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Either it's a slow news day or a producer at the BBC has finally come across the new method of multiplication and division now being taught in schools. Either way in a feat of sheer brilliance they decided to explain it to parents by cramming in a short session before the local news at 8:30; you know when parents are out dropping their kids off at school. Due to the short session they also did a poor job of explaining it; or to be precise why these new methods are being taught and why they might be better in the long run.
I'll guess that the majority reading this are familiar with the traditional method of multiplication
13
*12
130
+26
156
The new method is based on turning the problem into a "grid" and separating it out into simpler numbers
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
So we have a 10*10 grid, a 3*10 grid, a 10*2 grid and a 3*2 grid Multiply those simple numbers together and add. That's the first stage, but drawing a whole set of points is tedious and if done a person might as well count them as they went along; thus the next step is to simplify that as follows:
10| 3
10|100 30
2| 20 6
Note this is exactly the same as the previous grid, but in a far more compact form. The answer is found by adding all the values within the table. Now this may seem a long winded method of multiplication, but it has two major benefits. The first is that it becomes obvious how numbers multiply together. Consider the tradition method again step by step
13
*12
0
Why is there a zero there? One could answer that it's to allow us to reach the first "1" of "12" or that because the "1" is really a "10" it has to be added, but neither is an intuitive explanation. It becomes "what you need to do to multiply" a black box formula that you don't need to understand simply the ability to process. The grid method shows exactly why every number is where it is. The second reason is that this method can be used in exactly the same way for algebra. Try multiplying x+2 by x+4 using the traditional method
(x+2)
*(x+4)
Where would you even start? Now try it via "gridding"
x| 2
x|x² 2x
4|4x 8
and the answer of x²+6x+8 drops out instantly. This method is known as the distributive property (amusingly causing right-wing panic in the USA). Now while gridding is fine, it's still a little clumsy and thus the final step needs to be added:
13*12 = (10+3)(10+2)=(10*10)+(3*10)+(3*10)+(3*2)
That is remove the grid and, ironically, return to the same traditional method taught in algebra
(x+2)*(x+4) = (x*x)+(x*4)+(2*x)+(2*4)
So that's gridding what about "chunking"? The sum used as an example at the BBC was 987/7 in traditional full long diversion that would appear like so:
141
7|987
7
28
28
07
7
0
This seems obvious, but as with the initial "0" of multiplication there's a question of "why do you drop the numbers down?". The initial stage of "chunking" is to remove "chucks" from the number i.e. multiples of 7 like so
987
-700 (100)
-280 (40)
- 7 (1)
Then add the multiples together. This can also be shown via traditional long diversion as follows:
141
7|987
700
287
280
7
7
0
While this shows where every number comes from, unlike gridding this results in a set of far more difficult maths (287/7 rather than 28/7) It also causes problems when numbers don't even divide take 988/7 in traditional form
141.14
7|988.00
7
28
28
08
7
1 0
7
30
28
2
And so on. Try that using chunking
988
-700 (100)
-280 (40)
- 8 (1)
141 and 1 remaining. While simply stating and 1/7 is fine at the basic mathematics level how would that be expressed as a decimal if the only knowledge of division a person has is via chunking? Try chunking with algebra using our multiplication example; how many (x+2) "chunks" fit into x²+6x+8?
x²+6x+8
x+2 (?)
Try it using traditional long diversion:
x + 4
x+2|x²+6x+8
x²+2x
0+4x+8
4x+8
0
The same method can be used with minor modifications (how many x in x²) Okay it can be done via chunking with some additional lines :
x²+6x+8
x²+2x (x)
4x+8 (4)
Which again results in x+4, but is far more difficult to see.
As I see it "gridding" leads naturally to more advanced forms of alegbra; whilst "chunking", while being a better demonstrative method, may lead to complications further along.
25 Şubat 2013 Pazartesi
Mathematics - bases taught in schools?
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After my post regarding the new methods of multiplication and division I recalled an incident with the Bratii over the holiday period. As is often the case the entire family were discussing various topics one of which was Imperial vs. Metric measurements. My father was complaining regarding news and quiz questions that used the 'new' methods on the grounds that "no-one can understand them" this was quickly dismissed by presenting the point that it was not everyone merely himself and that this generation and the next would have no difficulty in imaging a "plume of dust 3km high". Also that this was far better for school work due to the reliance on base10 which was also used for basic mathematics. A point reinforced when I asked ho many feet there were in a mile and he came back with 1760 (which is yards) and then had to multiply by 3 in his head to provide an answer.
