Easy, yet interesting problem...

I think this is one of the rare problems where all the students solved it in the same way (whoever could solve it :)

While they were still working on this problem, I asked them about their opinion. Most of them guessed that the two areas will be equal. Couple of them thought the greener one might be bigger :)

Vaishnavi was the one who cracked it first...... She proved it beautifully that both the regions are of same size..... Though she could not explain and represent the solution clearly to me in the beginning, however, I was quite impressed with the way she pulled off this stint on the board to the class ....probably the after-effect of long interrogation session she had to go through :)

She said -

1) Radius of the smaller circles is half that of the bigger one. Hence the total area of four small circles is equal to the area of big circle... Note how she has explained this above!

2) Now, this bigger circle is made up of many smaller regions as shown above, labelled as a,b,c......j,k,l.

3) However, the 4 smaller circles include only 8 of these regions (overlapping parts considered twice) ...... ...Is she right? 

4) So, if we equate these two expressions, we get

a + b + c + d + e + f + g + h + i + j + k + l = (a + b + c + d) +  2( e + f + g + h)

This leads to   ( i + j + k + l )  =  ( e + f + g + h )

thus proving that the yellow and green regions have same areas.

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Jeetu explained this in similar yet little different way. He said -

1) Lets imagine that we have cut the four smaller circles out of paper.
2) Cut these circles further into pieces of shape and size as given in the diagram(12 pieces)
3) Place these pieces on their respective sections on the diagram. 
4) We will be able to place only 8 pieces on the diagram. The four petal shaped pieces will be left over. 

5) Now, because the total area of 4 circles = area of big circle, it implies that -

these spare petal-shaped pieces should be of the same size as that of the 4 sections between smaller & outer circles (i,j,k,l)

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How did you solve this problem?
What was your guess?
What's your view about the solutions of students?
How about you trying this with your children/ students?
If so, we (I and my students) will be happy to know your experience and their solutions :)

Thanks and Regards

Rupesh Gesota

PS: Students belong to marathi medium government school (class-7 and 8) based at Navi-Mumbai. To know more about their Maths Enrichment program, check this link:
www.supportmentor.weebly.com