Understand

Teachers for Tomorrow’ is a unique program for parents and teachers who wish to take their kids / students an extra mile in mathematical training. Cheenta uses modern tools (such as Latex, GeoGebra, STACK etc.) to deliver its courses. It also uses carefully experimented teaching methods developed in USSR, United States, and India. We firmly believe that these tools and methods are very valuable in stimulating creativity in young mind.

‘Teachers for tomorrow’ sessions are conducted online, biweekly. Email us at helpdesk@cheenta.com if you want to join the next session. As of now, they are free of cost.

The Training Program has two key components:

  • Tool Training (example: STACK, GeoGebra, Latex)
  • Experimental Learning Packets

Experimental Learning Packet: Invariance Principle

This document contains the following experimental learning packet: Invariance Principle. It can be readily implemented in the class. Students may range from Class 2 to Class 12 (or even in College).

The experimental learning packet consists of the following steps:

  • State the problem
  • Create miniature
  • Experiment to observe a key pattern.
  • Abstraction
  • Create an algorithm

Step 1: State the problem

Suppose n is a positive odd number. Write the numbers 1 to 2n on the board. Next erase any two numbers a and b and write down |a – b| on the board. Continue doing this until only one number is left on the board. Show that this last number is always odd.

Teacher Notes for Step 1, Step 2

Students of class 1 to 5 may not understand ‘odd’ or usage of symbols like ‘n’, ‘a’, b’ etc. This is a good place to introduce the notion of ‘odd-even’. Notice that it we divide any whole number by 2, the remainder is either 0 or 1. Hence all numbers can be split into 2 teams:

  • Those who give remainder 0 – even
  • Those who give remainder 1 – odd

This grouping of numbers is very significant. Later we may use 3 or 4 to create new groupings.

Once the student understands, what is odd and what is even, ask them to choose a favorite odd number. Suppose 7.

Next write down all the numbers from 1 to 14 on the board. Also ask the students to write these numbers in their notebook. It is very important that they are doing the experiment on their own.

Next, ask one them to come down to the board, and erase any two numbers. Suppose ne erases 6 and 9. Make sure that the students understand that one of these two numbers is small and the other one is large. This is a tacit way of talking about both the linear ordering of natural numbers and introducing the notion of absolute value.

Next, ask them to compute large – small. Go ahead and write these words on the board. Each student will choose a different pair of number. Hence each of them will have a different value for large and different value for small. This is highly desirable. All of them should not use 6 and 9. This gives us a way to introduce the notion of a variable (a ‘word’ that can assume different values).

Finally, ask them to erase small and large and instead write large – small. It is very important that each student has a different value for small, large and large – small.

This completes Step 1 and Step 2.

Step 3: Observe a key pattern

The key pattern is: sum of the numbers on the board is always odd and it remains odd at each step of the process.

Let the students compute the sum of the numbers in Step 2. Call this value SUM. If all of them started with 7 then the sum will be 105 (= 1+ 2+ 3 + … + 14). It is desirable that they create their own miniatures. In fact some of them may start with 7, some of them may start with 5, some may start with 9 etc. Then they will have different values for SUM.

Teacher Notes for Step 3

This is a great place to introduce Gauss Trick for computing (1+2+ … + n). The teacher may also introduce the triangular numbers, and a pictorial presentation, why Gauss Trick works.

Lead the student to the observation that at each step of the process (of erasing two numbers and writing down their difference), the value of SUM remains odd.

Step 4: Abstraction

Why do the SUM remain odd? It is a great place to understand that ‘remove two numbers and write their difference’ has the following effect on the value of SUM : SUM becomes SUM – twice of SMALL. Why?

After all this is what we are doing:

SUM – LARGE – SMALL + LARGE – SMALL = SUM – 2* SMALL

The original value of SUM is odd. Subtracting 2*SMALL is, in essence subtracting an even quantity from the original value of SUM (which was odd). And ODD – EVEN = ODD. This is a great place to discuss that ODD – EVEN = ODD.

Since at each step SUM remains odd, it stays odd in the last step. But there is only one number left in the last step. Hence the value of SUM in the last step is the same as the value of the number remaining in the last step. Hence it must be odd as well.

Step 5: Algorithm

The final step of the learning packet is the creation of an algorithm to compute the sum of the numbers from 1 to 2n. One may begin this process naively in the following manner:

Algorithm – First Pass

Step 1: Take the number 1

Step 2: Add 2 to 1 and record the SUM

Step 3: Add 3 to the previous value of SUM

Step 4: Add 4 to the previous value of SUM

etc.

Notice that we tacitly introduced ‘SUM’ whose value is getting updated. This is a perfectly good spot to talk about variables as ‘boxes whose value can be updated). Next, we will sharpen this algorithm in the second round.

Algorithm – Second Pass

Step 1: Make take boxes: ‘NUMBER’ and ‘SUM’

Step 2: Put 1 in both boxes.

Step 3: Add 1 to whatever is in ‘NUMBER’

Step 4: Add whatever is in ‘NUMBER’ to whatever is in the box ‘SUM’ (in this step 2 in ‘NUMBER’ and 3 is in ‘SUM’).

The student should repeat steps 3 and 4 several times to see how the values in the boxes ‘NUMBER’ and ‘SUM’ is getting updated.