Batch 2 - Class 28 - Pythagoras and Erastothenes

• Cutting a cube with a straight cut to get a hexagon

• Moscow - 104 - Place 10 lightbulbs in 5 rows with 4 bulbs in each row. 

  Answer: At vertices of a 5-Star12 lightbulbs in 6 rows with 4 in each row? (Answer: 5 different solutions)13 in 12 rows with 3 per row? (Answer: Hexagon, one at center, one at each vertex and midpoint)
• MartinShCol 5.2 - draw six line segments of equal length to form eight equilateral triangles

  Answer: 6 pointed star; Two parallel lines with two overlapping X's
• MartinShCol 5.11 - hidden cross. Find the figure on right in the pattern on the left.

• Explain the theorem with a right angle triangle, as well as with the squares below - Note that square of side is same as area of the square

• Illustrate simple proof as listed above through cutouts
• Another proof below

• 111 Proofs of Pythagoras theorem here - http://www.cut-the-knot.org/pythagoras/ 
• Practical Examples

  What is the length of ladder required to paint a 10 feet high wall, and avoiding an adjoining drain of 3 feet?Architecture...

• Let kids give ideas on how they would measure the circumference of Earth
• Describe Erastosthenes' method http://www.windows2universe.org/citizen_science/myw/w2u_eratosthenes_calc_earth_size.html 

  Syene - Sun was overhead (no shadow on well), but Alexandria had 7.2 degree shadow on a towerSyene and Alexandria were 800 km apartLet kids try and solveKey insight - the 7.2 degree angle of shadow is because circumference of the earth, and must be same as angle between Syene and Alexandria as observed from center of Earth. So circumference = 360/7.2*800 = 40,000 km.

• Moscow - 145 - Divide a regular hexagon into 12 equal quadrilaterals

• Moscow - 161 - Is it possible to cover a chess board using 15 L-shapes (3x2) and 1 2x2 square? Why?