Verbal counting. In the public school of S. A. Rachinsky

	N.P. Bogdanov-Belsky
	Verbal counting. At the public school of S. A. Rachinsky . 1895
	Canvas, oil. 107.4 × 79 cm
	State Tretyakov Gallery , Moscow

"Verbal counting. In the public school of S. A. Rachinsky ” - a painting by Russian artist N. P. Bogdanov-Belsky (1868-1945), painted in 1895 .

Description

The painting depicts a village school of the late 19th century during a lesson in arithmetic when solving fractions in the mind. The teacher is a real person, Sergei Alexandrovich Rachinsky (1833-1902), a botanist and mathematician, professor at Moscow University . In the wake of populism in 1872, Rachinsky returned to his native village of Tatevo , where he created a school with a hostel for peasant children, developed a unique method of teaching oral counting , instilling in the village children his skills and the basics of mathematical thinking. An episode from the life of the school with a creative atmosphere that reigned in the classroom was dedicated by Bogdanov-Belsky, himself a former student of Rachinsky.

An example is written on the chalkboard that students need to solve:

{\frac {10^{2}+11^{2}+12^{2}+13^{2}+14^{2}}{365}}.

{\ displaystyle {\ frac {10 ^ {2} + 11 ^ {2} + 12 ^ {2} + 13 ^ {2} + 14 ^ {2}} {365}}.}

{\frac {10^{2}+11^{2}+12^{2}+13^{2}+14^{2}}{365}}.

.

Solution to the problem posed in the picture

The terms written on the board have an interesting property:

10^{2}+11^{2}+12^{2}+13^{2}+14^{2}=10^{2}+(10+1)^{2}+(10+2)^{2}+(10+3)^{2}+(10+4)^{2}

{\ displaystyle 10 ^ {2} + 11 ^ {2} + 12 ^ {2} + 13 ^ {2} + 14 ^ {2} = 10 ^ {2} + (10 + 1) ^ {2} + ( 10 + 2) ^ {2} + (10 + 3) ^ {2} + (10 + 4) ^ {2}}

10^{2}+11^{2}+12^{2}+13^{2}+14^{2}=10^{2}+(10+1)^{2}+(10+2)^{2}+(10+3)^{2}+(10+4)^{2}

=10^{2}+(10^{2}+2\cdot 10\cdot 1+1^{2})+(10^{2}+2\cdot 10\cdot 2+2^{2})+(10^{2}+2\cdot 10\cdot 3+3^{2})+(10^{2}+2\cdot 10\cdot 4+4^{2})

{\ displaystyle = 10 ^ {2} + (10 ^ {2} +2 \ cdot 10 \ cdot 1 + 1 ^ {2}) + (10 ^ {2} +2 \ cdot 10 \ cdot 2 + 2 ^ { 2}) + (10 ^ {2} +2 \ cdot 10 \ cdot 3 + 3 ^ {2}) + (10 ^ {2} +2 \ cdot 10 \ cdot 4 + 4 ^ {2})}

=10^{2}+(10^{2}+2\cdot 10\cdot 1+1^{2})+(10^{2}+2\cdot 10\cdot 2+2^{2})+(10^{2}+2\cdot 10\cdot 3+3^{2})+(10^{2}+2\cdot 10\cdot 4+4^{2})

=5\cdot 100+2\cdot 10\cdot (1+2+3+4)+1^{2}+2^{2}+3^{2}+4^{2}=500+200+30=730=2\cdot 365.

{\ displaystyle = 5 \ cdot 100 + 2 \ cdot 10 \ cdot (1 + 2 + 3 + 4) + 1 ^ {2} + 2 ^ {2} + 3 ^ {2} + 4 ^ {2} = 500 + 200 + 30 = 730 = 2 \ cdot 365.}

=5\cdot 100+2\cdot 10\cdot (1+2+3+4)+1^{2}+2^{2}+3^{2}+4^{2}=500+200+30=730=2\cdot 365.

That is, the result of the calculation is 2.

Another calculation option:

10^{2}+11^{2}+12^{2}+13^{2}+14^{2}=(12-2)^{2}+(12-1)^{2}+12^{2}+(12+1)^{2}+(12+2)^{2}

{\ displaystyle 10 ^ {2} + 11 ^ {2} + 12 ^ {2} + 13 ^ {2} + 14 ^ {2} = (12-2) ^ {2} + (12-1) ^ { 2} + 12 ^ {2} + (12 + 1) ^ {2} + (12 + 2) ^ {2}}

=(12^{2}-2\cdot 12\cdot 2+2^{2})+(12^{2}-2\cdot 12\cdot 1+1^{2})+12^{2}+(12^{2}+2\cdot 12\cdot 2+2^{2})+(12^{2}+2\cdot 12\cdot 1+1^{2})

{\ displaystyle = (12 ^ {2} -2 \ cdot 12 \ cdot 2 + 2 ^ {2}) + (12 ^ {2} -2 \ cdot 12 \ cdot 1 + 1 ^ {2}) + 12 ^ {2} + (12 ^ {2} +2 \ cdot 12 \ cdot 2 + 2 ^ {2}) + (12 ^ {2} +2 \ cdot 12 \ cdot 1 + 1 ^ {2})}

=12^{2}+2^{2}+12^{2}+1^{2}+12^{2}+12^{2}+1^{2}+12^{2}+2^{2}=5\cdot 12^{2}+4+1+1+4=720+10=2\cdot 365.

{\ displaystyle = 12 ^ {2} + 2 ^ {2} + 12 ^ {2} + 1 ^ {2} + 12 ^ {2} + 12 ^ {2} + 1 ^ {2} + 12 ^ {2 } + 2 ^ {2} = 5 \ cdot 12 ^ {2} + 4 + 1 + 1 + 4 = 720 + 10 = 2 \ cdot 365.}

Sources

Faermark D. S. The task came from the picture: (About the painting by N. P. Bogdanov-Belsky “Oral Account” and the educator-educator S. A. Rachinsky ). - M .: Nauka , 1974. - 160, [4] p. - ( From the history of world culture ). - 90,000 copies. (region)
State Tretyakov Gallery. Art of the XII - beginning of the XX century. - M .: ScanRus, 2007. - S. 194. - ISBN 978-5-93221-120-5 ..
Example Solution (ORAL ACCOUNT ./ “Science and Life”, No. 7, 2006)
4 ways to solve the Rachinsky problem
Rachinsky sequences
Anna Bessarabova. Russia goes to the economy of the losers (the exercise depicted in the picture is given as an example)

Verbal counting. In the public school of S. A. Rachinsky

Description

Solution to the problem posed in the picture

Sources

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