The times tables – FIFTY NINE NUMBERS

WHAT'S THE FUSS??

Numeracy is about handling numbers and the times tables is a very useful window on
FIFTY NINE NUMBERS below 144, with thirty four under 50.   

What are these fifty nine numbers?  

THE PERFECT SQUARES are the only numbers that appear an odd number of times. That is twelve numbers. 36 appears five times, 4, 9 and 16 three appearances each and the rest just one.

All of the remaining 132 can be called 66 pairs – commutative means the product
 for 6 x 9 appears twice.

There are 5 prime numbers that appear twice (2,3,5,7,11) but from 13 up, primes and their multiples are not products, thereby excluding themselves.

Two numbers make 6 appearances – 12 and 24; ten make 4 appearances (the largest is 72)

Going up to 12 x 12 is a great introduction to three digit competency. Only seven numbers meet the criteria; three of which end in “0”.

The times tables is about getting to know thirty four numbers under 50 and fifty nine under 144. Barely enough of them to be able to reward speed and perseverance, once they have been learned.  By Grade 6, all learners should have passed a torture test (150 written questions – yes include duplicates – in 7 minutes, weekly till score beats 145. Allow them a couple of minutes to savour their accomplishment and the feelings that go with it. Then start the trek through the next topic.

Rote? Repetition? Tell McDonalds it won’t work.

3  *  12    =   4 * 9   =  6 * 6   =   9 * 4   =   12 * 3   = 36

NUMBERS UP TO 144 – COLLECTIONS AND EXCLUSIONS

	Inside the 12 Times Tables	Inside the 144
Used 6 times	Two:      (12 and 24)	None
Used 5 times	One: Only 36	None
Used 4 times	Ten: (6, 8, 10, 18, 20, 30, 40, 48, 60, 72	None
Used an odd number of times	Twelve:                all the perfect squares	144
Prime Numbers	Five:                      2,3,5,7, 11	Thirty Four:2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139
Numbers 100 and over	Seven: 100, 108, 110, 120, 121, 132, 144	Forty five: 100-144

THE TIMES TABLE GRID

	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

ERATOSTHENES SEIVE OF PRIME NUMBERS below 144

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100
101	102	103	104	105	106	107	108	109	110
111	112	113	114	115	116	117	118	119	120
121	122	123	124	125	126	127	128	129	130
131	132	133	134	135	136	137	138	139	140
141	142	143	144

44 PERFECT SQUARES Part 2

• ODD NUMBERS IN SQUARES
• SQUARES ENDING IN . . .44 AND . . . 444
• PYTHAGOREAN TRIPLES

What an odd series the perfect squares make.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Odd?  Just look at the differences or steps from one to the next.         1,3,5,7,9,11,13,15,17,19

If (n-1) ² is a square, n ² is a square too

and the step from (n-1) ²  to  n²  is the odd number 2n-1

Or using maths; 

n	n2	=	(n-1) 2	+	2n-1
3	9	=	2 x 2	+	5
7	49	=	36	+	13
8	64	=	49	+	15
10	100	=	81	+	19
100	10,000	=	9,801	+	199
101	10201	=	10,000	+	201

Or n^{2              -}                (n-1) ^{2               =                2n-1}

THE PERFECT END FOR A PERFECT SQUARE?

                                                10²   = 100

There is no perfect square below 100 that ends in a repeated numeral.

The next perfect square ending with a repeated numeral is 400.

Did you notice? Yes,  we skipped 144;  12 x 12  =  144

The only nonzero numeral that is repeated to end a perfect square is 4, so:

122	382	622	882	1122	1382	1622	1882	2122	2382
144	1444	3844	7744	12544	19044	26244	35344	44944	56644

In short, the pattern is (50n +/- 12)² = . . . 44

The other thing that caught your eye was 38² with its -444 ending. 

Other -444 endings can be found using (500n +/- 38)²  = . . . 444      

So the set is readily calculated; for a start  

                             (500  +  38 )²   =  (500  +  38 ) x (500  +  38 )   By       FOIL,

                                                =  500 x 500                                        First

     + 2x 500 x38                                  Outside, Inside,

     + 38 x 38                                        Last                

=  250,000       + 38,000  + 1444

                                                =   239,444      

Could this be a party trick?? "Go on, ask me the square of a number between 500 and 540"

When 500 x 500 gives so many zeros, the adding of even biggish numbers gets easy. Since you know a lot of squares already (especially if you know Pythagorean Triples as below), there are probably not many more  to learn. 

