Batch 2 - Class 87 - Little Boxes (Shapes to Infinity)

• (Dudeney - 54) A certain family party consisted of 1 grandfather, 1 grandmother, 2 fathers, 2 mothers, 4 children, 3 grandchildren, 1 brother, 2 sisters, 2 sons, 2 daughters, 1 father-in-law, 1 mother-in-law, and 1 daughter-in-law. Twenty-three people, you will say. No; there were only seven persons present. Can you show how this might be?

  The party consisted of two little girls and a boy, their father and mother, and their father's father and mother

• What is the area of rectangle labeled 1? What is the area of square labeled 2? How about the rectangle labeled 3? ... What is the pattern? Can older kids write down a general expression for the area of rectangle labeled n?

  1/2, 1/4, 1/8 ... 1/2^n
• What is the sum of areas of rectangles 1 and 2? What is the sum of areas of rectangles 1,2 and 3? How about 1,2,3 and 4? ... What is the pattern? Can older kids write down a general expression for the sum of areas 1 through n?

  3/4, 7/8,15/16 ... (2^n-1)/2^n = 1-1/2^n Note that remaining area is area of rectangle n which is 1/2^n as per previous problem
• If we added up areas of all rectangles, what would it be? Look at the picture and think. 

  How many rectangles would be need to get to 99%+ of the area? (Hint: think about previous problem)Seven rectangles because 2^7>100 and hence 1/2^7<1%
• How come the sum of infinite rectangles is finite?

  Zeno's paradox: You have to travel a mile. Before you get to a mile, you will get to half a mile. Before you walk half the mile, you will walk a quarter... It is an infinite sequence. How do you ever get to the finish line?
• Let us think about sum of all squares. What is the area of the squares labeled 2, 4, 6 and so on? Can older kids write down a general expression?

  1/4, 1/16, 1/64 ... 1/4^n
• What is the sum of areas of 2 and 4? 2, 4 and 6? 2, 4, 6 and 8? What is the pattern? Can older kids write down a general expression? 

  Teacher's Note: Can skip this part of general expression, or guide older kids through geometric sequence sum5/16, 21/64, 85/256 ... (4^n-1)/(3(4^n))
• If we added up the areas of all squares, what would it be?

  Teacher's Note: Kids may try to think about it algebraically - prompt them to think visually. If required, shade the squares that are being totalled1/3Teacher's Note: For older kids, could walk them through sum of infinite geometric series
• What is the sum of all odd-numbered rectangles in the first diagram?

  2/3Teacher's Note: One way to arrive at this is to subtract squares' area from total area. Another is to see that odd-rectangles are respectively twice the size of following rectangles
• Can you now find the sum of areas of squares labeled 4, 8, 16, 20 etc?

  1/15Teacher's Note: Again, one could do it by geometric series 1/16+1/16^2... Or visually by dividing rectangles 1-4 into 15 boxes of the same size as rectangle 4, and so on
• Can you find out sum of areas of rectangles 3, 6, 9 ...?

  1/7Teacher's Note: Motivate visual solutions at this point

• Assuming that the area of the entire triangle above is 1, what is the shaded area

  1/3
• Can you draw pictures of other shapes, and get to other ratios

  Teacher's Note: This has many possibilities. Encourage students to come up with more. Sphinx might be one

  

  
  

• Consider the stack of dice as shown above. What is the sum of all hidden faces (Note that faces at the back are not hidden)