At Isai Ambalam a couple of the children I was working intensely with programing graduated, it left a small class of 6th graders and a couple of 8th graders. The youth with AuraAuro took up various classes including my 6th grade. I was left shuttling between classes and the 8th graders. The 8th graders who are still here are not ready for their next grade both in language and math. Taking a leaf from my being independent book they were keen on taking the initiative and learning on their own and asking for help when they needed it. They were working on squares and square roots for a week on their own..., but I realized that they were not quite getting the sense of the numbers. They were doing much drill, making mistakes, but unable to notice their mistakes or having a ball park estimate.
They are hesitant to program themselves and had relied heavily on their partners who had left school. I asked them if they wanted to the computer as a calculator...they readily agreed (ha!).
Looking around and starting visualization
The sixth graders were drawing the representation of linear expressions like 5x+10 (and then changing the constant or the slope and getting various staircases for positive values of x. I asked the 8th graders if they would like to use scratch as a calculator and do the same for x^2 = x*x. They plotted it along with the linear curves and soon realized the much faster rate of its growth to exceed the screen size. They felt that x^3 should perhaps increase even faster and went ahead and implemented it to find out.
We talked about the possibility of using the computer to find the square roots of numbers by repeated calculating x^2 for numbers starting from 1 and stopping when they reached the numbers as given in the book.
Starting to look at the consecutive squares
We made a very simple program which started with x=1 kept incrementing x it by one and calculating a square (one multiplication per second). Scratch can automatically show the variables of your choice making it easier since we didn't need to program the printing.
They started looking at the results of squares of numbers and basic internalizations that the squares of numbers do increase as numbers increase and start to notice a few basic squares that they knew.
Efficiency of calculation
As we got to 4 digit numbers the time taken by the program was quite a lot and I asked them what could be done to speed up the program without changing 1 multiplication per second. My argument for the constraint was that that much time is required to register the square of a number and I did want them to connect a number to a square.
It took them a little while, but they concluded that we could start with squares of every 10 numbers 10, 20, 30, 40, etc and stop when the number was overshooting, go back one level and then increment by 1 to get to the result.
This reduced the square of 69 from 69 seconds to 7+9=16 seconds. They also started getting a clearer understanding of the ball park of the squares since these were the same as the squares single digit numbers in 100s.
Guessing the result
This brought us to the most interesting part of the book that talked about how you could guess the square root of a perfect square (of a 2 digit number). It talked about getting the 10s place of the number. They could see this from the program the optimized code a number like 2209 needed to be in the 40s as 40^2=1600 and 50^2=2500. It was then a matter of which of the 40s gives the result.
This brought us to the ones/units place and the fact that there is a clear mapping (two to one at times) between the units of the number and the units of its square, e.g. 2/8 have 4 in the units place, 3/7 have 9, 4/6 have 6, 5 is just 5 and 1/9 have 1. 2,3,7,8 do not come in the units place of a perfect square of an integer.
Looking at 2209, it could have a square of either 43 or 47. The next choice is determined by whether it is closer to 40^2 (1600) or 50^2 (2500).
Permeating
The fun with being able to guess something marginally larger than they thought they were originally capable of was that they showed what they could do to the 6th graders who were promptly curious on how this was all done and learnt it too.
It took them a little while, but they concluded that we could start with squares of every 10 numbers 10, 20, 30, 40, etc and stop when the number was overshooting, go back one level and then increment by 1 to get to the result.
This reduced the square of 69 from 69 seconds to 7+9=16 seconds. They also started getting a clearer understanding of the ball park of the squares since these were the same as the squares single digit numbers in 100s.
Guessing the result
This brought us to the most interesting part of the book that talked about how you could guess the square root of a perfect square (of a 2 digit number). It talked about getting the 10s place of the number. They could see this from the program the optimized code a number like 2209 needed to be in the 40s as 40^2=1600 and 50^2=2500. It was then a matter of which of the 40s gives the result.
This brought us to the ones/units place and the fact that there is a clear mapping (two to one at times) between the units of the number and the units of its square, e.g. 2/8 have 4 in the units place, 3/7 have 9, 4/6 have 6, 5 is just 5 and 1/9 have 1. 2,3,7,8 do not come in the units place of a perfect square of an integer.
Looking at 2209, it could have a square of either 43 or 47. The next choice is determined by whether it is closer to 40^2 (1600) or 50^2 (2500).
Permeating
The fun with being able to guess something marginally larger than they thought they were originally capable of was that they showed what they could do to the 6th graders who were promptly curious on how this was all done and learnt it too.
This gave them endless fun of resolving the solution and pitting themselves against their program and getting the solution before it did.
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