Batch 2 - Class 62 - 3D Cross sections
Preclass Exercise:
- For the Titanic problem, draw more correlations with some parameters besides gender and class, and come up with a better predictor. Upload it on the website to test the prediction accuracy
Attendance: Tishyaa, Smiti, Anisha, Kushaan, Aryan, Abhiram, Avy, Shubham, Praharsh, Nandini, Abhiram, Diya (skype)
Class Notes:
- What do human being see - 2D or 3D?
- Remember the cube and hexagon? Show the kids the same again to aid recall.
- What do you see in the picture below? How many cubes are there? Is it 6 or 10?
- Whats wrong with the following picture? Why is it impossible?
- Explanation of Cross Sections - Take an apple and with a straight cut, demonstrate a cross section
- What is the "cross section" here - its not the piece of apple we just cut; its the surface we formed
- Cross section of 3D objects are 2D surfaces
- It is no different from what our eyes really see - like the hexagon on a rubik cube above
- Cross sections of a sphere
- Show the class some balls, and ask them what the would be the cross sections if we were to cut it with a straight knife?
- Why does cutting it at an incline also lead to circle?
- Cross sections of a cube
- Can you create a square out of a cross section of a cube? (Teacher Notes: Explain kids how they would draw it. For rest of the class, let them draw the cross sections)
- How about a rectangle? (non square)
- Triangle?
- Trapezoid (explain a trapezoid)?
- A polygon with more than 4 sides?
- A polygon with more than 6 sides? If not, why not?
- Every side of the cross section is a result of intersection of a surface of cube with the plane being cut, and the cube has only 6 surfaces.
- Ask kids to take a rectangular pyramid and draw a cross section resulting from a vertical slice
- (Contributed by Avy) There is a potato which has 1% solids and 99% water. It is left to dry in the sun, till a point where it now has 98% water. How much does the potato weight now relative to original weight?
- Answer: 50%
- Lets reverse the process - Rotate 2D shapes into 3D shapes
- Take a right angle triangle and rotate it off the vertical axis. What will you get? (Explain notion of axis)
- Note that it is dependent on which axis you choose for rotation - what would happen if we rotated it along one of the edges?
- What would you get if you rotated a rectangle through the center dividing line parallel to an edge?
- What if you rotated it through a diagonal?
- What would happen if we rotated a circle through a line passing through center?
- What if the axis didn't pass through the center?
Homework:
- Draw all the cross sections that result from slicing a cylinder
- Answer:
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- (MC Diaries: Problem 7) We have three bottles: one has a circular opening, another a square opening, and the third has an opening in form of an isosceles triangle. See below. Design a bottle stopper that fits all the three bottles.
- What if we should be able to push the stopper into each of the bottles?
References:
Mathematical Circle Diaries, Year 1, by Anna Burago
Not Used
- A tetrahedron is a pyramid with a triangular base and triangular faces. Does a tetrahedron have a quadrilateral cross sections?
- How would you think about it? What should happen for a tetrahedron to have a quadrilateral cross section?
- Each surface would have to be cut with the same plane. Can you do that?