This led the conversation to the discussion of the various number bases used historically. Obvious base12 for inches into feet and pennies into shillings as well as base20 for shillings into pounds and others. Base10 seems only to have 'won' because that's the number of digits we normally possess. Although, it was pointed out, Base2 and Base16 is used in computers
"Why not just use Base2?" asked my mother "As that's the computer method"
"Because it's too large. If I wanted to represent 8b10 it would be as 1000b2" I replied
I then watched as both Bratus Pater and Bartus Major tilted their heads slightly in contemplation
"You're both working out if I've got that right" I laughed
Bratus Pater laughed and admitted to it but Major responded with "No! I've no idea what you're talking about"
This caused some wonderment and bemusement in both myself and his father. He tried some examples and neither Major nor Minor were any the wiser. I wondered if they'd been taught it under a different name and tried "Clock method" which was not understood by any including Bratus Pater until I explained it to him as modular arithmetic.
[Diversion - A simple method of base12 can be seen on an analogue clock (modular 12). Take a number such as 38b10 and travel around the clock subtracting 12 each time (similar to chunking) you go around 3 full times and need to travel 2 stops more (the modulus) the conversion becomes 3 mod 2 or 32b12]
At this point my father chimed in by asking how any of this was relevant or useful. Both myself and Bratus Pater (with our similar backgrounds) jumped in with the computer reference. We both explained that these (2 and 16) are the main bases that computers use; that although the majority of users have no requirement to ever know this it becomes more important at a lower programming level. Given that Bratus Major is moving towards an oil geology field, and that is a specialised subject, at some point he may well need to write or debug computer programmes to provide for something that cannot be obtained 'off the shelf'.
In terms of pure education I can now relate it to "gridding" as something that should be taught early on at the same time as basic base10 mathematics. Although base10 is the most used it appears to be taught in the same black box fashion as traditional multiplication. You start with the units and then move to the ten's the hundred's and so on. So what? Well why move up to hundred's? Consider if it were taught along side a base of 4.
We start with the units then move to the four's and then the sixteen's. Prompting the question why sixteen? So again in base10 we start with the unit's then the ten's then the ten-ten's then the ten-ten-ten's. In base4 again we start with the units and then move to the four's the four-four's and the four-four-four's. No need to get involved with exponents and powers just number multiplied by number.
To reiterate my father's question "Why does this matter?" Well it explains why the system works as it does and moreover demonstrates that the number 10 is not elevated to some special position. With a slight change in history we could all have been using base12 now.
This led the conversation to the discussion of the various number bases used historically. Obvious base12 for inches into feet and pennies into shillings as well as base20 for shillings into pounds and others. Base10 seems only to have 'won' because that's the number of digits we normally possess. Although, it was pointed out, Base2 and Base16 is used in computers
"Why not just use Base2?" asked my mother "As that's the computer method"
"Because it's too large. If I wanted to represent 8b10 it would be as 1000b2" I replied
I then watched as both Bratus Pater and Bartus Major tilted their heads slightly in contemplation
"You're both working out if I've got that right" I laughed
Bratus Pater laughed and admitted to it but Major responded with "No! I've no idea what you're talking about"
This caused some wonderment and bemusement in both myself and his father. He tried some examples and neither Major nor Minor were any the wiser. I wondered if they'd been taught it under a different name and tried "Clock method" which was not understood by any including Bratus Pater until I explained it to him as modular arithmetic.
[Diversion - A simple method of base12 can be seen on an analogue clock (modular 12). Take a number such as 38b10 and travel around the clock subtracting 12 each time (similar to chunking) you go around 3 full times and need to travel 2 stops more (the modulus) the conversion becomes 3 mod 2 or 32b12]
At this point my father chimed in by asking how any of this was relevant or useful. Both myself and Bratus Pater (with our similar backgrounds) jumped in with the computer reference. We both explained that these (2 and 16) are the main bases that computers use; that although the majority of users have no requirement to ever know this it becomes more important at a lower programming level. Given that Bratus Major is moving towards an oil geology field, and that is a specialised subject, at some point he may well need to write or debug computer programmes to provide for something that cannot be obtained 'off the shelf'.
In terms of pure education I can now relate it to "gridding" as something that should be taught early on at the same time as basic base10 mathematics. Although base10 is the most used it appears to be taught in the same black box fashion as traditional multiplication. You start with the units and then move to the ten's the hundred's and so on. So what? Well why move up to hundred's? Consider if it were taught along side a base of 4.
We start with the units then move to the four's and then the sixteen's. Prompting the question why sixteen? So again in base10 we start with the unit's then the ten's then the ten-ten's then the ten-ten-ten's. In base4 again we start with the units and then move to the four's the four-four's and the four-four-four's. No need to get involved with exponents and powers just number multiplied by number.
To reiterate my father's question "Why does this matter?" Well it explains why the system works as it does and moreover demonstrates that the number 10 is not elevated to some special position. With a slight change in history we could all have been using base12 now.
When the logical road solution isn't always the right one.
