Multiplication

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Multiplication

• Staff Tutorial

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• In this tutorial we run through some of the basic
  ideas and tricks in multiplying whole numbers.
• We do a bit of stuttering on the decimal system
  and the fact that 6 times 15 is the same as 15
  times 6 before we get under way.

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• Its perhaps a little surprising that although
  many civilisations counted using a decimal system
  it took a long while for everyone to realize that
  the decimal system was a good one to use for
  doing arithmetic. In the Western world it wasnt
  till 1202 that Fibonacci wrote his famous book on
  arithmetic and introduced Europe to a great way
  to do calculations. Its actually amazing that it
  took so long.

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• After all the Greeks, who were no mathematical
  slouches, had done a great deal Euclid, for
  instance, had written a book on geometry that
  sold for over 2000 years. But it took
  non-Europeans to see what a good idea it was to
  write three hundred and ninety-four as 394, with
  the right-most position standing for 4 ones (or
  units), the 9 just to the left standing for 90,
  and the 3 out the front standing for 300. That
  concise notation revolutionized the world.

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• Note too the way that the zero has come into
  action to keep the numbers apart and to
  distinguish 345 from 3045, 3405, and 3450.
• You might just like to rub over the next couple
  of numbers and see what the numbers in the
  different positions stand for.
• 2367 917305 890299.

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• That neat notation means that you never again
  have to multiply a number by ten using a
  calculator.
• You see 3 times 10 is three lots of 10 and so we
  only have to put a 3 in the tens column to give 3
  x 10 30.
• But the same trick works at higher levels. What
  is 53 x 10?
• Well 53 x 10 is the same as 50 x 10 plus 3 x 10.
  Now we already know that 3 x 10 30.

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• So what is 50 x 10? Surely we want to put 50 in
  the tens column? But its too big and the 5
  spills over into the hundreds column. Thats not
  a surprise. Think of it this way. Now 50 is 5
  lots of 10. So 50 x 10 is 5 lots of 10 times 10.
  Since ten tens is a hundred, 50 times 10 is
  surely 5 lots of 100. So 50 x 10 500.
• Then 53 x 10 50 x 10 3 x 10 500 30 530.
• And all that weve done is add a zero on the end.
  And that always works. Check it out for
• 28 x 10 123 x 10 39674 x 10.

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• I promised you that wed see that 6 times 15 is
  the same as 15 times 6. How can we do that?
• Which would you rather have? Six bags of sweets
  with 15 in each or fifteen bags of sweets with 6
  in each?

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• Lets take one bag of 15 sweets. We could divide
  the sweets into 6 6 3. Put the sixes into
  separate bags. Thatd give us two bags of 6
  sweets and 3 over. But wed be able to do that
  for all of the initial six bags. So now we have
  twelve bags of 6 sweets and six lots of 3 sweets.
• But 2 lots of 3 sweets gives another bag of 6.
  Whats more we get 3 of these 2 lots of 6. So now
  we have the twelve bags of 6 from above plus the
  3 bags of 6 from below. Thats 15 bags of 6 if
  ever I saw them.
• Why dont you show that 8 bags of 17 sweets
  contains as many sweets as 17 bags of 8 sweets?

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• But thats a bit tedious. Its not so easy to
  check out 392 bags of 563 sweets against 563 bags
  of 392 sweets. But that can be done. What you
  need to do is to is to get 563 different colour
  stickers (I jest of course). Then stick a
  different coloured on each of the 563 sweets in
  the first of the 392 bags. Then stick a different
  coloured sticker on each of the 563 sweets in the
  second of the 392 bags. Keep doing that until all
  the sweets have had a sticker put on them.

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• Now take all the sweets with a red sticker on
  them and put them into a new bag. That should
  give you 392 red-stickered sweets in the first
  bag. (I hope that you are doing this without
  touching the sweets. Someones going to want to
  eat them after youre finished.).
• Now take all the sweets with a blue sticker on
  them and put them into another new bag. That
  should give you 392 blue-stickered sweets in the
  second bag.
• Now take all the sweets with a green sticker on
  them and put them into another new bag. That
  should give you 392 green-stickered sweets in the
  third bag.

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• Can you see that you can keep going like this and
  eventually youll have 563 bags each with 392
  sweets in them?
• Now show that 25 bags with 179 in, is worth as
  much as 179 bags with 25 in. Can you think of a
  way to do this without using coloured stickers
  because the President of the Reserve Bank is
  getting a bit annoyed with people putting
  stickers all over his nice money.