  3             4             5

  6             8           10          2[3 - 4 - 5]

  5           12          13

  9           12          15          3[3 - 4 - 5]

  8           15          17

 12          16          20          4[3 - 4 - 5]

  7           24          25

 15          20          25          5[3 - 4 - 5]

 10          24          26          2[5 - 12 - 13]

 20          21          29

 18          24          30          6[3 - 4 - 5]

 16          30          34          2[8 - 15 - 17]

 21          28          35          7[3 - 4 - 5]

 12          35          37

 15          36          39          3[5 - 12 - 13]

 24          32          40          8[3 - 4 - 5]

  9           40          41

Ahh, perfect squares, all 44 of them  

44 PERFECT SQUARES . . . Part 1

• FACT – Forty  – the only number in the English Language where the letters are in alphabetic order
• Perfect squares – have 4 sides
• Let’s look at 44 perfect squares

THE SMALLEST                               and the sneakiest

144 = 12 x 12                  441 = 21 x 21

169 = 13 x 13                  961 = 31 x 31

                                      Similar, but an obvious fake

There is something odd about perfect squares. (even if half of them are even).

They have symmetry. All sorts.

There is symmetry in their numbers. We may have 10 numerals, but look at them.

1	2	3	4	5	6	7	8	9
	No square deal here		The square ends; 1, 9 and Zero too!			Or here

Only 1,4,5,6,9,0  can end a perfect square. 2,3,7,8 never do.

What an odd series the perfect squares make.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Odd?  Just look at the differences or steps from one to the next.         1,3,5,7,9,11,13,15,17,19

BILL BRYSON and SCIENCE

Bill Bryson is a good science writer.

Here are a few reasons: Dinosaurs, mining, Cholera

1.      It is general knowledge that the dinosaurs got wiped out by an asteroid.

Is it?
Since when?
The evidence?
Read "The Short History Of Nearly Everything".

I read this book before we “googled” to see if there was a story behind this “fact”

Bryson reports a geologist’s comment “I think we’ve found the crater”

This is the most obvious crater on the planet, so it was immediately the obvious suspect. Too bad that it was formed 65,000,000 years after the dinosaurs chilled out, give or take 55,000 years.

Bryson breezes through the story without being asked. The evidence points to Chicxulub ; for another good summary, try :  http://www.glencoe.com/sec/computered/col/chapter5/search_engines/yahoo/yahoo_6.html

2.      In “A Walk In The Woods” the coverage of the Centralia Coal mine fire made me check out how many mines are on fire, permanently. It does make coal mining appear to be as dangerous as gold or diamond, in spite of coal often being nearer the surface. Mining is high risk, everywhere.

3.      “At home”. John Snow was the hero in the struggle in against cholera. He came from the wrong side of the tracks, (this is a post on its own) but for 150 years he has been left off the honours board. Bryson simply restates the evidence – make up your own mind.   

Put your feet up !

THE HARDEST PLANET TO DISCOVER

THE HARDEST PLANET TO DISCOVER should be the one which shares the earth’s orbit, following Earth by exactly six months. Look for it with a telescope and, well, there would be the Sun in the way. So far we do not know if it is there.

BUT WE KNOW NOW THAT WE DO SHARE OUR ORBIT. We follow a lump of rock, just as Pacman pursues a pill or a pellet. The rock is 300m across, which makes it an asteroid; it would need to be under 50m to be a meteoroid. Its name is 2010 TK7

IT IS AT LAGRANGE POINT L4

Its orbit is very wobbly, because it is at lagrange point L4. But this makes it easier to find. Lagrange points are locations where the gravity from two neighbouring objects is equal, enabling objects to fall to them, or in and out of them for a wobble. Not only do asteroids fall into planets or moons, they fall into places such as L4.   

See http://en.wikipedia.org/wiki/List_of_objects_at_Lagrangian_points

This drawing shows the 5 lagrange points for
Sun (yellow) - Earth(blue)

THE QUESTION REMAINS

Is there anything PARKED AT L3 ? ? ?

New Moon and the Middle East; July to March

Last night I saw a thin banana moon; once again this waxing is underway.
What is there to look forward to this moonth?

The waxing is often a reminder that something anticipated is about to happen.

This month is survival. It is far too hot in Abu Dhabi to do much else?

The shadows from the lamps are exactly under the lamps. 

Abu Dhabi is on the Tropic of Cancer, the furthest from the equator where the midday sun passes exactly overhead.

The deck chairs are empty - Noel Coward would say there can't be any mad dogs or englishmen here

Yet there is a couple of people on the tennis court. Getting some stamina training for Wimbledon maybe (photo was taken 21st of last moonth)

Next new moon is the start of Ramadan. Happily we are allowed to slow down under the August sun this year.

In December the new moon will remind us that there is a Lunar Eclipse; from November to April we will start watching again at each New Moon to see the fastest thing in the sky rush up from the West past Venus and then on past Jupiter. Each month the gap between Venus and Jupiter will shrink, till there is a Conjunction in March. 

How would you know which of all those dots is Venus?

The brightest night sky objects are

1 the moon

2 Venus

3 Jupiter; 

Note that Venus and Jupiter are brighter than any star; that they are on the line taken by the moon makes them very easy to find.

In Abu Dhabi, the sky is hazy enough that these three are often the only objects that can be seen.