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Water pipe replacement work is being carried on in Stourport at the moment necessitating a set of controlled traffic lights to allow traffic through as these sit at a three-way junction it would seem logical to put up a set of three way traffic lights. Instead a set of two-way were placed at the junction itself; yesterday they were switched out for a three-way set and this morning back to a two-way. The question is why?
The junction itself in simplistic terms is a straight North-South run with a West side junction. While a two-way set of lights were in effect proceedings were as follows:
Traffic heading South on the main road continued only while the light was green; if they wished to make a turn West they did so at the end of the roadworks just before the stopped Northbound traffic. Traffic heading North on the main road were also required to wait for green; however should they be at the head of the queue and wish to turn West it was easily accomplished on a red light provided no Southbound traffic prevented them. For traffic from the West the rules were a little more complicated.
To head North was only possible on a green light, but would require beating or merging with any other Northbound traffic on the main road. To head South, however, could be accomplished on either a red or green light provided no traffic on the main road prevented it.
The lights were controlled but even with a timing switch the two main streams of traffic would have to wait perhaps a maximum of 2 minutes before the lights changed.
With the alteration to a three-way set of lights all traffic could only move on green. Traffic turning West from the South was now unable to reach the junction to turn on red and must wait also. If the lights were on a timer the wait now doubles as each take their turn. The natural flow of traffic from the West junction is halted. Logically the queues forming at this point should be reduced as they now have an absolute right to proceed at intervals, but the cost is extended queues on the main road. How extended?
Stourport has always had problems with its road network. Consider it a crossroads with the junctions connected by a narrow one-way system. In the mornings the majority flow is from the North, South and West exits outward to the North and East; while in the evening the flow reverses from North and East to North, South and West. In essence the lights blocked the South exit. In theory this should have no effect on the other exists, but as I said it's connected by a one-way system.
Traffic from the North wanting to travel West was unaffected. Traffic from the North to South or East to South of course was. However traffic from the East to North became blocked by the traffic trying to travel South. In turn this prevented lesser traffic from the West travelling either South or East.
This simple change resulted in evening rush hour traffic shifting from a queue stretching 0.1 miles along a single stretch of road into a a multi-tailed queue stretching out to at least 2 miles.
The lesson to learn is to understand and appreciate the traffic flows of wherever is being disrupted.
The junction itself in simplistic terms is a straight North-South run with a West side junction. While a two-way set of lights were in effect proceedings were as follows:
Traffic heading South on the main road continued only while the light was green; if they wished to make a turn West they did so at the end of the roadworks just before the stopped Northbound traffic. Traffic heading North on the main road were also required to wait for green; however should they be at the head of the queue and wish to turn West it was easily accomplished on a red light provided no Southbound traffic prevented them. For traffic from the West the rules were a little more complicated.
To head North was only possible on a green light, but would require beating or merging with any other Northbound traffic on the main road. To head South, however, could be accomplished on either a red or green light provided no traffic on the main road prevented it.
The lights were controlled but even with a timing switch the two main streams of traffic would have to wait perhaps a maximum of 2 minutes before the lights changed.
With the alteration to a three-way set of lights all traffic could only move on green. Traffic turning West from the South was now unable to reach the junction to turn on red and must wait also. If the lights were on a timer the wait now doubles as each take their turn. The natural flow of traffic from the West junction is halted. Logically the queues forming at this point should be reduced as they now have an absolute right to proceed at intervals, but the cost is extended queues on the main road. How extended?
Stourport has always had problems with its road network. Consider it a crossroads with the junctions connected by a narrow one-way system. In the mornings the majority flow is from the North, South and West exits outward to the North and East; while in the evening the flow reverses from North and East to North, South and West. In essence the lights blocked the South exit. In theory this should have no effect on the other exists, but as I said it's connected by a one-way system.
Traffic from the North wanting to travel West was unaffected. Traffic from the North to South or East to South of course was. However traffic from the East to North became blocked by the traffic trying to travel South. In turn this prevented lesser traffic from the West travelling either South or East.
This simple change resulted in evening rush hour traffic shifting from a queue stretching 0.1 miles along a single stretch of road into a a multi-tailed queue stretching out to at least 2 miles.
The lesson to learn is to understand and appreciate the traffic flows of wherever is being disrupted.
All surprises today
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The news has been so unpredictable today - first up is Nick Clegg denying that there was a cover-up over sexual misconduct. Personally I thought he was going to stand up and declare that he'd known all about this and did his best to make sure no-one else would.
The Academy award winners were also a big surprise. For the award to "Best film that makes America look great against evil foreigners" was a toss-up between the two French films and two films that made parts of America look bad and Argo. Best Pretentious Method Actor in a serious American film was also a toughie, but ended up going to the most pretentious method actor in a serious film.
