Question 637154
I assume that the blank spaces should have some pictures, but they did not show up on copying and pasting.
Maybe they looked something like this:
{{{drawing(300,300,-1,7,-1,7,
green(line(1,1,6,1)),green(line(2,2,6,2)),
green(line(2,3,6,3)),green(line(4,4,6,4)),
green(line(5,5,6,5)),green(line(1,1,1,6)),
green(line(2,1,2,6)),green(line(3,1,3,6)),
green(line(4,1,4,6)),green(line(5,1,5,6)),
red(rectangle(0,0,5.99,5.99)),red(rectangle(0,0,6.015,6.015)),
blue(rectangle(0,0,4.99,4.99)),blue(rectangle(0,0,5.01,5.01)),
rectangle(0,0,3.99,3.99),rectangle(0,0,4.015,4.015),
red(rectangle(0,0,2.99,2.99)),red(rectangle(0,0,3.015,3.015)),
blue(rectangle(0,0,1.985,1.985)),blue(rectangle(0,0,2.01,2.01)),
rectangle(0,0,0.99,0.99),rectangle(0,0,1.02,1.02)
)}}} I outlined the squares in different colors, showing the smaller squares on top of the larger one, and marking other tile borders in green.
 
1) You may notice that each lager square is formed by adding a border, 1 tile wide and L-shaped. That border is made up of two rectangles (an extra horizontal row of tiles and an extra vertical column of tiles) plus a corner square tile.
That tells you that
{{{(a+1)^2=a^2+2a+1}}}
{{{(a+1)^2}}} is the square of the next number, made of
the square of the number, {{{a^2}}} ,
the two rectangles, {2a}}} ,
and the one tile in the corner {{{1}}}
You could also notice that as the squares get larger, what is added is a smaller fraction of the original square (and also a smaller fraction compared to the enlarged square.
 
2) For 40 tiles, the previous full square contains {{{highlight(36)}}} tiles, making the square that is {{{highlight(6)}}} by {{{highlight(6)}}}. The extra 4 tiles are a fractional piece of the “L” that will make the next perfect square. What fraction of the “L-piece” is shaded? {{{highlight(4/13)}}} .
Adding the fraction to the previous full square side ({{{6}}}), you get
{{{6&4/13=6.30769}}} (rounded to 5 decimal places)
That is your estimate for {{{sqrt(40)}}}.
The calculator says that
{{{sqrt(40)=6.32456}}} (rounded to 5 decimal places)
The difference is {{{6.32456-6.30769=0.01687}}}
That shows you that your estimate was pretty accurate.
 
The table you are supposed to fill will show you that as the numbers get larger, the estimates get closer to the calculator result. The estimates get more accurate.
The table should look like this:
{{{drawing(800,250,0,8,0,6,
locate(0,6,number),
locate(0.7,6,side),
locate(1.3,6,fraction_of_L),locate(2.7,6,mixed_number),
locate(4.2,6,decimal_form),locate(6,6,sqrt(number)),
locate(7,6,difference),
locate(0.2,5,40),
locate(0.8,5,6),locate(1.8,5,4/13),
locate(3,5,6&4/13),locate(4.4,5,6.30769),
locate(6,5,6.35456),locate(7.1,5,0.01686),
locate(0.2,3.66,75),
locate(0.8,3.66,8),locate(1.8,3.66,11/17),
locate(3,3.66,8&11/17),locate(4.4,3.66,8.64706),
locate(6,3.66,8.66025),locate(7.1,3.66,0.01320),
locate(0.2,2.3,96),
locate(0.8,2.3,9),locate(1.8,2.3,15/19),
locate(3,2.3,9&15/19),locate(4.4,2.3,9.78947),
locate(6,2.3,9.79796),locate(7.1,2.3,0.00849),
locate(0.15,1,125),
locate(0.75,1,11),locate(1.8,1,4/23),
locate(2.9,1,11&4/23),locate(4.3,1,11.17391),
locate(6,1,6.35456),locate(7.1,1,0.00643)
)}}}