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 Post subject: VR or Maths?
PostPosted: Sun Apr 27, 2008 9:34 pm 
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Joined: Sun Apr 27, 2008 9:21 pm
Posts: 18
Location: Hertfordshire
Triangular Number - What is the technique of finding the answer quicker where you have a triangle with one dot. Add 2 dots to the 2nd triangle to make 3, add 3 to the 3rd triangle to make 6, add 4 to the 4th to make 10, add 5 to the 5th to make 15.

1 +2(1st), 3 +3 (2nd), 6 +4 (3rd), 10 +5 (4th), 15+ 6 (5th) etc

So what is the answer to the following triangular:

The 5th is 21 (is this corect?)
The 7th ...
The 9th ....
The 8th ...
The 10th ...

and so on

[/b]


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 Post subject: triangular numbers
PostPosted: Mon Apr 28, 2008 8:33 am 
You have the general idea but the fifth would actually be 15.

It goes 1st = 1
2nd = 3
3rd = 6
4th = 10
5th = 15
6th = 21

The formula for the nth term is 1/2n (n+1).

Thus if n was 7, 1/2 of 7 times 7+1
= 3 1/2 x 8
= 28

I think the above, however, is rather too complicated for a nine year old. You could either:
a) teach him to draw the triangles, adding one more dot each row, and then count them
b) or memorise the first 8 triangular numbers
c) or not worry about it because the time might be best used on something else.

Hope this helps.


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 Post subject:
PostPosted: Mon Apr 28, 2008 10:52 am 
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Joined: Mon Apr 28, 2008 10:28 am
Posts: 114
Location: Kent
My nine year old DD was asked to find a huge triangular no. for hwk-26th, I think!
We quickly realised adding it up (1+2+3 etc) was going to take forever and mistakes would probably be made on the way. Looking for an easier sum, we added the top and bottom rows of the triangle together


line 1 (1 dot) to line 26 (26 dots) = 27 dots
line 2 (2dots) to line 25 (25 dots) =27 dots
line 3 (3dots) to line 24 (24 dots) = 27 dots

continue doing this until you finish with
line 13 (13 dots) to line14 (14 dots) = 27 dots

This leaves you with 13 lines of 27 dots, and you can then quickly do this much more manageable multiplication to get your answer much more easily.

Sounds complex, but isn't if you sketch it on the bit of paper :) . DD loved the concept and quickly grasped pairing the largest and smallest lines. Works for odd triangular nos. too but need to add on the extra 'middle' line after doing the multiplication.

Hope this helps - not sure it conforms to any technical method but DD's teacher liked it. And aren't the children supposed to be being taught to play with and enjoy numbers?


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 Post subject:
PostPosted: Mon Apr 28, 2008 4:45 pm 
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Joined: Mon Feb 12, 2007 1:21 pm
Posts: 11954
1st = (1x2)/2

2nd = (2 x 3)/2


so 27th = (27 x 28 )/2


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 Post subject:
PostPosted: Wed Apr 30, 2008 9:25 am 
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Joined: Thu Jan 12, 2006 3:29 pm
Posts: 625
Hi

The formula for working out triangle numbers is

(n squared + n) divided by 2

This quadratic would appear in a GCSE paper at around a "B" grade.

However (27 x 27 + 27) divided by 2 requires a lower level of mathematics.

With the 1 3 6 10 15 number pattern it is reasonably easy to see that the pattern increases +2, +3, +4, +5, etc.

However, this is slow if you are trying to work out the 27th number.

One method for a nine year old is just to tell them how to work it out, without giving an explanation.

Then show them how
(1 x 1 + 1) divided by 2 = 1
(2 x 2 + 2) divided by 2 = 3
(3 x 3 + 3) divided by 2 = 6
(4 x 4 + 4) divided by 2 = 10
(5 x 5 + 5) divided by 2 = 15

Then check whether they understand the process with

(6 x 6 + 6) = 21, they can check there answer by using either a drawing or counting on method.

Once you are sure that the student does understand the process then he can progress to (27 x 27 + 27) divided by 2.

Regards

Mike

_________________
Mike Edwards is a co-author of The Tutors product range.


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 Post subject:
PostPosted: Thu May 01, 2008 1:18 pm 
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Joined: Mon Mar 12, 2007 11:49 am
Posts: 450
There's a great explanation of triangle numbers in 'More Murderous Maths' by Kjartan Poskitt

0 00000
00 0000
000 000
0000 00
00000 0

The diagram above represents two 'skewed' triangles next to one another. You can use this diagram to work out the triangle number of 5. You can see that by shunting the two triangles together, you end up with a rectangle. The rectangle has two sides of 6 units (where the two triangles are shunted together), and two sides of 5 units (where you've just got a side of a triangle). To work out the area of the rectangle, you just multiply the length by the width (5 x 6), which gives you 30 units. The area of one of the triangles is half of the area of the triangle (30 / 2) which is 15. So the triangle number of 5 (or perhaps it's the 5th triangle number - I'm not a mathematician) is 15.

You can demonstrate this with a large number of pennies or counters, and it at least explains why the formula works.

Y


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 Post subject:
PostPosted: Thu May 01, 2008 1:24 pm 
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Joined: Fri Nov 17, 2006 8:54 pm
Posts: 1770
Location: caversham
Y

I like that, very visual, easy to grasp and sticks in my mind. :lol:

steve


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