Best actress in a serious role dealing with serious issues looked like it was going to Jessica Chastain in Zero Dark Thirty except that had nasty bits in it that made the good guys look kinda bad so they switched to the serious film dealing with serious issues that didn't and thus Jennifer Lawrence in Silver Linings Playbook.
Best Actress in an uncomfortable role went to the prostitute Anne Hathway for pretending to be French; and Most famous person singing a song went to the most famous person to sing a song for a movie.
All in all a total shock
The Academy award winners were also a big surprise. For the award to "Best film that makes America look great against evil foreigners" was a toss-up between the two French films and two films that made parts of America look bad and Argo. Best Pretentious Method Actor in a serious American film was also a toughie, but ended up going to the most pretentious method actor in a serious film.
Best actress in a serious role dealing with serious issues looked like it was going to Jessica Chastain in Zero Dark Thirty except that had nasty bits in it that made the good guys look kinda bad so they switched to the serious film dealing with serious issues that didn't and thus Jennifer Lawrence in Silver Linings Playbook.
Best Actress in an uncomfortable role went to the prostitute Anne Hathway for pretending to be French; and Most famous person singing a song went to the most famous person to sing a song for a movie.
All in all a total shock
Nitpicky language
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I've been watching BBC's the Great British Menu in bulk 5, episodes at a time. Amazing how much rubbish can be removed - I've turned 2.5 hrs into 1.5 by fast-forwarding through the "this is what we're doing this year"'s the "and these are all the ingredients I'll be using"'s and the "and this is how how chefs did something regarding this year's brief"'s. Oh and the format's still a mess given the supposed goal, but I'll stop digressing.
Last week we had the North East; or Nor Feast as it was called. I've no problem with accents switching "th" for "f" or droppin' end sounds, they're simply diverse sounds and one is really no 'better' than another, but the question that kept being asked was
Yes, yes regionalism, but it's sloppy, causes an upset to my English parsing, and is unnecessary.
Oh I suppose it could have been worse; he could have been asking "Do you think this is a dish what you could make perfect for a hundred guests?"
Last week we had the North East; or Nor Feast as it was called. I've no problem with accents switching "th" for "f" or droppin' end sounds, they're simply diverse sounds and one is really no 'better' than another, but the question that kept being asked was
"Do you think this is a dish what will make the judges laugh?"Ouch. Try this "Do you know what is wrong with that sentence?". Both "Do" and "What" are questioning terms and I can split my query into "What is wrong with that sentence? Do you know?". Now try it with the chef's question "What will make the judges laugh? Do you think this is a dish?" Nonsensical. It could be saved by "What will make the judges laugh? Do you think this is such a dish?". But that's not really what the question was about. The chef was not asking what will make the judges laugh only if this was a dish that could cause them to do so. It wasn't two questions it was merely one presented in an inaccurate form.
Yes, yes regionalism, but it's sloppy, causes an upset to my English parsing, and is unnecessary.
Oh I suppose it could have been worse; he could have been asking "Do you think this is a dish what you could make perfect for a hundred guests?"
24 Şubat 2013 Pazar
Sony's Welcome Back Program Is Live
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The final piece of the PSN puzzle was put into place today, as the "Welcome Back" program content was released on the PSN. PS3 owners can make their selection of two games among the following:
The free 30 days of PlayStation Plus is also live for those that aren't already members. You are NOT automatically enrolled into this, you must "buy" it from the store by heading to the "Welcome Back" section of the store and choosing the "PlayStation Plus: 30 Days Free" option. Existing Plus members WILL automatically have their subscriptions extended by 60 days.
If you're a Home user... is anyone a Home user? Anyhow, if you are, you can find the free 100 virtual items in a kiosk in Home's Central Plaza.
Lastly, the free video content is up and available ONLY THIS WEEKEND.
So start downloading your games and watching your movies, you've got a lot to do this weekend thanks to Sony!
- Dead Nation
- inFamous
- LittleBigPlanet
- Super Stardust HD
- Wipeout HD + Fury
- LittleBigPlanet (PSP)
- ModNation Racers (PSP)
- Pursuit Force
- Killzone Liberation
The free 30 days of PlayStation Plus is also live for those that aren't already members. You are NOT automatically enrolled into this, you must "buy" it from the store by heading to the "Welcome Back" section of the store and choosing the "PlayStation Plus: 30 Days Free" option. Existing Plus members WILL automatically have their subscriptions extended by 60 days.
If you're a Home user... is anyone a Home user? Anyhow, if you are, you can find the free 100 virtual items in a kiosk in Home's Central Plaza.
Lastly, the free video content is up and available ONLY THIS WEEKEND.
So start downloading your games and watching your movies, you've got a lot to do this weekend thanks to Sony!
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