out how many groups of size B it contains.

This sounds like splitting hairs, but the former is known as ‘partitioning’ division, the latter ‘quotative’ division. These aren’t terms that Maths-Whizzers need to know, but they can help show that division isn’t about making things smaller.

Let’s go back to our teaser. If you were stuck, then think of the question this way – “How many groups of one quarter are in one half?” You should know that there are two halves in a whole and four quarters in a whole, so there are two quarters in one half, which means the answer is 2!

Simple, no? Maybe not yet, but it will be.

What if we reverse the question: “What is one quarter divided by one half?”. Here the issue isn’t so straightforward. One half does not ‘fit’ completely into one quarter, but if we ask ourselves how much of that half can we squeeze into that quarter, we see the answer is one half. In other words, there’s half of a half in a quarter.

One quarter divided by one half equals one half!

Now we’re rolling!

You need a simple rule for dividing fractions, not simply an understanding of the principles. So remember this process below and forever look upon division of fractions with confidence.

Step 1. Turn the second fraction (the one you want to divide by) upside-down (this is now a reciprocal).
Step 2. Multiply the first fraction by that reciprocal
Step 3. Simplify the fraction (if required)

The Mathisfun page below goes into greater detail, explaining why this method works, so check it out to get all the info.

Sources:

It is a truth not acknowledged universally enough* that Maths Is Good For You. By this I mean basic numeracy will give you skills for life and work, and develop mental rigour that will benefit you in ways you might never have appreciated.

Good outfit. Bad maths.

The God of Whizz is a cheerleader for maths, not least because of his role in making the fantastic online maths tutor called Maths-Whizz, but he’s inclined to be positive about maths and science skills in general, because he’s that kinda guy.

Something called ‘Hon-Sho‘ caught the God of Whizz’s attention last year, and to his dismay it’s still on the Interwebs in 2010. I’ll let their website explain:

Hon-Sho means “Your true character”. Rooted in mysticism and philosophy, Chinese oracle reading can be traced back over 4,000 years. Hon-Sho uses your personal and unique Digital-DNA to produce a character profile and daily oracle readings which enable you to make decisions regarding your fate.

Clear? Me neither. But it turns out that your ‘digital DNA’ is simply a set of numbers obtained from your date of birth, and the values of the letters in your name, where a=1 and z=26. So Samantha Smith, born on 17th September 1969, has values of 60, 77 and 69.

These magic numbers are used to create a personal ‘hexagram’ (a stack of six broken or unbroken lines), and a daily hexagram based on coincidences between your personal numbers and 64 daily random numbers. These hexagrams each link to particular nuggets of advice or information that can apparently be used to help you make decisions about your life.

Why ‘digital DNA’? Stay with me here – the hexagrams are divided into two ‘trigrams’ (three rows of broken or unbroken bars) which relate to binary counting methods. There are eight possible trigrams, and with two trigrams in a hexagram, 8 x 8 = 64.

It so happens that the genetic code of four nucleic acids (A,G,C,T) combine in threes to code for amino acids in 64 ways (4^3 = 64)! OMG.

Deep Breath.

The God of Whizz has, in theory, no objection to fools being parted from their money – it’s being going on since the beginning of time. The best schemes are those that seem just plausible enough, whilst promising incredible benefits that make the outlay worthwhile.

You say to yourself “I’m not sure if this is true, but if it is, it’s money well-spent!” It’s the precautionary principle, where the cost of not knowing your future outweighs the risk that you are throwing your money down the drain.

But the problem is that such things patently AREN’T true, and that a moment’s thought tells us that you would be better off throwing your money down the drain, after all – a sewer worker might score an unexpected bonus.

It doesn’t take a genius to realise that, according to Hon-Sho, your life is entirely bound in your date of birth and your given name. But what happens if you change your name? What’s the difference between the life chances of Jon and Jonathan, Alice and Alys? What if your mother had been induced a day early, or if her labour had lasted an extra day? Would your chances have forever changed when the clock in the maternity ward ticked past midnight? If the answer to any of the above is ‘yes’, then how?

All these scenarios would have yielded different ‘Digital DNA’, but would they have made for a fundamentally different person?

We are reassured that our actual genetic profiles are millions of base pairs long, that interpretation of our DNA still challenges some of the finest minds in science, and that it takes some serious kit to even read the stuff of life. So why might we willingly allow our lives to be boiled down to three short numbers?

Hon-Sho relies on our general lack of confidence with maths and our tendency to put value in ‘historic’ systems. The 64 coincidence seems magical enough to suggest that somehow the ancient Chinese anticipated modern science, just as the equally nonsensical Bible Code shows the Good Book anticipated the Second World War because some of its letters spelled ‘Hitler’.

But an old idea isn’t necessarily a good one, and a mathematical description isn’t necessarily a meaningful one. Ancient maths sounds doubly reliable (see fashions for vedic maths, mayan maths, etc).

But whilst many historic methods are still useful, Hon-Sho seems to involve throwing numbers at a wall and putting great, mystical, stock in the ones that stick. Daily Hon-Sho reports rely on the Barnum (or Forer) Effect, where vague descriptions and predictions are sufficiently specific to make the reader believe they apply to him or her directly.

Hon-Sho is the new horoscope. Which is all very well, it gives local newspapers a little extra revenue, and probably provides reassuring bromides to thousands of subscribers. Horoscope writers have been fleecing the gullible and wishful thinkers for decades. But the God of Whizz gets hot under the collar over this new abuse of science, maths, and rationality.

Numbers don’t mean anything. Maths has no intrinsic value, other than as a wonderful, challenging, and sometimes beautiful way of describing the world. It is the science of patterns, but nothing in those patterns holds any meaning, except where they relate to other things. The sooner we can all realise this, the sooner we can put our money towards things that will truly make us healthy, happy, and wise.

But if you still want help making decisions about the big things – toss a coin.

* With apologies to Austen, J.

The Gresham College is a venerable London institution devoted to providing free lectures and events for the public, in the best of traditions.

The venerable Gresham College

In the clip below, Barrow shows the maths behind bank numbers and what’s known as the ‘Luhn Test’.

View Barrow’s introduction to the lecture, and clips about the mathematical patterns behind everything – post codes, six degrees of separation, mobile phone IMEI numbers and all sorts of other things.

5...4...3...2...1.. Lift off!

Check out a gallery of this geektastic device (click the picture above), or watch one of the design team explain how the Lego Space Shuttle is put together, below.

[via Gizmodo]

Rather than serve only to be hooked, sliced, or shanked into the near distance, chased by a volley of insults, hundreds of red and blue golf balls have been put towards a magnificent three-dimensional Sierpinski Triangle (or tetrahedron, in this instance).

A use for golf balls - maths fractal pyramid

To those otherwise unversed in Sierpinski’s Triangle (also known as a ‘gasket’), it’s a beautifully elegant fractal.

See how successively small triangular ’subtractions’ from the main triangle produce a lovely, almost threadlike, fractal pattern. The rules for creating this are simple, and can be repeated ad infinitum.

Sierpinski Triangle (Wikimedia)

(Wikipedia)

You don’t even have to start with a triangle to end up with a triangular-shaped Sierpinski fractal. Try it!

Or, better, try the ‘Chaos Game‘, an exotic-sounding name for something you can play with just pencil, paper, ruler, and die:

(Wikipedia)

[Via Make Blog]