Batch 3 - Class 237 - Mini Mathematical Universe
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Pre-Class Exercise
How to arrange ten soldiers in five lines in such a way that each line contains four soldiers exactly?
- Answer: Since five lines each contain four soldiers, that means 20 points of presence, and hence each soldier appears on two lines. Thus, if we draw any 5 non parallel lines, they will each intersect at a point leading to 5C2=10 points where the soldiers may be placed. Some answers below
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Attendance Kabir, Vansh, Kushagra, Rhea, Rohan, Shikher, Aashvi, Anishka, Arnav, Arjun, Rehaan, SidharthM, Vivaan, Ayush, Muskaan, Smiti, Advay, Aarkin
Class puzzles (powerpoint available)
Portugese men o'war
We will show students certain diagrams and ask them to create the mathematical world hidden behind those numbers. To start, Portugese men o'war puzzle. Show the first diagram to students and ask them to guess numbers within each circle.
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Now let the students solve a few problems
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Discussion with students to capture the rules of this mini mathematical universe
- Children of a particular parent have consecutive numbers, left to right
- Leafs start with 1 as the leftmost one
- For a one level diagram, that can be any number (it is not counted as a leaf)
Can you think of any impossible diagrams? (Yes, for example where two leaf configurations don't lead to consecutive numbers at next level)
Starting from 1, can you draw a representation of each number - lets say till 20
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- Observations? (Powers of 2 are single nodes)
- Why does this happen? What does it mean? (Powers of 2 can not be represented as sum of consecutive whole numbers)
Euclid's Algorithm
Now we go to the next exercise. The students will say a pair of numbers, and the computer will produce a diagram. We are using the tool at https://undergroundmathematics.org/divisibility-and-induction/picture-this to generate the diagrams.
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Impose a cost on students to ask for a pair of numbers, so that they try to make intelligent guesses. After a few, ask students to guess (and draw the guesses), and let them compare with the correct answer.
Observations? Let kids make open-ended comments
- Multiples of the same set produces the same diagram
- Switching the numbers around transposes the diagram
- When same numbers are entered, its just one square
- The first number is horizontal dimension, the second is vertical
- The innermost shapes are all squares
- ...
Now push students to identify the mathematical pattern in this. What is actually being done?
- If we pick two numbers (a,b) such that b is smaller, then first we chop off as many b's from a as possible. If the remainder is c, now we solve for (c,b) by chopping off as many c's from b as possible. And so on.
- Will this process always terminate?
- If we gave you a diagram, can you now reverse compute the numbers?
- Note that the base number can be computed, but any multiples thereof are also other solutions to the same diagram
- What is the relationship between side length of smallest square to the two initial numbers
- Greatest common factor
- Can you prove that?
- How many steps can we have? What kind of pairs of integers give fewest and most steps?
- Can you translate the diagram into equations?
- This algorithm is called Euclid's Algorithm, and used to find greatest common factor of two numbers
- How else can you calculate the GCF of two numbers, say 329 and 49 - try and compare with above method (which is faster, easier, what if the numbers were larger)
Homework:
Try to create a mini mathematical universe of your own - a world driven by some mathematical rules, that others have to discover. If there are any interesting properties hidden in that world, the exploration becomes more interesting.
References: