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!

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.

Policy wonks have found that summer learning loss, an established side-effect of long school holidays, is particularly pronounced in some groups:

…children from the poorest backgrounds suffered most with ’summer learning loss’ because they were the least likely to practise reading and writing during the six-week break.

The Education Guardian has reported on plans from think-tank The Institute for Public Policy Research (IPPR) to shorten the long summer holidays. This should interest parents from any wealth bracket – without the right attention even the most expensively educated can suffer.

Further to this, long term times might be causing ‘burn out’ in some students.

A co-author of the IPPR study, Sonia Sodha, calls for a five-term system, with two-week breaks between eight-week terms, and a month for summer hols.

Many parents remember their long summer holidays with fondness, even though the original purpose of the long break – to allow children to return to farms for the picking season – has almost entirely been forgotten, certainly in practice.

Margaret Morrissey of the National Confederation of Parent Teacher Associations argues that children may also need “…a chance to recharge their batteries”; but one could well argue that if the terms weren’t so long, children’s batteries wouldn’t need recharging in the first place.

(As of 2010) Former Children’s Minister Kevin Brennan said the IPPR report (which also includes recommendations on increasing the role of play in classes for 5 and 6 year-olds and suggestions that secondary and primary schools employ in-house counsellors or use counselling networks) matches commitments made in the coming ‘Children’s Plan’.

Keeping children constructively entertained over summer is an issue, whether or not the Children’s Plan includes shortening the summer holidays. The finding that unstructured holidays can lead to ‘learning loss’ is still relevant, and is something we recognise. In 2007, Whizz Education launched our Summer Adventure – featuring the Professor lost in the jungle. Students were charged with helping him escape, by solving maths puzzles and playing jungle-themed games.

(a maths puzzle from the 2007 Summer Adventure)

The Summer Adventure was a huge success – Maths-Whizz students at Mead School, Kent, had a great day helping the Professor escape from the jungle when Toni Burkett and Monique Kleinschmidt from Whizz visited with prizes for their top-performing students.

(Our 2007 Summer Adventure in use at the Mead School, Tunbrdge Wells)

Our Summer (and Christmas) adventures encourage students to continue learning with Maths-Whizz; students must finish lessons before they can try the themed games, and they get to learn about jungle creatures with our colourful worksheets.

We’ll be bringing back the Prof and his rainforest antics each Summer, with brand new features. If you’re wondering how to keep your child occupied this summer, make sure he or she logs onto Maths-Whizz, because the Professor isn’t going to escape from the Mayan jungle all by himself…

[UPDATE: Read a study of 39 investigations into Summer Learning Loss]

[Read more about summer learning loss - Wikipedia]
 

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:

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.

As comedy maths puzzlers go, this one is a little forced, but nevertheless fun to share. I confess it took The God of Whizz a few goes to figure out that Pi = ‘pie’ (look at the scribbly reflected numbers).

Despite this neat coincidence, I somehow suspect that sharing this fancy formula in an exam or test won’t get you extra credit…

(via RedEye Puzzler)

Tim Muffett of BBC Breakfast interviewed students and teachers at north London’s Anson Primary School, which has a novel solution to the problem – teach the parents!

Teachers at Anson School have produced short video snippets outlining key principles that parents can watch with their offspring and so become a constructive part of the homework process.

The idea that Anson Primary School is teaching the wrong people is wide of the mark. As we know well at Maths-Whizz – the most engaged and motivated students have engaged and motivated parents.

A child whose mother enjoys a subject, or is confidently able to assist him with homework, will be more inclined to see value in the subject, to do well at school, and to ask for constructive help.

This is something we’ve been fostering with Maths-Whizz for some time.

The circle of learning with Maths-Whizz

Our home and schools maths tutoring services promote communication between parents, teachers, and students – parents experience our animated tutor with their children, teachers discuss student reports with parents, and kids tend to talk to one another about toys, pets, and our Challenge feature.

[BBC News]

The God of Whizz is behind on his reading, so I’ve come to this post late. But the trick was enough to impress the late, eminent, Richard Feynman, and will be enough to impress any internationally famous physicists that you know:

“When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch at calculating. For example, one time we were putting some numbers into a formula, and got to 48 squared. I reach for the Marchant calculator, and he says, ‘That’s 2300.’ I begin to push the buttons, and he says, ‘If you want it exactly, it’s 2304.’

The machine says 2304. ‘Gee! That’s pretty remarkable!’ I say.

‘Don’t you know how to square numbers near 50?’ he says. ‘You square 50 — that’s 2500 — and subtract 100 times the difference of your number from 50 (in this case it’s 2), so you have 2300. If you want the correction, square the difference and add it on. That makes 2304.’ ”

[From “Surely You’re Joking, Mr. Feynman!”, by R.P. Feynman]

Strogatz goes on to explain to us mortals how it works:

Bethe’s trick is based on the identity
(50 + x)2 = 2500 + 100x + x2.

He had memorized that equation and was applying it for the case where x is –2, corresponding to the number 48 = 50 – 2.

For an intuitive proof of this formula, imagine a square patch of carpet that measures 50 + x on each side.

Then its area is (50 + x) squared, which is what we’re looking for. But the diagram [below] shows that this area is made of a 50 by 50 square (this contributes the 2500 to the formula), two rectangles of dimensions 50 by x (each contributes an area of 50x, for a combined total of 100x), and finally the little x by x square gives an area of x squared, the final term in Bethe’s formula.

Geeky anecdote alert: As well as being a rather top scientist, Hans Bethe was also the missing link in a famous physics paper about the origin of elements in the early universe.

When Ralph Alpher and George Gamow co-authored a paper on the subject they decided, a little cheekily, to add the eminent Bethe to the list of authors, making it the Alpher, Bethe, Gamow paper about the very conditions after the Big Bang.

The joke is somewhat lost if you’ve not been up on your Greek alphabet (for shame!), which makes it sound as though the paper was written by the first three letters – alpha, beta, gamma.

Given that God tends to describe himself as the alpha and the omega, this would seem a pretty impressive lineup on a modest science paper…

Robo-Dog available in the Maths-Whizz Shop!

Feed him, play with him, watch him transform, and watch him sleep – it’s RoboDog!

RoboDog is the latest pet in the Maths-Whizz Bedroom. Get busy with our maths tutor and save up your hard-earned credits to add our shiny mechanical canine to your personal maths tutoring space.

Don’t forget to stock up on nuts and bolts – he’s on a stainless steel-only diet…