A question – 64 is a cube and a square; name another such number

NO CALCULATORS

 Should this question be in the “no calculators” section of a 7^th grade paper?

If Yes, it is a challenge to Budding Mathematician, who we can call BM. Best marks are usually won from a combination of discipline and simplicity.

            64  = 8^2 and 4^3
but the key to finding 
another qualifier 
is recognizing that 
            64 = 2^6

The rule is no calculators, so if BM actually gets to

           3^6 = 729,
it is not a number that is likely to jump to mind as a perfect square. 

Prime factors could be used to check, and once 3*3*81 is reached, confidence would creep in.

           4^6 = 4096; if BM remembered repeated doubling of 1 and reaching 1024,
it is just another two doublings,
but how many doublings was that altogether? 

Determination would suffer repeating the process, 

and again the confidence would allow some satisfaction.

Does this work have to be so hard? No!

1^2 is so obvious, that there is

no hesitation with 10^2 and 100^2

Finally BM sees that
1,000,000 = 10^6
is the simple answer,
so allows a big grin to spread over the BM face.
This is a first, and an empowering one. 

Never before had the number
1,000,000; 

big as it is, 

been the simple answer .

PLANETS - Wanderers - until 3000 years ago

At a glance "wanderer " might be a good word. Venus sitting on Jupiter with the moon off to Taurus

Four days pass; Mercury makes an appearance

Next Moon is rising, Venus nowhere near Jupiter. 

But take another look

Jupiter is still in Cancer; can this be called wandering?

And the moon, the sun, and the planets are all along a line 

there's Leo, Cancer, Gemini, Taurus, the constellations of the Ecliptic. 

The Planets walk this line, in the same direction, at speeds related to their distance from the sun. 

Is this movement straying, or erratic, ie wandering? The line with its revealing name  of "The Ecliptic" turns out to be a series circles/ellipses, with the sun at a centre, viewed from the edge, just as a coin viewed edge-on is a line. 

Further, the circular orbits so often taught, are a view never seen from earth! 

Phasing out "dial-up" phones started about 40 years ago, but we still "dial a number". 

Planets only wandered until the patterns in their movements were explained. If students are only taught that planets wander, the lesson is 3000 years out of date.  

Planets are wanderers . . . Is this first impression misleading?

Planets are wanderers . . .
Is this first impression misleading?

 I have just come across ANOTHER textbook which says the word "planet" comes from the Greek word meaning "wanderer".

I am often delighted by finding that the derivation of a word helps me understand it and use it better. This time I am not happy. 

www.brothersoft.com

It was useful to call a planet a wanderer a few thousand years ago, when the earth was so obviously flat. And yes, Sirius the Dog Star stayed faithful to Orion, while these planets roamed at different speeds, sometimes sharp eyes even see them go backwards.

But then there was an eclipse and they realised the Earth was round.

And over a few more months it was seen that wander was not such a good word. True, planets moved. But it was soon recognised that they were all on the same line and in the same direction AND the different speeds told us that they orbit at different distances, all much closer than any star. 

If “wander” means

1. To proceed in an irregular course; meander; or

2. To go astray,

The "wanderer" concept was out of date 3000 years ago.

This familiar view can not be seen from earth

From Earth, the planets make a procession along the path taken by the Sun. . The fastest is the closest , so the moon glides from planet to planet, to repeat its path a month later. The moon rises at sunset every 29 days. Mars does this every two years. Saturn takes 30 years to line up in front of eg Scorpio twice, when viewed from here. View a coin edge on and it is linear, like the Solar System. Irregular? Astray?
WE SET OUR CLOCKS BY THE SKY

http://freeware.intrastar.net

I’ll post a calendar ready for February, when Venus is seen starting a rush across the evening sky for a conjunction with Jupiter, with the dates of each time it passes a planet. This Highway of the Sky is called the “Ecliptic” because there are 4 eclipses on it most years.February to June next year is a great period for planet watching!

PRIME NUMBERS by the HUNDRED (to 2000)

This is a list of Primes under 2000.  The decreasing frequency of primes as numbers get larger is there to see, for example 25 primes from 0 to 100, but only 8 from 1500 to 1600

I wonder what is the first interval of 100 with no primes?
Are there any primes between 1811 and 1867? 

row 1 2 101 211 307 401 503 601 701 809 907 2 3 103 223 311 409 509 607 709 811 911 3 5 107 227 313 419 521 613 719 821 919 4 7 109 229 317 421 523 617 727 823 929 5 11 113 233 331 431 541 619 733 827 937 6 13 127 239 337 433 547 631 739 829 941 7 17 131 241 347 439 557 641 743 839 947 8 19 137 251 349 443 563 643 751 853 953 9 23 139 257 353 449 569 647 757 857 967 10 29 149 263 359 457 571 653 761 859 971 11 31 151 269 367 461 577 659 769 863 977 12 37 157 271 373 463 587 661 773 877 983 13 41 163 277 379 467 593 673 787 881 991 14 43 167 281 383 479 599 677 797 883 997 15 47 173 283 389 487 683 887 16 53 179 293 397 491 691 17 59 181 499 18 61 191 19 67 193 20 71 197 21 73 199 22 79 23 83 24 89 25 97

row 1 1009 1103 1201 1301 1427 1511 1601 1709 1801 1901 2 1013 1109 1213 1303 1429 1553 1607 1721 1811 1907 3 1019 1117 1217 1307 1433 1559 1609 1723 1867 1913 4 1021 1123 1223 1319 1439 1567 1613 1733 1871 1931 5 1031 1129 1229 1321 1447 1571 1619 1741 1873 1933 6 1033 1151 1231 1327 1451 1579 1621 1747 1877 1949 7 1039 1153 1237 1361 1453 1583 1627 1753 1879 1951 8 1049 1163 1249 1367 1459 1597 1637 1759 1889 1973 9 1051 1171 1259 1373 1471 1657 1777 1979 10 1061 1181 1277 1481 1697 1783 1987 11 1063 1187 1279 1483 1699 1787 12 1069 1193 1283 1487 1789 13 1087 1289 1489 14 1091 1291 1493 15 1093 1297 1499 16 1097 17 18

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