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• So you should be able to see now that 564 x 702
  is the same as 702 x 564. Whats more you can
  justify that the two are the same. But there is
  an easier way than the sticker route. Lets do
  some tiling.
• Get a whole stack of tiles and make a big
  rectangle of them. In fact make them so that you
  have 564 rows of tiles with 702 tiles in each row.

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• But now turn your big rectangle through 90º. Now
  you should see that you have 702 rows and 564
  tiles in each row.
• And that should convince you that 564 x 702 702
  x 564.

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• This business about the order of multiplication
  not mattering has a fancy mathematical name. Its
  called the commutative law. It just means that it
  doesnt matter which way we do it we get the same
  answer.
• Actually there are not too many things around
  that commute. Try interchanging the order of
  putting on your socks and your shoes. Shoes first
  give quite a different result. Can you think of
  anything but multiplication that is commutative?

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• OK, so now weve got some tools in place that
  might come in handy. We can multiply by 10 and we
  know that it doesnt matter what order we
  multiply numbers in, the answer is still the
  same.
• So what can we do to get 23 x 11? Well Im not
  convinced that 11 x 23 is any easier so lets try
  something else.
• We can all do 23 x 10. Thats 230. And 23 x 1 is
  23. So 23 x 11 230 23 253.

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• Actually we can see this using the tiling idea.
  If we had 23 tiles one way and 11 the other then
  the rectangle of tiles would be like the picture
  below.

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• If we cut the 11 side into a 10 and a 1 we get
  the picture below.

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• And we can do the odd calculation.
• Clearly 230 23 253 the number of tiles in
  the rectangle.

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• But can you find another way to represent 23 x
  11? How about using bags of sweets?
• And then have a crack at finding 47 x 12 and 13 x
  88.
• After that you might like to see what you can do
  with 19 x 91.

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• Actually 19 x 91 can be done in a number of ways.
  First you might break 19 up into 10 9. Then all
  you have to do is to multiply 10 x 91 and add it
  to 9 x 91.
• Of course you might prefer to do it the other way
  and first do 91 x 10 and add it to 91 x 9.
• But 19 20 1. So you could do 91 x 20 91 x
  1.
• And all of this could be done using tiles.
• Work out 19 x 91 using tiles in all of these
  ways.
• Can you think of another way?

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• But what about big numbers? How about 187 x 56?
  You can do all of that using tiles too. Look
  here.
• You can see that the product of 187 and 56 is the
  sum of the six smaller areas. Weve talked about
  multiplication by the tens and the units but not
  about the hundreds. But thats easy isnt it?

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• 1 x 100 is surely 100 and 5 x 100 and so on work
  the same way. But what about 10 x 100? Well here
  comes the commutative law to the rescue. 10 x 100
  100 x 10 and we know that 100 x 10 1000. So
  10 x 100 1000.
• What about 23 x 100 though? Well thats 20 x 100
  3 x 100. Obviously 3 x 100 300. And 20 x 100
  100 x 20 100 x 10 x 2 1000 x 2 2000. So
  23 x 100 2300.

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• So lets go back to 187 x 56. Weve seen how to
  do that the tiling way and it comes out like
  this
• 187 x 56 187 x 50 187 x 6.
• Now 187 x 6 100 x 6 80 x 6 7 x 6 600
  480 42 1122,
• And 187 x 50 (187 x 5) x 10 (100 x 5 80 x
  5 7 x 5) x 10 (500 400 35) x 10 (935) x
  10 9350.
• So 187 x 56 1122 9350 10472.

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• But you have just seen the good old
  multiplication algorithm done on its side. Lets
  do it the usual way up.
• 187
• 56
• 1122 this line is just 100 x 6 80 x 6 7 x
  6
• 9350 this line is just 100 x 50 80
  x 50 7 x 50
• 10472
• Notice that we get lazy in practice. The 50
  line we dont consciously multiply by 50 each
  time. We simply dump a 0 at the end and multiply
  187 by 5. Hopefully its now clear that the
  algorithm is exactly the same as the tiling
  method.

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• If you want to use the standard algorithm to
  multiply 2345 by 1234, then talk your way through
  why you put no zeros at the end of the first row,
  one zero at the end of the second row, two zeros
  at the end of the third row, and three zeros at
  the end of the last row. Then do the algorithm
  method and check it against the tiling method.

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• Please email us at derek_at_nzmaths.co.nz for any
  correspondence related to this workshop.