Batch 3 - Class 177 - Wrapping Gifts, Coloring Tetrahedrons

• There is a component made of solid copper sphere with a cylindrical bore through it. The length of the bore is 2 cm. How much copper is used in this component?

• Answer: 4/3(pi) - Note that you don't need any other dimensions to determine the answer

• You want to wrap a present. Lets start with the present having a square face of side w and height h. Lets say, our box of side 20 cm and height 6 cm. How much wrapping paper do you need to wrap the present?
• Motivate kids to now think about irregular shapes of wrapping paper
• For example, with a cross-shaped paper?
• What is the best you can do with a rectangular piece of paper? Or square wrapping paper?
• See if kids can come up with the diagonal method - some of them may have heard of it before.
• With the diagonal method, how much paper is required? Let kids calculate on their own.
• Walk through the calculation to get the perfect size

• For a 20x20x6 gift, this translates into 36.8 cm square paper
• One simple approximation we can make is diagonal+height*1.5, which in this case would give us 37.3 cm square paper
• Show the kids the actual wrapping
• What is the approximate saving of paper from the conventional folding method?
• What if we want to wrap a rectangular gift? Can we use the same technique?
• What size of paper do we need? Lets say our gift is 21x16x6? Let kids calculate on their own
• Walk through the calculation to get the perfect size

• For a 21x16x6 gift, this translates into 34.6 cm square paper
• One simple approximation we can make is diagonal+height*1.5, which in this case would give us 35.4 cm square paper
• Show the kids the actual wrapping
• What is the approximate saving of paper from the conventional folding method?
• What if we wanted to use a paper of slightly smaller width? Could we still do this by using longer sheet? What is the constraint on width - whats the minimum that would be required under this method?

• You have a triangular pyramid (also called tetrahedron) folded out of paper, and all the four faces are colored with different colors. How many different colorings of the pyramid are possible, provided that you should not be able to rotate from one distinct coloring to another.
• Answer: 2
• Can you figure out which one of the following is the different one relative to others?

• What if you wanted to color the tetrahedron vertices with four different colors (one each on a vertex)?
• This is the same problem as the first. Only 2 solutions.
• Can you map the two problems to reason that they are really the same?

• What if you wanted to color the tetrahedron vertices with two different colors. How many ways of doing that are there?
• 5 - Only the number of vertices with a certain color matter. All the "different" configurations with (say) 2 red and 2 blue vertices are rotationally the same