A while ago I pointed out that while I’ve never found creating a simple 3×3 magic square very useful in the classroom there are plenty of spinoff activities that really do work well. I devised Polo Squares when I was giving a session for 300 children at the Royal Institution in the middle of London. The children were from a dozen or more schools and I wanted to offer something that was easily accessible and could be instantly usable no matter how big the groups were or whether they arrived early or with just five minutes to spare.
Every child had a set of ten digit cards, from 0 to 9. I asked them to make a hollow square using eight of the cards, so the cards on each side of the square add up to the same total. I provided lots of blank record sheets; clearly a lot of work was done at home after the session, because the postman was delivering solutions for weeks afterwards.
I’ve no idea how many different Polo Squares there are, but I know it’s several dozen, so no-one’s going to say “I’m finished, what do I do next?” Each solution uses eight of the ten cards, so there’s an element of Trial and Improvement.
And there are plenty of challenges – “What’s the highest / lowest Polo total?”, “Can you make all totals in between?”, “Which totals have the highest number of solutions?”, …..
Polo Squares offer a nice challenge – even wider than I’d realised; George was just five, and the younger brother of someone who’d attended the session.
Using digit cards helps the activity and also makes it feel like fun. The cards invite the use of some problem-solving skills (look at L & A’s) , and give some meaty mental arithmetic practice.
And in just the same way I could use them with children waiting for my session to start, teachers can have some digit cards and answer blanks available on parents’ evenings!
One of the blogs I follow is Find The Factors. It’s very different from this one. For a start, it’s frighteningly efficient, and appears on a daily basis – not like this one, which happens now and again when I get around to it.
Iva Sallay gives us a new puzzle every day, where you’re given a few entries in a multiplication square and have to figure out the multipliers heading the columns and rows. So clearly we’re in an area highlighting multiplication facts – in the puzzle below the 42 in the top row has to be the product of 6 and 7 – but we can’t immediately do anything with this because we don’t know if it the 42 comes from 6×7 or 7×6. And the 20 will offer more problems, since we’ll have to discover whether it comes from the product of 2 and 10, or 4 and 5; even then, we must decide on the orientation.
A Find The Factors puzzle will be full of challenges like this, and the beauty of the blog is that they’re graded from level 1 all the way to the most challenging level 6. The puzzle below is level 4:
This puzzle is number 177, and you’ll usually get some facts about the puzzle number – its factorisation, and in this case, you’re also given a magic square made up of prime numbers; the magic total is 177.
The obvious comparison is to Sudoku puzzles. I rarely do Sudoku; I usually find I get so far and then have no option other than guessing between several possibilities until I find the right one. Just occasionally I’ve had to do this with Find The Factors, but generally speaking even at level 6 I can eventually reason the whole thing out. Mind you, sometimes I have to work hard, but that’s surely the whole idea!
I’ve shown one of Iva’s 1-12 puzzles, but she also gives 1-10 puzzles as well, and you can have a week’s worth of puzzles to download in Excel format.
You can find Iva’s blog at http://findthefactors.wordpress.com
I do the puzzles for my own enjoyment, but my pupils would find them daunting and indeed too time-consuming. However, the basic idea is one I’m happy to challenge them with, so I may offer them a 4×4 square constructed on the same theme. Once they’d tackled one of those, we’re likely to use the interactive version on the NRICH website (http://nrich.maths.org/7382) where you can find a variety of challenges all based on the same starting point).
Ladies and Gentlemen: Roll up, Roll up, because I’m going to give you, at positively no charge, my all-time winner of the “Sir’s busy and must not be interrupted for any reason whatsoever” award. You can use it with virtually any group and every child will get lots of lovely arithmetic practice; there’s pattern and symmetry, exploration, hypothesis, searching for completeness, and no doubt more bedsides.
The previous post gives you a 4×4 magic square, so we know that each of the four rows, the four columns, and the two diagonals all add up to the magic number. For example:
But what we often fail to notice is that these aren’t the only sets of four numbers that add up to 34. Try, for example, the numbers in the corners. Then look at some of these configurations (and won’t each one suggest some more arrangements to try?).
How many patterns do you think there are? (I give children a sheet with 35 grids – but that may or may not give you a clue.) This is, incidentally, another wonderful activity to have available for parents to try on an Open Evening. Give them plenty of grids, pens, a large expanse of wall to fill, and let them get on with it.
I can make you a better maths teacher in ten seconds. It doesn’t matter what age group you teach or how your school organises its teaching groups. You won’t need any special training and or any equipment, any electronic wizardry, any textbooks or software.
Better still, you can do the same and make all of your colleagues better maths teachers in the same time, and your school won’t have to pay a consultant a thousand pounds to do it.
It’s widely quoted that teachers typically expect an answer to a question in 0.9 seconds. What kind of considered response can you give in less than a second? For many children the most likely reaction is panic.
On the other hand, if you give children some thinking time then they’re got the chance to assess the question and frame a suitable response. So the strategy I’ll offer you is that all you need to do is to count to ten before you accept an answer to any question.
Whether or not this is spelt out explicitly doesn’t worry me one bit; the point is you’re giving children a little bit of time to think about the answer they’re going to give you. At one session one teacher was rather sceptical that such a simple technique could have any significant effect, but she agreed a little reluctantly to give it a try. A couple of weeks later I had an e-mail: “I have been amazed how helpful it is to wait ten seconds before asking children to answer questions, so simple, yet effective.”
Share this technique with your colleagues and you’ve already earned your consultancy fee, but you can certainly take things further. Are there questions which genuinely do merit an instant response? And what does a 30-minute question look like?
*** In a Y4 classroom I asked children to put 8 into their calculator and add on the number which would make it display 20. I then asked them to work in pairs and use their own numbers. Every pair was able to choose the target that suited them – within 15 minutes one pair was playing to 1000, with the rule that their numbers must have at least three places of decimals!
*** If all the number keys except the 3 and 4 have fallen off your calculator how can you make it display numbers like 10?
*** Paul, aged 6, and not of exceptional ability, told me “I do experiments with my calculator; I see what happens when I multiply numbers by 99”. I set him the challenge of finding two two-digit numbers whose product is 7326, and he refused to go out to break until he’d found them.
*** I have a special calculator that the manufacturers call the “Wrongulator”. If you ask it the answer to 4×4 it shows 11. For 22-8 it shows 9. What do you think it displays for 27+19?
The obvious thing that calculators do is give the answer to sums. From there it’s a very short step to say that they impede mental and written arithmetic. But in each of the cases above, ask yourself where the arithmetic is actually being done, and then you get a different perspective. All the arithmetic is being done by the children, and being done mentally; the calculator’s rôle is providing feedback on the work already done in the head. In the Y4 classroom no teacher can possibly give instant feedback to fifteen pairs of children each working at a different problem (and what teacher has ever dared ask children to do six- or seven-digit mental arithmetic?). Similarly, without the stimulus of the calculator could I ever have expected a 6-yearold to be able to handle multiplication by 99? The “Broken Calculator” activity is so well-established that a Google search locates more than 47 000 000 hits. And my Wrongulator turns the conventional situation around so that you have to think what the answer should be, rather than what answer it actually displays.
Each of these activities makes it clear that the calculator has the potential to stimulate and enhance children’s mental arithmetic rather than stunt it. It’s now approaching thirty years since since Hilary Shuard’s CAN project demonstrated that when primary children had unrestricted use of a calculator their arithmetical understanding and fluency were not handicapped but greatly improved. In an allied project in Hertfordshire we found this was true not just for junior school children, but for all ages from Early Years through secondary.
Policy-makers are happy to encourage schools to spend thousands and thousands on computers and allied equipment, but ignore the proven benefits of the simplest piece of IT, which every child has in their pocket on their mobile phone. We have a quasi-religious attachment to traditional algorithms for long division, etc. Why stop there? After all, at school I had to learn an algorithm for extracting square roots, and even once encountered the corresponding method for cube roots!
I often read superficial arguments that calculators necessarily reduce mental exercise (not true – see the examples above) and that mental exercise is analogous to physical exercise. Maybe it is, but anyone who advances that view to me is liable to be asked whether they do their washing by taking it down to the river bank. Everybody knows that the traditional way of washing is by banging it with stones, but having a washing machine makes it possible to wash more things more efficiently, and make more time to do other things.
Surely there’s no game so rich in mental arithmetic as Darts? Agreed, you’d be very nervous about children actually throwing the things, and few of them have much idea of the rules. However, working with a printed target (close your eyes and stab with a pencil) makes a good learning model which allows you to start with one dart, bring in doubling, move to two and three darts, and incorporate the subtraction element as well, not forgetting the final touch that you need to finish on a double.
Two resources I wouldn’t want to be without are:
* a printout of the standard board; the one I use is at
* Chris Farmer’s software. I’ve tried to persuade Chris to offer a few more options, but it’s very cheap and it works well
Here’s a game that’s almost infinitely flexible and is one of my very favourites for working with children. There are two parts of the board; one section consists of a grid of hexagons (though you’re welcome to use squares):
Write a number into each hexagon:
When it is your turn to play you choose two numbers you wish to add together and cover them with counters:
I use these a lot. They’re not threatening, they’re fun, and they’re hugely informative. The set I use most often is a simple set of a couple of dozen cards, each with a
multiplication statement, e.g. 6×6=36. Some of the statements are true, but some, such as 8×8=63, are false.
I ask children to sort them into three categories, true, false, and don’t know or not sure. What’s particularly revealing is when children tell you why they put cards into the false pile. One child will say 8×3=25 can’t be true because the correct answer must be even; another will say it’s because neither number is 5.
It’s such a valuable activity that I’ve got plenty of other sets as well, with different numbers and operations.
I’m guessing, but I imagine that if you’re reading this there’s a good chance that you see a multiplication square as a beautifully structured array, logical and full of intriguing patterns, some known and some waiting to be discovered. For most of the children I work with the situation is very different. For them a tables square can be an intimidating affair, rather like one of those medieval maps of a largely unexplored continent where only the outer fringes are known and the inner areas are largely unfamiliar – “Here be dragons” indeed. In the last couple of years I’ve made much less use of a multiplication square; instead I’ve very often used a set of multiplication strips – a tables square cut into ten separate strips, one for each multiplication table. The strips are much less daunting and far more accessible, and they introduce a tactile and exploratory aspect. Give a child a multiplication square and they’ll look at it – and that’s about it. Give them a set of multiplication strips and they’ll pick them up and start to do something. Perhaps they’ll put them together and reassemble the tables square:
or often they’ll start to pair them up or sort them into groups. Sometimes they’ll sort them into two groups, the odd-numbered tables, and the even-numbered. Straight away there are interesting discoveries to be made. Choose one of the even-table strips. Can you find some odd numbers on it? No, they’re all even. So what do you expect if we look at the odd-table strips? Here’s an observation that tends to come as a complete surprise – three times as many of the numbers within a tables square are even as odd, and there are no odd numbers to be found anywhere within any even table.
More often, the initial sort puts strips into doubling pairs, x3 with x6, and x5 with x10. Immediately all sorts of questions arise. If you pair x2 with x4, then what do you do with the x8, and the x1? So in fact some of your sorting puts the strips into families rather than pairs. Once you’ve got the x1, x2, x4, x8 family isn’t it tempting to wonder if there’s not a further member waiting to be included – so you can incorporate the x16, and … ? And when you sort into pairs, what about those strips which don’t have a partner, like the x7 ? Perhaps we really ought to have a x14 strip? Or could we have a x1½ strip to provide an alternative partner for the x3 ? Of course none of these questions arise every time, but all of them are questions which have arisen with my pupils, and I find it useful to have some empty strips on hand. There’s no need actually to write out the full x14 table onto the strip, simply labelling it x14 makes the point perfectly well.