document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=637154>Question 637154</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Read the following lesson estimating square roots.  Complete the table at the end and turn in to your instructor.<BR>\n" );
document.write( "1. Study the following geometric pattern of squares. What do you notice?<BR>\n" );
document.write( "1x1	2x2=4	3x3=9		4x4=16				5x5=25				6x6=36		<BR>\n" );
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document.write( "			6x6 + 4 = 40													 	 	 	 	 	 <BR>\n" );
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document.write( "Did you see that each square in the first row is formed by adding the “L-shaped” piece to the previous smaller square?\r<BR>\n" );
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document.write( "2.  For 40 tiles, the previous full square contains _______ tiles, making the square that is _____ by ____. 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?  ________<BR>\n" );
document.write( "We know  lies between 6 and 7. Let’s use 4/13 in estimating 6  as an approximate value for  . Use this method to complete the table, rounding as shown.\r<BR>\n" );
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document.write( "Number	Side of<BR>\n" );
document.write( "last sq	Fractional<BR>\n" );
document.write( " L-piece	Mixed numeral form	Decimal form	Sq Root from TI-73	Difference<BR>\n" );
document.write( "in values<BR>\n" );
document.write( "40	6	4/13	6  <BR>\n" );
document.write( "6.30769	6.32456	.01687<BR>\n" );
document.write( "75						<BR>\n" );
document.write( "96						<BR>\n" );
document.write( "125						\r<BR>\n" );
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document.write( "1x1	2x2=4	3x3=9		4x4=16				5x5=25				6x6=36		<BR>\n" );
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document.write( "			6x6 + 4 = 40													 	 	 	 	 	 <BR>\n" );
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document.write( "																							 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #401544 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=401544\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume that the blank spaces should have some pictures, but they did not show up on copying and pasting.<BR>\n" );
document.write( "Maybe they looked something like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C7%2C-1%2C7%2C%0D%0Agreen%28line%281%2C1%2C6%2C1%29%29%2Cgreen%28line%282%2C2%2C6%2C2%29%29%2C%0D%0Agreen%28line%282%2C3%2C6%2C3%29%29%2Cgreen%28line%284%2C4%2C6%2C4%29%29%2C%0D%0Agreen%28line%285%2C5%2C6%2C5%29%29%2Cgreen%28line%281%2C1%2C1%2C6%29%29%2C%0D%0Agreen%28line%282%2C1%2C2%2C6%29%29%2Cgreen%28line%283%2C1%2C3%2C6%29%29%2C%0D%0Agreen%28line%284%2C1%2C4%2C6%29%29%2Cgreen%28line%285%2C1%2C5%2C6%29%29%2C%0D%0Ared%28rectangle%280%2C0%2C5.99%2C5.99%29%29%2Cred%28rectangle%280%2C0%2C6.015%2C6.015%29%29%2C%0D%0Ablue%28rectangle%280%2C0%2C4.99%2C4.99%29%29%2Cblue%28rectangle%280%2C0%2C5.01%2C5.01%29%29%2C%0D%0Arectangle%280%2C0%2C3.99%2C3.99%29%2Crectangle%280%2C0%2C4.015%2C4.015%29%2C%0D%0Ared%28rectangle%280%2C0%2C2.99%2C2.99%29%29%2Cred%28rectangle%280%2C0%2C3.015%2C3.015%29%29%2C%0D%0Ablue%28rectangle%280%2C0%2C1.985%2C1.985%29%29%2Cblue%28rectangle%280%2C0%2C2.01%2C2.01%29%29%2C%0D%0Arectangle%280%2C0%2C0.99%2C0.99%29%2Crectangle%280%2C0%2C1.02%2C1.02%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > I outlined the squares in different colors, showing the smaller squares on top of the larger one, and marking other tile borders in green.<BR>\n" );
document.write( " <BR>\n" );
document.write( "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.<BR>\n" );
document.write( "That tells you that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%5E2=a%5E2%2B2a%2B1 ALIGN=MIDDLE  ALT=\"%28a%2B1%29%5E2=a%5E2%2B2a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"%28a%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the square of the next number, made of<BR>\n" );
document.write( "the square of the number, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the two rectangles, {2a}}} ,<BR>\n" );
document.write( "and the one tile in the corner <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "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.<BR>\n" );
document.write( " <BR>\n" );
document.write( "2) For 40 tiles, the previous full square contains <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2836%29 ALIGN=MIDDLE  ALT=\"highlight%2836%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > tiles, making the square that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286%29 ALIGN=MIDDLE  ALT=\"highlight%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286%29 ALIGN=MIDDLE  ALT=\"highlight%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. 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? <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%284%2F13%29 ALIGN=MIDDLE  ALT=\"highlight%284%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Adding the fraction to the previous full square side (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >), you get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%264%2F13=6.30769 ALIGN=MIDDLE  ALT=\"6%264%2F13=6.30769\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded to 5 decimal places)<BR>\n" );
document.write( "That is your estimate for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2840%29 ALIGN=MIDDLE  ALT=\"sqrt%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The calculator says that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2840%29=6.32456 ALIGN=MIDDLE  ALT=\"sqrt%2840%29=6.32456\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded to 5 decimal places)<BR>\n" );
document.write( "The difference is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.32456-6.30769=0.01687 ALIGN=MIDDLE  ALT=\"6.32456-6.30769=0.01687\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That shows you that your estimate was pretty accurate.<BR>\n" );
document.write( " <BR>\n" );
document.write( "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.<BR>\n" );
document.write( "The table should look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C250%2C0%2C8%2C0%2C6%2C%0D%0Alocate%280%2C6%2Cnumber%29%2C%0D%0Alocate%280.7%2C6%2Cside%29%2C%0D%0Alocate%281.3%2C6%2Cfraction_of_L%29%2Clocate%282.7%2C6%2Cmixed_number%29%2C%0D%0Alocate%284.2%2C6%2Cdecimal_form%29%2Clocate%286%2C6%2Csqrt%28number%29%29%2C%0D%0Alocate%287%2C6%2Cdifference%29%2C%0D%0Alocate%280.2%2C5%2C40%29%2C%0D%0Alocate%280.8%2C5%2C6%29%2Clocate%281.8%2C5%2C4%2F13%29%2C%0D%0Alocate%283%2C5%2C6%264%2F13%29%2Clocate%284.4%2C5%2C6.30769%29%2C%0D%0Alocate%286%2C5%2C6.35456%29%2Clocate%287.1%2C5%2C0.01686%29%2C%0D%0Alocate%280.2%2C3.66%2C75%29%2C%0D%0Alocate%280.8%2C3.66%2C8%29%2Clocate%281.8%2C3.66%2C11%2F17%29%2C%0D%0Alocate%283%2C3.66%2C8%2611%2F17%29%2Clocate%284.4%2C3.66%2C8.64706%29%2C%0D%0Alocate%286%2C3.66%2C8.66025%29%2Clocate%287.1%2C3.66%2C0.01320%29%2C%0D%0Alocate%280.2%2C2.3%2C96%29%2C%0D%0Alocate%280.8%2C2.3%2C9%29%2Clocate%281.8%2C2.3%2C15%2F19%29%2C%0D%0Alocate%283%2C2.3%2C9%2615%2F19%29%2Clocate%284.4%2C2.3%2C9.78947%29%2C%0D%0Alocate%286%2C2.3%2C9.79796%29%2Clocate%287.1%2C2.3%2C0.00849%29%2C%0D%0Alocate%280.15%2C1%2C125%29%2C%0D%0Alocate%280.75%2C1%2C11%29%2Clocate%281.8%2C1%2C4%2F23%29%2C%0D%0Alocate%282.9%2C1%2C11%264%2F23%29%2Clocate%284.3%2C1%2C11.17391%29%2C%0D%0Alocate%286%2C1%2C6.35456%29%2Clocate%287.1%2C1%2C0.00643%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );